Octave 3.8, jcobi/4

Percentage Accurate: 16.3% → 83.0%

Time: 5.7s

Alternatives: 11

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: 16.3% 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: 83.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(\alpha, \beta\right) + \mathsf{max}\left(\alpha, \beta\right)\\ t_1 := t\_0 + 2 \cdot i\\ t_2 := t\_1 \cdot t\_1\\ t_3 := i \cdot \left(t\_0 + i\right)\\ t_4 := \mathsf{max}\left(\alpha, \beta\right) + \mathsf{min}\left(\alpha, \beta\right)\\ t_5 := t\_4 + i\\ t_6 := \mathsf{max}\left(\alpha, \beta\right) \cdot \mathsf{min}\left(\alpha, \beta\right)\\ t_7 := \mathsf{fma}\left(2, i, t\_4\right)\\ \mathbf{if}\;\frac{\frac{t\_3 \cdot \left(t\_6 + t\_3\right)}{t\_2}}{t\_2 - 1} \leq \infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_5, i, t\_6\right) \cdot \frac{t\_5 \cdot i}{t\_7 \cdot t\_7}}{\mathsf{fma}\left(t\_7, t\_7, -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(0.0625 + 0.125 \cdot \frac{\mathsf{min}\left(\alpha, \beta\right)}{i}\right) \cdot \left(1 - \frac{1}{\frac{0.5}{\frac{t\_4}{i}} - -1}\right)\\ \end{array} \]

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 (+ t_4 i))
        (t_6 (* (fmax alpha beta) (fmin alpha beta)))
        (t_7 (fma 2.0 i t_4)))
   (if (<= (/ (/ (* t_3 (+ t_6 t_3)) t_2) (- t_2 1.0)) INFINITY)
     (/ (* (fma t_5 i t_6) (/ (* t_5 i) (* t_7 t_7))) (fma t_7 t_7 -1.0))
     (*
      (+ 0.0625 (* 0.125 (/ (fmin alpha beta) i)))
      (- 1.0 (/ 1.0 (- (/ 0.5 (/ t_4 i)) -1.0)))))))

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 = t_4 + i;
	double t_6 = fmax(alpha, beta) * fmin(alpha, beta);
	double t_7 = fma(2.0, i, t_4);
	double tmp;
	if ((((t_3 * (t_6 + t_3)) / t_2) / (t_2 - 1.0)) <= ((double) INFINITY)) {
		tmp = (fma(t_5, i, t_6) * ((t_5 * i) / (t_7 * t_7))) / fma(t_7, t_7, -1.0);
	} else {
		tmp = (0.0625 + (0.125 * (fmin(alpha, beta) / i))) * (1.0 - (1.0 / ((0.5 / (t_4 / i)) - -1.0)));
	}
	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 = Float64(t_4 + i)
	t_6 = Float64(fmax(alpha, beta) * fmin(alpha, beta))
	t_7 = fma(2.0, i, t_4)
	tmp = 0.0
	if (Float64(Float64(Float64(t_3 * Float64(t_6 + t_3)) / t_2) / Float64(t_2 - 1.0)) <= Inf)
		tmp = Float64(Float64(fma(t_5, i, t_6) * Float64(Float64(t_5 * i) / Float64(t_7 * t_7))) / fma(t_7, t_7, -1.0));
	else
		tmp = Float64(Float64(0.0625 + Float64(0.125 * Float64(fmin(alpha, beta) / i))) * Float64(1.0 - Float64(1.0 / Float64(Float64(0.5 / Float64(t_4 / i)) - -1.0))));
	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[(t$95$4 + i), $MachinePrecision]}, Block[{t$95$6 = N[(N[Max[alpha, beta], $MachinePrecision] * N[Min[alpha, beta], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$7 = N[(2.0 * i + t$95$4), $MachinePrecision]}, If[LessEqual[N[(N[(N[(t$95$3 * N[(t$95$6 + t$95$3), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision] / N[(t$95$2 - 1.0), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[(t$95$5 * i + t$95$6), $MachinePrecision] * N[(N[(t$95$5 * i), $MachinePrecision] / N[(t$95$7 * t$95$7), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$7 * t$95$7 + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(0.0625 + N[(0.125 * N[(N[Min[alpha, beta], $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(1.0 / N[(N[(0.5 / N[(t$95$4 / i), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 16.3%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. associate-/l* N/A

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

   3. Applied rewrites 36.3%

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

   5. Applied rewrites 38.2%

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

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

   1. Initial program 16.3%

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

   2. Taylor expanded in i around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 76.7

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

   4. Applied rewrites 76.7%

      \[\leadsto \color{blue}{\left(0.0625 + 0.0625 \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i}} \]
   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. sub-to-mult N/A

