Octave 3.8, jcobi/4

Percentage Accurate: 16.4% → 99.7%

Time: 10.5s

Alternatives: 11

Speedup: 115.0×

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: 99.7% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 13.3%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. times-frac N/A

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

   7. lift--.f64 N/A

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

   8. lift-*.f64 N/A

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

4. Applied rewrites 39.2%

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

   1. lift-/.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. div-add N/A

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

   4. associate-*r/ N/A

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

   5. lift-/.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. associate-/l* N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 99.3

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

6. Applied rewrites 99.3%

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

Alternative 2: 85.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(2, i, \beta + \alpha\right)\\ \frac{\mathsf{fma}\left(\frac{i}{t\_0}, \left(\beta + \alpha\right) + i, \beta \cdot \frac{\alpha}{t\_0}\right)}{t\_0 - 1} \cdot \left(\frac{\beta + i}{\mathsf{fma}\left(2, i, \beta\right)} \cdot \frac{i}{1 + \mathsf{fma}\left(2, i, \beta\right)}\right) \end{array} \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 13.3%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. times-frac N/A

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

   7. lift--.f64 N/A

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

   8. lift-*.f64 N/A

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

4. Applied rewrites 39.2%

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

   1. lift-/.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. div-add N/A

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

   4. associate-*r/ N/A

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

   5. lift-/.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. associate-/l* N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 99.3

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

6. Applied rewrites 99.3%

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

7. Taylor expanded in alpha around 0

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

   1. times-frac N/A

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

   2. div-add-rev N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-+.f64 N/A

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

   6. lower-+.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. div-add-rev N/A

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

   9. lower-/.f64 N/A

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

   10. lower-+.f64 N/A

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

   11. lower-+.f64 N/A

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

   12. lower-*.f64 87.1

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

9. Applied rewrites 87.1%

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

11. Applied rewrites 87.1%

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

Alternative 3: 78.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \beta + 2 \cdot i\\ \mathbf{if}\;\beta \leq 10^{+160}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + i}{\mathsf{fma}\left(2, i, \beta + \alpha\right) - 1} \cdot \left(\frac{i}{1 + t\_0} \cdot \frac{\beta + i}{t\_0}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ beta (* 2.0 i))))
   (if (<= beta 1e+160)
     0.0625
     (*
      (/ (+ alpha i) (- (fma 2.0 i (+ beta alpha)) 1.0))
      (* (/ i (+ 1.0 t_0)) (/ (+ beta i) t_0))))))

C

double code(double alpha, double beta, double i) {
	double t_0 = beta + (2.0 * i);
	double tmp;
	if (beta <= 1e+160) {
		tmp = 0.0625;
	} else {
		tmp = ((alpha + i) / (fma(2.0, i, (beta + alpha)) - 1.0)) * ((i / (1.0 + t_0)) * ((beta + i) / t_0));
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(beta + Float64(2.0 * i))
	tmp = 0.0
	if (beta <= 1e+160)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(Float64(alpha + i) / Float64(fma(2.0, i, Float64(beta + alpha)) - 1.0)) * Float64(Float64(i / Float64(1.0 + t_0)) * Float64(Float64(beta + i) / t_0)));
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(beta + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 1e+160], 0.0625, N[(N[(N[(alpha + i), $MachinePrecision] / N[(N[(2.0 * i + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(i / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision] * N[(N[(beta + i), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.00000000000000001e160

   1. Initial program 16.6%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 83.1%

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

   if 1.00000000000000001e160 < 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. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. times-frac N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

   4. Applied rewrites 27.2%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. div-add N/A

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

      4. associate-*r/ N/A

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

      5. lift-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 99.3

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

   6. Applied rewrites 99.3%

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

   7. Taylor expanded in alpha around 0

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

      1. times-frac N/A

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

      2. div-add-rev N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. div-add-rev N/A

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

      9. lower-/.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-*.f64 94.0

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

   9. Applied rewrites 94.0%

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

   10. Taylor expanded in beta around inf

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

       1. lower-+.f64 74.5

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

   12. Applied rewrites 74.5%

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

Alternative 4: 78.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(2, i, \beta + \alpha\right)\\ \mathbf{if}\;\beta \leq 2.8 \cdot 10^{+160}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + i}{\beta} \cdot \frac{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{t\_0}}{1 + t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (fma 2.0 i (+ beta alpha))))
   (if (<= beta 2.8e+160)
     0.0625
     (*
      (/ (+ alpha i) beta)
      (/ (* (+ (+ beta alpha) i) (/ i t_0)) (+ 1.0 t_0))))))

