Octave 3.8, jcobi/4

Percentage Accurate: 16.5% → 83.7%

Time: 5.1s

Alternatives: 12

Speedup: 75.4×

Specification

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

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 16.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 83.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(\alpha, \beta\right) \cdot \mathsf{min}\left(\alpha, \beta\right)\\ t_1 := \mathsf{min}\left(\alpha, \beta\right) + \mathsf{max}\left(\alpha, \beta\right)\\ t_2 := t\_1 + 2 \cdot i\\ t_3 := i \cdot \left(t\_1 + i\right)\\ t_4 := \mathsf{max}\left(\alpha, \beta\right) + \mathsf{min}\left(\alpha, \beta\right)\\ t_5 := \mathsf{fma}\left(2, i, t\_4\right)\\ t_6 := t\_2 \cdot t\_2\\ t_7 := t\_4 + i\\ \mathbf{if}\;\frac{\frac{t\_3 \cdot \left(t\_0 + t\_3\right)}{t\_6}}{t\_6 - 1} \leq \infty:\\ \;\;\;\;\frac{1}{\frac{t\_5 \cdot t\_5}{\left(\frac{\mathsf{fma}\left(t\_7, i, t\_0\right)}{\mathsf{fma}\left(t\_5, t\_5, -1\right)} \cdot t\_7\right) \cdot i}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{max}\left(\alpha, \beta\right), 0.125, 0.0625 \cdot i\right) - \mathsf{max}\left(\alpha, \beta\right) \cdot 0.125}{i}\\ \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (* (fmax alpha beta) (fmin alpha beta)))
        (t_1 (+ (fmin alpha beta) (fmax alpha beta)))
        (t_2 (+ t_1 (* 2.0 i)))
        (t_3 (* i (+ t_1 i)))
        (t_4 (+ (fmax alpha beta) (fmin alpha beta)))
        (t_5 (fma 2.0 i t_4))
        (t_6 (* t_2 t_2))
        (t_7 (+ t_4 i)))
   (if (<= (/ (/ (* t_3 (+ t_0 t_3)) t_6) (- t_6 1.0)) INFINITY)
     (/
      1.0
      (/ (* t_5 t_5) (* (* (/ (fma t_7 i t_0) (fma t_5 t_5 -1.0)) t_7) i)))
     (/
      (-
       (fma (fmax alpha beta) 0.125 (* 0.0625 i))
       (* (fmax alpha beta) 0.125))
      i))))

C

double code(double alpha, double beta, double i) {
	double t_0 = fmax(alpha, beta) * fmin(alpha, beta);
	double t_1 = fmin(alpha, beta) + fmax(alpha, beta);
	double t_2 = t_1 + (2.0 * i);
	double t_3 = i * (t_1 + i);
	double t_4 = fmax(alpha, beta) + fmin(alpha, beta);
	double t_5 = fma(2.0, i, t_4);
	double t_6 = t_2 * t_2;
	double t_7 = t_4 + i;
	double tmp;
	if ((((t_3 * (t_0 + t_3)) / t_6) / (t_6 - 1.0)) <= ((double) INFINITY)) {
		tmp = 1.0 / ((t_5 * t_5) / (((fma(t_7, i, t_0) / fma(t_5, t_5, -1.0)) * t_7) * i));
	} else {
		tmp = (fma(fmax(alpha, beta), 0.125, (0.0625 * i)) - (fmax(alpha, beta) * 0.125)) / i;
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(fmax(alpha, beta) * fmin(alpha, beta))
	t_1 = Float64(fmin(alpha, beta) + fmax(alpha, beta))
	t_2 = Float64(t_1 + Float64(2.0 * i))
	t_3 = Float64(i * Float64(t_1 + i))
	t_4 = Float64(fmax(alpha, beta) + fmin(alpha, beta))
	t_5 = fma(2.0, i, t_4)
	t_6 = Float64(t_2 * t_2)
	t_7 = Float64(t_4 + i)
	tmp = 0.0
	if (Float64(Float64(Float64(t_3 * Float64(t_0 + t_3)) / t_6) / Float64(t_6 - 1.0)) <= Inf)
		tmp = Float64(1.0 / Float64(Float64(t_5 * t_5) / Float64(Float64(Float64(fma(t_7, i, t_0) / fma(t_5, t_5, -1.0)) * t_7) * i)));
	else
		tmp = Float64(Float64(fma(fmax(alpha, beta), 0.125, Float64(0.0625 * i)) - Float64(fmax(alpha, beta) * 0.125)) / i);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[Max[alpha, beta], $MachinePrecision] * N[Min[alpha, beta], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Min[alpha, beta], $MachinePrecision] + N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(i * N[(t$95$1 + i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Max[alpha, beta], $MachinePrecision] + N[Min[alpha, beta], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(2.0 * i + t$95$4), $MachinePrecision]}, Block[{t$95$6 = N[(t$95$2 * t$95$2), $MachinePrecision]}, Block[{t$95$7 = N[(t$95$4 + i), $MachinePrecision]}, If[LessEqual[N[(N[(N[(t$95$3 * N[(t$95$0 + t$95$3), $MachinePrecision]), $MachinePrecision] / t$95$6), $MachinePrecision] / N[(t$95$6 - 1.0), $MachinePrecision]), $MachinePrecision], Infinity], N[(1.0 / N[(N[(t$95$5 * t$95$5), $MachinePrecision] / N[(N[(N[(N[(t$95$7 * i + t$95$0), $MachinePrecision] / N[(t$95$5 * t$95$5 + -1.0), $MachinePrecision]), $MachinePrecision] * t$95$7), $MachinePrecision] * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Max[alpha, beta], $MachinePrecision] * 0.125 + N[(0.0625 * i), $MachinePrecision]), $MachinePrecision] - N[(N[Max[alpha, beta], $MachinePrecision] * 0.125), $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 16.5%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

   3. Applied rewrites 38.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

   5. Applied rewrites 38.3%

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

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

   1. Initial program 16.5%

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

   2. 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 77.0%

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

   4. Applied rewrites 77.0%

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

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. add-to-fraction N/A

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

      6. lower-/.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. distribute-lft-out N/A

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

      10. lift-+.f64 N/A

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

      11. associate-*r* N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 77.0%

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lift-+.f64 77.0%

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

   6. Applied rewrites 77.0%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-+.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. sub-div N/A

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

      12. lower-/.f64 N/A

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

   8. Applied rewrites 77.1%

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

   9. Taylor expanded in alpha around 0

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

   11. Applied rewrites 72.7%

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

   12. Taylor expanded in alpha around 0

       \[\leadsto \frac{\mathsf{fma}\left(\beta, 0.125, 0.0625 \cdot i\right) - \beta \cdot 0.125}{i} \]
   13. Step-by-step derivation

