Octave 3.8, jcobi/4

Percentage Accurate: 15.7% → 83.3%

Time: 5.3s

Alternatives: 11

Speedup: 75.0×

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: 15.7% 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: 83.3% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

function code(alpha, beta, i)
	t_0 = Float64(fmax(alpha, beta) * fmin(alpha, beta))
	t_1 = Float64(fmin(alpha, beta) + fmax(alpha, beta))
	t_2 = Float64(t_1 + Float64(2.0 * i))
	t_3 = Float64(t_2 * t_2)
	t_4 = Float64(i * Float64(t_1 + i))
	t_5 = Float64(fmax(alpha, beta) + i)
	t_6 = fma(2.0, i, fmax(alpha, beta))
	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_5, i, t_0) / Float64(t_6 * t_6)) * Float64(Float64(t_5 * i) / fma(t_6, t_6, -1.0)));
	else
		tmp = Float64(Float64(fma(0.0625, i, Float64(Float64(fmax(alpha, beta) + fmin(alpha, beta)) * 0.125)) / i) - Float64(0.125 * Float64(t_1 / i)));
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[Max[alpha, beta], $MachinePrecision] * N[Min[alpha, beta], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Min[alpha, beta], $MachinePrecision] + N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(i * N[(t$95$1 + i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[Max[alpha, beta], $MachinePrecision] + i), $MachinePrecision]}, Block[{t$95$6 = N[(2.0 * i + N[Max[alpha, beta], $MachinePrecision]), $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$5 * i + t$95$0), $MachinePrecision] / N[(t$95$6 * t$95$6), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$5 * i), $MachinePrecision] / N[(t$95$6 * t$95$6 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.0625 * i + N[(N[(N[Max[alpha, beta], $MachinePrecision] + N[Min[alpha, beta], $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision] - N[(0.125 * N[(t$95$1 / i), $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 15.7%

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

   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 14.8%

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

   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 15.9%

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

   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 16.0%

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

   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 16.1%

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

   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 14.9%

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

   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 14.3%

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

       1. lift-/.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-/l/ N/A

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

   21. Applied rewrites 32.9%

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

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

   1. Initial program 15.7%

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

   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 76.4%

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

   4. Applied rewrites 76.4%

      \[\leadsto \color{blue}{\left(0.0625 + 0.0625 \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i}} \]
   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 76.4%

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

      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 76.4%

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

   6. Applied rewrites 76.4%

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

Alternative 2: 82.9% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[Min[alpha, beta], $MachinePrecision] + N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision]}, 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] + i), $MachinePrecision]}, Block[{t$95$5 = N[(2.0 * i + N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(t$95$3 * N[(N[(N[Max[alpha, beta], $MachinePrecision] * N[Min[alpha, beta], $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision] / N[(t$95$2 - 1.0), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[(i * t$95$4), $MachinePrecision] / N[(t$95$5 * t$95$5), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$4 * i), $MachinePrecision] / N[(t$95$5 * t$95$5 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.0625 * i + N[(N[(N[Max[alpha, beta], $MachinePrecision] + N[Min[alpha, beta], $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision] - N[(0.125 * N[(t$95$0 / i), $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 15.7%

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

   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 14.8%

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

   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 15.9%

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

   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 16.0%

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

   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 16.1%

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

   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 14.9%

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

   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 14.3%

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

       1. lift-/.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-/l/ N/A

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

   21. Applied rewrites 32.9%

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

   22. Taylor expanded in alpha around 0

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

       1. lower-*.f64 N/A

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

       2. lower-+.f64 33.1%

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

   24. Applied rewrites 33.1%

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

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

   1. Initial program 15.7%

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

   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 76.4%

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

   4. Applied rewrites 76.4%

      \[\leadsto \color{blue}{\left(0.0625 + 0.0625 \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i}} \]
   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 76.4%

