1/2(abs(p)+abs(r) + sqrt((p-r)^2 + 4q^2))

Percentage Accurate: 45.8% → 72.4%

Time: 3.5s

Alternatives: 7

Speedup: 3.4×

Specification

Math

\[\begin{array}{l} \\ \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \end{array} \]

FPCore

(FPCore (p r q)
 :precision binary64
 (*
  (/ 1.0 2.0)
  (+ (+ (fabs p) (fabs r)) (sqrt (+ (pow (- p r) 2.0) (* 4.0 (pow q 2.0)))))))

C

double code(double p, double r, double q) {
	return (1.0 / 2.0) * ((fabs(p) + fabs(r)) + sqrt((pow((p - r), 2.0) + (4.0 * pow(q, 2.0)))));
}

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(p, r, q)
use fmin_fmax_functions
    real(8), intent (in) :: p
    real(8), intent (in) :: r
    real(8), intent (in) :: q
    code = (1.0d0 / 2.0d0) * ((abs(p) + abs(r)) + sqrt((((p - r) ** 2.0d0) + (4.0d0 * (q ** 2.0d0)))))
end function

Java

public static double code(double p, double r, double q) {
	return (1.0 / 2.0) * ((Math.abs(p) + Math.abs(r)) + Math.sqrt((Math.pow((p - r), 2.0) + (4.0 * Math.pow(q, 2.0)))));
}

Python

def code(p, r, q):
	return (1.0 / 2.0) * ((math.fabs(p) + math.fabs(r)) + math.sqrt((math.pow((p - r), 2.0) + (4.0 * math.pow(q, 2.0)))))

Julia

function code(p, r, q)
	return Float64(Float64(1.0 / 2.0) * Float64(Float64(abs(p) + abs(r)) + sqrt(Float64((Float64(p - r) ^ 2.0) + Float64(4.0 * (q ^ 2.0))))))
end

MATLAB

function tmp = code(p, r, q)
	tmp = (1.0 / 2.0) * ((abs(p) + abs(r)) + sqrt((((p - r) ^ 2.0) + (4.0 * (q ^ 2.0)))));
end

Wolfram

code[p_, r_, q_] := N[(N[(1.0 / 2.0), $MachinePrecision] * N[(N[(N[Abs[p], $MachinePrecision] + N[Abs[r], $MachinePrecision]), $MachinePrecision] + N[Sqrt[N[(N[Power[N[(p - r), $MachinePrecision], 2.0], $MachinePrecision] + N[(4.0 * N[Power[q, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 45.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \end{array} \]

FPCore

(FPCore (p r q)
 :precision binary64
 (*
  (/ 1.0 2.0)
  (+ (+ (fabs p) (fabs r)) (sqrt (+ (pow (- p r) 2.0) (* 4.0 (pow q 2.0)))))))

C

double code(double p, double r, double q) {
	return (1.0 / 2.0) * ((fabs(p) + fabs(r)) + sqrt((pow((p - r), 2.0) + (4.0 * pow(q, 2.0)))));
}

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(p, r, q)
use fmin_fmax_functions
    real(8), intent (in) :: p
    real(8), intent (in) :: r
    real(8), intent (in) :: q
    code = (1.0d0 / 2.0d0) * ((abs(p) + abs(r)) + sqrt((((p - r) ** 2.0d0) + (4.0d0 * (q ** 2.0d0)))))
end function

Java

public static double code(double p, double r, double q) {
	return (1.0 / 2.0) * ((Math.abs(p) + Math.abs(r)) + Math.sqrt((Math.pow((p - r), 2.0) + (4.0 * Math.pow(q, 2.0)))));
}

Python

def code(p, r, q):
	return (1.0 / 2.0) * ((math.fabs(p) + math.fabs(r)) + math.sqrt((math.pow((p - r), 2.0) + (4.0 * math.pow(q, 2.0)))))

Julia

function code(p, r, q)
	return Float64(Float64(1.0 / 2.0) * Float64(Float64(abs(p) + abs(r)) + sqrt(Float64((Float64(p - r) ^ 2.0) + Float64(4.0 * (q ^ 2.0))))))
end

MATLAB

function tmp = code(p, r, q)
	tmp = (1.0 / 2.0) * ((abs(p) + abs(r)) + sqrt((((p - r) ^ 2.0) + (4.0 * (q ^ 2.0)))));
end

Wolfram

code[p_, r_, q_] := N[(N[(1.0 / 2.0), $MachinePrecision] * N[(N[(N[Abs[p], $MachinePrecision] + N[Abs[r], $MachinePrecision]), $MachinePrecision] + N[Sqrt[N[(N[Power[N[(p - r), $MachinePrecision], 2.0], $MachinePrecision] + N[(4.0 * N[Power[q, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 72.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} [p, r, q] = \mathsf{sort}([p, r, q])\\ \\ \begin{array}{l} t_0 := \left|p\right| + \left|r\right|\\ t_1 := \frac{1}{2} \cdot \left(t\_0 + \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right)\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{+152}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;r \cdot \left(0.5 - -0.5 \cdot \frac{t\_0 + \left(-p\right)}{r}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: p, r, and q should be sorted in increasing order before calling this function.
(FPCore (p r q)
 :precision binary64
 (let* ((t_0 (+ (fabs p) (fabs r)))
        (t_1
         (*
          (/ 1.0 2.0)
          (+ t_0 (sqrt (+ (pow (- p r) 2.0) (* 4.0 (pow q 2.0))))))))
   (if (<= t_1 5e+152) t_1 (* r (- 0.5 (* -0.5 (/ (+ t_0 (- p)) r)))))))

C

assert(p < r && r < q);
double code(double p, double r, double q) {
	double t_0 = fabs(p) + fabs(r);
	double t_1 = (1.0 / 2.0) * (t_0 + sqrt((pow((p - r), 2.0) + (4.0 * pow(q, 2.0)))));
	double tmp;
	if (t_1 <= 5e+152) {
		tmp = t_1;
	} else {
		tmp = r * (0.5 - (-0.5 * ((t_0 + -p) / r)));
	}
	return tmp;
}