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

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

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

   6. Applied rewrites 76.7%

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

   8. Applied rewrites 76.7%

      \[\leadsto \mathsf{fma}\left(\beta + \alpha, \frac{0.125}{i}, 0.0625\right) \cdot \color{blue}{\left(1 - \frac{1}{\frac{0.5}{\frac{\beta + \alpha}{i}} - -1}\right)} \]

   9. Taylor expanded in beta around 0

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

       1. lower-+.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-/.f64 76.4

          \[\leadsto \left(0.0625 + 0.125 \cdot \frac{\alpha}{i}\right) \cdot \left(1 - \frac{1}{\frac{0.5}{\frac{\beta + \alpha}{i}} - -1}\right) \]

   11. Applied rewrites 76.4%

       \[\leadsto \left(0.0625 + 0.125 \cdot \frac{\alpha}{i}\right) \cdot \left(\color{blue}{1} - \frac{1}{\frac{0.5}{\frac{\beta + \alpha}{i}} - -1}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 83.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(\alpha, \beta\right) \cdot \mathsf{min}\left(\alpha, \beta\right)\\ t_1 := \mathsf{min}\left(\alpha, \beta\right) + \mathsf{max}\left(\alpha, \beta\right)\\ t_2 := t\_1 + 2 \cdot i\\ t_3 := t\_2 \cdot t\_2\\ t_4 := i \cdot \left(t\_1 + i\right)\\ t_5 := \mathsf{max}\left(\alpha, \beta\right) + \mathsf{min}\left(\alpha, \beta\right)\\ t_6 := t\_5 + i\\ t_7 := \mathsf{fma}\left(2, i, t\_5\right)\\ \mathbf{if}\;\frac{\frac{t\_4 \cdot \left(t\_0 + t\_4\right)}{t\_3}}{t\_3 - 1} \leq \infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_6, i, t\_0\right)}{\mathsf{fma}\left(t\_7, t\_7, -1\right)} \cdot \frac{t\_6 \cdot i}{t\_7 \cdot t\_7}\\ \mathbf{else}:\\ \;\;\;\;\left(0.0625 + 0.125 \cdot \frac{\mathsf{min}\left(\alpha, \beta\right)}{i}\right) \cdot \left(1 - \frac{1}{\frac{0.5}{\frac{t\_5}{i}} - -1}\right)\\ \end{array} \]

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)))
     (*
      (+ 0.0625 (* 0.125 (/ (fmin alpha beta) i)))
      (- 1.0 (/ 1.0 (- (/ 0.5 (/ t_5 i)) -1.0)))))))

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 = (0.0625 + (0.125 * (fmin(alpha, beta) / i))) * (1.0 - (1.0 / ((0.5 / (t_5 / i)) - -1.0)));
	}
	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(0.0625 + Float64(0.125 * Float64(fmin(alpha, beta) / i))) * Float64(1.0 - Float64(1.0 / Float64(Float64(0.5 / Float64(t_5 / i)) - -1.0))));
	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[(0.0625 + N[(0.125 * N[(N[Min[alpha, beta], $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(1.0 / N[(N[(0.5 / N[(t$95$5 / i), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 16.3%

      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   2. 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 38.2%

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

   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 16.3%

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

   2. Taylor expanded in i around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 76.7

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

   4. Applied rewrites 76.7%

      \[\leadsto \color{blue}{\left(0.0625 + 0.0625 \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i}} \]
   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. sub-to-mult N/A

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

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

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

   6. Applied rewrites 76.7%

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

   8. Applied rewrites 76.7%

      \[\leadsto \mathsf{fma}\left(\beta + \alpha, \frac{0.125}{i}, 0.0625\right) \cdot \color{blue}{\left(1 - \frac{1}{\frac{0.5}{\frac{\beta + \alpha}{i}} - -1}\right)} \]

   9. Taylor expanded in beta around 0

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

       1. lower-+.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-/.f64 76.4

          \[\leadsto \left(0.0625 + 0.125 \cdot \frac{\alpha}{i}\right) \cdot \left(1 - \frac{1}{\frac{0.5}{\frac{\beta + \alpha}{i}} - -1}\right) \]