   14. Applied rewrites 73.9%

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

Alternative 2: 83.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(\alpha, \beta\right) \cdot \mathsf{min}\left(\alpha, \beta\right)\\ t_1 := \mathsf{min}\left(\alpha, \beta\right) + \mathsf{max}\left(\alpha, \beta\right)\\ t_2 := t\_1 + 2 \cdot i\\ t_3 := t\_2 \cdot t\_2\\ t_4 := i \cdot \left(t\_1 + i\right)\\ t_5 := \mathsf{max}\left(\alpha, \beta\right) + \mathsf{min}\left(\alpha, \beta\right)\\ t_6 := t\_5 + i\\ t_7 := \mathsf{fma}\left(2, i, t\_5\right)\\ \mathbf{if}\;\frac{\frac{t\_4 \cdot \left(t\_0 + t\_4\right)}{t\_3}}{t\_3 - 1} \leq \infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_6, i, t\_0\right)}{\mathsf{fma}\left(t\_7, t\_7, -1\right)} \cdot \frac{t\_6 \cdot i}{t\_7 \cdot t\_7}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{max}\left(\alpha, \beta\right), 0.125, 0.0625 \cdot i\right) - \mathsf{max}\left(\alpha, \beta\right) \cdot 0.125}{i}\\ \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (* (fmax alpha beta) (fmin alpha beta)))
        (t_1 (+ (fmin alpha beta) (fmax alpha beta)))
        (t_2 (+ t_1 (* 2.0 i)))
        (t_3 (* t_2 t_2))
        (t_4 (* i (+ t_1 i)))
        (t_5 (+ (fmax alpha beta) (fmin alpha beta)))
        (t_6 (+ t_5 i))
        (t_7 (fma 2.0 i t_5)))
   (if (<= (/ (/ (* t_4 (+ t_0 t_4)) t_3) (- t_3 1.0)) INFINITY)
     (* (/ (fma t_6 i t_0) (fma t_7 t_7 -1.0)) (/ (* t_6 i) (* t_7 t_7)))
     (/
      (-
       (fma (fmax alpha beta) 0.125 (* 0.0625 i))
       (* (fmax alpha beta) 0.125))
      i))))

C

double code(double alpha, double beta, double i) {
	double t_0 = fmax(alpha, beta) * fmin(alpha, beta);
	double t_1 = fmin(alpha, beta) + fmax(alpha, beta);
	double t_2 = t_1 + (2.0 * i);
	double t_3 = t_2 * t_2;
	double t_4 = i * (t_1 + i);
	double t_5 = fmax(alpha, beta) + fmin(alpha, beta);
	double t_6 = t_5 + i;
	double t_7 = fma(2.0, i, t_5);
	double tmp;
	if ((((t_4 * (t_0 + t_4)) / t_3) / (t_3 - 1.0)) <= ((double) INFINITY)) {
		tmp = (fma(t_6, i, t_0) / fma(t_7, t_7, -1.0)) * ((t_6 * i) / (t_7 * t_7));
	} else {
		tmp = (fma(fmax(alpha, beta), 0.125, (0.0625 * i)) - (fmax(alpha, beta) * 0.125)) / i;
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(fmax(alpha, beta) * fmin(alpha, beta))
	t_1 = Float64(fmin(alpha, beta) + fmax(alpha, beta))
	t_2 = Float64(t_1 + Float64(2.0 * i))
	t_3 = Float64(t_2 * t_2)
	t_4 = Float64(i * Float64(t_1 + i))
	t_5 = Float64(fmax(alpha, beta) + fmin(alpha, beta))
	t_6 = Float64(t_5 + i)
	t_7 = fma(2.0, i, t_5)
	tmp = 0.0
	if (Float64(Float64(Float64(t_4 * Float64(t_0 + t_4)) / t_3) / Float64(t_3 - 1.0)) <= Inf)
		tmp = Float64(Float64(fma(t_6, i, t_0) / fma(t_7, t_7, -1.0)) * Float64(Float64(t_6 * i) / Float64(t_7 * t_7)));
	else
		tmp = Float64(Float64(fma(fmax(alpha, beta), 0.125, Float64(0.0625 * i)) - Float64(fmax(alpha, beta) * 0.125)) / i);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[Max[alpha, beta], $MachinePrecision] * N[Min[alpha, beta], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Min[alpha, beta], $MachinePrecision] + N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(i * N[(t$95$1 + i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[Max[alpha, beta], $MachinePrecision] + N[Min[alpha, beta], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(t$95$5 + i), $MachinePrecision]}, Block[{t$95$7 = N[(2.0 * i + t$95$5), $MachinePrecision]}, If[LessEqual[N[(N[(N[(t$95$4 * N[(t$95$0 + t$95$4), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision] / N[(t$95$3 - 1.0), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[(t$95$6 * i + t$95$0), $MachinePrecision] / N[(t$95$7 * t$95$7 + -1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$6 * i), $MachinePrecision] / N[(t$95$7 * t$95$7), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Max[alpha, beta], $MachinePrecision] * 0.125 + N[(0.0625 * i), $MachinePrecision]), $MachinePrecision] - N[(N[Max[alpha, beta], $MachinePrecision] * 0.125), $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 16.5%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

   3. Applied rewrites 38.3%

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

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

   1. Initial program 16.5%

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

   2. 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 77.0%

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

   4. Applied rewrites 77.0%

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

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. add-to-fraction N/A

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

      6. lower-/.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. distribute-lft-out N/A

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

      10. lift-+.f64 N/A

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

      11. associate-*r* N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 77.0%

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lift-+.f64 77.0%

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

   6. Applied rewrites 77.0%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-+.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. sub-div N/A

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

      12. lower-/.f64 N/A

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

   8. Applied rewrites 77.1%

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

   9. Taylor expanded in alpha around 0

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

   11. Applied rewrites 72.7%

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

   12. Taylor expanded in alpha around 0

       \[\leadsto \frac{\mathsf{fma}\left(\beta, 0.125, 0.0625 \cdot i\right) - \beta \cdot 0.125}{i} \]
   13. Step-by-step derivation