Fortran

NOTE: p, r, and q should be sorted in increasing order before calling this function.
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(p, r, q)
use fmin_fmax_functions
    real(8), intent (in) :: p
    real(8), intent (in) :: r
    real(8), intent (in) :: q
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = abs(p) + abs(r)
    t_1 = (1.0d0 / 2.0d0) * (t_0 + sqrt((((p - r) ** 2.0d0) + (4.0d0 * (q ** 2.0d0)))))
    if (t_1 <= 5d+152) then
        tmp = t_1
    else
        tmp = r * (0.5d0 - ((-0.5d0) * ((t_0 + -p) / r)))
    end if
    code = tmp
end function

Java

assert p < r && r < q;
public static double code(double p, double r, double q) {
	double t_0 = Math.abs(p) + Math.abs(r);
	double t_1 = (1.0 / 2.0) * (t_0 + Math.sqrt((Math.pow((p - r), 2.0) + (4.0 * Math.pow(q, 2.0)))));
	double tmp;
	if (t_1 <= 5e+152) {
		tmp = t_1;
	} else {
		tmp = r * (0.5 - (-0.5 * ((t_0 + -p) / r)));
	}
	return tmp;
}

Python

[p, r, q] = sort([p, r, q])
def code(p, r, q):
	t_0 = math.fabs(p) + math.fabs(r)
	t_1 = (1.0 / 2.0) * (t_0 + math.sqrt((math.pow((p - r), 2.0) + (4.0 * math.pow(q, 2.0)))))
	tmp = 0
	if t_1 <= 5e+152:
		tmp = t_1
	else:
		tmp = r * (0.5 - (-0.5 * ((t_0 + -p) / r)))
	return tmp

Julia

p, r, q = sort([p, r, q])
function code(p, r, q)
	t_0 = Float64(abs(p) + abs(r))
	t_1 = Float64(Float64(1.0 / 2.0) * Float64(t_0 + sqrt(Float64((Float64(p - r) ^ 2.0) + Float64(4.0 * (q ^ 2.0))))))
	tmp = 0.0
	if (t_1 <= 5e+152)
		tmp = t_1;
	else
		tmp = Float64(r * Float64(0.5 - Float64(-0.5 * Float64(Float64(t_0 + Float64(-p)) / r))));
	end
	return tmp
end

MATLAB

p, r, q = num2cell(sort([p, r, q])){:}
function tmp_2 = code(p, r, q)
	t_0 = abs(p) + abs(r);
	t_1 = (1.0 / 2.0) * (t_0 + sqrt((((p - r) ^ 2.0) + (4.0 * (q ^ 2.0)))));
	tmp = 0.0;
	if (t_1 <= 5e+152)
		tmp = t_1;
	else
		tmp = r * (0.5 - (-0.5 * ((t_0 + -p) / r)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: p, r, and q should be sorted in increasing order before calling this function.
code[p_, r_, q_] := Block[{t$95$0 = N[(N[Abs[p], $MachinePrecision] + N[Abs[r], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(1.0 / 2.0), $MachinePrecision] * N[(t$95$0 + N[Sqrt[N[(N[Power[N[(p - r), $MachinePrecision], 2.0], $MachinePrecision] + N[(4.0 * N[Power[q, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 5e+152], t$95$1, N[(r * N[(0.5 - N[(-0.5 * N[(N[(t$95$0 + (-p)), $MachinePrecision] / r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (+.f64 (+.f64 (fabs.f64 p) (fabs.f64 r)) (sqrt.f64 (+.f64 (pow.f64 (-.f64 p r) #s(literal 2 binary64)) (*.f64 #s(literal 4 binary64) (pow.f64 q #s(literal 2 binary64))))))) < 5e152

   1. Initial program 97.3%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]

   if 5e152 < (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (+.f64 (+.f64 (fabs.f64 p) (fabs.f64 r)) (sqrt.f64 (+.f64 (pow.f64 (-.f64 p r) #s(literal 2 binary64)) (*.f64 #s(literal 4 binary64) (pow.f64 q #s(literal 2 binary64)))))))

   1. Initial program 7.8%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]

   2. Taylor expanded in p around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(\mathsf{neg}\left(p\right)\right)\right) \]

      2. lower-neg.f64 39.7

         \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(-p\right)\right) \]

   4. Applied rewrites 39.7%

      \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \color{blue}{\left(-p\right)}\right) \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2}} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(-p\right)\right) \]

      2. metadata-eval 39.7

         \[\leadsto \color{blue}{0.5} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(-p\right)\right) \]

   6. Applied rewrites 39.7%

      \[\leadsto \color{blue}{0.5} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(-p\right)\right) \]

   7. Taylor expanded in r around inf

      \[\leadsto \color{blue}{r \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{\left|p\right| + \left(\left|r\right| + -1 \cdot p\right)}{r}\right)} \]
   8. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto r \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{\left|p\right| + \left(\left|r\right| + -1 \cdot p\right)}{r}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto r \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{\left|p\right| + \left(\left|r\right| + -1 \cdot p\right)}{r}\right)} \]

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

         \[\leadsto r \cdot \left(\frac{1}{2} - \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{\left|p\right| + \left(\left|r\right| + -1 \cdot p\right)}{r}}\right) \]

      4. lower--.f64 N/A

         \[\leadsto r \cdot \left(\frac{1}{2} - \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{\left|p\right| + \left(\left|r\right| + -1 \cdot p\right)}{r}}\right) \]

      5. metadata-eval N/A

         \[\leadsto r \cdot \left(\frac{1}{2} - \frac{-1}{2} \cdot \frac{\color{blue}{\left|p\right| + \left(\left|r\right| + -1 \cdot p\right)}}{r}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto r \cdot \left(\frac{1}{2} - \frac{-1}{2} \cdot \color{blue}{\frac{\left|p\right| + \left(\left|r\right| + -1 \cdot p\right)}{r}}\right) \]

      7. lower-/.f64 N/A

         \[\leadsto r \cdot \left(\frac{1}{2} - \frac{-1}{2} \cdot \frac{\left|p\right| + \left(\left|r\right| + -1 \cdot p\right)}{\color{blue}{r}}\right) \]