C

double code(double alpha, double beta, double i) {
	double t_0 = fma(2.0, i, (beta + alpha));
	double tmp;
	if (beta <= 2.8e+160) {
		tmp = 0.0625;
	} else {
		tmp = ((alpha + i) / beta) * ((((beta + alpha) + i) * (i / t_0)) / (1.0 + t_0));
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = fma(2.0, i, Float64(beta + alpha))
	tmp = 0.0
	if (beta <= 2.8e+160)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(Float64(alpha + i) / beta) * Float64(Float64(Float64(Float64(beta + alpha) + i) * Float64(i / t_0)) / Float64(1.0 + t_0)));
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(2.0 * i + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 2.8e+160], 0.0625, N[(N[(N[(alpha + i), $MachinePrecision] / beta), $MachinePrecision] * N[(N[(N[(N[(beta + alpha), $MachinePrecision] + i), $MachinePrecision] * N[(i / t$95$0), $MachinePrecision]), $MachinePrecision] / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.8e160

   1. Initial program 16.6%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 82.7%

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

   if 2.8e160 < 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. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. times-frac N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

   4. Applied rewrites 27.8%

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

   5. Taylor expanded in beta around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. distribute-lft-out N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 73.6

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

   7. Applied rewrites 73.6%

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

   9. Applied rewrites 73.6%

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

Alternative 5: 78.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.8 \cdot 10^{+160}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\alpha + i\right) \cdot \frac{-1}{-\beta}\right) \cdot \frac{i}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 2.8e+160)
   0.0625
   (* (* (+ alpha i) (/ -1.0 (- beta))) (/ i beta))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 2.8e+160) {
		tmp = 0.0625;
	} else {
		tmp = ((alpha + i) * (-1.0 / -beta)) * (i / 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 <= 2.8d+160) then
        tmp = 0.0625d0
    else
        tmp = ((alpha + i) * ((-1.0d0) / -beta)) * (i / beta)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 2.8e+160) {
		tmp = 0.0625;
	} else {
		tmp = ((alpha + i) * (-1.0 / -beta)) * (i / beta);
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if beta <= 2.8e+160:
		tmp = 0.0625
	else:
		tmp = ((alpha + i) * (-1.0 / -beta)) * (i / beta)
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 2.8e+160)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(Float64(alpha + i) * Float64(-1.0 / Float64(-beta))) * Float64(i / beta));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 2.8e+160)
		tmp = 0.0625;
	else
		tmp = ((alpha + i) * (-1.0 / -beta)) * (i / beta);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 2.8e+160], 0.0625, N[(N[(N[(alpha + i), $MachinePrecision] * N[(-1.0 / (-beta)), $MachinePrecision]), $MachinePrecision] * N[(i / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.8e160

   1. Initial program 16.6%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 82.7%

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

   if 2.8e160 < 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. Add Preprocessing

   3. Taylor expanded in beta around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 73.2

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

   5. Applied rewrites 73.2%

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

   7. Applied rewrites 37.8%

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

   9. Applied rewrites 73.3%

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

Alternative 6: 78.3% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.8 \cdot 10^{+160}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 2.8e+160) 0.0625 (* (/ (+ alpha i) beta) (/ i beta))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 2.8e+160) {
		tmp = 0.0625;
	} else {
		tmp = ((alpha + i) / beta) * (i / 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 <= 2.8d+160) then
        tmp = 0.0625d0
    else
        tmp = ((alpha + i) / beta) * (i / beta)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 2.8e+160) {
		tmp = 0.0625;
	} else {
		tmp = ((alpha + i) / beta) * (i / beta);
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if beta <= 2.8e+160:
		tmp = 0.0625
	else:
		tmp = ((alpha + i) / beta) * (i / beta)
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 2.8e+160)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(Float64(alpha + i) / beta) * Float64(i / beta));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 2.8e+160)
		tmp = 0.0625;
	else
		tmp = ((alpha + i) / beta) * (i / beta);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 2.8e+160], 0.0625, N[(N[(N[(alpha + i), $MachinePrecision] / beta), $MachinePrecision] * N[(i / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.8e160

   1. Initial program 16.6%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 82.7%

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

   if 2.8e160 < 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. Add Preprocessing

   3. Taylor expanded in beta around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 73.2

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

   5. Applied rewrites 73.2%

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

   7. Applied rewrites 37.8%

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

   9. Applied rewrites 73.2%

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

Alternative 7: 77.5% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.1 \cdot 10^{+161}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 1.1e+161) 0.0625 (* (/ i beta) (/ i beta))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.1e+161) {
		tmp = 0.0625;
	} else {
		tmp = (i / beta) * (i / 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 <= 1.1d+161) then
        tmp = 0.0625d0
    else
        tmp = (i / beta) * (i / beta)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.1e+161) {
		tmp = 0.0625;
	} else {
		tmp = (i / beta) * (i / beta);
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if beta <= 1.1e+161:
		tmp = 0.0625
	else:
		tmp = (i / beta) * (i / beta)
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 1.1e+161)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(i / beta) * Float64(i / beta));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 1.1e+161)
		tmp = 0.0625;
	else
		tmp = (i / beta) * (i / beta);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 1.1e+161], 0.0625, N[(N[(i / beta), $MachinePrecision] * N[(i / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.1e161