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

      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 76.4%

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

   6. Applied rewrites 76.4%

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

Alternative 3: 82.9% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(2.0 * i + N[Max[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[(N[Max[alpha, beta], $MachinePrecision] + i), $MachinePrecision] * i), $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], Infinity], N[(N[(t$95$5 * N[(t$95$5 / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.0625 * i + N[(N[(N[Max[alpha, beta], $MachinePrecision] + N[Min[alpha, beta], $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision] - N[(0.125 * N[(t$95$1 / i), $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 15.7%

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

   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 14.8%

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

   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 15.9%

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

   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 16.0%

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

   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 16.1%

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

   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 14.9%

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

   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 14.3%

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

       1. lift-/.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-/l/ N/A

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

   21. Applied rewrites 32.9%

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

   22. Taylor expanded in alpha around 0

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

       1. lower-*.f64 N/A

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

       2. lower-+.f64 33.1%

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

   24. Applied rewrites 33.1%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-*r/ N/A

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

       4. lower-/.f64 N/A

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

   26. Applied rewrites 33.1%

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

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

   1. Initial program 15.7%

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

   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 76.4%

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

   4. Applied rewrites 76.4%

      \[\leadsto \color{blue}{\left(0.0625 + 0.0625 \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i}} \]
   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 76.4%

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

      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 76.4%

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

   6. Applied rewrites 76.4%

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

Alternative 4: 79.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_0 := 0.125 \cdot \mathsf{min}\left(\alpha, \beta\right)\\ t_1 := \mathsf{fma}\left(2, i, \mathsf{max}\left(\alpha, \beta\right) + \mathsf{min}\left(\alpha, \beta\right)\right)\\ t_2 := t\_1 - -1\\ \mathbf{if}\;i \leq 1.22 \cdot 10^{+112}:\\ \;\;\;\;\frac{\frac{\left(i \cdot i\right) \cdot 0.25}{t\_2}}{t\_1 - 1}\\ \mathbf{elif}\;i \leq 7.8 \cdot 10^{+126}:\\ \;\;\;\;\frac{\left(\mathsf{min}\left(\alpha, \beta\right) + i\right) \cdot i}{t\_2} \cdot \frac{-1}{1 - t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.0625, i, t\_0\right) - t\_0}{i}\\ \end{array} \]

FPCore

(FPCore (alpha beta i)
  :precision binary64
  (let* ((t_0 (* 0.125 (fmin alpha beta)))
       (t_1 (fma 2.0 i (+ (fmax alpha beta) (fmin alpha beta))))
       (t_2 (- t_1 -1.0)))
  (if (<= i 1.22e+112)
    (/ (/ (* (* i i) 0.25) t_2) (- t_1 1.0))
    (if (<= i 7.8e+126)
      (* (/ (* (+ (fmin alpha beta) i) i) t_2) (/ -1.0 (- 1.0 t_1)))
      (/ (- (fma 0.0625 i t_0) t_0) i)))))