   9. Applied rewrites 54.0%

      \[\leadsto \color{blue}{r \cdot \left(0.5 - -0.5 \cdot \frac{\left(\left|p\right| + \left|r\right|\right) + \left(-p\right)}{r}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 66.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} [p, r, q] = \mathsf{sort}([p, r, q])\\ \\ \begin{array}{l} \mathbf{if}\;q \leq 9.5 \cdot 10^{+141}:\\ \;\;\;\;r \cdot \left(0.5 - -0.5 \cdot \frac{\left(\left|p\right| + \left|r\right|\right) + \left(-p\right)}{r}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, r + \left(q + q\right), p \cdot \left(0.5 - -0.125 \cdot \frac{p}{q}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: p, r, and q should be sorted in increasing order before calling this function.
(FPCore (p r q)
 :precision binary64
 (if (<= q 9.5e+141)
   (* r (- 0.5 (* -0.5 (/ (+ (+ (fabs p) (fabs r)) (- p)) r))))
   (fma 0.5 (+ r (+ q q)) (* p (- 0.5 (* -0.125 (/ p q)))))))

C

assert(p < r && r < q);
double code(double p, double r, double q) {
	double tmp;
	if (q <= 9.5e+141) {
		tmp = r * (0.5 - (-0.5 * (((fabs(p) + fabs(r)) + -p) / r)));
	} else {
		tmp = fma(0.5, (r + (q + q)), (p * (0.5 - (-0.125 * (p / q)))));
	}
	return tmp;
}

Julia

p, r, q = sort([p, r, q])
function code(p, r, q)
	tmp = 0.0
	if (q <= 9.5e+141)
		tmp = Float64(r * Float64(0.5 - Float64(-0.5 * Float64(Float64(Float64(abs(p) + abs(r)) + Float64(-p)) / r))));
	else
		tmp = fma(0.5, Float64(r + Float64(q + q)), Float64(p * Float64(0.5 - Float64(-0.125 * Float64(p / q)))));
	end
	return tmp
end

Wolfram

NOTE: p, r, and q should be sorted in increasing order before calling this function.
code[p_, r_, q_] := If[LessEqual[q, 9.5e+141], N[(r * N[(0.5 - N[(-0.5 * N[(N[(N[(N[Abs[p], $MachinePrecision] + N[Abs[r], $MachinePrecision]), $MachinePrecision] + (-p)), $MachinePrecision] / r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(r + N[(q + q), $MachinePrecision]), $MachinePrecision] + N[(p * N[(0.5 - N[(-0.125 * N[(p / q), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if q < 9.49999999999999974e141

   1. Initial program 50.9%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]

   2. Taylor expanded in p around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(\mathsf{neg}\left(p\right)\right)\right) \]

      2. lower-neg.f64 43.9

         \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(-p\right)\right) \]

   4. Applied rewrites 43.9%

      \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \color{blue}{\left(-p\right)}\right) \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2}} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(-p\right)\right) \]

      2. metadata-eval 43.9

         \[\leadsto \color{blue}{0.5} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(-p\right)\right) \]

   6. Applied rewrites 43.9%

      \[\leadsto \color{blue}{0.5} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(-p\right)\right) \]

   7. Taylor expanded in r around inf

      \[\leadsto \color{blue}{r \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{\left|p\right| + \left(\left|r\right| + -1 \cdot p\right)}{r}\right)} \]
   8. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto r \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{\left|p\right| + \left(\left|r\right| + -1 \cdot p\right)}{r}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto r \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{\left|p\right| + \left(\left|r\right| + -1 \cdot p\right)}{r}\right)} \]

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

         \[\leadsto r \cdot \left(\frac{1}{2} - \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{\left|p\right| + \left(\left|r\right| + -1 \cdot p\right)}{r}}\right) \]

      4. lower--.f64 N/A

         \[\leadsto r \cdot \left(\frac{1}{2} - \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{\left|p\right| + \left(\left|r\right| + -1 \cdot p\right)}{r}}\right) \]

      5. metadata-eval N/A

         \[\leadsto r \cdot \left(\frac{1}{2} - \frac{-1}{2} \cdot \frac{\color{blue}{\left|p\right| + \left(\left|r\right| + -1 \cdot p\right)}}{r}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto r \cdot \left(\frac{1}{2} - \frac{-1}{2} \cdot \color{blue}{\frac{\left|p\right| + \left(\left|r\right| + -1 \cdot p\right)}{r}}\right) \]

      7. lower-/.f64 N/A

         \[\leadsto r \cdot \left(\frac{1}{2} - \frac{-1}{2} \cdot \frac{\left|p\right| + \left(\left|r\right| + -1 \cdot p\right)}{\color{blue}{r}}\right) \]

   9. Applied rewrites 64.2%

      \[\leadsto \color{blue}{r \cdot \left(0.5 - -0.5 \cdot \frac{\left(\left|p\right| + \left|r\right|\right) + \left(-p\right)}{r}\right)} \]

   if 9.49999999999999974e141 < q

   1. Initial program 12.4%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]

   2. Taylor expanded in r around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\left|p\right| + \left(\left|r\right| + \sqrt{4 \cdot {q}^{2} + {p}^{2}}\right)\right)} \]
   3. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{\left|p\right|} + \left(\left|r\right| + \sqrt{4 \cdot {q}^{2} + {p}^{2}}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\left|p\right| + \left(\left|r\right| + \sqrt{4 \cdot {q}^{2} + {p}^{2}}\right)\right) \cdot \color{blue}{\frac{1}{2}} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\left|p\right| + \left(\left|r\right| + \sqrt{4 \cdot {q}^{2} + {p}^{2}}\right)\right) \cdot \color{blue}{\frac{1}{2}} \]

   4. Applied rewrites 12.4%

      \[\leadsto \color{blue}{\left(\left(\sqrt{\mathsf{fma}\left(q \cdot q, 4, p \cdot p\right)} + r\right) + p\right) \cdot 0.5} \]

   5. Taylor expanded in p around 0

      \[\leadsto \frac{1}{2} \cdot \left(r + 2 \cdot q\right) + \color{blue}{p \cdot \left(\frac{1}{2} + \frac{1}{8} \cdot \frac{p}{q}\right)} \]
   6. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, r + \color{blue}{2 \cdot q}, p \cdot \left(\frac{1}{2} + \frac{1}{8} \cdot \frac{p}{q}\right)\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, r + 2 \cdot \color{blue}{q}, p \cdot \left(\frac{1}{2} + \frac{1}{8} \cdot \frac{p}{q}\right)\right) \]