   14. Applied rewrites 73.9%

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

Alternative 3: 83.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(\alpha, \beta\right) \cdot \mathsf{min}\left(\alpha, \beta\right)\\ t_1 := \mathsf{min}\left(\alpha, \beta\right) + \mathsf{max}\left(\alpha, \beta\right)\\ t_2 := t\_1 + 2 \cdot i\\ t_3 := t\_2 \cdot t\_2\\ t_4 := i \cdot \left(t\_1 + i\right)\\ t_5 := \mathsf{max}\left(\alpha, \beta\right) + \mathsf{min}\left(\alpha, \beta\right)\\ t_6 := t\_5 + i\\ t_7 := \mathsf{fma}\left(2, i, t\_5\right)\\ \mathbf{if}\;\frac{\frac{t\_4 \cdot \left(t\_0 + t\_4\right)}{t\_3}}{t\_3 - 1} \leq \infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_6, i, t\_0\right)}{\mathsf{fma}\left(t\_7, t\_7, -1\right)} \cdot \left(i \cdot \frac{t\_6}{t\_7 \cdot t\_7}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{max}\left(\alpha, \beta\right), 0.125, 0.0625 \cdot i\right) - \mathsf{max}\left(\alpha, \beta\right) \cdot 0.125}{i}\\ \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (* (fmax alpha beta) (fmin alpha beta)))
        (t_1 (+ (fmin alpha beta) (fmax alpha beta)))
        (t_2 (+ t_1 (* 2.0 i)))
        (t_3 (* t_2 t_2))
        (t_4 (* i (+ t_1 i)))
        (t_5 (+ (fmax alpha beta) (fmin alpha beta)))
        (t_6 (+ t_5 i))
        (t_7 (fma 2.0 i t_5)))
   (if (<= (/ (/ (* t_4 (+ t_0 t_4)) t_3) (- t_3 1.0)) INFINITY)
     (* (/ (fma t_6 i t_0) (fma t_7 t_7 -1.0)) (* i (/ t_6 (* t_7 t_7))))
     (/
      (-
       (fma (fmax alpha beta) 0.125 (* 0.0625 i))
       (* (fmax alpha beta) 0.125))
      i))))

C

double code(double alpha, double beta, double i) {
	double t_0 = fmax(alpha, beta) * fmin(alpha, beta);
	double t_1 = fmin(alpha, beta) + fmax(alpha, beta);
	double t_2 = t_1 + (2.0 * i);
	double t_3 = t_2 * t_2;
	double t_4 = i * (t_1 + i);
	double t_5 = fmax(alpha, beta) + fmin(alpha, beta);
	double t_6 = t_5 + i;
	double t_7 = fma(2.0, i, t_5);
	double tmp;
	if ((((t_4 * (t_0 + t_4)) / t_3) / (t_3 - 1.0)) <= ((double) INFINITY)) {
		tmp = (fma(t_6, i, t_0) / fma(t_7, t_7, -1.0)) * (i * (t_6 / (t_7 * t_7)));
	} else {
		tmp = (fma(fmax(alpha, beta), 0.125, (0.0625 * i)) - (fmax(alpha, beta) * 0.125)) / i;
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(fmax(alpha, beta) * fmin(alpha, beta))
	t_1 = Float64(fmin(alpha, beta) + fmax(alpha, beta))
	t_2 = Float64(t_1 + Float64(2.0 * i))
	t_3 = Float64(t_2 * t_2)
	t_4 = Float64(i * Float64(t_1 + i))
	t_5 = Float64(fmax(alpha, beta) + fmin(alpha, beta))
	t_6 = Float64(t_5 + i)
	t_7 = fma(2.0, i, t_5)
	tmp = 0.0
	if (Float64(Float64(Float64(t_4 * Float64(t_0 + t_4)) / t_3) / Float64(t_3 - 1.0)) <= Inf)
		tmp = Float64(Float64(fma(t_6, i, t_0) / fma(t_7, t_7, -1.0)) * Float64(i * Float64(t_6 / Float64(t_7 * t_7))));
	else
		tmp = Float64(Float64(fma(fmax(alpha, beta), 0.125, Float64(0.0625 * i)) - Float64(fmax(alpha, beta) * 0.125)) / i);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[Max[alpha, beta], $MachinePrecision] * N[Min[alpha, beta], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Min[alpha, beta], $MachinePrecision] + N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(i * N[(t$95$1 + i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[Max[alpha, beta], $MachinePrecision] + N[Min[alpha, beta], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(t$95$5 + i), $MachinePrecision]}, Block[{t$95$7 = N[(2.0 * i + t$95$5), $MachinePrecision]}, If[LessEqual[N[(N[(N[(t$95$4 * N[(t$95$0 + t$95$4), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision] / N[(t$95$3 - 1.0), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[(t$95$6 * i + t$95$0), $MachinePrecision] / N[(t$95$7 * t$95$7 + -1.0), $MachinePrecision]), $MachinePrecision] * N[(i * N[(t$95$6 / N[(t$95$7 * t$95$7), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Max[alpha, beta], $MachinePrecision] * 0.125 + N[(0.0625 * i), $MachinePrecision]), $MachinePrecision] - N[(N[Max[alpha, beta], $MachinePrecision] * 0.125), $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 16.5%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

   3. Applied rewrites 38.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-+.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-+.f64 N/A

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

      15. lift-fma.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lift-+.f64 N/A

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

   5. Applied rewrites 38.1%

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

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

   1. Initial program 16.5%

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

   2. 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 77.0%

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

   4. Applied rewrites 77.0%

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

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. add-to-fraction N/A

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

      6. lower-/.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. distribute-lft-out N/A

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

      10. lift-+.f64 N/A

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

      11. associate-*r* N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 77.0%

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lift-+.f64 77.0%

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

   6. Applied rewrites 77.0%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-+.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. sub-div N/A

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

      12. lower-/.f64 N/A

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

   8. Applied rewrites 77.1%

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

   9. Taylor expanded in alpha around 0

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

   11. Applied rewrites 72.7%

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

   12. Taylor expanded in alpha around 0

       \[\leadsto \frac{\mathsf{fma}\left(\beta, 0.125, 0.0625 \cdot i\right) - \beta \cdot 0.125}{i} \]
   13. Step-by-step derivation

