Octave 3.8, jcobi/4

Percentage Accurate: 16.4% → 84.4%

Time: 4.2s

Alternatives: 11

Speedup: 75.4×

Specification

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

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 16.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 84.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 - 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(i * N[(N[(alpha + beta), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(alpha + beta), $MachinePrecision] - N[(-2.0 * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(t$95$3 * N[(N[(beta * alpha), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / t$95$2), $MachinePrecision], Infinity], N[(N[(N[(t$95$3 / t$95$4), $MachinePrecision] * N[(N[(beta * alpha + t$95$3), $MachinePrecision] / t$95$4), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], N[(N[(0.0625 - N[(-0.0625 * N[(N[(2.0 * alpha + N[(2.0 * beta), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.125 * N[(N[(alpha + beta), $MachinePrecision] / 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 46.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lift-*.f64 N/A

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

   3. Applied rewrites 99.6%

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

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

   1. Initial program 0.0%

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

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

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

      3. lower--.f64 N/A

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

      4. metadata-eval 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-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 76.1

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

   4. Applied rewrites 76.1%

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

Alternative 2: 78.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := t\_0 \cdot t\_0\\ t_2 := i \cdot \left(\left(\alpha + \beta\right) + i\right)\\ \mathbf{if}\;\frac{\frac{t\_2 \cdot \left(\beta \cdot \alpha + t\_2\right)}{t\_1}}{t\_1 - 1} \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(i, \alpha + i, \frac{i \cdot \left(2 \cdot \left(i \cdot i\right)\right)}{\beta}\right) - \frac{i \cdot \left(\left(\alpha + i\right) \cdot \mathsf{fma}\left(4, \alpha, 8 \cdot i\right)\right)}{\beta}}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\left(0.0625 - -0.0625 \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i)))
        (t_1 (* t_0 t_0))
        (t_2 (* i (+ (+ alpha beta) i))))
   (if (<= (/ (/ (* t_2 (+ (* beta alpha) t_2)) t_1) (- t_1 1.0)) 5e-5)
     (/
      (-
       (fma i (+ alpha i) (/ (* i (* 2.0 (* i i))) beta))
       (/ (* i (* (+ alpha i) (fma 4.0 alpha (* 8.0 i)))) beta))
      (* beta beta))
     (-
      (- 0.0625 (* -0.0625 (/ (fma 2.0 alpha (* 2.0 beta)) i)))
      (* 0.125 (/ (+ alpha beta) i))))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double t_1 = t_0 * t_0;
	double t_2 = i * ((alpha + beta) + i);
	double tmp;
	if ((((t_2 * ((beta * alpha) + t_2)) / t_1) / (t_1 - 1.0)) <= 5e-5) {
		tmp = (fma(i, (alpha + i), ((i * (2.0 * (i * i))) / beta)) - ((i * ((alpha + i) * fma(4.0, alpha, (8.0 * i)))) / beta)) / (beta * beta);
	} else {
		tmp = (0.0625 - (-0.0625 * (fma(2.0, alpha, (2.0 * beta)) / i))) - (0.125 * ((alpha + beta) / i));
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_1 = Float64(t_0 * t_0)
	t_2 = Float64(i * Float64(Float64(alpha + beta) + i))
	tmp = 0.0
	if (Float64(Float64(Float64(t_2 * Float64(Float64(beta * alpha) + t_2)) / t_1) / Float64(t_1 - 1.0)) <= 5e-5)
		tmp = Float64(Float64(fma(i, Float64(alpha + i), Float64(Float64(i * Float64(2.0 * Float64(i * i))) / beta)) - Float64(Float64(i * Float64(Float64(alpha + i) * fma(4.0, alpha, Float64(8.0 * i)))) / beta)) / Float64(beta * beta));
	else
		tmp = Float64(Float64(0.0625 - Float64(-0.0625 * Float64(fma(2.0, alpha, Float64(2.0 * beta)) / i))) - Float64(0.125 * Float64(Float64(alpha + beta) / i)));
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(i * N[(N[(alpha + beta), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(t$95$2 * N[(N[(beta * alpha), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(t$95$1 - 1.0), $MachinePrecision]), $MachinePrecision], 5e-5], N[(N[(N[(i * N[(alpha + i), $MachinePrecision] + N[(N[(i * N[(2.0 * N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision] - N[(N[(i * N[(N[(alpha + i), $MachinePrecision] * N[(4.0 * alpha + N[(8.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision] / N[(beta * beta), $MachinePrecision]), $MachinePrecision], N[(N[(0.0625 - N[(-0.0625 * N[(N[(2.0 * alpha + N[(2.0 * beta), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.125 * N[(N[(alpha + beta), $MachinePrecision] / 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))) < 5.00000000000000024e-5

   1. Initial program 98.8%

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

   2. Taylor expanded in beta around inf

      \[\leadsto \color{blue}{\frac{\left(i \cdot \left(\alpha + i\right) + \frac{i \cdot \left(i \cdot \left(\alpha + i\right) + {\left(\alpha + i\right)}^{2}\right)}{\beta}\right) - \frac{i \cdot \left(\left(\alpha + i\right) \cdot \left(4 \cdot \alpha + 8 \cdot i\right)\right)}{\beta}}{{\beta}^{2}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\left(i \cdot \left(\alpha + i\right) + \frac{i \cdot \left(i \cdot \left(\alpha + i\right) + {\left(\alpha + i\right)}^{2}\right)}{\beta}\right) - \frac{i \cdot \left(\left(\alpha + i\right) \cdot \left(4 \cdot \alpha + 8 \cdot i\right)\right)}{\beta}}{\color{blue}{{\beta}^{2}}} \]

   4. Applied rewrites 51.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(i, \alpha + i, \frac{i \cdot \mathsf{fma}\left(i, \alpha + i, \left(\alpha + i\right) \cdot \left(\alpha + i\right)\right)}{\beta}\right) - \frac{i \cdot \left(\left(\alpha + i\right) \cdot \mathsf{fma}\left(4, \alpha, 8 \cdot i\right)\right)}{\beta}}{\beta \cdot \beta}} \]