   11. Applied rewrites 76.4%

       \[\leadsto \left(0.0625 + 0.125 \cdot \frac{\alpha}{i}\right) \cdot \left(\color{blue}{1} - \frac{1}{\frac{0.5}{\frac{\beta + \alpha}{i}} - -1}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 82.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(\alpha, \beta\right) + \mathsf{max}\left(\alpha, \beta\right)\\ t_1 := t\_0 + 2 \cdot i\\ t_2 := t\_1 \cdot t\_1\\ t_3 := i \cdot \left(t\_0 + i\right)\\ t_4 := \mathsf{max}\left(\alpha, \beta\right) + \mathsf{min}\left(\alpha, \beta\right)\\ t_5 := t\_4 + i\\ t_6 := \mathsf{max}\left(\alpha, \beta\right) \cdot \mathsf{min}\left(\alpha, \beta\right)\\ t_7 := \mathsf{fma}\left(2, i, t\_4\right)\\ \mathbf{if}\;\frac{\frac{t\_3 \cdot \left(t\_6 + t\_3\right)}{t\_2}}{t\_2 - 1} \leq \infty:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(t\_5, i, t\_6\right)}{t\_7 \cdot t\_7} \cdot t\_5}{\mathsf{fma}\left(t\_7, t\_7, -1\right)} \cdot i\\ \mathbf{else}:\\ \;\;\;\;\left(0.0625 + 0.125 \cdot \frac{\mathsf{min}\left(\alpha, \beta\right)}{i}\right) \cdot \left(1 - \frac{1}{\frac{0.5}{\frac{t\_4}{i}} - -1}\right)\\ \end{array} \]

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 (+ t_4 i))
        (t_6 (* (fmax alpha beta) (fmin alpha beta)))
        (t_7 (fma 2.0 i t_4)))
   (if (<= (/ (/ (* t_3 (+ t_6 t_3)) t_2) (- t_2 1.0)) INFINITY)
     (* (/ (* (/ (fma t_5 i t_6) (* t_7 t_7)) t_5) (fma t_7 t_7 -1.0)) i)
     (*
      (+ 0.0625 (* 0.125 (/ (fmin alpha beta) i)))
      (- 1.0 (/ 1.0 (- (/ 0.5 (/ t_4 i)) -1.0)))))))

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 = t_4 + i;
	double t_6 = fmax(alpha, beta) * fmin(alpha, beta);
	double t_7 = fma(2.0, i, t_4);
	double tmp;
	if ((((t_3 * (t_6 + t_3)) / t_2) / (t_2 - 1.0)) <= ((double) INFINITY)) {
		tmp = (((fma(t_5, i, t_6) / (t_7 * t_7)) * t_5) / fma(t_7, t_7, -1.0)) * i;
	} else {
		tmp = (0.0625 + (0.125 * (fmin(alpha, beta) / i))) * (1.0 - (1.0 / ((0.5 / (t_4 / i)) - -1.0)));
	}
	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 = Float64(t_4 + i)
	t_6 = Float64(fmax(alpha, beta) * fmin(alpha, beta))
	t_7 = fma(2.0, i, t_4)
	tmp = 0.0
	if (Float64(Float64(Float64(t_3 * Float64(t_6 + t_3)) / t_2) / Float64(t_2 - 1.0)) <= Inf)
		tmp = Float64(Float64(Float64(Float64(fma(t_5, i, t_6) / Float64(t_7 * t_7)) * t_5) / fma(t_7, t_7, -1.0)) * i);
	else
		tmp = Float64(Float64(0.0625 + Float64(0.125 * Float64(fmin(alpha, beta) / i))) * Float64(1.0 - Float64(1.0 / Float64(Float64(0.5 / Float64(t_4 / i)) - -1.0))));
	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[(t$95$4 + i), $MachinePrecision]}, Block[{t$95$6 = N[(N[Max[alpha, beta], $MachinePrecision] * N[Min[alpha, beta], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$7 = N[(2.0 * i + t$95$4), $MachinePrecision]}, If[LessEqual[N[(N[(N[(t$95$3 * N[(t$95$6 + t$95$3), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision] / N[(t$95$2 - 1.0), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[(N[(N[(t$95$5 * i + t$95$6), $MachinePrecision] / N[(t$95$7 * t$95$7), $MachinePrecision]), $MachinePrecision] * t$95$5), $MachinePrecision] / N[(t$95$7 * t$95$7 + -1.0), $MachinePrecision]), $MachinePrecision] * i), $MachinePrecision], N[(N[(0.0625 + N[(0.125 * N[(N[Min[alpha, beta], $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(1.0 / N[(N[(0.5 / N[(t$95$4 / i), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 16.3%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. associate-/l* N/A