C

double code(double alpha, double beta, double i) {
	double t_0 = 0.125 * fmin(alpha, beta);
	double t_1 = fma(2.0, i, (fmax(alpha, beta) + fmin(alpha, beta)));
	double t_2 = t_1 - -1.0;
	double tmp;
	if (i <= 1.22e+112) {
		tmp = (((i * i) * 0.25) / t_2) / (t_1 - 1.0);
	} else if (i <= 7.8e+126) {
		tmp = (((fmin(alpha, beta) + i) * i) / t_2) * (-1.0 / (1.0 - t_1));
	} else {
		tmp = (fma(0.0625, i, t_0) - t_0) / i;
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(0.125 * fmin(alpha, beta))
	t_1 = fma(2.0, i, Float64(fmax(alpha, beta) + fmin(alpha, beta)))
	t_2 = Float64(t_1 - -1.0)
	tmp = 0.0
	if (i <= 1.22e+112)
		tmp = Float64(Float64(Float64(Float64(i * i) * 0.25) / t_2) / Float64(t_1 - 1.0));
	elseif (i <= 7.8e+126)
		tmp = Float64(Float64(Float64(Float64(fmin(alpha, beta) + i) * i) / t_2) * Float64(-1.0 / Float64(1.0 - t_1)));
	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[Min[alpha, beta], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 * i + N[(N[Max[alpha, beta], $MachinePrecision] + N[Min[alpha, beta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 - -1.0), $MachinePrecision]}, If[LessEqual[i, 1.22e+112], N[(N[(N[(N[(i * i), $MachinePrecision] * 0.25), $MachinePrecision] / t$95$2), $MachinePrecision] / N[(t$95$1 - 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 7.8e+126], N[(N[(N[(N[(N[Min[alpha, beta], $MachinePrecision] + i), $MachinePrecision] * i), $MachinePrecision] / t$95$2), $MachinePrecision] * N[(-1.0 / N[(1.0 - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.0625 * i + t$95$0), $MachinePrecision] - t$95$0), $MachinePrecision] / i), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if i < 1.22e112

   1. Initial program 15.7%

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

   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 13.2%

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

   4. Applied rewrites 13.2%

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

   5. Taylor expanded in i around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 35.6%

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

   7. Applied rewrites 35.6%

      \[\leadsto \frac{\color{blue}{0.25 \cdot {i}^{2}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

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

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

   9. Applied rewrites 37.9%

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

   if 1.22e112 < i < 7.7999999999999999e126

   1. Initial program 15.7%

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

   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 13.2%

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

   4. Applied rewrites 13.2%

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

   6. Applied rewrites 13.2%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

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

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. lift-+.f64 N/A

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

   8. Applied rewrites 17.2%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lower-*.f64 N/A

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

      4. frac-2neg N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lift--.f64 N/A

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

      8. sub-negate-rev N/A

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

      9. lower--.f64 17.2%

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

   10. Applied rewrites 17.2%

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

   if 7.7999999999999999e126 < i

   1. Initial program 15.7%

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

   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 76.4%

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

   4. Applied rewrites 76.4%

      \[\leadsto \color{blue}{\left(0.0625 + 0.0625 \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i}} \]
   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 76.4%

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

      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 76.4%

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

   6. Applied rewrites 76.4%

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

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

   8. Applied rewrites 76.4%

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

   9. Taylor expanded in alpha around inf

      \[\leadsto \frac{\mathsf{fma}\left(0.0625, i, \frac{1}{8} \cdot \alpha\right) - 0.125 \cdot \left(\beta + \alpha\right)}{i} \]
   10. Step-by-step derivation

       1. lower-*.f64 72.4%

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

   11. Applied rewrites 72.4%

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

   12. Taylor expanded in alpha around inf

       \[\leadsto \frac{\mathsf{fma}\left(0.0625, i, 0.125 \cdot \alpha\right) - \frac{1}{8} \cdot \alpha}{i} \]
   13. Step-by-step derivation

       1. lower-*.f64 73.7%

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

   14. Applied rewrites 73.7%

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

Alternative 5: 79.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} t_0 := 0.125 \cdot \mathsf{min}\left(\alpha, \beta\right)\\ t_1 := \mathsf{fma}\left(2, i, \mathsf{max}\left(\alpha, \beta\right) + \mathsf{min}\left(\alpha, \beta\right)\right)\\ t_2 := \mathsf{fma}\left(2, i, \mathsf{max}\left(\alpha, \beta\right)\right)\\ \mathbf{if}\;i \leq 1.22 \cdot 10^{+112}:\\ \;\;\;\;\frac{\frac{\left(i \cdot i\right) \cdot 0.25}{t\_1 - -1}}{t\_1 - 1}\\ \mathbf{elif}\;i \leq 7.8 \cdot 10^{+126}:\\ \;\;\;\;\frac{\frac{\left(\mathsf{min}\left(\alpha, \beta\right) + i\right) \cdot i}{t\_2 - -1}}{t\_2 - 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.0625, i, t\_0\right) - t\_0}{i}\\ \end{array} \]