   14. Applied rewrites 73.9%

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

Alternative 4: 83.2% accurate, 0.4× speedup

Math

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

FPCore

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

C

double code(double alpha, double beta, double i) {
	double t_0 = fmin(alpha, beta) + fmax(alpha, beta);
	double t_1 = i * (t_0 + i);
	double t_2 = t_0 + (2.0 * i);
	double t_3 = t_2 * t_2;
	double t_4 = fmax(alpha, beta) + fmin(alpha, beta);
	double t_5 = fma(2.0, i, t_4);
	double tmp;
	if ((((t_1 * ((fmax(alpha, beta) * fmin(alpha, beta)) + t_1)) / t_3) / (t_3 - 1.0)) <= ((double) INFINITY)) {
		tmp = ((i * (fmax(alpha, beta) + i)) / (pow((fmax(alpha, beta) + (2.0 * i)), 2.0) - 1.0)) * (((t_4 + i) * i) / (t_5 * t_5));
	} else {
		tmp = (fma(fmax(alpha, beta), 0.125, (0.0625 * i)) - (fmax(alpha, beta) * 0.125)) / i;
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(fmin(alpha, beta) + fmax(alpha, beta))
	t_1 = Float64(i * Float64(t_0 + i))
	t_2 = Float64(t_0 + Float64(2.0 * i))
	t_3 = Float64(t_2 * t_2)
	t_4 = Float64(fmax(alpha, beta) + fmin(alpha, beta))
	t_5 = fma(2.0, i, t_4)
	tmp = 0.0
	if (Float64(Float64(Float64(t_1 * Float64(Float64(fmax(alpha, beta) * fmin(alpha, beta)) + t_1)) / t_3) / Float64(t_3 - 1.0)) <= Inf)
		tmp = Float64(Float64(Float64(i * Float64(fmax(alpha, beta) + i)) / Float64((Float64(fmax(alpha, beta) + Float64(2.0 * i)) ^ 2.0) - 1.0)) * Float64(Float64(Float64(t_4 + i) * i) / Float64(t_5 * t_5)));
	else
		tmp = Float64(Float64(fma(fmax(alpha, beta), 0.125, Float64(0.0625 * i)) - Float64(fmax(alpha, beta) * 0.125)) / i);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[Min[alpha, beta], $MachinePrecision] + N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(i * N[(t$95$0 + i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(N[Max[alpha, beta], $MachinePrecision] + N[Min[alpha, beta], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(2.0 * i + t$95$4), $MachinePrecision]}, If[LessEqual[N[(N[(N[(t$95$1 * N[(N[(N[Max[alpha, beta], $MachinePrecision] * N[Min[alpha, beta], $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision] / N[(t$95$3 - 1.0), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[(i * N[(N[Max[alpha, beta], $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision] / N[(N[Power[N[(N[Max[alpha, beta], $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(t$95$4 + i), $MachinePrecision] * i), $MachinePrecision] / N[(t$95$5 * t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Max[alpha, beta], $MachinePrecision] * 0.125 + N[(0.0625 * i), $MachinePrecision]), $MachinePrecision] - N[(N[Max[alpha, beta], $MachinePrecision] * 0.125), $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 16.5%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

   3. Applied rewrites 38.3%

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

   4. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-*.f64 35.5%

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

   6. Applied rewrites 35.5%

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

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

   1. Initial program 16.5%

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

   2. 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 77.0%

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

   4. Applied rewrites 77.0%

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

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. add-to-fraction N/A

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

      6. lower-/.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. distribute-lft-out N/A

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

      10. lift-+.f64 N/A

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

      11. associate-*r* N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 77.0%

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lift-+.f64 77.0%

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

   6. Applied rewrites 77.0%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-+.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. sub-div N/A

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

      12. lower-/.f64 N/A

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

   8. Applied rewrites 77.1%

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

   9. Taylor expanded in alpha around 0

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

   11. Applied rewrites 72.7%

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

   12. Taylor expanded in alpha around 0

       \[\leadsto \frac{\mathsf{fma}\left(\beta, 0.125, 0.0625 \cdot i\right) - \beta \cdot 0.125}{i} \]
   13. Step-by-step derivation

   14. Applied rewrites 73.9%

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

Alternative 5: 80.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(\alpha, \beta\right) + 2 \cdot i\\ t_1 := \mathsf{max}\left(\alpha, \beta\right) \cdot \mathsf{min}\left(\alpha, \beta\right)\\ t_2 := \mathsf{min}\left(\alpha, \beta\right) + \mathsf{max}\left(\alpha, \beta\right)\\ t_3 := t\_2 + 2 \cdot i\\ t_4 := t\_3 \cdot t\_3\\ t_5 := i \cdot \left(t\_2 + i\right)\\ t_6 := i \cdot \left(\mathsf{max}\left(\alpha, \beta\right) + i\right)\\ t_7 := t\_0 \cdot t\_0\\ \mathbf{if}\;\frac{\frac{t\_5 \cdot \left(t\_1 + t\_5\right)}{t\_4}}{t\_4 - 1} \leq 0.1:\\ \;\;\;\;\frac{\frac{t\_6 \cdot \left(t\_1 + t\_6\right)}{t\_7}}{t\_7 - 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{max}\left(\alpha, \beta\right), 0.125, 0.0625 \cdot i\right) - \mathsf{max}\left(\alpha, \beta\right) \cdot 0.125}{i}\\ \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (fmax alpha beta) (* 2.0 i)))
        (t_1 (* (fmax alpha beta) (fmin alpha beta)))
        (t_2 (+ (fmin alpha beta) (fmax alpha beta)))
        (t_3 (+ t_2 (* 2.0 i)))
        (t_4 (* t_3 t_3))
        (t_5 (* i (+ t_2 i)))
        (t_6 (* i (+ (fmax alpha beta) i)))
        (t_7 (* t_0 t_0)))
   (if (<= (/ (/ (* t_5 (+ t_1 t_5)) t_4) (- t_4 1.0)) 0.1)
     (/ (/ (* t_6 (+ t_1 t_6)) t_7) (- t_7 1.0))
     (/
      (-
       (fma (fmax alpha beta) 0.125 (* 0.0625 i))
       (* (fmax alpha beta) 0.125))
      i))))