      3. count-2-rev N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, r + \left(q + q\right), p \cdot \left(\frac{1}{2} + \frac{1}{8} \cdot \frac{p}{q}\right)\right) \]

      4. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, r + \left(q + q\right), p \cdot \left(\frac{1}{2} + \frac{1}{8} \cdot \frac{p}{q}\right)\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, r + \left(q + q\right), p \cdot \left(\frac{1}{2} + \frac{1}{8} \cdot \frac{p}{q}\right)\right) \]

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

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, r + \left(q + q\right), p \cdot \left(\frac{1}{2} - \left(\mathsf{neg}\left(\frac{1}{8}\right)\right) \cdot \frac{p}{q}\right)\right) \]

      7. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, r + \left(q + q\right), p \cdot \left(\frac{1}{2} - \left(\mathsf{neg}\left(\frac{1}{8}\right)\right) \cdot \frac{p}{q}\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, r + \left(q + q\right), p \cdot \left(\frac{1}{2} - \frac{-1}{8} \cdot \frac{p}{q}\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, r + \left(q + q\right), p \cdot \left(\frac{1}{2} - \frac{-1}{8} \cdot \frac{p}{q}\right)\right) \]

      10. lower-/.f64 78.5

          \[\leadsto \mathsf{fma}\left(0.5, r + \left(q + q\right), p \cdot \left(0.5 - -0.125 \cdot \frac{p}{q}\right)\right) \]

   7. Applied rewrites 78.5%

      \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{r + \left(q + q\right)}, p \cdot \left(0.5 - -0.125 \cdot \frac{p}{q}\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 66.0% accurate, 2.2× speedup

Math

\[\begin{array}{l} [p, r, q] = \mathsf{sort}([p, r, q])\\ \\ \begin{array}{l} \mathbf{if}\;q \leq 9.5 \cdot 10^{+141}:\\ \;\;\;\;r \cdot \left(0.5 - -0.5 \cdot \frac{\left(\left|p\right| + \left|r\right|\right) + \left(-p\right)}{r}\right)\\ \mathbf{else}:\\ \;\;\;\;q - -0.5 \cdot \left(p + r\right)\\ \end{array} \end{array} \]

FPCore

NOTE: p, r, and q should be sorted in increasing order before calling this function.
(FPCore (p r q)
 :precision binary64
 (if (<= q 9.5e+141)
   (* r (- 0.5 (* -0.5 (/ (+ (+ (fabs p) (fabs r)) (- p)) r))))
   (- q (* -0.5 (+ p r)))))

C

assert(p < r && r < q);
double code(double p, double r, double q) {
	double tmp;
	if (q <= 9.5e+141) {
		tmp = r * (0.5 - (-0.5 * (((fabs(p) + fabs(r)) + -p) / r)));
	} else {
		tmp = q - (-0.5 * (p + r));
	}
	return tmp;
}

Fortran

NOTE: p, r, and q should be sorted in increasing order before calling this function.
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(p, r, q)
use fmin_fmax_functions
    real(8), intent (in) :: p
    real(8), intent (in) :: r
    real(8), intent (in) :: q
    real(8) :: tmp
    if (q <= 9.5d+141) then
        tmp = r * (0.5d0 - ((-0.5d0) * (((abs(p) + abs(r)) + -p) / r)))
    else
        tmp = q - ((-0.5d0) * (p + r))
    end if
    code = tmp
end function

Java

assert p < r && r < q;
public static double code(double p, double r, double q) {
	double tmp;
	if (q <= 9.5e+141) {
		tmp = r * (0.5 - (-0.5 * (((Math.abs(p) + Math.abs(r)) + -p) / r)));
	} else {
		tmp = q - (-0.5 * (p + r));
	}
	return tmp;
}

Python

[p, r, q] = sort([p, r, q])
def code(p, r, q):
	tmp = 0
	if q <= 9.5e+141:
		tmp = r * (0.5 - (-0.5 * (((math.fabs(p) + math.fabs(r)) + -p) / r)))
	else:
		tmp = q - (-0.5 * (p + r))
	return tmp

Julia

p, r, q = sort([p, r, q])
function code(p, r, q)
	tmp = 0.0
	if (q <= 9.5e+141)
		tmp = Float64(r * Float64(0.5 - Float64(-0.5 * Float64(Float64(Float64(abs(p) + abs(r)) + Float64(-p)) / r))));
	else
		tmp = Float64(q - Float64(-0.5 * Float64(p + r)));
	end
	return tmp
end

MATLAB

p, r, q = num2cell(sort([p, r, q])){:}
function tmp_2 = code(p, r, q)
	tmp = 0.0;
	if (q <= 9.5e+141)
		tmp = r * (0.5 - (-0.5 * (((abs(p) + abs(r)) + -p) / r)));
	else
		tmp = q - (-0.5 * (p + r));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: p, r, and q should be sorted in increasing order before calling this function.
code[p_, r_, q_] := If[LessEqual[q, 9.5e+141], N[(r * N[(0.5 - N[(-0.5 * N[(N[(N[(N[Abs[p], $MachinePrecision] + N[Abs[r], $MachinePrecision]), $MachinePrecision] + (-p)), $MachinePrecision] / r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(q - N[(-0.5 * N[(p + r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if q < 9.49999999999999974e141

   1. Initial program 50.9%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]

   2. Taylor expanded in p around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(\mathsf{neg}\left(p\right)\right)\right) \]

      2. lower-neg.f64 43.9

         \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(-p\right)\right) \]

   4. Applied rewrites 43.9%

      \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \color{blue}{\left(-p\right)}\right) \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2}} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(-p\right)\right) \]

      2. metadata-eval 43.9

         \[\leadsto \color{blue}{0.5} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(-p\right)\right) \]

   6. Applied rewrites 43.9%

      \[\leadsto \color{blue}{0.5} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(-p\right)\right) \]

   7. Taylor expanded in r around inf

      \[\leadsto \color{blue}{r \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{\left|p\right| + \left(\left|r\right| + -1 \cdot p\right)}{r}\right)} \]
   8. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto r \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{\left|p\right| + \left(\left|r\right| + -1 \cdot p\right)}{r}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto r \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{\left|p\right| + \left(\left|r\right| + -1 \cdot p\right)}{r}\right)} \]