   5. Taylor expanded in alpha around 0

      \[\leadsto \frac{\mathsf{fma}\left(i, \alpha + i, \frac{i \cdot \left(2 \cdot {i}^{2}\right)}{\beta}\right) - \frac{i \cdot \left(\left(\alpha + i\right) \cdot \mathsf{fma}\left(4, \alpha, 8 \cdot i\right)\right)}{\beta}}{\beta \cdot \beta} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(i, \alpha + i, \frac{i \cdot \left(2 \cdot {i}^{2}\right)}{\beta}\right) - \frac{i \cdot \left(\left(\alpha + i\right) \cdot \mathsf{fma}\left(4, \alpha, 8 \cdot i\right)\right)}{\beta}}{\beta \cdot \beta} \]

      2. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(i, \alpha + i, \frac{i \cdot \left(2 \cdot \left(i \cdot i\right)\right)}{\beta}\right) - \frac{i \cdot \left(\left(\alpha + i\right) \cdot \mathsf{fma}\left(4, \alpha, 8 \cdot i\right)\right)}{\beta}}{\beta \cdot \beta} \]

      3. lower-*.f64 52.1

         \[\leadsto \frac{\mathsf{fma}\left(i, \alpha + i, \frac{i \cdot \left(2 \cdot \left(i \cdot i\right)\right)}{\beta}\right) - \frac{i \cdot \left(\left(\alpha + i\right) \cdot \mathsf{fma}\left(4, \alpha, 8 \cdot i\right)\right)}{\beta}}{\beta \cdot \beta} \]

   7. Applied rewrites 52.1%

      \[\leadsto \frac{\mathsf{fma}\left(i, \alpha + i, \frac{i \cdot \left(2 \cdot \left(i \cdot i\right)\right)}{\beta}\right) - \frac{i \cdot \left(\left(\alpha + i\right) \cdot \mathsf{fma}\left(4, \alpha, 8 \cdot i\right)\right)}{\beta}}{\beta \cdot \beta} \]

   if 5.00000000000000024e-5 < (/.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 13.7%

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

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

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

      3. lower--.f64 N/A

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

      4. metadata-eval 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-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 79.8

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

   4. Applied rewrites 79.8%

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

Alternative 3: 78.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := t\_0 \cdot t\_0\\ t_2 := i \cdot \left(\left(\alpha + \beta\right) + i\right)\\ \mathbf{if}\;\frac{\frac{t\_2 \cdot \left(\beta \cdot \alpha + t\_2\right)}{t\_1}}{t\_1 - 1} \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\frac{\left(i \cdot i\right) \cdot \left(1 + -6 \cdot \frac{i}{\beta}\right)}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\left(0.0625 - -0.0625 \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i)))
        (t_1 (* t_0 t_0))
        (t_2 (* i (+ (+ alpha beta) i))))
   (if (<= (/ (/ (* t_2 (+ (* beta alpha) t_2)) t_1) (- t_1 1.0)) 5e-5)
     (/ (* (* i i) (+ 1.0 (* -6.0 (/ i beta)))) (* beta beta))
     (-
      (- 0.0625 (* -0.0625 (/ (fma 2.0 alpha (* 2.0 beta)) i)))
      (* 0.125 (/ (+ alpha beta) i))))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double t_1 = t_0 * t_0;
	double t_2 = i * ((alpha + beta) + i);
	double tmp;
	if ((((t_2 * ((beta * alpha) + t_2)) / t_1) / (t_1 - 1.0)) <= 5e-5) {
		tmp = ((i * i) * (1.0 + (-6.0 * (i / beta)))) / (beta * beta);
	} else {
		tmp = (0.0625 - (-0.0625 * (fma(2.0, alpha, (2.0 * beta)) / i))) - (0.125 * ((alpha + beta) / i));
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_1 = Float64(t_0 * t_0)
	t_2 = Float64(i * Float64(Float64(alpha + beta) + i))
	tmp = 0.0
	if (Float64(Float64(Float64(t_2 * Float64(Float64(beta * alpha) + t_2)) / t_1) / Float64(t_1 - 1.0)) <= 5e-5)
		tmp = Float64(Float64(Float64(i * i) * Float64(1.0 + Float64(-6.0 * Float64(i / beta)))) / Float64(beta * beta));
	else
		tmp = Float64(Float64(0.0625 - Float64(-0.0625 * Float64(fma(2.0, alpha, Float64(2.0 * beta)) / i))) - Float64(0.125 * Float64(Float64(alpha + beta) / i)));
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(i * N[(N[(alpha + beta), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(t$95$2 * N[(N[(beta * alpha), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(t$95$1 - 1.0), $MachinePrecision]), $MachinePrecision], 5e-5], N[(N[(N[(i * i), $MachinePrecision] * N[(1.0 + N[(-6.0 * N[(i / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(beta * beta), $MachinePrecision]), $MachinePrecision], N[(N[(0.0625 - N[(-0.0625 * N[(N[(2.0 * alpha + N[(2.0 * beta), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.125 * N[(N[(alpha + beta), $MachinePrecision] / 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))) < 5.00000000000000024e-5

   1. Initial program 98.8%

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

   2. Taylor expanded in beta around inf

      \[\leadsto \color{blue}{\frac{\left(i \cdot \left(\alpha + i\right) + \frac{i \cdot \left(i \cdot \left(\alpha + i\right) + {\left(\alpha + i\right)}^{2}\right)}{\beta}\right) - \frac{i \cdot \left(\left(\alpha + i\right) \cdot \left(4 \cdot \alpha + 8 \cdot i\right)\right)}{\beta}}{{\beta}^{2}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\left(i \cdot \left(\alpha + i\right) + \frac{i \cdot \left(i \cdot \left(\alpha + i\right) + {\left(\alpha + i\right)}^{2}\right)}{\beta}\right) - \frac{i \cdot \left(\left(\alpha + i\right) \cdot \left(4 \cdot \alpha + 8 \cdot i\right)\right)}{\beta}}{\color{blue}{{\beta}^{2}}} \]