C

double code(double alpha, double beta, double i) {
	double t_0 = fmax(alpha, beta) + (2.0 * i);
	double t_1 = fmax(alpha, beta) * fmin(alpha, beta);
	double t_2 = fmin(alpha, beta) + fmax(alpha, beta);
	double t_3 = t_2 + (2.0 * i);
	double t_4 = t_3 * t_3;
	double t_5 = i * (t_2 + i);
	double t_6 = i * (fmax(alpha, beta) + i);
	double t_7 = t_0 * t_0;
	double tmp;
	if ((((t_5 * (t_1 + t_5)) / t_4) / (t_4 - 1.0)) <= 0.1) {
		tmp = ((t_6 * (t_1 + t_6)) / t_7) / (t_7 - 1.0);
	} else {
		tmp = (fma(fmax(alpha, beta), 0.125, (0.0625 * i)) - (fmax(alpha, beta) * 0.125)) / i;
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(fmax(alpha, beta) + Float64(2.0 * i))
	t_1 = Float64(fmax(alpha, beta) * fmin(alpha, beta))
	t_2 = Float64(fmin(alpha, beta) + fmax(alpha, beta))
	t_3 = Float64(t_2 + Float64(2.0 * i))
	t_4 = Float64(t_3 * t_3)
	t_5 = Float64(i * Float64(t_2 + i))
	t_6 = Float64(i * Float64(fmax(alpha, beta) + i))
	t_7 = Float64(t_0 * t_0)
	tmp = 0.0
	if (Float64(Float64(Float64(t_5 * Float64(t_1 + t_5)) / t_4) / Float64(t_4 - 1.0)) <= 0.1)
		tmp = Float64(Float64(Float64(t_6 * Float64(t_1 + t_6)) / t_7) / Float64(t_7 - 1.0));
	else
		tmp = Float64(Float64(fma(fmax(alpha, beta), 0.125, Float64(0.0625 * i)) - Float64(fmax(alpha, beta) * 0.125)) / i);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[Max[alpha, beta], $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Max[alpha, beta], $MachinePrecision] * N[Min[alpha, beta], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Min[alpha, beta], $MachinePrecision] + N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 * t$95$3), $MachinePrecision]}, Block[{t$95$5 = N[(i * N[(t$95$2 + i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(i * N[(N[Max[alpha, beta], $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$7 = N[(t$95$0 * t$95$0), $MachinePrecision]}, If[LessEqual[N[(N[(N[(t$95$5 * N[(t$95$1 + t$95$5), $MachinePrecision]), $MachinePrecision] / t$95$4), $MachinePrecision] / N[(t$95$4 - 1.0), $MachinePrecision]), $MachinePrecision], 0.1], N[(N[(N[(t$95$6 * N[(t$95$1 + t$95$6), $MachinePrecision]), $MachinePrecision] / t$95$7), $MachinePrecision] / N[(t$95$7 - 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Max[alpha, beta], $MachinePrecision] * 0.125 + N[(0.0625 * i), $MachinePrecision]), $MachinePrecision] - N[(N[Max[alpha, beta], $MachinePrecision] * 0.125), $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))) < 0.10000000000000001

   1. Initial program 16.5%

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

   2. 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 16.1%

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

   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 17.5%

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

   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 17.6%

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

   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 17.7%

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

   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 16.0%

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

   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 15.4%

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

   if 0.10000000000000001 < (/.f64 (/.f64 (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i)))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)))) (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) #s(literal 1 binary64)))

   1. Initial program 16.5%

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

   2. 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 77.0%

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

   4. Applied rewrites 77.0%

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

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. add-to-fraction N/A

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

      6. lower-/.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. distribute-lft-out N/A

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

      10. lift-+.f64 N/A

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

      11. associate-*r* N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 77.0%

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lift-+.f64 77.0%

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

   6. Applied rewrites 77.0%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-+.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. sub-div N/A

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

      12. lower-/.f64 N/A

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

   8. Applied rewrites 77.1%

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

   9. Taylor expanded in alpha around 0

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

   11. Applied rewrites 72.7%

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

   12. Taylor expanded in alpha around 0

       \[\leadsto \frac{\mathsf{fma}\left(\beta, 0.125, 0.0625 \cdot i\right) - \beta \cdot 0.125}{i} \]
   13. Step-by-step derivation

   14. Applied rewrites 73.9%

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

Alternative 6: 79.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(\alpha, \beta\right) \cdot \mathsf{min}\left(\alpha, \beta\right)\\ t_1 := \mathsf{min}\left(\alpha, \beta\right) + \mathsf{max}\left(\alpha, \beta\right)\\ t_2 := t\_1 + 2 \cdot i\\ t_3 := t\_2 \cdot t\_2\\ t_4 := i \cdot \left(t\_1 + i\right)\\ t_5 := \mathsf{max}\left(\alpha, \beta\right) + \mathsf{min}\left(\alpha, \beta\right)\\ t_6 := \mathsf{fma}\left(2, i, t\_5\right)\\ \mathbf{if}\;\frac{\frac{t\_4 \cdot \left(t\_0 + t\_4\right)}{t\_3}}{t\_3 - 1} \leq 5 \cdot 10^{-10}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_5 + i, i, t\_0\right)}{\mathsf{fma}\left(t\_6, t\_6, -1\right)} \cdot \frac{i}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{max}\left(\alpha, \beta\right), 0.125, 0.0625 \cdot i\right) - \mathsf{max}\left(\alpha, \beta\right) \cdot 0.125}{i}\\ \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (* (fmax alpha beta) (fmin alpha beta)))
        (t_1 (+ (fmin alpha beta) (fmax alpha beta)))
        (t_2 (+ t_1 (* 2.0 i)))
        (t_3 (* t_2 t_2))
        (t_4 (* i (+ t_1 i)))
        (t_5 (+ (fmax alpha beta) (fmin alpha beta)))
        (t_6 (fma 2.0 i t_5)))
   (if (<= (/ (/ (* t_4 (+ t_0 t_4)) t_3) (- t_3 1.0)) 5e-10)
     (* (/ (fma (+ t_5 i) i t_0) (fma t_6 t_6 -1.0)) (/ i t_1))
     (/
      (-
       (fma (fmax alpha beta) 0.125 (* 0.0625 i))
       (* (fmax alpha beta) 0.125))
      i))))

C

double code(double alpha, double beta, double i) {
	double t_0 = fmax(alpha, beta) * fmin(alpha, beta);
	double t_1 = fmin(alpha, beta) + fmax(alpha, beta);
	double t_2 = t_1 + (2.0 * i);
	double t_3 = t_2 * t_2;
	double t_4 = i * (t_1 + i);
	double t_5 = fmax(alpha, beta) + fmin(alpha, beta);
	double t_6 = fma(2.0, i, t_5);
	double tmp;
	if ((((t_4 * (t_0 + t_4)) / t_3) / (t_3 - 1.0)) <= 5e-10) {
		tmp = (fma((t_5 + i), i, t_0) / fma(t_6, t_6, -1.0)) * (i / t_1);
	} else {
		tmp = (fma(fmax(alpha, beta), 0.125, (0.0625 * i)) - (fmax(alpha, beta) * 0.125)) / i;
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(fmax(alpha, beta) * fmin(alpha, beta))
	t_1 = Float64(fmin(alpha, beta) + fmax(alpha, beta))
	t_2 = Float64(t_1 + Float64(2.0 * i))
	t_3 = Float64(t_2 * t_2)
	t_4 = Float64(i * Float64(t_1 + i))
	t_5 = Float64(fmax(alpha, beta) + fmin(alpha, beta))
	t_6 = fma(2.0, i, t_5)
	tmp = 0.0
	if (Float64(Float64(Float64(t_4 * Float64(t_0 + t_4)) / t_3) / Float64(t_3 - 1.0)) <= 5e-10)
		tmp = Float64(Float64(fma(Float64(t_5 + i), i, t_0) / fma(t_6, t_6, -1.0)) * Float64(i / t_1));
	else
		tmp = Float64(Float64(fma(fmax(alpha, beta), 0.125, Float64(0.0625 * i)) - Float64(fmax(alpha, beta) * 0.125)) / i);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[Max[alpha, beta], $MachinePrecision] * N[Min[alpha, beta], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Min[alpha, beta], $MachinePrecision] + N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(i * N[(t$95$1 + i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[Max[alpha, beta], $MachinePrecision] + N[Min[alpha, beta], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(2.0 * i + t$95$5), $MachinePrecision]}, If[LessEqual[N[(N[(N[(t$95$4 * N[(t$95$0 + t$95$4), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision] / N[(t$95$3 - 1.0), $MachinePrecision]), $MachinePrecision], 5e-10], N[(N[(N[(N[(t$95$5 + i), $MachinePrecision] * i + t$95$0), $MachinePrecision] / N[(t$95$6 * t$95$6 + -1.0), $MachinePrecision]), $MachinePrecision] * N[(i / t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Max[alpha, beta], $MachinePrecision] * 0.125 + N[(0.0625 * i), $MachinePrecision]), $MachinePrecision] - N[(N[Max[alpha, beta], $MachinePrecision] * 0.125), $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))) < 5.00000000000000031e-10