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

         \[\leadsto r \cdot \left(\frac{1}{2} - \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{\left|p\right| + \left(\left|r\right| + -1 \cdot p\right)}{r}}\right) \]

      4. lower--.f64 N/A

         \[\leadsto r \cdot \left(\frac{1}{2} - \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{\left|p\right| + \left(\left|r\right| + -1 \cdot p\right)}{r}}\right) \]

      5. metadata-eval N/A

         \[\leadsto r \cdot \left(\frac{1}{2} - \frac{-1}{2} \cdot \frac{\color{blue}{\left|p\right| + \left(\left|r\right| + -1 \cdot p\right)}}{r}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto r \cdot \left(\frac{1}{2} - \frac{-1}{2} \cdot \color{blue}{\frac{\left|p\right| + \left(\left|r\right| + -1 \cdot p\right)}{r}}\right) \]

      7. lower-/.f64 N/A

         \[\leadsto r \cdot \left(\frac{1}{2} - \frac{-1}{2} \cdot \frac{\left|p\right| + \left(\left|r\right| + -1 \cdot p\right)}{\color{blue}{r}}\right) \]

   9. Applied rewrites 64.2%

      \[\leadsto \color{blue}{r \cdot \left(0.5 - -0.5 \cdot \frac{\left(\left|p\right| + \left|r\right|\right) + \left(-p\right)}{r}\right)} \]

   if 9.49999999999999974e141 < q

   1. Initial program 12.4%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]

   2. Taylor expanded in q around inf

      \[\leadsto \color{blue}{q \cdot \left(1 + \frac{1}{2} \cdot \frac{\left|p\right| + \left|r\right|}{q}\right)} \]
   3. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto q \cdot \left(1 + \frac{1}{2} \cdot \frac{\color{blue}{\left|p\right| + \left|r\right|}}{q}\right) \]

      2. *-commutative N/A

         \[\leadsto \left(1 + \frac{1}{2} \cdot \frac{\left|p\right| + \left|r\right|}{q}\right) \cdot \color{blue}{q} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(1 + \frac{1}{2} \cdot \frac{\left|p\right| + \left|r\right|}{q}\right) \cdot \color{blue}{q} \]

   4. Applied rewrites 77.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{r + p}{q}, 0.5, 1\right) \cdot q} \]

   5. Taylor expanded in q around 0

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

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

         \[\leadsto q - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \color{blue}{\left(p + r\right)} \]

      2. lower--.f64 N/A

         \[\leadsto q - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \color{blue}{\left(p + r\right)} \]

      3. metadata-eval N/A

         \[\leadsto q - \frac{-1}{2} \cdot \left(p + r\right) \]

      4. lower-*.f64 N/A

         \[\leadsto q - \frac{-1}{2} \cdot \left(p + \color{blue}{r}\right) \]

      5. lower-+.f64 77.6

         \[\leadsto q - -0.5 \cdot \left(p + r\right) \]

   7. Applied rewrites 77.6%

      \[\leadsto q - \color{blue}{-0.5 \cdot \left(p + r\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 55.9% accurate, 2.5× speedup

Math

\[\begin{array}{l} [p, r, q] = \mathsf{sort}([p, r, q])\\ \\ \begin{array}{l} \mathbf{if}\;r \leq 1.15 \cdot 10^{-205}:\\ \;\;\;\;\left(\left|p\right| + \left(\left|r\right| - p\right)\right) \cdot 0.5\\ \mathbf{elif}\;r \leq 4.2 \cdot 10^{+52}:\\ \;\;\;\;q - -0.5 \cdot \left(p + r\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + r\right)\\ \end{array} \end{array} \]

FPCore

NOTE: p, r, and q should be sorted in increasing order before calling this function.
(FPCore (p r q)
 :precision binary64
 (if (<= r 1.15e-205)
   (* (+ (fabs p) (- (fabs r) p)) 0.5)
   (if (<= r 4.2e+52)
     (- q (* -0.5 (+ p r)))
     (* (/ 1.0 2.0) (+ (+ (fabs p) (fabs r)) r)))))

C

assert(p < r && r < q);
double code(double p, double r, double q) {
	double tmp;
	if (r <= 1.15e-205) {
		tmp = (fabs(p) + (fabs(r) - p)) * 0.5;
	} else if (r <= 4.2e+52) {
		tmp = q - (-0.5 * (p + r));
	} else {
		tmp = (1.0 / 2.0) * ((fabs(p) + fabs(r)) + r);
	}
	return tmp;
}

Fortran

NOTE: p, r, and q should be sorted in increasing order before calling this function.
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(p, r, q)
use fmin_fmax_functions
    real(8), intent (in) :: p
    real(8), intent (in) :: r
    real(8), intent (in) :: q
    real(8) :: tmp
    if (r <= 1.15d-205) then
        tmp = (abs(p) + (abs(r) - p)) * 0.5d0
    else if (r <= 4.2d+52) then
        tmp = q - ((-0.5d0) * (p + r))
    else
        tmp = (1.0d0 / 2.0d0) * ((abs(p) + abs(r)) + r)
    end if
    code = tmp
end function

Java

assert p < r && r < q;
public static double code(double p, double r, double q) {
	double tmp;
	if (r <= 1.15e-205) {
		tmp = (Math.abs(p) + (Math.abs(r) - p)) * 0.5;
	} else if (r <= 4.2e+52) {
		tmp = q - (-0.5 * (p + r));
	} else {
		tmp = (1.0 / 2.0) * ((Math.abs(p) + Math.abs(r)) + r);
	}
	return tmp;
}

Python

[p, r, q] = sort([p, r, q])
def code(p, r, q):
	tmp = 0
	if r <= 1.15e-205:
		tmp = (math.fabs(p) + (math.fabs(r) - p)) * 0.5
	elif r <= 4.2e+52:
		tmp = q - (-0.5 * (p + r))
	else:
		tmp = (1.0 / 2.0) * ((math.fabs(p) + math.fabs(r)) + r)
	return tmp