   4. Applied rewrites 51.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(i, \alpha + i, \frac{i \cdot \mathsf{fma}\left(i, \alpha + i, \left(\alpha + i\right) \cdot \left(\alpha + i\right)\right)}{\beta}\right) - \frac{i \cdot \left(\left(\alpha + i\right) \cdot \mathsf{fma}\left(4, \alpha, 8 \cdot i\right)\right)}{\beta}}{\beta \cdot \beta}} \]

   5. Taylor expanded in alpha around 0

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

      1. lower--.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \frac{{i}^{3}}{\beta} + {i}^{2}\right) - 8 \cdot \frac{{i}^{3}}{\beta}}{\beta \cdot \beta} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(2, \frac{{i}^{3}}{\beta}, {i}^{2}\right) - 8 \cdot \frac{{i}^{3}}{\beta}}{\beta \cdot \beta} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(2, \frac{{i}^{3}}{\beta}, {i}^{2}\right) - 8 \cdot \frac{{i}^{3}}{\beta}}{\beta \cdot \beta} \]

      4. unpow3 N/A

         \[\leadsto \frac{\mathsf{fma}\left(2, \frac{\left(i \cdot i\right) \cdot i}{\beta}, {i}^{2}\right) - 8 \cdot \frac{{i}^{3}}{\beta}}{\beta \cdot \beta} \]

      5. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(2, \frac{{i}^{2} \cdot i}{\beta}, {i}^{2}\right) - 8 \cdot \frac{{i}^{3}}{\beta}}{\beta \cdot \beta} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(2, \frac{{i}^{2} \cdot i}{\beta}, {i}^{2}\right) - 8 \cdot \frac{{i}^{3}}{\beta}}{\beta \cdot \beta} \]

      7. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(2, \frac{\left(i \cdot i\right) \cdot i}{\beta}, {i}^{2}\right) - 8 \cdot \frac{{i}^{3}}{\beta}}{\beta \cdot \beta} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(2, \frac{\left(i \cdot i\right) \cdot i}{\beta}, {i}^{2}\right) - 8 \cdot \frac{{i}^{3}}{\beta}}{\beta \cdot \beta} \]

      9. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(2, \frac{\left(i \cdot i\right) \cdot i}{\beta}, i \cdot i\right) - 8 \cdot \frac{{i}^{3}}{\beta}}{\beta \cdot \beta} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(2, \frac{\left(i \cdot i\right) \cdot i}{\beta}, i \cdot i\right) - 8 \cdot \frac{{i}^{3}}{\beta}}{\beta \cdot \beta} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(2, \frac{\left(i \cdot i\right) \cdot i}{\beta}, i \cdot i\right) - 8 \cdot \frac{{i}^{3}}{\beta}}{\beta \cdot \beta} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(2, \frac{\left(i \cdot i\right) \cdot i}{\beta}, i \cdot i\right) - 8 \cdot \frac{{i}^{3}}{\beta}}{\beta \cdot \beta} \]

      13. unpow3 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 47.4

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

   7. Applied rewrites 47.4%

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

   8. Taylor expanded in i around 0

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

      1. lower-*.f64 N/A

         \[\leadsto \frac{{i}^{2} \cdot \left(1 + -6 \cdot \frac{i}{\beta}\right)}{\beta \cdot \beta} \]

      2. pow2 N/A

         \[\leadsto \frac{\left(i \cdot i\right) \cdot \left(1 + -6 \cdot \frac{i}{\beta}\right)}{\beta \cdot \beta} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\left(i \cdot i\right) \cdot \left(1 + -6 \cdot \frac{i}{\beta}\right)}{\beta \cdot \beta} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{\left(i \cdot i\right) \cdot \left(1 + -6 \cdot \frac{i}{\beta}\right)}{\beta \cdot \beta} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\left(i \cdot i\right) \cdot \left(1 + -6 \cdot \frac{i}{\beta}\right)}{\beta \cdot \beta} \]

      6. lower-/.f64 47.7

         \[\leadsto \frac{\left(i \cdot i\right) \cdot \left(1 + -6 \cdot \frac{i}{\beta}\right)}{\beta \cdot \beta} \]

   10. Applied rewrites 47.7%

       \[\leadsto \frac{\left(i \cdot i\right) \cdot \left(1 + -6 \cdot \frac{i}{\beta}\right)}{\beta \cdot \beta} \]

   if 5.00000000000000024e-5 < (/.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 13.7%

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

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

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

      3. lower--.f64 N/A

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

      4. metadata-eval 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-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 79.8