   1. Initial program 16.5%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

   3. Applied rewrites 38.3%

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

   4. Taylor expanded in i around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 10.3%

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

   6. Applied rewrites 10.3%

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

   if 5.00000000000000031e-10 < (/.f64 (/.f64 (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i)))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)))) (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) #s(literal 1 binary64)))

   1. Initial program 16.5%

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

   2. 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 77.0%

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

   4. Applied rewrites 77.0%

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

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. add-to-fraction N/A

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

      6. lower-/.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. distribute-lft-out N/A

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

      10. lift-+.f64 N/A

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

      11. associate-*r* N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 77.0%

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lift-+.f64 77.0%

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

   6. Applied rewrites 77.0%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-+.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. sub-div N/A

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

      12. lower-/.f64 N/A

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

   8. Applied rewrites 77.1%

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

   9. Taylor expanded in alpha around 0

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

   11. Applied rewrites 72.7%

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

   12. Taylor expanded in alpha around 0

       \[\leadsto \frac{\mathsf{fma}\left(\beta, 0.125, 0.0625 \cdot i\right) - \beta \cdot 0.125}{i} \]
   13. Step-by-step derivation

   14. Applied rewrites 73.9%

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

Alternative 7: 79.5% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (fmin alpha beta) (fmax alpha beta)))
        (t_1 (+ t_0 (* 2.0 i)))
        (t_2 (* t_1 t_1))
        (t_3 (- t_2 1.0))
        (t_4 (* i (+ t_0 i))))
   (if (<=
        (/ (/ (* t_4 (+ (* (fmax alpha beta) (fmin alpha beta)) t_4)) t_2) t_3)
        5e-10)
     (/ (* -1.0 (* i (fma -1.0 (fmin alpha beta) (* -1.0 i)))) t_3)
     (/
      (-
       (fma (fmax alpha beta) 0.125 (* 0.0625 i))
       (* (fmax alpha beta) 0.125))
      i))))

C

double code(double alpha, double beta, double i) {
	double t_0 = fmin(alpha, beta) + fmax(alpha, beta);
	double t_1 = t_0 + (2.0 * i);
	double t_2 = t_1 * t_1;
	double t_3 = t_2 - 1.0;
	double t_4 = i * (t_0 + i);
	double tmp;
	if ((((t_4 * ((fmax(alpha, beta) * fmin(alpha, beta)) + t_4)) / t_2) / t_3) <= 5e-10) {
		tmp = (-1.0 * (i * fma(-1.0, fmin(alpha, beta), (-1.0 * i)))) / t_3;
	} else {
		tmp = (fma(fmax(alpha, beta), 0.125, (0.0625 * i)) - (fmax(alpha, beta) * 0.125)) / i;
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(fmin(alpha, beta) + fmax(alpha, beta))
	t_1 = Float64(t_0 + Float64(2.0 * i))
	t_2 = Float64(t_1 * t_1)
	t_3 = Float64(t_2 - 1.0)
	t_4 = Float64(i * Float64(t_0 + i))
	tmp = 0.0
	if (Float64(Float64(Float64(t_4 * Float64(Float64(fmax(alpha, beta) * fmin(alpha, beta)) + t_4)) / t_2) / t_3) <= 5e-10)
		tmp = Float64(Float64(-1.0 * Float64(i * fma(-1.0, fmin(alpha, beta), Float64(-1.0 * i)))) / t_3);
	else
		tmp = Float64(Float64(fma(fmax(alpha, beta), 0.125, Float64(0.0625 * i)) - Float64(fmax(alpha, beta) * 0.125)) / i);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[Min[alpha, beta], $MachinePrecision] + N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 - 1.0), $MachinePrecision]}, Block[{t$95$4 = N[(i * N[(t$95$0 + i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(t$95$4 * N[(N[(N[Max[alpha, beta], $MachinePrecision] * N[Min[alpha, beta], $MachinePrecision]), $MachinePrecision] + t$95$4), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision] / t$95$3), $MachinePrecision], 5e-10], N[(N[(-1.0 * N[(i * N[(-1.0 * N[Min[alpha, beta], $MachinePrecision] + N[(-1.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision], N[(N[(N[(N[Max[alpha, beta], $MachinePrecision] * 0.125 + N[(0.0625 * i), $MachinePrecision]), $MachinePrecision] - N[(N[Max[alpha, beta], $MachinePrecision] * 0.125), $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))) < 5.00000000000000031e-10

   1. Initial program 16.5%

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

   2. Taylor expanded in beta around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 13.7%

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

   4. Applied rewrites 13.7%

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

   if 5.00000000000000031e-10 < (/.f64 (/.f64 (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i)))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)))) (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) #s(literal 1 binary64)))

   1. Initial program 16.5%

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

   2. 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 77.0%

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

   4. Applied rewrites 77.0%

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

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. add-to-fraction N/A

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

      6. lower-/.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. distribute-lft-out N/A

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

      10. lift-+.f64 N/A

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

      11. associate-*r* N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 77.0%

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lift-+.f64 77.0%

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

   6. Applied rewrites 77.0%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-+.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. sub-div N/A

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

      12. lower-/.f64 N/A

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

   8. Applied rewrites 77.1%

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

   9. Taylor expanded in alpha around 0

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

   11. Applied rewrites 72.7%

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

   12. Taylor expanded in alpha around 0

       \[\leadsto \frac{\mathsf{fma}\left(\beta, 0.125, 0.0625 \cdot i\right) - \beta \cdot 0.125}{i} \]
   13. Step-by-step derivation