Julia

p, r, q = sort([p, r, q])
function code(p, r, q)
	tmp = 0.0
	if (r <= 1.15e-205)
		tmp = Float64(Float64(abs(p) + Float64(abs(r) - p)) * 0.5);
	elseif (r <= 4.2e+52)
		tmp = Float64(q - Float64(-0.5 * Float64(p + r)));
	else
		tmp = Float64(Float64(1.0 / 2.0) * Float64(Float64(abs(p) + abs(r)) + r));
	end
	return tmp
end

MATLAB

p, r, q = num2cell(sort([p, r, q])){:}
function tmp_2 = code(p, r, q)
	tmp = 0.0;
	if (r <= 1.15e-205)
		tmp = (abs(p) + (abs(r) - p)) * 0.5;
	elseif (r <= 4.2e+52)
		tmp = q - (-0.5 * (p + r));
	else
		tmp = (1.0 / 2.0) * ((abs(p) + abs(r)) + r);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: p, r, and q should be sorted in increasing order before calling this function.
code[p_, r_, q_] := If[LessEqual[r, 1.15e-205], N[(N[(N[Abs[p], $MachinePrecision] + N[(N[Abs[r], $MachinePrecision] - p), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[r, 4.2e+52], N[(q - N[(-0.5 * N[(p + r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / 2.0), $MachinePrecision] * N[(N[(N[Abs[p], $MachinePrecision] + N[Abs[r], $MachinePrecision]), $MachinePrecision] + r), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if r < 1.15e-205

   1. Initial program 48.9%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]

   2. Taylor expanded in p around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(\mathsf{neg}\left(p\right)\right)\right) \]

      2. lower-neg.f64 60.6

         \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(-p\right)\right) \]

   4. Applied rewrites 60.6%

      \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \color{blue}{\left(-p\right)}\right) \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2}} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(-p\right)\right) \]

      2. metadata-eval 60.6

         \[\leadsto \color{blue}{0.5} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(-p\right)\right) \]

   6. Applied rewrites 60.6%

      \[\leadsto \color{blue}{0.5} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(-p\right)\right) \]

   7. Taylor expanded in r around inf

      \[\leadsto \color{blue}{r \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{\left|p\right| + \left(\left|r\right| + -1 \cdot p\right)}{r}\right)} \]
   8. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto r \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{\left|p\right| + \left(\left|r\right| + -1 \cdot p\right)}{r}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto r \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{\left|p\right| + \left(\left|r\right| + -1 \cdot p\right)}{r}\right)} \]

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

         \[\leadsto r \cdot \left(\frac{1}{2} - \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{\left|p\right| + \left(\left|r\right| + -1 \cdot p\right)}{r}}\right) \]

      4. lower--.f64 N/A

         \[\leadsto r \cdot \left(\frac{1}{2} - \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{\left|p\right| + \left(\left|r\right| + -1 \cdot p\right)}{r}}\right) \]

      5. metadata-eval N/A

         \[\leadsto r \cdot \left(\frac{1}{2} - \frac{-1}{2} \cdot \frac{\color{blue}{\left|p\right| + \left(\left|r\right| + -1 \cdot p\right)}}{r}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto r \cdot \left(\frac{1}{2} - \frac{-1}{2} \cdot \color{blue}{\frac{\left|p\right| + \left(\left|r\right| + -1 \cdot p\right)}{r}}\right) \]

      7. lower-/.f64 N/A

         \[\leadsto r \cdot \left(\frac{1}{2} - \frac{-1}{2} \cdot \frac{\left|p\right| + \left(\left|r\right| + -1 \cdot p\right)}{\color{blue}{r}}\right) \]

   9. Applied rewrites 41.7%

      \[\leadsto \color{blue}{r \cdot \left(0.5 - -0.5 \cdot \frac{\left(\left|p\right| + \left|r\right|\right) + \left(-p\right)}{r}\right)} \]

   10. Taylor expanded in r around 0

       \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\left(\left|p\right| + \left|r\right|\right) - p\right)} \]
   11. Step-by-step derivation

       1. metadata-eval N/A

          \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) - p\right) \]

       2. *-commutative N/A

          \[\leadsto \left(\left(\left|p\right| + \left|r\right|\right) - p\right) \cdot \frac{1}{\color{blue}{2}} \]

       3. lower-*.f64 N/A

          \[\leadsto \left(\left(\left|p\right| + \left|r\right|\right) - p\right) \cdot \frac{1}{\color{blue}{2}} \]

       4. associate--l+ N/A

          \[\leadsto \left(\left|p\right| + \left(\left|r\right| - p\right)\right) \cdot \frac{1}{2} \]

       5. lower-+.f64 N/A

          \[\leadsto \left(\left|p\right| + \left(\left|r\right| - p\right)\right) \cdot \frac{1}{2} \]

       6. lift-fabs.f64 N/A

          \[\leadsto \left(\left|p\right| + \left(\left|r\right| - p\right)\right) \cdot \frac{1}{2} \]

       7. lower--.f64 N/A

          \[\leadsto \left(\left|p\right| + \left(\left|r\right| - p\right)\right) \cdot \frac{1}{2} \]

       8. lift-fabs.f64 N/A

          \[\leadsto \left(\left|p\right| + \left(\left|r\right| - p\right)\right) \cdot \frac{1}{2} \]

       9. metadata-eval 60.6

          \[\leadsto \left(\left|p\right| + \left(\left|r\right| - p\right)\right) \cdot 0.5 \]

   12. Applied rewrites 60.6%

       \[\leadsto \left(\left|p\right| + \left(\left|r\right| - p\right)\right) \cdot \color{blue}{0.5} \]

   if 1.15e-205 < r < 4.2e52

   1. Initial program 62.5%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]

   2. Taylor expanded in q around inf

      \[\leadsto \color{blue}{q \cdot \left(1 + \frac{1}{2} \cdot \frac{\left|p\right| + \left|r\right|}{q}\right)} \]
   3. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto q \cdot \left(1 + \frac{1}{2} \cdot \frac{\color{blue}{\left|p\right| + \left|r\right|}}{q}\right) \]

      2. *-commutative N/A

         \[\leadsto \left(1 + \frac{1}{2} \cdot \frac{\left|p\right| + \left|r\right|}{q}\right) \cdot \color{blue}{q} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(1 + \frac{1}{2} \cdot \frac{\left|p\right| + \left|r\right|}{q}\right) \cdot \color{blue}{q} \]