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

   3. Applied rewrites 36.3%

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

   5. Applied rewrites 18.1%

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

   7. Applied rewrites 38.1%

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

   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 16.3%

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

   2. Taylor expanded in i around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 76.7

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

   4. Applied rewrites 76.7%

      \[\leadsto \color{blue}{\left(0.0625 + 0.0625 \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i}} \]
   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. sub-to-mult N/A

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

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

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

   6. Applied rewrites 76.7%

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

   8. Applied rewrites 76.7%

      \[\leadsto \mathsf{fma}\left(\beta + \alpha, \frac{0.125}{i}, 0.0625\right) \cdot \color{blue}{\left(1 - \frac{1}{\frac{0.5}{\frac{\beta + \alpha}{i}} - -1}\right)} \]

   9. Taylor expanded in beta around 0

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

       1. lower-+.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-/.f64 76.4

          \[\leadsto \left(0.0625 + 0.125 \cdot \frac{\alpha}{i}\right) \cdot \left(1 - \frac{1}{\frac{0.5}{\frac{\beta + \alpha}{i}} - -1}\right) \]

   11. Applied rewrites 76.4%

       \[\leadsto \left(0.0625 + 0.125 \cdot \frac{\alpha}{i}\right) \cdot \left(\color{blue}{1} - \frac{1}{\frac{0.5}{\frac{\beta + \alpha}{i}} - -1}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 81.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := i \cdot \left(\left(\alpha + \beta\right) + i\right)\\ t_1 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_2 := t\_1 \cdot t\_1\\ t_3 := \mathsf{fma}\left(2, i, \beta + \alpha\right)\\ \mathbf{if}\;\frac{\frac{t\_0 \cdot \left(\beta \cdot \alpha + t\_0\right)}{t\_2}}{t\_2 - 1} \leq \infty:\\ \;\;\;\;\left(\left(\left(\beta + \alpha\right) + i\right) \cdot i\right) \cdot \frac{\frac{\mathsf{fma}\left(\alpha, \beta + i, i \cdot \left(\beta + i\right)\right)}{t\_3 \cdot t\_3}}{\mathsf{fma}\left(t\_3, t\_3, -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(0.0625 + 0.125 \cdot \frac{\alpha}{i}\right) \cdot \left(1 - \frac{1}{\frac{0.5}{\frac{\beta + \alpha}{i}} - -1}\right)\\ \end{array} \]

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_3 (fma 2.0 i (+ beta alpha))))
   (if (<= (/ (/ (* t_0 (+ (* beta alpha) t_0)) t_2) (- t_2 1.0)) INFINITY)
     (*
      (* (+ (+ beta alpha) i) i)
      (/
       (/ (fma alpha (+ beta i) (* i (+ beta i))) (* t_3 t_3))
       (fma t_3 t_3 -1.0)))
     (*
      (+ 0.0625 (* 0.125 (/ alpha i)))
      (- 1.0 (/ 1.0 (- (/ 0.5 (/ (+ beta alpha) i)) -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;
	double t_3 = fma(2.0, i, (beta + alpha));
	double tmp;
	if ((((t_0 * ((beta * alpha) + t_0)) / t_2) / (t_2 - 1.0)) <= ((double) INFINITY)) {
		tmp = (((beta + alpha) + i) * i) * ((fma(alpha, (beta + i), (i * (beta + i))) / (t_3 * t_3)) / fma(t_3, t_3, -1.0));
	} else {
		tmp = (0.0625 + (0.125 * (alpha / i))) * (1.0 - (1.0 / ((0.5 / ((beta + alpha) / i)) - -1.0)));
	}
	return tmp;
}

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)
	t_3 = fma(2.0, i, Float64(beta + alpha))
	tmp = 0.0
	if (Float64(Float64(Float64(t_0 * Float64(Float64(beta * alpha) + t_0)) / t_2) / Float64(t_2 - 1.0)) <= Inf)
		tmp = Float64(Float64(Float64(Float64(beta + alpha) + i) * i) * Float64(Float64(fma(alpha, Float64(beta + i), Float64(i * Float64(beta + i))) / Float64(t_3 * t_3)) / fma(t_3, t_3, -1.0)));
	else
		tmp = Float64(Float64(0.0625 + Float64(0.125 * Float64(alpha / i))) * Float64(1.0 - Float64(1.0 / Float64(Float64(0.5 / Float64(Float64(beta + alpha) / i)) - -1.0))));
	end
	return tmp
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]}, Block[{t$95$3 = N[(2.0 * i + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[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], Infinity], N[(N[(N[(N[(beta + alpha), $MachinePrecision] + i), $MachinePrecision] * i), $MachinePrecision] * N[(N[(N[(alpha * N[(beta + i), $MachinePrecision] + N[(i * N[(beta + i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$3 * t$95$3), $MachinePrecision]), $MachinePrecision] / N[(t$95$3 * t$95$3 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.0625 + N[(0.125 * N[(alpha / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(1.0 / N[(N[(0.5 / N[(N[(beta + alpha), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 16.3%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. associate-/l* N/A