   14. Applied rewrites 73.9%

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

Alternative 8: 79.5% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (fmin alpha beta) (fmax alpha beta)))
        (t_1 (+ t_0 (* 2.0 i)))
        (t_2 (* t_1 t_1))
        (t_3 (* i (+ t_0 i)))
        (t_4 (+ (fmax alpha beta) (fmin alpha beta)))
        (t_5 (fma 2.0 i t_4)))
   (if (<=
        (/
         (/ (* t_3 (+ (* (fmax alpha beta) (fmin alpha beta)) t_3)) t_2)
         (- t_2 1.0))
        5e-10)
     (*
      (/ (+ (fmin alpha beta) i) (fmax alpha beta))
      (/ (* (+ t_4 i) i) (* t_5 t_5)))
     (/
      (-
       (fma (fmax alpha beta) 0.125 (* 0.0625 i))
       (* (fmax alpha beta) 0.125))
      i))))

C

double code(double alpha, double beta, double i) {
	double t_0 = fmin(alpha, beta) + fmax(alpha, beta);
	double t_1 = t_0 + (2.0 * i);
	double t_2 = t_1 * t_1;
	double t_3 = i * (t_0 + i);
	double t_4 = fmax(alpha, beta) + fmin(alpha, beta);
	double t_5 = fma(2.0, i, t_4);
	double tmp;
	if ((((t_3 * ((fmax(alpha, beta) * fmin(alpha, beta)) + t_3)) / t_2) / (t_2 - 1.0)) <= 5e-10) {
		tmp = ((fmin(alpha, beta) + i) / fmax(alpha, beta)) * (((t_4 + i) * i) / (t_5 * t_5));
	} else {
		tmp = (fma(fmax(alpha, beta), 0.125, (0.0625 * i)) - (fmax(alpha, beta) * 0.125)) / i;
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(fmin(alpha, beta) + fmax(alpha, beta))
	t_1 = Float64(t_0 + Float64(2.0 * i))
	t_2 = Float64(t_1 * t_1)
	t_3 = Float64(i * Float64(t_0 + i))
	t_4 = Float64(fmax(alpha, beta) + fmin(alpha, beta))
	t_5 = fma(2.0, i, t_4)
	tmp = 0.0
	if (Float64(Float64(Float64(t_3 * Float64(Float64(fmax(alpha, beta) * fmin(alpha, beta)) + t_3)) / t_2) / Float64(t_2 - 1.0)) <= 5e-10)
		tmp = Float64(Float64(Float64(fmin(alpha, beta) + i) / fmax(alpha, beta)) * Float64(Float64(Float64(t_4 + i) * i) / Float64(t_5 * t_5)));
	else
		tmp = Float64(Float64(fma(fmax(alpha, beta), 0.125, Float64(0.0625 * i)) - Float64(fmax(alpha, beta) * 0.125)) / i);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[Min[alpha, beta], $MachinePrecision] + N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(i * N[(t$95$0 + i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Max[alpha, beta], $MachinePrecision] + N[Min[alpha, beta], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(2.0 * i + t$95$4), $MachinePrecision]}, If[LessEqual[N[(N[(N[(t$95$3 * N[(N[(N[Max[alpha, beta], $MachinePrecision] * N[Min[alpha, beta], $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision] / N[(t$95$2 - 1.0), $MachinePrecision]), $MachinePrecision], 5e-10], N[(N[(N[(N[Min[alpha, beta], $MachinePrecision] + i), $MachinePrecision] / N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(t$95$4 + i), $MachinePrecision] * i), $MachinePrecision] / N[(t$95$5 * t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Max[alpha, beta], $MachinePrecision] * 0.125 + N[(0.0625 * i), $MachinePrecision]), $MachinePrecision] - N[(N[Max[alpha, beta], $MachinePrecision] * 0.125), $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))) < 5.00000000000000031e-10

   1. Initial program 16.5%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

   3. Applied rewrites 38.3%

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

   4. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 6.5%

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

   6. Applied rewrites 6.5%

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

   if 5.00000000000000031e-10 < (/.f64 (/.f64 (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i)))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)))) (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) #s(literal 1 binary64)))

   1. Initial program 16.5%

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

   2. 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 77.0%

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

   4. Applied rewrites 77.0%

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

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. add-to-fraction N/A

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

      6. lower-/.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. distribute-lft-out N/A

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

      10. lift-+.f64 N/A

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

      11. associate-*r* N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 77.0%

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lift-+.f64 77.0%

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

   6. Applied rewrites 77.0%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-+.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. sub-div N/A

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

      12. lower-/.f64 N/A

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

   8. Applied rewrites 77.1%

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

   9. Taylor expanded in alpha around 0

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

   11. Applied rewrites 72.7%

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

   12. Taylor expanded in alpha around 0

       \[\leadsto \frac{\mathsf{fma}\left(\beta, 0.125, 0.0625 \cdot i\right) - \beta \cdot 0.125}{i} \]
   13. Step-by-step derivation

   14. Applied rewrites 73.9%

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

Alternative 9: 79.5% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (fmin alpha beta) (fmax alpha beta)))
        (t_1 (+ t_0 (* 2.0 i)))
        (t_2 (* t_1 t_1))
        (t_3 (* i (+ t_0 i))))
   (if (<=
        (/
         (/ (* t_3 (+ (* (fmax alpha beta) (fmin alpha beta)) t_3)) t_2)
         (- t_2 1.0))
        5e-10)
     (/ (* i (+ (fmin alpha beta) i)) (pow (fmax alpha beta) 2.0))
     (/
      (-
       (fma (fmax alpha beta) 0.125 (* 0.0625 i))
       (* (fmax alpha beta) 0.125))
      i))))