FPCore

(FPCore (alpha beta i)
  :precision binary64
  (let* ((t_0 (* 0.125 (fmin alpha beta)))
       (t_1 (fma 2.0 i (+ (fmax alpha beta) (fmin alpha beta))))
       (t_2 (fma 2.0 i (fmax alpha beta))))
  (if (<= i 1.22e+112)
    (/ (/ (* (* i i) 0.25) (- t_1 -1.0)) (- t_1 1.0))
    (if (<= i 7.8e+126)
      (/ (/ (* (+ (fmin alpha beta) i) i) (- t_2 -1.0)) (- t_2 1.0))
      (/ (- (fma 0.0625 i t_0) t_0) i)))))

C

double code(double alpha, double beta, double i) {
	double t_0 = 0.125 * fmin(alpha, beta);
	double t_1 = fma(2.0, i, (fmax(alpha, beta) + fmin(alpha, beta)));
	double t_2 = fma(2.0, i, fmax(alpha, beta));
	double tmp;
	if (i <= 1.22e+112) {
		tmp = (((i * i) * 0.25) / (t_1 - -1.0)) / (t_1 - 1.0);
	} else if (i <= 7.8e+126) {
		tmp = (((fmin(alpha, beta) + i) * i) / (t_2 - -1.0)) / (t_2 - 1.0);
	} else {
		tmp = (fma(0.0625, i, t_0) - t_0) / i;
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(0.125 * fmin(alpha, beta))
	t_1 = fma(2.0, i, Float64(fmax(alpha, beta) + fmin(alpha, beta)))
	t_2 = fma(2.0, i, fmax(alpha, beta))
	tmp = 0.0
	if (i <= 1.22e+112)
		tmp = Float64(Float64(Float64(Float64(i * i) * 0.25) / Float64(t_1 - -1.0)) / Float64(t_1 - 1.0));
	elseif (i <= 7.8e+126)
		tmp = Float64(Float64(Float64(Float64(fmin(alpha, beta) + i) * i) / Float64(t_2 - -1.0)) / Float64(t_2 - 1.0));
	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[Min[alpha, beta], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 * i + N[(N[Max[alpha, beta], $MachinePrecision] + N[Min[alpha, beta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 * i + N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, 1.22e+112], N[(N[(N[(N[(i * i), $MachinePrecision] * 0.25), $MachinePrecision] / N[(t$95$1 - -1.0), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 - 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 7.8e+126], N[(N[(N[(N[(N[Min[alpha, beta], $MachinePrecision] + i), $MachinePrecision] * i), $MachinePrecision] / N[(t$95$2 - -1.0), $MachinePrecision]), $MachinePrecision] / N[(t$95$2 - 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.0625 * i + t$95$0), $MachinePrecision] - t$95$0), $MachinePrecision] / i), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if i < 1.22e112

   1. Initial program 15.7%

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

   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 13.2%

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

   4. Applied rewrites 13.2%

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

   5. Taylor expanded in i around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 35.6%

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

   7. Applied rewrites 35.6%

      \[\leadsto \frac{\color{blue}{0.25 \cdot {i}^{2}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

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

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

   9. Applied rewrites 37.9%

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

   if 1.22e112 < i < 7.7999999999999999e126

   1. Initial program 15.7%

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

   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 13.2%

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

   4. Applied rewrites 13.2%

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

   6. Applied rewrites 13.2%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

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

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. lift-+.f64 N/A

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

   8. Applied rewrites 17.2%

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

   9. Taylor expanded in alpha around 0

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

   11. Applied rewrites 17.1%

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

   12. Taylor expanded in alpha around 0

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

   14. Applied rewrites 17.0%

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

   if 7.7999999999999999e126 < i

   1. Initial program 15.7%

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

   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 76.4%

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

   4. Applied rewrites 76.4%

      \[\leadsto \color{blue}{\left(0.0625 + 0.0625 \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i}} \]
   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 76.4%