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

   3. Applied rewrites 36.3%

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

   4. Taylor expanded in alpha around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 36.3

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

   6. Applied rewrites 36.3%

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

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

   1. Initial program 16.3%

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

   2. Taylor expanded in i around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 76.7

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

   4. Applied rewrites 76.7%

      \[\leadsto \color{blue}{\left(0.0625 + 0.0625 \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i}} \]
   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. sub-to-mult N/A

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

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

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

   6. Applied rewrites 76.7%

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

   8. Applied rewrites 76.7%

      \[\leadsto \mathsf{fma}\left(\beta + \alpha, \frac{0.125}{i}, 0.0625\right) \cdot \color{blue}{\left(1 - \frac{1}{\frac{0.5}{\frac{\beta + \alpha}{i}} - -1}\right)} \]

   9. Taylor expanded in beta around 0

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

       1. lower-+.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-/.f64 76.4

          \[\leadsto \left(0.0625 + 0.125 \cdot \frac{\alpha}{i}\right) \cdot \left(1 - \frac{1}{\frac{0.5}{\frac{\beta + \alpha}{i}} - -1}\right) \]

   11. Applied rewrites 76.4%

       \[\leadsto \left(0.0625 + 0.125 \cdot \frac{\alpha}{i}\right) \cdot \left(\color{blue}{1} - \frac{1}{\frac{0.5}{\frac{\beta + \alpha}{i}} - -1}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 80.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(\alpha, \beta\right) + 2 \cdot i\\ t_1 := \mathsf{max}\left(\alpha, \beta\right) \cdot \mathsf{min}\left(\alpha, \beta\right)\\ t_2 := 0.125 \cdot \left(\mathsf{max}\left(\alpha, \beta\right) + \mathsf{min}\left(\alpha, \beta\right)\right)\\ t_3 := \mathsf{min}\left(\alpha, \beta\right) + \mathsf{max}\left(\alpha, \beta\right)\\ t_4 := t\_3 + 2 \cdot i\\ t_5 := t\_4 \cdot t\_4\\ t_6 := i \cdot \left(t\_3 + i\right)\\ t_7 := i \cdot \left(\mathsf{max}\left(\alpha, \beta\right) + i\right)\\ t_8 := t\_0 \cdot t\_0\\ \mathbf{if}\;\frac{\frac{t\_6 \cdot \left(t\_1 + t\_6\right)}{t\_5}}{t\_5 - 1} \leq 0.1:\\ \;\;\;\;\frac{\frac{t\_7 \cdot \left(t\_1 + t\_7\right)}{t\_8}}{t\_8 - 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.0625, i, t\_2\right) - t\_2}{i}\\ \end{array} \]