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

   4. Applied rewrites 79.8%

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

Alternative 4: 78.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := t\_0 \cdot t\_0\\ t_2 := i \cdot \left(\left(\alpha + \beta\right) + i\right)\\ \mathbf{if}\;\frac{\frac{t\_2 \cdot \left(\beta \cdot \alpha + t\_2\right)}{t\_1}}{t\_1 - 1} \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\frac{\left(i \cdot i\right) \cdot \left(1 + 2 \cdot \frac{i}{\beta}\right) - 8 \cdot \frac{\left(i \cdot i\right) \cdot i}{\beta}}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\left(0.0625 - -0.0625 \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i)))
        (t_1 (* t_0 t_0))
        (t_2 (* i (+ (+ alpha beta) i))))
   (if (<= (/ (/ (* t_2 (+ (* beta alpha) t_2)) t_1) (- t_1 1.0)) 5e-5)
     (/
      (- (* (* i i) (+ 1.0 (* 2.0 (/ i beta)))) (* 8.0 (/ (* (* i i) i) beta)))
      (* beta beta))
     (-
      (- 0.0625 (* -0.0625 (/ (fma 2.0 alpha (* 2.0 beta)) i)))
      (* 0.125 (/ (+ alpha beta) i))))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double t_1 = t_0 * t_0;
	double t_2 = i * ((alpha + beta) + i);
	double tmp;
	if ((((t_2 * ((beta * alpha) + t_2)) / t_1) / (t_1 - 1.0)) <= 5e-5) {
		tmp = (((i * i) * (1.0 + (2.0 * (i / beta)))) - (8.0 * (((i * i) * i) / beta))) / (beta * beta);
	} else {
		tmp = (0.0625 - (-0.0625 * (fma(2.0, alpha, (2.0 * beta)) / i))) - (0.125 * ((alpha + beta) / i));
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_1 = Float64(t_0 * t_0)
	t_2 = Float64(i * Float64(Float64(alpha + beta) + i))
	tmp = 0.0
	if (Float64(Float64(Float64(t_2 * Float64(Float64(beta * alpha) + t_2)) / t_1) / Float64(t_1 - 1.0)) <= 5e-5)
		tmp = Float64(Float64(Float64(Float64(i * i) * Float64(1.0 + Float64(2.0 * Float64(i / beta)))) - Float64(8.0 * Float64(Float64(Float64(i * i) * i) / beta))) / Float64(beta * beta));
	else
		tmp = Float64(Float64(0.0625 - Float64(-0.0625 * Float64(fma(2.0, alpha, Float64(2.0 * beta)) / i))) - Float64(0.125 * Float64(Float64(alpha + beta) / i)));
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(i * N[(N[(alpha + beta), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(t$95$2 * N[(N[(beta * alpha), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(t$95$1 - 1.0), $MachinePrecision]), $MachinePrecision], 5e-5], N[(N[(N[(N[(i * i), $MachinePrecision] * N[(1.0 + N[(2.0 * N[(i / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(8.0 * N[(N[(N[(i * i), $MachinePrecision] * i), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(beta * beta), $MachinePrecision]), $MachinePrecision], N[(N[(0.0625 - N[(-0.0625 * N[(N[(2.0 * alpha + N[(2.0 * beta), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.125 * N[(N[(alpha + beta), $MachinePrecision] / 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))) < 5.00000000000000024e-5

   1. Initial program 98.8%

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

   2. Taylor expanded in beta around inf

      \[\leadsto \color{blue}{\frac{\left(i \cdot \left(\alpha + i\right) + \frac{i \cdot \left(i \cdot \left(\alpha + i\right) + {\left(\alpha + i\right)}^{2}\right)}{\beta}\right) - \frac{i \cdot \left(\left(\alpha + i\right) \cdot \left(4 \cdot \alpha + 8 \cdot i\right)\right)}{\beta}}{{\beta}^{2}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\left(i \cdot \left(\alpha + i\right) + \frac{i \cdot \left(i \cdot \left(\alpha + i\right) + {\left(\alpha + i\right)}^{2}\right)}{\beta}\right) - \frac{i \cdot \left(\left(\alpha + i\right) \cdot \left(4 \cdot \alpha + 8 \cdot i\right)\right)}{\beta}}{\color{blue}{{\beta}^{2}}} \]

   4. Applied rewrites 51.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(i, \alpha + i, \frac{i \cdot \mathsf{fma}\left(i, \alpha + i, \left(\alpha + i\right) \cdot \left(\alpha + i\right)\right)}{\beta}\right) - \frac{i \cdot \left(\left(\alpha + i\right) \cdot \mathsf{fma}\left(4, \alpha, 8 \cdot i\right)\right)}{\beta}}{\beta \cdot \beta}} \]

   5. Taylor expanded in alpha around 0

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

      1. lower--.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \frac{{i}^{3}}{\beta} + {i}^{2}\right) - 8 \cdot \frac{{i}^{3}}{\beta}}{\beta \cdot \beta} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(2, \frac{{i}^{3}}{\beta}, {i}^{2}\right) - 8 \cdot \frac{{i}^{3}}{\beta}}{\beta \cdot \beta} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(2, \frac{{i}^{3}}{\beta}, {i}^{2}\right) - 8 \cdot \frac{{i}^{3}}{\beta}}{\beta \cdot \beta} \]

      4. unpow3 N/A

         \[\leadsto \frac{\mathsf{fma}\left(2, \frac{\left(i \cdot i\right) \cdot i}{\beta}, {i}^{2}\right) - 8 \cdot \frac{{i}^{3}}{\beta}}{\beta \cdot \beta} \]

      5. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(2, \frac{{i}^{2} \cdot i}{\beta}, {i}^{2}\right) - 8 \cdot \frac{{i}^{3}}{\beta}}{\beta \cdot \beta} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(2, \frac{{i}^{2} \cdot i}{\beta}, {i}^{2}\right) - 8 \cdot \frac{{i}^{3}}{\beta}}{\beta \cdot \beta} \]

      7. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(2, \frac{\left(i \cdot i\right) \cdot i}{\beta}, {i}^{2}\right) - 8 \cdot \frac{{i}^{3}}{\beta}}{\beta \cdot \beta} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(2, \frac{\left(i \cdot i\right) \cdot i}{\beta}, {i}^{2}\right) - 8 \cdot \frac{{i}^{3}}{\beta}}{\beta \cdot \beta} \]

      9. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(2, \frac{\left(i \cdot i\right) \cdot i}{\beta}, i \cdot i\right) - 8 \cdot \frac{{i}^{3}}{\beta}}{\beta \cdot \beta} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(2, \frac{\left(i \cdot i\right) \cdot i}{\beta}, i \cdot i\right) - 8 \cdot \frac{{i}^{3}}{\beta}}{\beta \cdot \beta} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(2, \frac{\left(i \cdot i\right) \cdot i}{\beta}, i \cdot i\right) - 8 \cdot \frac{{i}^{3}}{\beta}}{\beta \cdot \beta} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(2, \frac{\left(i \cdot i\right) \cdot i}{\beta}, i \cdot i\right) - 8 \cdot \frac{{i}^{3}}{\beta}}{\beta \cdot \beta} \]

      13. unpow3 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 47.4

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

   7. Applied rewrites 47.4%

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

   8. Taylor expanded in i around 0

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 47.4

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

   10. Applied rewrites 47.4%

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

   if 5.00000000000000024e-5 < (/.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 13.7%

