Octave 3.8, jcobi/4

Percentage Accurate: 15.9% → 84.0%

Time: 5.4s

Alternatives: 12

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: 15.9% 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: 84.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(2, i, \beta + \alpha\right)\\ t_1 := \left(\beta + \alpha\right) + i\\ \mathbf{if}\;i \leq 2.4 \cdot 10^{+124}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(t\_1, i, \beta \cdot \alpha\right)}{t\_0}}{t\_0 - 1} \cdot \frac{t\_1 \cdot \frac{i}{t\_0}}{t\_0 - -1}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (fma 2.0 i (+ beta alpha))) (t_1 (+ (+ beta alpha) i)))
   (if (<= i 2.4e+124)
     (*
      (/ (/ (fma t_1 i (* beta alpha)) t_0) (- t_0 1.0))
      (/ (* t_1 (/ i t_0)) (- t_0 -1.0)))
     0.0625)))

C

double code(double alpha, double beta, double i) {
	double t_0 = fma(2.0, i, (beta + alpha));
	double t_1 = (beta + alpha) + i;
	double tmp;
	if (i <= 2.4e+124) {
		tmp = ((fma(t_1, i, (beta * alpha)) / t_0) / (t_0 - 1.0)) * ((t_1 * (i / t_0)) / (t_0 - -1.0));
	} else {
		tmp = 0.0625;
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = fma(2.0, i, Float64(beta + alpha))
	t_1 = Float64(Float64(beta + alpha) + i)
	tmp = 0.0
	if (i <= 2.4e+124)
		tmp = Float64(Float64(Float64(fma(t_1, i, Float64(beta * alpha)) / t_0) / Float64(t_0 - 1.0)) * Float64(Float64(t_1 * Float64(i / t_0)) / Float64(t_0 - -1.0)));
	else
		tmp = 0.0625;
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(2.0 * i + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(beta + alpha), $MachinePrecision] + i), $MachinePrecision]}, If[LessEqual[i, 2.4e+124], N[(N[(N[(N[(t$95$1 * i + N[(beta * alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 - 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$1 * N[(i / t$95$0), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0625]]]

Derivation

1. Split input into 2 regimes

2. if i < 2.40000000000000006e124

   1. Initial program 15.9%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. times-frac N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

   3. Applied rewrites 43.1%

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

   if 2.40000000000000006e124 < i

   1. Initial program 15.9%

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

   2. Taylor expanded in i around inf

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

   4. Applied rewrites 71.6%

      \[\leadsto \color{blue}{0.0625} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 81.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double t_1 = t_0 * t_0;
	double t_2 = i * ((alpha + beta) + i);
	double t_3 = (beta + alpha) + i;
	double t_4 = fma(2.0, i, (beta + alpha));
	double tmp;
	if ((((t_2 * ((beta * alpha) + t_2)) / t_1) / (t_1 - 1.0)) <= ((double) INFINITY)) {
		tmp = (fma(t_3, i, (beta * alpha)) / t_4) * ((t_3 * (i / t_4)) / fma(t_4, t_4, -1.0));
	} else {
		tmp = fma(beta, -0.125, fma(beta, 0.125, (0.0625 * i))) / i;
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_1 = Float64(t_0 * t_0)
	t_2 = Float64(i * Float64(Float64(alpha + beta) + i))
	t_3 = Float64(Float64(beta + alpha) + i)
	t_4 = fma(2.0, i, Float64(beta + alpha))
	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(t_3, i, Float64(beta * alpha)) / t_4) * Float64(Float64(t_3 * Float64(i / t_4)) / fma(t_4, t_4, -1.0)));
	else
		tmp = Float64(fma(beta, -0.125, fma(beta, 0.125, Float64(0.0625 * i))) / i);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(i * N[(N[(alpha + beta), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(beta + alpha), $MachinePrecision] + i), $MachinePrecision]}, Block[{t$95$4 = N[(2.0 * i + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, 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[(t$95$3 * i + N[(beta * alpha), $MachinePrecision]), $MachinePrecision] / t$95$4), $MachinePrecision] * N[(N[(t$95$3 * N[(i / t$95$4), $MachinePrecision]), $MachinePrecision] / N[(t$95$4 * t$95$4 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(beta * -0.125 + N[(beta * 0.125 + N[(0.0625 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $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 15.9%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. associate-/l/ N/A