   1. Initial program 16.6%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 82.7%

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

   if 1.1e161 < 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. Add Preprocessing

   3. Taylor expanded in beta around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 73.2

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

   5. Applied rewrites 73.2%

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

   6. Taylor expanded in alpha around 0

      \[\leadsto \frac{i}{\beta} \cdot \frac{\color{blue}{i}}{\beta} \]
   7. Step-by-step derivation

   8. Applied rewrites 67.7%

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

Alternative 8: 77.4% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.08 \cdot 10^{+161}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\beta} \cdot i}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 1.08e+161) 0.0625 (/ (* (/ i beta) i) beta)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.08e+161) {
		tmp = 0.0625;
	} else {
		tmp = ((i / beta) * i) / 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 <= 1.08d+161) then
        tmp = 0.0625d0
    else
        tmp = ((i / beta) * i) / beta
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.08e+161) {
		tmp = 0.0625;
	} else {
		tmp = ((i / beta) * i) / beta;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if beta <= 1.08e+161:
		tmp = 0.0625
	else:
		tmp = ((i / beta) * i) / beta
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 1.08e+161)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(Float64(i / beta) * i) / beta);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 1.08e+161)
		tmp = 0.0625;
	else
		tmp = ((i / beta) * i) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 1.08e+161], 0.0625, N[(N[(N[(i / beta), $MachinePrecision] * i), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.08e161

   1. Initial program 16.6%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 82.7%

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

   if 1.08e161 < 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. Add Preprocessing

   3. Taylor expanded in beta around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 73.2

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

   5. Applied rewrites 73.2%

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

   6. Taylor expanded in alpha around 0

      \[\leadsto \frac{{i}^{2}}{\color{blue}{{\beta}^{2}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 25.3%

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

   10. Applied rewrites 67.6%

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

Alternative 9: 74.6% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.08 \cdot 10^{+161}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\beta}}{\beta} \cdot i\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 1.08e+161) 0.0625 (* (/ (/ i beta) beta) i)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.08e+161) {
		tmp = 0.0625;
	} else {
		tmp = ((i / beta) / 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 <= 1.08d+161) then
        tmp = 0.0625d0
    else
        tmp = ((i / beta) / beta) * i
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.08e+161) {
		tmp = 0.0625;
	} else {
		tmp = ((i / beta) / beta) * i;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if beta <= 1.08e+161:
		tmp = 0.0625
	else:
		tmp = ((i / beta) / beta) * i
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 1.08e+161)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(Float64(i / beta) / beta) * i);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 1.08e+161)
		tmp = 0.0625;
	else
		tmp = ((i / beta) / beta) * i;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 1.08e+161], 0.0625, N[(N[(N[(i / beta), $MachinePrecision] / beta), $MachinePrecision] * i), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.08e161

   1. Initial program 16.6%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 82.7%

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

   if 1.08e161 < 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. Add Preprocessing

   3. Taylor expanded in beta around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 73.2

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

   5. Applied rewrites 73.2%

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

   6. Taylor expanded in alpha around 0

      \[\leadsto \frac{{i}^{2}}{\color{blue}{{\beta}^{2}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 25.3%

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

   10. Applied rewrites 53.8%

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

Alternative 10: 72.8% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.15 \cdot 10^{+220}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\alpha \cdot \frac{i}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 2.15e+220) 0.0625 (* alpha (/ i (* beta beta)))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 2.15e+220) {
		tmp = 0.0625;
	} else {
		tmp = alpha * (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 <= 2.15d+220) then
        tmp = 0.0625d0
    else
        tmp = alpha * (i / (beta * beta))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 2.15e+220) {
		tmp = 0.0625;
	} else {
		tmp = alpha * (i / (beta * beta));
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if beta <= 2.15e+220:
		tmp = 0.0625
	else:
		tmp = alpha * (i / (beta * beta))
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 2.15e+220)
		tmp = 0.0625;
	else
		tmp = Float64(alpha * Float64(i / Float64(beta * beta)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 2.15e+220)
		tmp = 0.0625;
	else
		tmp = alpha * (i / (beta * beta));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 2.15e+220], 0.0625, N[(alpha * N[(i / N[(beta * beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.15e220

   1. Initial program 15.0%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 78.1%

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

   if 2.15e220 < 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. Add Preprocessing

   3. Taylor expanded in beta around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 82.3

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

   5. Applied rewrites 82.3%

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

   6. Taylor expanded in alpha around inf

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

   8. Applied rewrites 36.0%

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

Alternative 11: 71.3% accurate, 115.0× 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 13.3%

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

3. Taylor expanded in i around inf

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

5. Applied rewrites 71.6%

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

Reproduce

herbie shell --seed 2024350 
(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)))