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

      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 76.4%

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

   6. Applied rewrites 76.4%

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

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

   8. Applied rewrites 76.4%

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

   9. Taylor expanded in alpha around inf

      \[\leadsto \frac{\mathsf{fma}\left(0.0625, i, \frac{1}{8} \cdot \alpha\right) - 0.125 \cdot \left(\beta + \alpha\right)}{i} \]
   10. Step-by-step derivation

       1. lower-*.f64 72.4%

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

   11. Applied rewrites 72.4%

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

   12. Taylor expanded in alpha around inf

       \[\leadsto \frac{\mathsf{fma}\left(0.0625, i, 0.125 \cdot \alpha\right) - \frac{1}{8} \cdot \alpha}{i} \]
   13. Step-by-step derivation

       1. lower-*.f64 73.7%

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

   14. Applied rewrites 73.7%

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

Alternative 6: 76.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(\alpha, \beta\right) + \mathsf{max}\left(\alpha, \beta\right)\\ t_1 := i \cdot \left(t\_0 + i\right)\\ t_2 := \mathsf{fma}\left(2, i, \mathsf{max}\left(\alpha, \beta\right)\right)\\ t_3 := t\_0 + 2 \cdot i\\ t_4 := t\_3 \cdot t\_3\\ t_5 := 0.125 \cdot \left(\mathsf{max}\left(\alpha, \beta\right) + \mathsf{min}\left(\alpha, \beta\right)\right)\\ \mathbf{if}\;\frac{\frac{t\_1 \cdot \left(\mathsf{max}\left(\alpha, \beta\right) \cdot \mathsf{min}\left(\alpha, \beta\right) + t\_1\right)}{t\_4}}{t\_4 - 1} \leq 10^{-27}:\\ \;\;\;\;\frac{i \cdot \left(\mathsf{min}\left(\alpha, \beta\right) + i\right)}{\mathsf{fma}\left(t\_2, t\_2, -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.0625, i, t\_5\right) - t\_5}{i}\\ \end{array} \]

FPCore

(FPCore (alpha beta i)
  :precision binary64
  (let* ((t_0 (+ (fmin alpha beta) (fmax alpha beta)))
       (t_1 (* i (+ t_0 i)))
       (t_2 (fma 2.0 i (fmax alpha beta)))
       (t_3 (+ t_0 (* 2.0 i)))
       (t_4 (* t_3 t_3))
       (t_5 (* 0.125 (+ (fmax alpha beta) (fmin alpha beta)))))
  (if (<=
       (/
        (/
         (* t_1 (+ (* (fmax alpha beta) (fmin alpha beta)) t_1))
         t_4)
        (- t_4 1.0))
       1e-27)
    (/ (* i (+ (fmin alpha beta) i)) (fma t_2 t_2 -1.0))
    (/ (- (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 = fma(2.0, i, fmax(alpha, beta));
	double t_3 = t_0 + (2.0 * i);
	double t_4 = t_3 * t_3;
	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_4) / (t_4 - 1.0)) <= 1e-27) {
		tmp = (i * (fmin(alpha, beta) + i)) / fma(t_2, t_2, -1.0);
	} 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 = fma(2.0, i, fmax(alpha, beta))
	t_3 = Float64(t_0 + Float64(2.0 * i))
	t_4 = Float64(t_3 * t_3)
	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_4) / Float64(t_4 - 1.0)) <= 1e-27)
		tmp = Float64(Float64(i * Float64(fmin(alpha, beta) + i)) / fma(t_2, t_2, -1.0));
	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[(2.0 * i + N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$0 + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 * t$95$3), $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$4), $MachinePrecision] / N[(t$95$4 - 1.0), $MachinePrecision]), $MachinePrecision], 1e-27], N[(N[(i * N[(N[Min[alpha, beta], $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision] / N[(t$95$2 * t$95$2 + -1.0), $MachinePrecision]), $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))) < 1e-27