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

   3. Applied rewrites 38.2%

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

   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 15.9%

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

   2. 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 77.6

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

   4. Applied rewrites 77.6%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. metadata-eval 77.6

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

   6. Applied rewrites 77.6%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-*.f64 N/A

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

      9. add-to-fraction-rev N/A

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

      10. lift-fma.f64 N/A

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

      11. div-add-rev N/A

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

      12. lower-/.f64 N/A

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

   8. Applied rewrites 77.7%

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

   9. Taylor expanded in alpha around 0

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

   11. Applied rewrites 74.2%

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

   12. Taylor expanded in alpha around 0

       \[\leadsto \frac{\mathsf{fma}\left(\beta, \frac{-1}{8}, \mathsf{fma}\left(\beta, \frac{1}{8}, \frac{1}{16} \cdot i\right)\right)}{i} \]
   13. Step-by-step derivation

   14. Applied rewrites 75.0%

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

Alternative 3: 80.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double t_1 = t_0 * t_0;
	double t_2 = i * ((alpha + beta) + i);
	double t_3 = (beta + alpha) + i;
	double t_4 = fma(2.0, i, (beta + alpha));
	double tmp;
	if ((((t_2 * ((beta * alpha) + t_2)) / t_1) / (t_1 - 1.0)) <= ((double) INFINITY)) {
		tmp = (t_3 * i) * ((fma(t_3, i, (beta * alpha)) / (t_4 * t_4)) / fma(t_4, t_4, -1.0));
	} else {
		tmp = fma(beta, -0.125, fma(beta, 0.125, (0.0625 * i))) / i;
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_1 = Float64(t_0 * t_0)
	t_2 = Float64(i * Float64(Float64(alpha + beta) + i))
	t_3 = Float64(Float64(beta + alpha) + i)
	t_4 = fma(2.0, i, Float64(beta + alpha))
	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 * i) * Float64(Float64(fma(t_3, i, Float64(beta * alpha)) / Float64(t_4 * t_4)) / fma(t_4, t_4, -1.0)));
	else
		tmp = Float64(fma(beta, -0.125, fma(beta, 0.125, Float64(0.0625 * i))) / i);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(i * N[(N[(alpha + beta), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(beta + alpha), $MachinePrecision] + i), $MachinePrecision]}, Block[{t$95$4 = N[(2.0 * i + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, 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 * i), $MachinePrecision] * N[(N[(N[(t$95$3 * i + N[(beta * alpha), $MachinePrecision]), $MachinePrecision] / N[(t$95$4 * t$95$4), $MachinePrecision]), $MachinePrecision] / N[(t$95$4 * t$95$4 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(beta * -0.125 + N[(beta * 0.125 + N[(0.0625 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $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 15.9%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. associate-/l* N/A

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

   3. Applied rewrites 36.3%

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

   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 15.9%

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

   2. 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 77.6

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

   4. Applied rewrites 77.6%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. metadata-eval 77.6

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

   6. Applied rewrites 77.6%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-*.f64 N/A

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

      9. add-to-fraction-rev N/A

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

      10. lift-fma.f64 N/A

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

      11. div-add-rev N/A

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

      12. lower-/.f64 N/A

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

   8. Applied rewrites 77.7%

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

   9. Taylor expanded in alpha around 0

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

   11. Applied rewrites 74.2%

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

   12. Taylor expanded in alpha around 0

       \[\leadsto \frac{\mathsf{fma}\left(\beta, \frac{-1}{8}, \mathsf{fma}\left(\beta, \frac{1}{8}, \frac{1}{16} \cdot i\right)\right)}{i} \]
   13. Step-by-step derivation