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

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

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

      3. lower--.f64 N/A

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

      4. metadata-eval 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-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 79.8

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

   4. Applied rewrites 79.8%

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

Alternative 5: 74.1% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 1.56 \cdot 10^{+19}:\\ \;\;\;\;\frac{i \cdot \left(\alpha + i\right)}{\mathsf{fma}\left(i, \mathsf{fma}\left(4, \beta, 4 \cdot i\right), \beta \cdot \beta\right) - 1}\\ \mathbf{elif}\;i \leq 3.3 \cdot 10^{+152}:\\ \;\;\;\;\frac{0.25 \cdot \left(i \cdot i\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= i 1.56e+19)
   (/ (* i (+ alpha i)) (- (fma i (fma 4.0 beta (* 4.0 i)) (* beta beta)) 1.0))
   (if (<= i 3.3e+152)
     (/ (* 0.25 (* i i)) (- (* (+ (+ alpha beta) (* 2.0 i)) (* 2.0 i)) 1.0))
     0.0625)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (i <= 1.56e+19) {
		tmp = (i * (alpha + i)) / (fma(i, fma(4.0, beta, (4.0 * i)), (beta * beta)) - 1.0);
	} else if (i <= 3.3e+152) {
		tmp = (0.25 * (i * i)) / ((((alpha + beta) + (2.0 * i)) * (2.0 * i)) - 1.0);
	} else {
		tmp = 0.0625;
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (i <= 1.56e+19)
		tmp = Float64(Float64(i * Float64(alpha + i)) / Float64(fma(i, fma(4.0, beta, Float64(4.0 * i)), Float64(beta * beta)) - 1.0));
	elseif (i <= 3.3e+152)
		tmp = Float64(Float64(0.25 * Float64(i * i)) / Float64(Float64(Float64(Float64(alpha + beta) + Float64(2.0 * i)) * Float64(2.0 * i)) - 1.0));
	else
		tmp = 0.0625;
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[i, 1.56e+19], N[(N[(i * N[(alpha + i), $MachinePrecision]), $MachinePrecision] / N[(N[(i * N[(4.0 * beta + N[(4.0 * i), $MachinePrecision]), $MachinePrecision] + N[(beta * beta), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 3.3e+152], N[(N[(0.25 * N[(i * i), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision] * N[(2.0 * i), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision], 0.0625]]

Derivation

1. Split input into 3 regimes

2. if i < 1.56e19

   1. Initial program 69.8%

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

   2. Taylor expanded in beta around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 45.3

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

   4. Applied rewrites 45.3%

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

   5. Taylor expanded in alpha around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lift-*.f64 36.0

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

   7. Applied rewrites 36.0%

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

   8. Taylor expanded in i around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

         \[\leadsto \frac{i \cdot \left(\alpha + i\right)}{\mathsf{fma}\left(i, \mathsf{fma}\left(4, \beta, 4 \cdot i\right), \beta \cdot \beta\right) - 1} \]

      5. lower-*.f64 36.0

         \[\leadsto \frac{i \cdot \left(\alpha + i\right)}{\mathsf{fma}\left(i, \mathsf{fma}\left(4, \beta, 4 \cdot i\right), \beta \cdot \beta\right) - 1} \]

   10. Applied rewrites 36.0%

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

   if 1.56e19 < i < 3.3000000000000001e152

   1. Initial program 27.8%

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

   2. Taylor expanded in i around inf

      \[\leadsto \frac{\color{blue}{\frac{1}{4} \cdot {i}^{2}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   3. 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. unpow2 N/A

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

      3. lower-*.f64 71.6

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

   4. Applied rewrites 71.6%

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

   5. Taylor expanded in i around inf

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

      1. lift-*.f64 67.2

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

   7. Applied rewrites 67.2%

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

   if 3.3000000000000001e152 < i

   1. Initial program 0.0%

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

   2. Taylor expanded in i around inf

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

   4. Applied rewrites 84.8%

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

Alternative 6: 74.1% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \beta + 2 \cdot i\\ \mathbf{if}\;i \leq 1.56 \cdot 10^{+19}:\\ \;\;\;\;\frac{i \cdot \left(\alpha + i\right)}{t\_0 \cdot t\_0 - 1}\\ \mathbf{elif}\;i \leq 3.3 \cdot 10^{+152}:\\ \;\;\;\;\frac{0.25 \cdot \left(i \cdot i\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ beta (* 2.0 i))))
   (if (<= i 1.56e+19)
     (/ (* i (+ alpha i)) (- (* t_0 t_0) 1.0))
     (if (<= i 3.3e+152)
       (/ (* 0.25 (* i i)) (- (* (+ (+ alpha beta) (* 2.0 i)) (* 2.0 i)) 1.0))
       0.0625))))

C

double code(double alpha, double beta, double i) {
	double t_0 = beta + (2.0 * i);
	double tmp;
	if (i <= 1.56e+19) {
		tmp = (i * (alpha + i)) / ((t_0 * t_0) - 1.0);
	} else if (i <= 3.3e+152) {
		tmp = (0.25 * (i * i)) / ((((alpha + beta) + (2.0 * i)) * (2.0 * i)) - 1.0);
	} else {
		tmp = 0.0625;
	}
	return tmp;
}

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) :: tmp
    t_0 = beta + (2.0d0 * i)
    if (i <= 1.56d+19) then
        tmp = (i * (alpha + i)) / ((t_0 * t_0) - 1.0d0)
    else if (i <= 3.3d+152) then
        tmp = (0.25d0 * (i * i)) / ((((alpha + beta) + (2.0d0 * i)) * (2.0d0 * i)) - 1.0d0)
    else
        tmp = 0.0625d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = beta + (2.0 * i);
	double tmp;
	if (i <= 1.56e+19) {
		tmp = (i * (alpha + i)) / ((t_0 * t_0) - 1.0);
	} else if (i <= 3.3e+152) {
		tmp = (0.25 * (i * i)) / ((((alpha + beta) + (2.0 * i)) * (2.0 * i)) - 1.0);
	} else {
		tmp = 0.0625;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = beta + (2.0 * i)
	tmp = 0
	if i <= 1.56e+19:
		tmp = (i * (alpha + i)) / ((t_0 * t_0) - 1.0)
	elif i <= 3.3e+152:
		tmp = (0.25 * (i * i)) / ((((alpha + beta) + (2.0 * i)) * (2.0 * i)) - 1.0)
	else:
		tmp = 0.0625
	return tmp