   4. Applied rewrites 27.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{r + p}{q}, 0.5, 1\right) \cdot q} \]

   5. Taylor expanded in q around 0

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

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

         \[\leadsto q - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \color{blue}{\left(p + r\right)} \]

      2. lower--.f64 N/A

         \[\leadsto q - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \color{blue}{\left(p + r\right)} \]

      3. metadata-eval N/A

         \[\leadsto q - \frac{-1}{2} \cdot \left(p + r\right) \]

      4. lower-*.f64 N/A

         \[\leadsto q - \frac{-1}{2} \cdot \left(p + \color{blue}{r}\right) \]

      5. lower-+.f64 27.2

         \[\leadsto q - -0.5 \cdot \left(p + r\right) \]

   7. Applied rewrites 27.2%

      \[\leadsto q - \color{blue}{-0.5 \cdot \left(p + r\right)} \]

   if 4.2e52 < r

   1. Initial program 29.9%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]

   2. Taylor expanded in r around inf

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

   4. Applied rewrites 73.9%

      \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \color{blue}{r}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 55.7% accurate, 3.4× speedup

Math

\[\begin{array}{l} [p, r, q] = \mathsf{sort}([p, r, q])\\ \\ \begin{array}{l} \mathbf{if}\;r \leq 1.15 \cdot 10^{-205}:\\ \;\;\;\;\left(\left|p\right| + \left(\left|r\right| - p\right)\right) \cdot 0.5\\ \mathbf{elif}\;r \leq 4.5 \cdot 10^{+52}:\\ \;\;\;\;q - -0.5 \cdot \left(p + r\right)\\ \mathbf{else}:\\ \;\;\;\;r\\ \end{array} \end{array} \]

FPCore

NOTE: p, r, and q should be sorted in increasing order before calling this function.
(FPCore (p r q)
 :precision binary64
 (if (<= r 1.15e-205)
   (* (+ (fabs p) (- (fabs r) p)) 0.5)
   (if (<= r 4.5e+52) (- q (* -0.5 (+ p r))) r)))

C

assert(p < r && r < q);
double code(double p, double r, double q) {
	double tmp;
	if (r <= 1.15e-205) {
		tmp = (fabs(p) + (fabs(r) - p)) * 0.5;
	} else if (r <= 4.5e+52) {
		tmp = q - (-0.5 * (p + r));
	} else {
		tmp = r;
	}
	return tmp;
}

Fortran

NOTE: p, r, and q should be sorted in increasing order before calling this function.
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(p, r, q)
use fmin_fmax_functions
    real(8), intent (in) :: p
    real(8), intent (in) :: r
    real(8), intent (in) :: q
    real(8) :: tmp
    if (r <= 1.15d-205) then
        tmp = (abs(p) + (abs(r) - p)) * 0.5d0
    else if (r <= 4.5d+52) then
        tmp = q - ((-0.5d0) * (p + r))
    else
        tmp = r
    end if
    code = tmp
end function

Java

assert p < r && r < q;
public static double code(double p, double r, double q) {
	double tmp;
	if (r <= 1.15e-205) {
		tmp = (Math.abs(p) + (Math.abs(r) - p)) * 0.5;
	} else if (r <= 4.5e+52) {
		tmp = q - (-0.5 * (p + r));
	} else {
		tmp = r;
	}
	return tmp;
}

Python

[p, r, q] = sort([p, r, q])
def code(p, r, q):
	tmp = 0
	if r <= 1.15e-205:
		tmp = (math.fabs(p) + (math.fabs(r) - p)) * 0.5
	elif r <= 4.5e+52:
		tmp = q - (-0.5 * (p + r))
	else:
		tmp = r
	return tmp

Julia

p, r, q = sort([p, r, q])
function code(p, r, q)
	tmp = 0.0
	if (r <= 1.15e-205)
		tmp = Float64(Float64(abs(p) + Float64(abs(r) - p)) * 0.5);
	elseif (r <= 4.5e+52)
		tmp = Float64(q - Float64(-0.5 * Float64(p + r)));
	else
		tmp = r;
	end
	return tmp
end

MATLAB

p, r, q = num2cell(sort([p, r, q])){:}
function tmp_2 = code(p, r, q)
	tmp = 0.0;
	if (r <= 1.15e-205)
		tmp = (abs(p) + (abs(r) - p)) * 0.5;
	elseif (r <= 4.5e+52)
		tmp = q - (-0.5 * (p + r));
	else
		tmp = r;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: p, r, and q should be sorted in increasing order before calling this function.
code[p_, r_, q_] := If[LessEqual[r, 1.15e-205], N[(N[(N[Abs[p], $MachinePrecision] + N[(N[Abs[r], $MachinePrecision] - p), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[r, 4.5e+52], N[(q - N[(-0.5 * N[(p + r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], r]]

Derivation

1. Split input into 3 regimes

2. if r < 1.15e-205

   1. Initial program 48.9%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]

   2. Taylor expanded in p around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(\mathsf{neg}\left(p\right)\right)\right) \]

      2. lower-neg.f64 60.6

         \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(-p\right)\right) \]

   4. Applied rewrites 60.6%

      \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \color{blue}{\left(-p\right)}\right) \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2}} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(-p\right)\right) \]

      2. metadata-eval 60.6

         \[\leadsto \color{blue}{0.5} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(-p\right)\right) \]

   6. Applied rewrites 60.6%

      \[\leadsto \color{blue}{0.5} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(-p\right)\right) \]

   7. Taylor expanded in r around inf

      \[\leadsto \color{blue}{r \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{\left|p\right| + \left(\left|r\right| + -1 \cdot p\right)}{r}\right)} \]
   8. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto r \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{\left|p\right| + \left(\left|r\right| + -1 \cdot p\right)}{r}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto r \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{\left|p\right| + \left(\left|r\right| + -1 \cdot p\right)}{r}\right)} \]

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

         \[\leadsto r \cdot \left(\frac{1}{2} - \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{\left|p\right| + \left(\left|r\right| + -1 \cdot p\right)}{r}}\right) \]

      4. lower--.f64 N/A

         \[\leadsto r \cdot \left(\frac{1}{2} - \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{\left|p\right| + \left(\left|r\right| + -1 \cdot p\right)}{r}}\right) \]