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 (* 0.125 (+ (fmax alpha beta) (fmin alpha beta))))
        (t_3 (+ (fmin alpha beta) (fmax alpha beta)))
        (t_4 (+ t_3 (* 2.0 i)))
        (t_5 (* t_4 t_4))
        (t_6 (* i (+ t_3 i)))
        (t_7 (* i (+ (fmax alpha beta) i)))
        (t_8 (* t_0 t_0)))
   (if (<= (/ (/ (* t_6 (+ t_1 t_6)) t_5) (- t_5 1.0)) 0.1)
     (/ (/ (* t_7 (+ t_1 t_7)) t_8) (- t_8 1.0))
     (/ (- (fma 0.0625 i t_2) t_2) 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 = 0.125 * (fmax(alpha, beta) + fmin(alpha, beta));
	double t_3 = fmin(alpha, beta) + fmax(alpha, beta);
	double t_4 = t_3 + (2.0 * i);
	double t_5 = t_4 * t_4;
	double t_6 = i * (t_3 + i);
	double t_7 = i * (fmax(alpha, beta) + i);
	double t_8 = t_0 * t_0;
	double tmp;
	if ((((t_6 * (t_1 + t_6)) / t_5) / (t_5 - 1.0)) <= 0.1) {
		tmp = ((t_7 * (t_1 + t_7)) / t_8) / (t_8 - 1.0);
	} else {
		tmp = (fma(0.0625, i, t_2) - t_2) / 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(0.125 * Float64(fmax(alpha, beta) + fmin(alpha, beta)))
	t_3 = Float64(fmin(alpha, beta) + fmax(alpha, beta))
	t_4 = Float64(t_3 + Float64(2.0 * i))
	t_5 = Float64(t_4 * t_4)
	t_6 = Float64(i * Float64(t_3 + i))
	t_7 = Float64(i * Float64(fmax(alpha, beta) + i))
	t_8 = Float64(t_0 * t_0)
	tmp = 0.0
	if (Float64(Float64(Float64(t_6 * Float64(t_1 + t_6)) / t_5) / Float64(t_5 - 1.0)) <= 0.1)
		tmp = Float64(Float64(Float64(t_7 * Float64(t_1 + t_7)) / t_8) / Float64(t_8 - 1.0));
	else
		tmp = Float64(Float64(fma(0.0625, i, t_2) - t_2) / 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[(0.125 * N[(N[Max[alpha, beta], $MachinePrecision] + N[Min[alpha, beta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Min[alpha, beta], $MachinePrecision] + N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$4 * t$95$4), $MachinePrecision]}, Block[{t$95$6 = N[(i * N[(t$95$3 + i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$7 = N[(i * N[(N[Max[alpha, beta], $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$8 = N[(t$95$0 * t$95$0), $MachinePrecision]}, If[LessEqual[N[(N[(N[(t$95$6 * N[(t$95$1 + t$95$6), $MachinePrecision]), $MachinePrecision] / t$95$5), $MachinePrecision] / N[(t$95$5 - 1.0), $MachinePrecision]), $MachinePrecision], 0.1], N[(N[(N[(t$95$7 * N[(t$95$1 + t$95$7), $MachinePrecision]), $MachinePrecision] / t$95$8), $MachinePrecision] / N[(t$95$8 - 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.0625 * i + t$95$2), $MachinePrecision] - t$95$2), $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 16.3%

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

   2. Taylor expanded in alpha around 0

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

   4. Applied rewrites 15.5%

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

   5. Taylor expanded in alpha around 0

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

   7. Applied rewrites 16.5%

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

   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 16.6%

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

   11. Taylor expanded in alpha around 0

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

   13. Applied rewrites 16.7%

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

   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 15.2%

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

   17. Taylor expanded in alpha around 0

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

   19. Applied rewrites 14.9%

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

   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 16.3%

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

   2. Taylor expanded in i around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 76.7

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

   4. Applied rewrites 76.7%

      \[\leadsto \color{blue}{\left(0.0625 + 0.0625 \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i}} \]
   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 76.7

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

      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 76.7

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

   6. Applied rewrites 76.7%

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

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      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

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

      13. lower--.f64 76.7

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 76.7

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 76.7

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

   8. Applied rewrites 76.7%

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

Alternative 6: 79.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(\alpha, \beta\right) + \mathsf{max}\left(\alpha, \beta\right)\\ t_1 := i \cdot \left(t\_0 + i\right)\\ t_2 := t\_0 + 2 \cdot i\\ t_3 := t\_2 \cdot t\_2\\ t_4 := t\_3 - 1\\ t_5 := 0.125 \cdot \left(\mathsf{max}\left(\alpha, \beta\right) + \mathsf{min}\left(\alpha, \beta\right)\right)\\ \mathbf{if}\;\frac{\frac{t\_1 \cdot \left(\mathsf{max}\left(\alpha, \beta\right) \cdot \mathsf{min}\left(\alpha, \beta\right) + t\_1\right)}{t\_3}}{t\_4} \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\frac{-1 \cdot \left(i \cdot \mathsf{fma}\left(-1, \mathsf{min}\left(\alpha, \beta\right), -1 \cdot i\right)\right)}{t\_4}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.0625, i, t\_5\right) - t\_5}{i}\\ \end{array} \]