Julia

function code(alpha, beta, i)
	t_0 = Float64(beta + Float64(2.0 * i))
	tmp = 0.0
	if (i <= 1.56e+19)
		tmp = Float64(Float64(i * Float64(alpha + i)) / Float64(Float64(t_0 * t_0) - 1.0));
	elseif (i <= 3.3e+152)
		tmp = Float64(Float64(0.25 * Float64(i * i)) / Float64(Float64(Float64(Float64(alpha + beta) + Float64(2.0 * i)) * Float64(2.0 * i)) - 1.0));
	else
		tmp = 0.0625;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	t_0 = beta + (2.0 * i);
	tmp = 0.0;
	if (i <= 1.56e+19)
		tmp = (i * (alpha + i)) / ((t_0 * t_0) - 1.0);
	elseif (i <= 3.3e+152)
		tmp = (0.25 * (i * i)) / ((((alpha + beta) + (2.0 * i)) * (2.0 * i)) - 1.0);
	else
		tmp = 0.0625;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(beta + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, 1.56e+19], N[(N[(i * N[(alpha + i), $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$0 * t$95$0), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 3.3e+152], N[(N[(0.25 * N[(i * i), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision] * N[(2.0 * i), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision], 0.0625]]]

Derivation

1. Split input into 3 regimes

2. if i < 1.56e19

   1. Initial program 69.8%

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

   2. Taylor expanded in beta around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 45.3

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

   4. Applied rewrites 45.3%

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

   5. Taylor expanded in alpha around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lift-*.f64 36.0

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

   7. Applied rewrites 36.0%

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

   if 1.56e19 < i < 3.3000000000000001e152

   1. Initial program 27.8%

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

   2. Taylor expanded in i around inf

      \[\leadsto \frac{\color{blue}{\frac{1}{4} \cdot {i}^{2}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   3. 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. unpow2 N/A

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

      3. lower-*.f64 71.6

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

   4. Applied rewrites 71.6%

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

   5. Taylor expanded in i around inf

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

      1. lift-*.f64 67.2

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

   7. Applied rewrites 67.2%

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

   if 3.3000000000000001e152 < i

   1. Initial program 0.0%

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

   2. Taylor expanded in i around inf

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

   4. Applied rewrites 84.8%

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

Alternative 7: 72.8% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \beta + 2 \cdot i\\ \mathbf{if}\;i \leq 1.56 \cdot 10^{+19}:\\ \;\;\;\;\frac{i \cdot \left(\alpha + i\right)}{t\_0 \cdot t\_0 - 1}\\ \mathbf{elif}\;i \leq 3.3 \cdot 10^{+152}:\\ \;\;\;\;\frac{0.25 \cdot \left(i \cdot i\right)}{t\_0 \cdot \left(2 \cdot i\right) - 1}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ beta (* 2.0 i))))
   (if (<= i 1.56e+19)
     (/ (* i (+ alpha i)) (- (* t_0 t_0) 1.0))
     (if (<= i 3.3e+152)
       (/ (* 0.25 (* i i)) (- (* t_0 (* 2.0 i)) 1.0))
       0.0625))))

C

double code(double alpha, double beta, double i) {
	double t_0 = beta + (2.0 * i);
	double tmp;
	if (i <= 1.56e+19) {
		tmp = (i * (alpha + i)) / ((t_0 * t_0) - 1.0);
	} else if (i <= 3.3e+152) {
		tmp = (0.25 * (i * i)) / ((t_0 * (2.0 * i)) - 1.0);
	} else {
		tmp = 0.0625;
	}
	return tmp;
}

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) :: tmp
    t_0 = beta + (2.0d0 * i)
    if (i <= 1.56d+19) then
        tmp = (i * (alpha + i)) / ((t_0 * t_0) - 1.0d0)
    else if (i <= 3.3d+152) then
        tmp = (0.25d0 * (i * i)) / ((t_0 * (2.0d0 * i)) - 1.0d0)
    else
        tmp = 0.0625d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = beta + (2.0 * i);
	double tmp;
	if (i <= 1.56e+19) {
		tmp = (i * (alpha + i)) / ((t_0 * t_0) - 1.0);
	} else if (i <= 3.3e+152) {
		tmp = (0.25 * (i * i)) / ((t_0 * (2.0 * i)) - 1.0);
	} else {
		tmp = 0.0625;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = beta + (2.0 * i)
	tmp = 0
	if i <= 1.56e+19:
		tmp = (i * (alpha + i)) / ((t_0 * t_0) - 1.0)
	elif i <= 3.3e+152:
		tmp = (0.25 * (i * i)) / ((t_0 * (2.0 * i)) - 1.0)
	else:
		tmp = 0.0625
	return tmp

Julia

function code(alpha, beta, i)
	t_0 = Float64(beta + Float64(2.0 * i))
	tmp = 0.0
	if (i <= 1.56e+19)
		tmp = Float64(Float64(i * Float64(alpha + i)) / Float64(Float64(t_0 * t_0) - 1.0));
	elseif (i <= 3.3e+152)
		tmp = Float64(Float64(0.25 * Float64(i * i)) / Float64(Float64(t_0 * Float64(2.0 * i)) - 1.0));
	else
		tmp = 0.0625;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	t_0 = beta + (2.0 * i);
	tmp = 0.0;
	if (i <= 1.56e+19)
		tmp = (i * (alpha + i)) / ((t_0 * t_0) - 1.0);
	elseif (i <= 3.3e+152)
		tmp = (0.25 * (i * i)) / ((t_0 * (2.0 * i)) - 1.0);
	else
		tmp = 0.0625;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(beta + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, 1.56e+19], N[(N[(i * N[(alpha + i), $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$0 * t$95$0), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 3.3e+152], N[(N[(0.25 * N[(i * i), $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$0 * N[(2.0 * i), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision], 0.0625]]]