   14. Applied rewrites 75.0%

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

Alternative 4: 77.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := t\_0 \cdot t\_0\\ t_2 := i \cdot \left(\left(\alpha + \beta\right) + i\right)\\ t_3 := i \cdot \left(\beta + i\right)\\ t_4 := \beta + 2 \cdot i\\ t_5 := t\_4 \cdot t\_4\\ \mathbf{if}\;\frac{\frac{t\_2 \cdot \left(\beta \cdot \alpha + t\_2\right)}{t\_1}}{t\_1 - 1} \leq 10^{-9}:\\ \;\;\;\;\frac{\frac{t\_3 \cdot \left(\beta \cdot \alpha + t\_3\right)}{t\_5}}{t\_5 - 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\beta, -0.125, \mathsf{fma}\left(\beta, 0.125, 0.0625 \cdot i\right)\right)}{i}\\ \end{array} \end{array} \]

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 (* i (+ beta i)))
        (t_4 (+ beta (* 2.0 i)))
        (t_5 (* t_4 t_4)))
   (if (<= (/ (/ (* t_2 (+ (* beta alpha) t_2)) t_1) (- t_1 1.0)) 1e-9)
     (/ (/ (* t_3 (+ (* beta alpha) t_3)) t_5) (- t_5 1.0))
     (/ (fma beta -0.125 (fma beta 0.125 (* 0.0625 i))) i))))

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 = i * (beta + i);
	double t_4 = beta + (2.0 * i);
	double t_5 = t_4 * t_4;
	double tmp;
	if ((((t_2 * ((beta * alpha) + t_2)) / t_1) / (t_1 - 1.0)) <= 1e-9) {
		tmp = ((t_3 * ((beta * alpha) + t_3)) / t_5) / (t_5 - 1.0);
	} else {
		tmp = fma(beta, -0.125, fma(beta, 0.125, (0.0625 * i))) / i;
	}
	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(i * Float64(beta + i))
	t_4 = Float64(beta + Float64(2.0 * i))
	t_5 = Float64(t_4 * t_4)
	tmp = 0.0
	if (Float64(Float64(Float64(t_2 * Float64(Float64(beta * alpha) + t_2)) / t_1) / Float64(t_1 - 1.0)) <= 1e-9)
		tmp = Float64(Float64(Float64(t_3 * Float64(Float64(beta * alpha) + t_3)) / t_5) / Float64(t_5 - 1.0));
	else
		tmp = Float64(fma(beta, -0.125, fma(beta, 0.125, Float64(0.0625 * i))) / i);
	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[(i * N[(beta + i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(beta + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$4 * t$95$4), $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], 1e-9], N[(N[(N[(t$95$3 * N[(N[(beta * alpha), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision] / t$95$5), $MachinePrecision] / N[(t$95$5 - 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(beta * -0.125 + N[(beta * 0.125 + N[(0.0625 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $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))) < 1.00000000000000006e-9

   1. Initial program 15.9%

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

   2. 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 15.0%

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

   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 16.2%

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

   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 16.4%

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

   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 16.4%

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

   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 14.6%

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

   if 1.00000000000000006e-9 < (/.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 15.9%

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

   2. 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 77.6

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

   4. Applied rewrites 77.6%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. metadata-eval 77.6

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

   6. Applied rewrites 77.6%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-*.f64 N/A

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

      9. add-to-fraction-rev N/A

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

      10. lift-fma.f64 N/A

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

      11. div-add-rev N/A

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

      12. lower-/.f64 N/A

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

   8. Applied rewrites 77.7%

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

   9. Taylor expanded in alpha around 0

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

   11. Applied rewrites 74.2%

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

   12. Taylor expanded in alpha around 0

       \[\leadsto \frac{\mathsf{fma}\left(\beta, \frac{-1}{8}, \mathsf{fma}\left(\beta, \frac{1}{8}, \frac{1}{16} \cdot i\right)\right)}{i} \]
   13. Step-by-step derivation