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 (- t_3 1.0))
        (t_5 (* 0.125 (+ (fmax alpha beta) (fmin alpha beta)))))
   (if (<=
        (/ (/ (* t_1 (+ (* (fmax alpha beta) (fmin alpha beta)) t_1)) t_3) t_4)
        2e-6)
     (/ (* -1.0 (* i (fma -1.0 (fmin alpha beta) (* -1.0 i)))) t_4)
     (/ (- (fma 0.0625 i t_5) t_5) 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 = t_3 - 1.0;
	double t_5 = 0.125 * (fmax(alpha, beta) + fmin(alpha, beta));
	double tmp;
	if ((((t_1 * ((fmax(alpha, beta) * fmin(alpha, beta)) + t_1)) / t_3) / t_4) <= 2e-6) {
		tmp = (-1.0 * (i * fma(-1.0, fmin(alpha, beta), (-1.0 * i)))) / t_4;
	} else {
		tmp = (fma(0.0625, i, t_5) - t_5) / 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(t_3 - 1.0)
	t_5 = Float64(0.125 * Float64(fmax(alpha, beta) + fmin(alpha, beta)))
	tmp = 0.0
	if (Float64(Float64(Float64(t_1 * Float64(Float64(fmax(alpha, beta) * fmin(alpha, beta)) + t_1)) / t_3) / t_4) <= 2e-6)
		tmp = Float64(Float64(-1.0 * Float64(i * fma(-1.0, fmin(alpha, beta), Float64(-1.0 * i)))) / t_4);
	else
		tmp = Float64(Float64(fma(0.0625, i, t_5) - t_5) / 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[(t$95$3 - 1.0), $MachinePrecision]}, Block[{t$95$5 = N[(0.125 * N[(N[Max[alpha, beta], $MachinePrecision] + N[Min[alpha, beta], $MachinePrecision]), $MachinePrecision]), $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] / t$95$4), $MachinePrecision], 2e-6], N[(N[(-1.0 * N[(i * N[(-1.0 * N[Min[alpha, beta], $MachinePrecision] + N[(-1.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$4), $MachinePrecision], N[(N[(N[(0.0625 * i + t$95$5), $MachinePrecision] - t$95$5), $MachinePrecision] / i), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i)))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)))) (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) #s(literal 1 binary64))) < 1.99999999999999991e-6

   1. Initial program 16.3%

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

   2. 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 13.8

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

   4. Applied rewrites 13.8%

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

   if 1.99999999999999991e-6 < (/.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 16.3%

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

   2. Taylor expanded in i around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 76.7

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

   4. Applied rewrites 76.7%

      \[\leadsto \color{blue}{\left(0.0625 + 0.0625 \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i}} \]
   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 76.7

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

      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 76.7

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

   6. Applied rewrites 76.7%

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

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      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

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

      13. lower--.f64 76.7

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 76.7

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 76.7

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

   8. Applied rewrites 76.7%

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

Alternative 7: 79.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := 0.125 \cdot \left(\mathsf{max}\left(\alpha, \beta\right) + \mathsf{min}\left(\alpha, \beta\right)\right)\\ t_1 := \mathsf{min}\left(\alpha, \beta\right) + \mathsf{max}\left(\alpha, \beta\right)\\ t_2 := t\_1 + 2 \cdot i\\ t_3 := t\_2 \cdot t\_2\\ t_4 := i \cdot \left(t\_1 + i\right)\\ \mathbf{if}\;\frac{\frac{t\_4 \cdot \left(\mathsf{max}\left(\alpha, \beta\right) \cdot \mathsf{min}\left(\alpha, \beta\right) + t\_4\right)}{t\_3}}{t\_3 - 1} \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\frac{i \cdot \left(\mathsf{min}\left(\alpha, \beta\right) + i\right)}{{\left(\mathsf{max}\left(\alpha, \beta\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.0625, i, t\_0\right) - t\_0}{i}\\ \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (* 0.125 (+ (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))))
   (if (<=
        (/
         (/ (* t_4 (+ (* (fmax alpha beta) (fmin alpha beta)) t_4)) t_3)
         (- t_3 1.0))
        2e-6)
     (/ (* i (+ (fmin alpha beta) i)) (pow (fmax alpha beta) 2.0))
     (/ (- (fma 0.0625 i t_0) t_0) i))))

C

double code(double alpha, double beta, double i) {
	double t_0 = 0.125 * (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 tmp;
	if ((((t_4 * ((fmax(alpha, beta) * fmin(alpha, beta)) + t_4)) / t_3) / (t_3 - 1.0)) <= 2e-6) {
		tmp = (i * (fmin(alpha, beta) + i)) / pow(fmax(alpha, beta), 2.0);
	} else {
		tmp = (fma(0.0625, i, t_0) - t_0) / i;
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(0.125 * 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))
	tmp = 0.0
	if (Float64(Float64(Float64(t_4 * Float64(Float64(fmax(alpha, beta) * fmin(alpha, beta)) + t_4)) / t_3) / Float64(t_3 - 1.0)) <= 2e-6)
		tmp = Float64(Float64(i * Float64(fmin(alpha, beta) + i)) / (fmax(alpha, beta) ^ 2.0));
	else
		tmp = Float64(Float64(fma(0.0625, i, t_0) - t_0) / i);
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i)))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)))) (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) #s(literal 1 binary64))) < 1.99999999999999991e-6