Derivation

1. Split input into 3 regimes

2. if i < 1.56e19

   1. Initial program 69.8%

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

   2. Taylor expanded in beta around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 45.3

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

   4. Applied rewrites 45.3%

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

   5. Taylor expanded in alpha around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lift-*.f64 36.0

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

   7. Applied rewrites 36.0%

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

   if 1.56e19 < i < 3.3000000000000001e152

   1. Initial program 27.8%

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

   2. Taylor expanded in i around inf

      \[\leadsto \frac{\color{blue}{\frac{1}{4} \cdot {i}^{2}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   3. 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. unpow2 N/A

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

      3. lower-*.f64 71.6

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

   4. Applied rewrites 71.6%

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

   5. Taylor expanded in i around inf

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

      1. lift-*.f64 67.2

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

   7. Applied rewrites 67.2%

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

   8. Taylor expanded in alpha around 0

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

   10. Applied rewrites 63.5%

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

   if 3.3000000000000001e152 < i

   1. Initial program 0.0%

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

   2. Taylor expanded in i around inf

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

   4. Applied rewrites 84.8%

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

Alternative 8: 72.7% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6.5 \cdot 10^{+225}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;0.125 \cdot \frac{\beta}{i} - 0.125 \cdot \frac{\alpha + \beta}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 6.5e+225)
   0.0625
   (- (* 0.125 (/ beta i)) (* 0.125 (/ (+ alpha beta) i)))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 6.5e+225) {
		tmp = 0.0625;
	} else {
		tmp = (0.125 * (beta / i)) - (0.125 * ((alpha + beta) / i));
	}
	return tmp;
}

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) :: tmp
    if (beta <= 6.5d+225) then
        tmp = 0.0625d0
    else
        tmp = (0.125d0 * (beta / i)) - (0.125d0 * ((alpha + beta) / i))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 6.5e+225) {
		tmp = 0.0625;
	} else {
		tmp = (0.125 * (beta / i)) - (0.125 * ((alpha + beta) / i));
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if beta <= 6.5e+225:
		tmp = 0.0625
	else:
		tmp = (0.125 * (beta / i)) - (0.125 * ((alpha + beta) / i))
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 6.5e+225)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(0.125 * Float64(beta / i)) - Float64(0.125 * Float64(Float64(alpha + beta) / i)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 6.5e+225)
		tmp = 0.0625;
	else
		tmp = (0.125 * (beta / i)) - (0.125 * ((alpha + beta) / i));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 6.5e+225], 0.0625, N[(N[(0.125 * N[(beta / i), $MachinePrecision]), $MachinePrecision] - N[(0.125 * N[(N[(alpha + beta), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 6.5000000000000006e225

   1. Initial program 18.0%

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

   2. Taylor expanded in i around inf

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

   4. Applied rewrites 76.6%

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

   if 6.5000000000000006e225 < beta

   1. Initial program 0.0%

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

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

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

      3. lower--.f64 N/A

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

      4. metadata-eval 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-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 44.7

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

   4. Applied rewrites 44.7%

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

   5. Taylor expanded in beta around inf

      \[\leadsto \frac{1}{8} \cdot \frac{\beta}{i} - \color{blue}{\frac{1}{8}} \cdot \frac{\alpha + \beta}{i} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{8} \cdot \frac{\beta}{i} - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

      2. lower-/.f64 32.2

         \[\leadsto 0.125 \cdot \frac{\beta}{i} - 0.125 \cdot \frac{\alpha + \beta}{i} \]

   7. Applied rewrites 32.2%

      \[\leadsto 0.125 \cdot \frac{\beta}{i} - \color{blue}{0.125} \cdot \frac{\alpha + \beta}{i} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 72.6% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6.5 \cdot 10^{+225}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \alpha}{\left(\beta + 2 \cdot i\right) \cdot \beta - 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 6.5e+225)
   0.0625
   (/ (* i alpha) (- (* (+ beta (* 2.0 i)) beta) 1.0))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 6.5e+225) {
		tmp = 0.0625;
	} else {
		tmp = (i * alpha) / (((beta + (2.0 * i)) * beta) - 1.0);
	}
	return tmp;
}

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) :: tmp
    if (beta <= 6.5d+225) then
        tmp = 0.0625d0
    else
        tmp = (i * alpha) / (((beta + (2.0d0 * i)) * beta) - 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 6.5e+225) {
		tmp = 0.0625;
	} else {
		tmp = (i * alpha) / (((beta + (2.0 * i)) * beta) - 1.0);
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if beta <= 6.5e+225:
		tmp = 0.0625
	else:
		tmp = (i * alpha) / (((beta + (2.0 * i)) * beta) - 1.0)
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 6.5e+225)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(i * alpha) / Float64(Float64(Float64(beta + Float64(2.0 * i)) * beta) - 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 6.5e+225)
		tmp = 0.0625;
	else
		tmp = (i * alpha) / (((beta + (2.0 * i)) * beta) - 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 6.5e+225], 0.0625, N[(N[(i * alpha), $MachinePrecision] / N[(N[(N[(beta + N[(2.0 * i), $MachinePrecision]), $MachinePrecision] * beta), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 6.5000000000000006e225

   1. Initial program 18.0%

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

   2. Taylor expanded in i around inf

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

   4. Applied rewrites 76.6%

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

   if 6.5000000000000006e225 < beta

   1. Initial program 0.0%

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

   2. Taylor expanded in beta around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 30.1

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

   4. Applied rewrites 30.1%

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

   5. Taylor expanded in alpha around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lift-*.f64 30.1

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

   7. Applied rewrites 30.1%

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

   8. Taylor expanded in beta around inf

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

   10. Applied rewrites 30.1%

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

   11. Taylor expanded in alpha around inf

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

   13. Applied rewrites 31.7%

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

Alternative 10: 72.5% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.02 \cdot 10^{+226}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot i}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 1.02e+226) 0.0625 (/ (* i i) (* beta beta))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.02e+226) {
		tmp = 0.0625;
	} else {
		tmp = (i * i) / (beta * beta);
	}
	return tmp;
}