C

double code(double alpha, double beta, double i) {
	double t_0 = fmin(alpha, beta) + fmax(alpha, beta);
	double t_1 = t_0 + (2.0 * i);
	double t_2 = t_1 * t_1;
	double t_3 = i * (t_0 + i);
	double tmp;
	if ((((t_3 * ((fmax(alpha, beta) * fmin(alpha, beta)) + t_3)) / t_2) / (t_2 - 1.0)) <= 5e-10) {
		tmp = (i * (fmin(alpha, beta) + i)) / pow(fmax(alpha, beta), 2.0);
	} else {
		tmp = (fma(fmax(alpha, beta), 0.125, (0.0625 * i)) - (fmax(alpha, beta) * 0.125)) / i;
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(fmin(alpha, beta) + fmax(alpha, beta))
	t_1 = Float64(t_0 + Float64(2.0 * i))
	t_2 = Float64(t_1 * t_1)
	t_3 = Float64(i * Float64(t_0 + i))
	tmp = 0.0
	if (Float64(Float64(Float64(t_3 * Float64(Float64(fmax(alpha, beta) * fmin(alpha, beta)) + t_3)) / t_2) / Float64(t_2 - 1.0)) <= 5e-10)
		tmp = Float64(Float64(i * Float64(fmin(alpha, beta) + i)) / (fmax(alpha, beta) ^ 2.0));
	else
		tmp = Float64(Float64(fma(fmax(alpha, beta), 0.125, Float64(0.0625 * i)) - Float64(fmax(alpha, beta) * 0.125)) / i);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[Min[alpha, beta], $MachinePrecision] + N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(i * N[(t$95$0 + i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(t$95$3 * N[(N[(N[Max[alpha, beta], $MachinePrecision] * N[Min[alpha, beta], $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision] / N[(t$95$2 - 1.0), $MachinePrecision]), $MachinePrecision], 5e-10], N[(N[(i * N[(N[Min[alpha, beta], $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision] / N[Power[N[Max[alpha, beta], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Max[alpha, beta], $MachinePrecision] * 0.125 + N[(0.0625 * i), $MachinePrecision]), $MachinePrecision] - N[(N[Max[alpha, beta], $MachinePrecision] * 0.125), $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))) < 5.00000000000000031e-10

   1. Initial program 16.5%

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

   2. 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 5.00000000000000031e-10 < (/.f64 (/.f64 (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i)))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)))) (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) #s(literal 1 binary64)))

   1. Initial program 16.5%

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

   2. 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 77.0%

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

   4. Applied rewrites 77.0%

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

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. add-to-fraction N/A

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

      6. lower-/.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. distribute-lft-out N/A

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

      10. lift-+.f64 N/A

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

      11. associate-*r* N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 77.0%

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lift-+.f64 77.0%

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

   6. Applied rewrites 77.0%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-+.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. sub-div N/A

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

      12. lower-/.f64 N/A

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

   8. Applied rewrites 77.1%

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

   9. Taylor expanded in alpha around 0

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

   11. Applied rewrites 72.7%

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

   12. Taylor expanded in alpha around 0

       \[\leadsto \frac{\mathsf{fma}\left(\beta, 0.125, 0.0625 \cdot i\right) - \beta \cdot 0.125}{i} \]
   13. Step-by-step derivation

   14. Applied rewrites 73.9%

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

Alternative 10: 77.1% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 16.5%

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

2. 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 77.0%

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

4. Applied rewrites 77.0%

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

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

   3. lift-/.f64 N/A

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

   4. associate-*r/ N/A

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

   5. add-to-fraction N/A

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

   6. lower-/.f64 N/A

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

   7. lift-fma.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. distribute-lft-out N/A

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

   10. lift-+.f64 N/A

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

   11. associate-*r* N/A

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

   12. metadata-eval N/A

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

   13. lower-fma.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 77.0%

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

   16. lift-+.f64 N/A

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

   17. +-commutative N/A

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

   18. lift-+.f64 77.0%

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

6. Applied rewrites 77.0%

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

   1. lift--.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. associate-*r/ N/A

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

   6. lift-+.f64 N/A

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

   7. +-commutative N/A

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

   8. lift-+.f64 N/A

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

   9. *-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. sub-div N/A

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

   12. lower-/.f64 N/A

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

8. Applied rewrites 77.1%

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

9. Taylor expanded in alpha around 0

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

11. Applied rewrites 72.7%

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

12. Taylor expanded in alpha around 0

    \[\leadsto \frac{\mathsf{fma}\left(\beta, 0.125, 0.0625 \cdot i\right) - \beta \cdot 0.125}{i} \]
13. Step-by-step derivation

14. Applied rewrites 73.9%

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

Alternative 11: 73.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(\alpha, \beta\right) + \mathsf{max}\left(\alpha, \beta\right)\\ \mathbf{if}\;\mathsf{max}\left(\alpha, \beta\right) \leq 3.6 \cdot 10^{+215}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.125, t\_0, 0.125 \cdot t\_0\right)}{i}\\ \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (fmin alpha beta) (fmax alpha beta))))
   (if (<= (fmax alpha beta) 3.6e+215)
     0.0625
     (/ (fma -0.125 t_0 (* 0.125 t_0)) i))))

C

double code(double alpha, double beta, double i) {
	double t_0 = fmin(alpha, beta) + fmax(alpha, beta);
	double tmp;
	if (fmax(alpha, beta) <= 3.6e+215) {
		tmp = 0.0625;
	} else {
		tmp = fma(-0.125, t_0, (0.125 * t_0)) / i;
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(fmin(alpha, beta) + fmax(alpha, beta))
	tmp = 0.0
	if (fmax(alpha, beta) <= 3.6e+215)
		tmp = 0.0625;
	else
		tmp = Float64(fma(-0.125, t_0, Float64(0.125 * t_0)) / i);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[Min[alpha, beta], $MachinePrecision] + N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Max[alpha, beta], $MachinePrecision], 3.6e+215], 0.0625, N[(N[(-0.125 * t$95$0 + N[(0.125 * t$95$0), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.59999999999999974e215

   1. Initial program 16.5%

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

   2. Taylor expanded in i around inf

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

   4. Applied rewrites 70.3%

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

   if 3.59999999999999974e215 < beta

   1. Initial program 16.5%

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

   2. 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 77.0%

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

   4. Applied rewrites 77.0%

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

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. add-to-fraction N/A

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

      6. lower-/.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. distribute-lft-out N/A

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

      10. lift-+.f64 N/A

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

      11. associate-*r* N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 77.0%

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lift-+.f64 77.0%

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

   6. Applied rewrites 77.0%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. lift-/.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. div-add N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-+.f64 N/A

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

      11. associate-*r/ N/A

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

      12. lift-/.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-+l+ N/A

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

      15. *-commutative N/A

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

      16. associate-/l* N/A

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

      17. lower-fma.f64 N/A

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

   8. Applied rewrites 70.2%

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

   9. Taylor expanded in i around 0

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

       1. lower-/.f64 N/A

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

       2. lower-fma.f64 N/A

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

       3. lower-+.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. lower-+.f64 10.2%

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

   11. Applied rewrites 10.2%

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

Alternative 12: 70.3% accurate, 75.4× speedup

Math

\[0.0625 \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[alpha_, beta_, i_] := 0.0625

Derivation

1. Initial program 16.5%

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

2. Taylor expanded in i around inf

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

4. Applied rewrites 70.3%

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

Reproduce

herbie shell --seed 2025182 
(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)))