   1. Initial program 16.3%

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

   2. 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 9.4

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

   4. Applied rewrites 9.4%

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

   if 1.99999999999999991e-6 < (/.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 16.3%

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

   2. Taylor expanded in i around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 76.7

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

   4. Applied rewrites 76.7%

      \[\leadsto \color{blue}{\left(0.0625 + 0.0625 \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i}} \]
   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 76.7

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

      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 76.7

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

   6. Applied rewrites 76.7%

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

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      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

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

      13. lower--.f64 76.7

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 76.7

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 76.7

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

   8. Applied rewrites 76.7%

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

Alternative 8: 76.7% accurate, 3.2× speedup

Math

\[\begin{array}{l} t_0 := 0.125 \cdot \left(\beta + \alpha\right)\\ \frac{\mathsf{fma}\left(0.0625, i, t\_0\right) - t\_0}{i} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (* 0.125 (+ beta alpha)))) (/ (- (fma 0.0625 i t_0) t_0) i)))

C

double code(double alpha, double beta, double i) {
	double t_0 = 0.125 * (beta + alpha);
	return (fma(0.0625, i, t_0) - t_0) / i;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(0.125 * Float64(beta + alpha))
	return Float64(Float64(fma(0.0625, i, t_0) - t_0) / i)
end

Wolfram

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

Derivation

1. Initial program 16.3%

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

2. Taylor expanded in i around inf

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

   1. lower--.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-+.f64 76.7

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

4. Applied rewrites 76.7%

   \[\leadsto \color{blue}{\left(0.0625 + 0.0625 \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i}} \]
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 76.7

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

   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 76.7

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

6. Applied rewrites 76.7%

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

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

   5. lift-/.f64 N/A

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

   6. associate-*r/ N/A

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

   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

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

   13. lower--.f64 76.7

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

   14. lift-*.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 76.7

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

   17. lift-*.f64 N/A

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

   18. *-commutative N/A

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

   19. lower-*.f64 76.7

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

8. Applied rewrites 76.7%

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

Alternative 9: 76.7% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 16.3%

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

2. Taylor expanded in i around inf

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

   1. lower--.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-+.f64 76.7

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

4. Applied rewrites 76.7%

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

5. Taylor expanded in alpha around 0

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 72.5

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

7. Applied rewrites 72.5%

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

Alternative 10: 73.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\mathsf{max}\left(\alpha, \beta\right) \leq 5.8 \cdot 10^{+223}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\left(1 - 1\right) \cdot \mathsf{fma}\left(\frac{\mathsf{max}\left(\alpha, \beta\right) + \mathsf{min}\left(\alpha, \beta\right)}{i}, 0.125, 0.0625\right)\\ \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= (fmax alpha beta) 5.8e+223)
   0.0625
   (*
    (- 1.0 1.0)
    (fma (/ (+ (fmax alpha beta) (fmin alpha beta)) i) 0.125 0.0625))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (fmax(alpha, beta) <= 5.8e+223) {
		tmp = 0.0625;
	} else {
		tmp = (1.0 - 1.0) * fma(((fmax(alpha, beta) + fmin(alpha, beta)) / i), 0.125, 0.0625);
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (fmax(alpha, beta) <= 5.8e+223)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(1.0 - 1.0) * fma(Float64(Float64(fmax(alpha, beta) + fmin(alpha, beta)) / i), 0.125, 0.0625));
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[N[Max[alpha, beta], $MachinePrecision], 5.8e+223], 0.0625, N[(N[(1.0 - 1.0), $MachinePrecision] * N[(N[(N[(N[Max[alpha, beta], $MachinePrecision] + N[Min[alpha, beta], $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision] * 0.125 + 0.0625), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.8000000000000004e223

   1. Initial program 16.3%

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

   2. Taylor expanded in i around inf

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

   4. Applied rewrites 69.9%

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

   if 5.8000000000000004e223 < beta

   1. Initial program 16.3%

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

   2. Taylor expanded in i around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 76.7

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

   4. Applied rewrites 76.7%

      \[\leadsto \color{blue}{\left(0.0625 + 0.0625 \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i}} \]
   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. sub-to-mult N/A

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

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

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

   6. Applied rewrites 76.7%

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

   7. Taylor expanded in alpha around inf

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

   9. Applied rewrites 10.4%

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

Alternative 11: 69.9% 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 16.3%

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

2. Taylor expanded in i around inf

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

4. Applied rewrites 69.9%

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

Reproduce

herbie shell --seed 2025174 
(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)))