   14. Applied rewrites 75.0%

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

Alternative 5: 77.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := t\_0 \cdot t\_0\\ t_2 := i \cdot \left(\left(\alpha + \beta\right) + i\right)\\ t_3 := \mathsf{fma}\left(2, i, \beta + \alpha\right)\\ \mathbf{if}\;\frac{\frac{t\_2 \cdot \left(\beta \cdot \alpha + t\_2\right)}{t\_1}}{t\_1 - 1} \leq 10^{-9}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\beta + \alpha\right) + i, i, \beta \cdot \alpha\right)}{t\_3} \cdot \frac{i}{\mathsf{fma}\left(t\_3, t\_3, -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\beta, -0.125, \mathsf{fma}\left(\beta, 0.125, 0.0625 \cdot i\right)\right)}{i}\\ \end{array} \end{array} \]

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 2.0 i (+ beta alpha))))
   (if (<= (/ (/ (* t_2 (+ (* beta alpha) t_2)) t_1) (- t_1 1.0)) 1e-9)
     (*
      (/ (fma (+ (+ beta alpha) i) i (* beta alpha)) t_3)
      (/ i (fma t_3 t_3 -1.0)))
     (/ (fma beta -0.125 (fma beta 0.125 (* 0.0625 i))) i))))

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(2.0, i, (beta + alpha));
	double tmp;
	if ((((t_2 * ((beta * alpha) + t_2)) / t_1) / (t_1 - 1.0)) <= 1e-9) {
		tmp = (fma(((beta + alpha) + i), i, (beta * alpha)) / t_3) * (i / fma(t_3, t_3, -1.0));
	} else {
		tmp = fma(beta, -0.125, fma(beta, 0.125, (0.0625 * i))) / i;
	}
	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(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)) <= 1e-9)
		tmp = Float64(Float64(fma(Float64(Float64(beta + alpha) + i), i, Float64(beta * alpha)) / t_3) * Float64(i / fma(t_3, t_3, -1.0)));
	else
		tmp = Float64(fma(beta, -0.125, fma(beta, 0.125, Float64(0.0625 * i))) / i);
	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[(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], 1e-9], N[(N[(N[(N[(N[(beta + alpha), $MachinePrecision] + i), $MachinePrecision] * i + N[(beta * alpha), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision] * N[(i / N[(t$95$3 * t$95$3 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(beta * -0.125 + N[(beta * 0.125 + N[(0.0625 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $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))) < 1.00000000000000006e-9

   1. Initial program 15.9%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. associate-/l/ N/A

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

   3. Applied rewrites 38.2%

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

   4. Taylor expanded in alpha around inf

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

   6. Applied rewrites 14.8%

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

   if 1.00000000000000006e-9 < (/.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 15.9%

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

   2. 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 77.6

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

   4. Applied rewrites 77.6%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. metadata-eval 77.6

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

   6. Applied rewrites 77.6%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-*.f64 N/A

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

      9. add-to-fraction-rev N/A

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

      10. lift-fma.f64 N/A

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

      11. div-add-rev N/A

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

      12. lower-/.f64 N/A

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

   8. Applied rewrites 77.7%

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

   9. Taylor expanded in alpha around 0

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

   11. Applied rewrites 74.2%

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

   12. Taylor expanded in alpha around 0

       \[\leadsto \frac{\mathsf{fma}\left(\beta, \frac{-1}{8}, \mathsf{fma}\left(\beta, \frac{1}{8}, \frac{1}{16} \cdot i\right)\right)}{i} \]
   13. Step-by-step derivation

   14. Applied rewrites 75.0%

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

Alternative 6: 76.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := t\_0 \cdot t\_0\\ t_2 := t\_1 - 1\\ t_3 := i \cdot \left(\left(\alpha + \beta\right) + i\right)\\ \mathbf{if}\;\frac{\frac{t\_3 \cdot \left(\beta \cdot \alpha + t\_3\right)}{t\_1}}{t\_2} \leq 10^{-9}:\\ \;\;\;\;\frac{-1 \cdot \left(i \cdot \left(-1 \cdot i\right)\right)}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\beta, -0.125, \mathsf{fma}\left(\beta, 0.125, 0.0625 \cdot i\right)\right)}{i}\\ \end{array} \end{array} \]

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) 1e-9)
     (/ (* -1.0 (* i (* -1.0 i))) t_2)
     (/ (fma beta -0.125 (fma beta 0.125 (* 0.0625 i))) i))))