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) :: tmp
    if (beta <= 1.02d+226) then
        tmp = 0.0625d0
    else
        tmp = (i * i) / (beta * beta)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.02e+226) {
		tmp = 0.0625;
	} else {
		tmp = (i * i) / (beta * beta);
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if beta <= 1.02e+226:
		tmp = 0.0625
	else:
		tmp = (i * i) / (beta * beta)
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 1.02e+226)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(i * i) / Float64(beta * beta));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 1.02e+226)
		tmp = 0.0625;
	else
		tmp = (i * i) / (beta * beta);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 1.02e+226], 0.0625, N[(N[(i * i), $MachinePrecision] / N[(beta * beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.02e226

   1. Initial program 18.0%

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

   2. Taylor expanded in i around inf

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

   4. Applied rewrites 76.6%

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

   if 1.02e226 < beta

   1. Initial program 0.0%

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

   2. Taylor expanded in beta around inf

      \[\leadsto \color{blue}{\frac{\left(i \cdot \left(\alpha + i\right) + \frac{i \cdot \left(i \cdot \left(\alpha + i\right) + {\left(\alpha + i\right)}^{2}\right)}{\beta}\right) - \frac{i \cdot \left(\left(\alpha + i\right) \cdot \left(4 \cdot \alpha + 8 \cdot i\right)\right)}{\beta}}{{\beta}^{2}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\left(i \cdot \left(\alpha + i\right) + \frac{i \cdot \left(i \cdot \left(\alpha + i\right) + {\left(\alpha + i\right)}^{2}\right)}{\beta}\right) - \frac{i \cdot \left(\left(\alpha + i\right) \cdot \left(4 \cdot \alpha + 8 \cdot i\right)\right)}{\beta}}{\color{blue}{{\beta}^{2}}} \]

   4. Applied rewrites 24.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(i, \alpha + i, \frac{i \cdot \mathsf{fma}\left(i, \alpha + i, \left(\alpha + i\right) \cdot \left(\alpha + i\right)\right)}{\beta}\right) - \frac{i \cdot \left(\left(\alpha + i\right) \cdot \mathsf{fma}\left(4, \alpha, 8 \cdot i\right)\right)}{\beta}}{\beta \cdot \beta}} \]

   5. Taylor expanded in alpha around 0

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

      1. lower--.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \frac{{i}^{3}}{\beta} + {i}^{2}\right) - 8 \cdot \frac{{i}^{3}}{\beta}}{\beta \cdot \beta} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(2, \frac{{i}^{3}}{\beta}, {i}^{2}\right) - 8 \cdot \frac{{i}^{3}}{\beta}}{\beta \cdot \beta} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(2, \frac{{i}^{3}}{\beta}, {i}^{2}\right) - 8 \cdot \frac{{i}^{3}}{\beta}}{\beta \cdot \beta} \]

      4. unpow3 N/A

         \[\leadsto \frac{\mathsf{fma}\left(2, \frac{\left(i \cdot i\right) \cdot i}{\beta}, {i}^{2}\right) - 8 \cdot \frac{{i}^{3}}{\beta}}{\beta \cdot \beta} \]

      5. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(2, \frac{{i}^{2} \cdot i}{\beta}, {i}^{2}\right) - 8 \cdot \frac{{i}^{3}}{\beta}}{\beta \cdot \beta} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(2, \frac{{i}^{2} \cdot i}{\beta}, {i}^{2}\right) - 8 \cdot \frac{{i}^{3}}{\beta}}{\beta \cdot \beta} \]

      7. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(2, \frac{\left(i \cdot i\right) \cdot i}{\beta}, {i}^{2}\right) - 8 \cdot \frac{{i}^{3}}{\beta}}{\beta \cdot \beta} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(2, \frac{\left(i \cdot i\right) \cdot i}{\beta}, {i}^{2}\right) - 8 \cdot \frac{{i}^{3}}{\beta}}{\beta \cdot \beta} \]

      9. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(2, \frac{\left(i \cdot i\right) \cdot i}{\beta}, i \cdot i\right) - 8 \cdot \frac{{i}^{3}}{\beta}}{\beta \cdot \beta} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(2, \frac{\left(i \cdot i\right) \cdot i}{\beta}, i \cdot i\right) - 8 \cdot \frac{{i}^{3}}{\beta}}{\beta \cdot \beta} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(2, \frac{\left(i \cdot i\right) \cdot i}{\beta}, i \cdot i\right) - 8 \cdot \frac{{i}^{3}}{\beta}}{\beta \cdot \beta} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(2, \frac{\left(i \cdot i\right) \cdot i}{\beta}, i \cdot i\right) - 8 \cdot \frac{{i}^{3}}{\beta}}{\beta \cdot \beta} \]

      13. unpow3 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 25.6

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

   7. Applied rewrites 25.6%

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

   8. Taylor expanded in beta around inf

      \[\leadsto \frac{{i}^{2}}{\beta \cdot \beta} \]
   9. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto \frac{i \cdot i}{\beta \cdot \beta} \]

      2. lift-*.f64 30.5

         \[\leadsto \frac{i \cdot i}{\beta \cdot \beta} \]

   10. Applied rewrites 30.5%

       \[\leadsto \frac{i \cdot i}{\beta \cdot \beta} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 71.4% accurate, 75.4× speedup

Math

\[\begin{array}{l} \\ 0.0625 \end{array} \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[alpha_, beta_, i_] := 0.0625

Derivation

1. Initial program 16.4%

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

2. Taylor expanded in i around inf

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

4. Applied rewrites 71.4%

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

Reproduce

herbie shell --seed 2025110 
(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)))