   1. Initial program 15.7%

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

   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 13.2%

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

   4. Applied rewrites 13.2%

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

   6. Applied rewrites 13.2%

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

   7. Taylor expanded in alpha around 0

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

   9. Applied rewrites 11.9%

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

   10. Taylor expanded in alpha around 0

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

   12. Applied rewrites 11.8%

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

   if 1e-27 < (/.f64 (/.f64 (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i)))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)))) (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) #s(literal 1 binary64)))

   1. Initial program 15.7%

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

   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 76.4%

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

   4. Applied rewrites 76.4%

      \[\leadsto \color{blue}{\left(0.0625 + 0.0625 \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i}} \]
   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 76.4%

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

      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 76.4%

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

   6. Applied rewrites 76.4%

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

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

   8. Applied rewrites 76.4%

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

Alternative 7: 76.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(\alpha, \beta\right) + \mathsf{max}\left(\alpha, \beta\right)\\ t_1 := t\_0 + 2 \cdot i\\ t_2 := t\_1 \cdot t\_1\\ t_3 := i \cdot \left(t\_0 + i\right)\\ t_4 := \mathsf{max}\left(\alpha, \beta\right) + \mathsf{min}\left(\alpha, \beta\right)\\ t_5 := 0.125 \cdot t\_4\\ \mathbf{if}\;\frac{\frac{t\_3 \cdot \left(\mathsf{max}\left(\alpha, \beta\right) \cdot \mathsf{min}\left(\alpha, \beta\right) + t\_3\right)}{t\_2}}{t\_2 - 1} \leq 10^{-27}:\\ \;\;\;\;\frac{\frac{i \cdot \left(\mathsf{min}\left(\alpha, \beta\right) + i\right)}{\mathsf{max}\left(\alpha, \beta\right)}}{\mathsf{fma}\left(2, i, t\_4\right) - 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.0625, i, t\_5\right) - t\_5}{i}\\ \end{array} \]

FPCore

(FPCore (alpha beta i)
  :precision binary64
  (let* ((t_0 (+ (fmin alpha beta) (fmax alpha beta)))
       (t_1 (+ 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 (* 0.125 t_4)))
  (if (<=
       (/
        (/
         (* t_3 (+ (* (fmax alpha beta) (fmin alpha beta)) t_3))
         t_2)
        (- t_2 1.0))
       1e-27)
    (/
     (/ (* i (+ (fmin alpha beta) i)) (fmax alpha beta))
     (- (fma 2.0 i t_4) 1.0))
    (/ (- (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 = 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 = 0.125 * t_4;
	double tmp;
	if ((((t_3 * ((fmax(alpha, beta) * fmin(alpha, beta)) + t_3)) / t_2) / (t_2 - 1.0)) <= 1e-27) {
		tmp = ((i * (fmin(alpha, beta) + i)) / fmax(alpha, beta)) / (fma(2.0, i, t_4) - 1.0);
	} 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(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(0.125 * t_4)
	tmp = 0.0
	if (Float64(Float64(Float64(t_3 * Float64(Float64(fmax(alpha, beta) * fmin(alpha, beta)) + t_3)) / t_2) / Float64(t_2 - 1.0)) <= 1e-27)
		tmp = Float64(Float64(Float64(i * Float64(fmin(alpha, beta) + i)) / fmax(alpha, beta)) / Float64(fma(2.0, i, t_4) - 1.0));
	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[(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[(0.125 * t$95$4), $MachinePrecision]}, If[LessEqual[N[(N[(N[(t$95$3 * N[(N[(N[Max[alpha, beta], $MachinePrecision] * N[Min[alpha, beta], $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision] / N[(t$95$2 - 1.0), $MachinePrecision]), $MachinePrecision], 1e-27], N[(N[(N[(i * N[(N[Min[alpha, beta], $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision] / N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision] / N[(N[(2.0 * i + t$95$4), $MachinePrecision] - 1.0), $MachinePrecision]), $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))) < 1e-27