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) <= 1e-9) {
		tmp = (-1.0 * (i * (-1.0 * i))) / t_2;
	} else {
		tmp = fma(beta, -0.125, fma(beta, 0.125, (0.0625 * i))) / i;
	}
	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) <= 1e-9)
		tmp = Float64(Float64(-1.0 * Float64(i * Float64(-1.0 * i))) / t_2);
	else
		tmp = Float64(fma(beta, -0.125, fma(beta, 0.125, Float64(0.0625 * i))) / i);
	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], 1e-9], N[(N[(-1.0 * N[(i * N[(-1.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], N[(N[(beta * -0.125 + N[(beta * 0.125 + N[(0.0625 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $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))) < 1.00000000000000006e-9

   1. Initial program 15.9%

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

   2. Taylor expanded in alpha around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 13.4

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

   4. Applied rewrites 13.4%

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

   5. Taylor expanded in beta around 0

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

      1. lower-*.f64 17.4

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

   7. Applied rewrites 17.4%

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

   if 1.00000000000000006e-9 < (/.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 15.9%

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

   2. 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 77.6

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

   4. Applied rewrites 77.6%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. metadata-eval 77.6

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

   6. Applied rewrites 77.6%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-*.f64 N/A

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

      9. add-to-fraction-rev N/A

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

      10. lift-fma.f64 N/A

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

      11. div-add-rev N/A

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

      12. lower-/.f64 N/A

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

   8. Applied rewrites 77.7%

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

   9. Taylor expanded in alpha around 0

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

   11. Applied rewrites 74.2%

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

   12. Taylor expanded in alpha around 0

       \[\leadsto \frac{\mathsf{fma}\left(\beta, \frac{-1}{8}, \mathsf{fma}\left(\beta, \frac{1}{8}, \frac{1}{16} \cdot i\right)\right)}{i} \]
   13. Step-by-step derivation

   14. Applied rewrites 75.0%

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

Alternative 7: 76.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := t\_0 \cdot t\_0\\ t_2 := i \cdot \left(\left(\alpha + \beta\right) + i\right)\\ \mathbf{if}\;\frac{\frac{t\_2 \cdot \left(\beta \cdot \alpha + t\_2\right)}{t\_1}}{t\_1 - 1} \leq 10^{-9}:\\ \;\;\;\;i \cdot \frac{\alpha + i}{{\beta}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\beta, -0.125, \mathsf{fma}\left(\beta, 0.125, 0.0625 \cdot i\right)\right)}{i}\\ \end{array} \end{array} \]

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)) 1e-9)
     (* i (/ (+ alpha i) (pow beta 2.0)))
     (/ (fma beta -0.125 (fma beta 0.125 (* 0.0625 i))) i))))

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)) <= 1e-9) {
		tmp = i * ((alpha + i) / pow(beta, 2.0));
	} else {
		tmp = fma(beta, -0.125, fma(beta, 0.125, (0.0625 * i))) / i;
	}
	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)) <= 1e-9)
		tmp = Float64(i * Float64(Float64(alpha + i) / (beta ^ 2.0)));
	else
		tmp = Float64(fma(beta, -0.125, fma(beta, 0.125, Float64(0.0625 * i))) / i);
	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], 1e-9], N[(i * N[(N[(alpha + i), $MachinePrecision] / N[Power[beta, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(beta * -0.125 + N[(beta * 0.125 + N[(0.0625 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $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))) < 1.00000000000000006e-9

   1. Initial program 15.9%

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

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

   3. Applied rewrites 13.8%

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

   4. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-pow.f64 9.7

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

   6. Applied rewrites 9.7%

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

   if 1.00000000000000006e-9 < (/.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 15.9%

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

   2. 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 77.6

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

   4. Applied rewrites 77.6%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. metadata-eval 77.6

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

   6. Applied rewrites 77.6%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-*.f64 N/A

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

      9. add-to-fraction-rev N/A

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

      10. lift-fma.f64 N/A

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

      11. div-add-rev N/A

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

      12. lower-/.f64 N/A

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

   8. Applied rewrites 77.7%

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

   9. Taylor expanded in alpha around 0

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

   11. Applied rewrites 74.2%

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

   12. Taylor expanded in alpha around 0

       \[\leadsto \frac{\mathsf{fma}\left(\beta, \frac{-1}{8}, \mathsf{fma}\left(\beta, \frac{1}{8}, \frac{1}{16} \cdot i\right)\right)}{i} \]
   13. Step-by-step derivation

   14. Applied rewrites 75.0%

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

Alternative 8: 76.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := t\_0 \cdot t\_0\\ t_2 := i \cdot \left(\left(\alpha + \beta\right) + i\right)\\ \mathbf{if}\;\frac{\frac{t\_2 \cdot \left(\beta \cdot \alpha + t\_2\right)}{t\_1}}{t\_1 - 1} \leq 10^{-9}:\\ \;\;\;\;\frac{i \cdot \left(\alpha + i\right)}{{\beta}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\beta, -0.125, \mathsf{fma}\left(\beta, 0.125, 0.0625 \cdot i\right)\right)}{i}\\ \end{array} \end{array} \]

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)) 1e-9)
     (/ (* i (+ alpha i)) (pow beta 2.0))
     (/ (fma beta -0.125 (fma beta 0.125 (* 0.0625 i))) i))))

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)) <= 1e-9) {
		tmp = (i * (alpha + i)) / pow(beta, 2.0);
	} else {
		tmp = fma(beta, -0.125, fma(beta, 0.125, (0.0625 * i))) / i;
	}
	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)) <= 1e-9)
		tmp = Float64(Float64(i * Float64(alpha + i)) / (beta ^ 2.0));
	else
		tmp = Float64(fma(beta, -0.125, fma(beta, 0.125, Float64(0.0625 * i))) / i);
	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], 1e-9], N[(N[(i * N[(alpha + i), $MachinePrecision]), $MachinePrecision] / N[Power[beta, 2.0], $MachinePrecision]), $MachinePrecision], N[(N[(beta * -0.125 + N[(beta * 0.125 + N[(0.0625 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $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))) < 1.00000000000000006e-9

   1. Initial program 15.9%

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

   2. 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 9.2

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

   4. Applied rewrites 9.2%

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

   if 1.00000000000000006e-9 < (/.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 15.9%

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

   2. 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 77.6

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

   4. Applied rewrites 77.6%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. metadata-eval 77.6

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

   6. Applied rewrites 77.6%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-*.f64 N/A

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

      9. add-to-fraction-rev N/A

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

      10. lift-fma.f64 N/A

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

      11. div-add-rev N/A

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

      12. lower-/.f64 N/A

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

   8. Applied rewrites 77.7%

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

   9. Taylor expanded in alpha around 0

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

   11. Applied rewrites 74.2%

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

   12. Taylor expanded in alpha around 0

       \[\leadsto \frac{\mathsf{fma}\left(\beta, \frac{-1}{8}, \mathsf{fma}\left(\beta, \frac{1}{8}, \frac{1}{16} \cdot i\right)\right)}{i} \]
   13. Step-by-step derivation

   14. Applied rewrites 75.0%

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

Alternative 9: 75.0% accurate, 4.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 15.9%

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

2. 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 77.6

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

4. Applied rewrites 77.6%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

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

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. +-commutative N/A

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

   9. lift-+.f64 N/A

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

   10. metadata-eval 77.6

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

6. Applied rewrites 77.6%

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

   1. lift-fma.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. lift-fma.f64 N/A

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

   5. +-commutative N/A

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

   6. lift-/.f64 N/A

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

   7. associate-*l/ N/A

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

   8. lift-*.f64 N/A

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

   9. add-to-fraction-rev N/A

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

   10. lift-fma.f64 N/A

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

   11. div-add-rev N/A

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

   12. lower-/.f64 N/A

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

8. Applied rewrites 77.7%

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

9. Taylor expanded in alpha around 0

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

11. Applied rewrites 74.2%

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

12. Taylor expanded in alpha around 0

    \[\leadsto \frac{\mathsf{fma}\left(\beta, \frac{-1}{8}, \mathsf{fma}\left(\beta, \frac{1}{8}, \frac{1}{16} \cdot i\right)\right)}{i} \]
13. Step-by-step derivation