   1. Initial program 15.7%

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

   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 13.2%

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

   4. Applied rewrites 13.2%

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

   6. Applied rewrites 13.2%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

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

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. lift-+.f64 N/A

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

   8. Applied rewrites 17.2%

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

   9. Taylor expanded in beta around inf

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

       1. lower-/.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-+.f64 12.6%

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

   11. Applied rewrites 12.6%

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

   if 1e-27 < (/.f64 (/.f64 (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i)))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)))) (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) #s(literal 1 binary64)))

   1. Initial program 15.7%

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

   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 76.4%

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

   4. Applied rewrites 76.4%

      \[\leadsto \color{blue}{\left(0.0625 + 0.0625 \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i}} \]
   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 76.4%

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

      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 76.4%

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

   6. Applied rewrites 76.4%

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

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

   8. Applied rewrites 76.4%

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

Alternative 8: 76.0% 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 15.7%

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

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 76.4%

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

4. Applied rewrites 76.4%

   \[\leadsto \color{blue}{\left(0.0625 + 0.0625 \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i}} \]
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 76.4%

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

   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 76.4%

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

6. Applied rewrites 76.4%

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

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

8. Applied rewrites 76.4%

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

Alternative 9: 76.0% accurate, 3.1× speedup

Math

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

FPCore

(FPCore (alpha beta i)
  :precision binary64
  (let* ((t_0 (* 0.125 (fmax alpha beta))))
  (/ (- (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);
	return (fma(0.0625, i, t_0) - t_0) / i;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(0.125 * fmax(alpha, beta))
	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[Max[alpha, beta], $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(0.0625 * i + t$95$0), $MachinePrecision] - t$95$0), $MachinePrecision] / i), $MachinePrecision]]

Derivation

1. Initial program 15.7%

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

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 76.4%

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

4. Applied rewrites 76.4%

   \[\leadsto \color{blue}{\left(0.0625 + 0.0625 \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i}} \]
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 76.4%

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

   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 76.4%

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

6. Applied rewrites 76.4%

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

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

8. Applied rewrites 76.4%

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

9. Taylor expanded in alpha around 0

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

11. Applied rewrites 72.4%

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

12. Taylor expanded in alpha around 0

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

14. Applied rewrites 73.6%

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

Alternative 10: 73.7% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 15.7%

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

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 76.4%

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

4. Applied rewrites 76.4%

   \[\leadsto \color{blue}{\left(0.0625 + 0.0625 \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i}} \]
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 76.4%

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

   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 76.4%

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

6. Applied rewrites 76.4%

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

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

8. Applied rewrites 76.4%

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

9. Taylor expanded in alpha around inf

   \[\leadsto \frac{\mathsf{fma}\left(0.0625, i, \frac{1}{8} \cdot \alpha\right) - 0.125 \cdot \left(\beta + \alpha\right)}{i} \]
10. Step-by-step derivation

    1. lower-*.f64 72.4%

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

11. Applied rewrites 72.4%

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

12. Taylor expanded in alpha around inf

    \[\leadsto \frac{\mathsf{fma}\left(0.0625, i, 0.125 \cdot \alpha\right) - \frac{1}{8} \cdot \alpha}{i} \]
13. Step-by-step derivation

    1. lower-*.f64 73.7%

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

14. Applied rewrites 73.7%

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

Alternative 11: 70.0% accurate, 75.0× 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 15.7%

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

2. Taylor expanded in i around inf

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

4. Applied rewrites 70.0%

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

Reproduce

herbie shell --seed 2025212 
(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)))