14. Applied rewrites 75.0%

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

Alternative 10: 73.0% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 5.8 \cdot 10^{+253}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\alpha}{i}, -0.125, 0.125 \cdot \frac{\alpha}{i}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 5.8e+253) 0.0625 (fma (/ alpha i) -0.125 (* 0.125 (/ alpha i)))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 5.8e+253) {
		tmp = 0.0625;
	} else {
		tmp = fma((alpha / i), -0.125, (0.125 * (alpha / i)));
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 5.8e+253)
		tmp = 0.0625;
	else
		tmp = fma(Float64(alpha / i), -0.125, Float64(0.125 * Float64(alpha / i)));
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 5.8e+253], 0.0625, N[(N[(alpha / i), $MachinePrecision] * -0.125 + N[(0.125 * N[(alpha / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.79999999999999976e253

   1. Initial program 15.9%

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

   2. Taylor expanded in i around inf

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

   4. Applied rewrites 71.6%

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

   if 5.79999999999999976e253 < beta

   1. Initial program 15.9%

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

   2. 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 77.6

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

   4. Applied rewrites 77.6%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. metadata-eval 77.6

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

   6. Applied rewrites 77.6%

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

   7. Taylor expanded in alpha around inf

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

      1. lower-/.f64 74.1

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

   9. Applied rewrites 74.1%

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

   10. Taylor expanded in alpha around inf

       \[\leadsto \mathsf{fma}\left(\frac{\alpha}{i}, \frac{-1}{8}, \mathsf{fma}\left(\frac{\alpha}{i}, \frac{1}{8}, \frac{1}{16}\right)\right) \]
   11. Step-by-step derivation

       1. lower-/.f64 74.9

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

   12. Applied rewrites 74.9%

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

   13. Taylor expanded in alpha around inf

       \[\leadsto \mathsf{fma}\left(\frac{\alpha}{i}, \frac{-1}{8}, \frac{1}{8} \cdot \frac{\alpha}{i}\right) \]
   14. Step-by-step derivation

       1. lower-*.f64 N/A

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

       2. lower-/.f64 9.6

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

   15. Applied rewrites 9.6%

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

Alternative 11: 73.0% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 5.8 \cdot 10^{+253}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\beta + \alpha, -0.125, 0.125 \cdot \beta\right)}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 5.8e+253)
   0.0625
   (/ (fma (+ beta alpha) -0.125 (* 0.125 beta)) i)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 5.8e+253) {
		tmp = 0.0625;
	} else {
		tmp = fma((beta + alpha), -0.125, (0.125 * beta)) / i;
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 5.8e+253)
		tmp = 0.0625;
	else
		tmp = Float64(fma(Float64(beta + alpha), -0.125, Float64(0.125 * beta)) / i);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 5.8e+253], 0.0625, N[(N[(N[(beta + alpha), $MachinePrecision] * -0.125 + N[(0.125 * beta), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.79999999999999976e253

   1. Initial program 15.9%

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

   2. Taylor expanded in i around inf

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

   4. Applied rewrites 71.6%

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

   if 5.79999999999999976e253 < beta

   1. Initial program 15.9%

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

   2. 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 77.6

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

   4. Applied rewrites 77.6%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. metadata-eval 77.6

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

   6. Applied rewrites 77.6%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-*.f64 N/A

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

      9. add-to-fraction-rev N/A

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

      10. lift-fma.f64 N/A

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

      11. div-add-rev N/A

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

      12. lower-/.f64 N/A

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

   8. Applied rewrites 77.7%

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

   9. Taylor expanded in beta around inf

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

       1. lower-*.f64 6.1

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

   11. Applied rewrites 6.1%

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

Alternative 12: 71.6% 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 15.9%

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

2. Taylor expanded in i around inf

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

4. Applied rewrites 71.6%

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

Reproduce

herbie shell --seed 2025164 
(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)))