      5. metadata-eval N/A

         \[\leadsto r \cdot \left(\frac{1}{2} - \frac{-1}{2} \cdot \frac{\color{blue}{\left|p\right| + \left(\left|r\right| + -1 \cdot p\right)}}{r}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto r \cdot \left(\frac{1}{2} - \frac{-1}{2} \cdot \color{blue}{\frac{\left|p\right| + \left(\left|r\right| + -1 \cdot p\right)}{r}}\right) \]

      7. lower-/.f64 N/A

         \[\leadsto r \cdot \left(\frac{1}{2} - \frac{-1}{2} \cdot \frac{\left|p\right| + \left(\left|r\right| + -1 \cdot p\right)}{\color{blue}{r}}\right) \]

   9. Applied rewrites 41.7%

      \[\leadsto \color{blue}{r \cdot \left(0.5 - -0.5 \cdot \frac{\left(\left|p\right| + \left|r\right|\right) + \left(-p\right)}{r}\right)} \]

   10. Taylor expanded in r around 0

       \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\left(\left|p\right| + \left|r\right|\right) - p\right)} \]
   11. Step-by-step derivation

       1. metadata-eval N/A

          \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) - p\right) \]

       2. *-commutative N/A

          \[\leadsto \left(\left(\left|p\right| + \left|r\right|\right) - p\right) \cdot \frac{1}{\color{blue}{2}} \]

       3. lower-*.f64 N/A

          \[\leadsto \left(\left(\left|p\right| + \left|r\right|\right) - p\right) \cdot \frac{1}{\color{blue}{2}} \]

       4. associate--l+ N/A

          \[\leadsto \left(\left|p\right| + \left(\left|r\right| - p\right)\right) \cdot \frac{1}{2} \]

       5. lower-+.f64 N/A

          \[\leadsto \left(\left|p\right| + \left(\left|r\right| - p\right)\right) \cdot \frac{1}{2} \]

       6. lift-fabs.f64 N/A

          \[\leadsto \left(\left|p\right| + \left(\left|r\right| - p\right)\right) \cdot \frac{1}{2} \]

       7. lower--.f64 N/A

          \[\leadsto \left(\left|p\right| + \left(\left|r\right| - p\right)\right) \cdot \frac{1}{2} \]

       8. lift-fabs.f64 N/A

          \[\leadsto \left(\left|p\right| + \left(\left|r\right| - p\right)\right) \cdot \frac{1}{2} \]

       9. metadata-eval 60.6

          \[\leadsto \left(\left|p\right| + \left(\left|r\right| - p\right)\right) \cdot 0.5 \]

   12. Applied rewrites 60.6%

       \[\leadsto \left(\left|p\right| + \left(\left|r\right| - p\right)\right) \cdot \color{blue}{0.5} \]

   if 1.15e-205 < r < 4.5e52

   1. Initial program 62.5%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]

   2. Taylor expanded in q around inf

      \[\leadsto \color{blue}{q \cdot \left(1 + \frac{1}{2} \cdot \frac{\left|p\right| + \left|r\right|}{q}\right)} \]
   3. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto q \cdot \left(1 + \frac{1}{2} \cdot \frac{\color{blue}{\left|p\right| + \left|r\right|}}{q}\right) \]

      2. *-commutative N/A

         \[\leadsto \left(1 + \frac{1}{2} \cdot \frac{\left|p\right| + \left|r\right|}{q}\right) \cdot \color{blue}{q} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(1 + \frac{1}{2} \cdot \frac{\left|p\right| + \left|r\right|}{q}\right) \cdot \color{blue}{q} \]

   4. Applied rewrites 27.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{r + p}{q}, 0.5, 1\right) \cdot q} \]

   5. Taylor expanded in q around 0

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

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

         \[\leadsto q - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \color{blue}{\left(p + r\right)} \]

      2. lower--.f64 N/A

         \[\leadsto q - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \color{blue}{\left(p + r\right)} \]

      3. metadata-eval N/A

         \[\leadsto q - \frac{-1}{2} \cdot \left(p + r\right) \]

      4. lower-*.f64 N/A

         \[\leadsto q - \frac{-1}{2} \cdot \left(p + \color{blue}{r}\right) \]

      5. lower-+.f64 27.2

         \[\leadsto q - -0.5 \cdot \left(p + r\right) \]

   7. Applied rewrites 27.2%

      \[\leadsto q - \color{blue}{-0.5 \cdot \left(p + r\right)} \]

   if 4.5e52 < r

   1. Initial program 29.9%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]

   2. Taylor expanded in p around inf

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

   4. Applied rewrites 15.1%

      \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \color{blue}{p}\right) \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + p\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2}} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + p\right) \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\left|p\right| + \left|r\right|\right) + p\right) \cdot \frac{1}{2}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left|p\right| + \left|r\right|\right) + p\right) \cdot \frac{1}{2}} \]

   6. Applied rewrites 14.8%

      \[\leadsto \color{blue}{\left(p + \left(r + p\right)\right) \cdot 0.5} \]

   7. Taylor expanded in p around -inf

      \[\leadsto \color{blue}{r} \]
   8. Step-by-step derivation

   9. Applied rewrites 73.6%

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

Alternative 6: 41.5% accurate, 11.9× speedup

Math

\[\begin{array}{l} [p, r, q] = \mathsf{sort}([p, r, q])\\ \\ \begin{array}{l} \mathbf{if}\;r \leq 3.5 \cdot 10^{+52}:\\ \;\;\;\;q\\ \mathbf{else}:\\ \;\;\;\;r\\ \end{array} \end{array} \]

FPCore

NOTE: p, r, and q should be sorted in increasing order before calling this function.
(FPCore (p r q) :precision binary64 (if (<= r 3.5e+52) q r))

C

assert(p < r && r < q);
double code(double p, double r, double q) {
	double tmp;
	if (r <= 3.5e+52) {
		tmp = q;
	} else {
		tmp = r;
	}
	return tmp;
}

Fortran

NOTE: p, r, and q should be sorted in increasing order before calling this function.
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(p, r, q)
use fmin_fmax_functions
    real(8), intent (in) :: p
    real(8), intent (in) :: r
    real(8), intent (in) :: q
    real(8) :: tmp
    if (r <= 3.5d+52) then
        tmp = q
    else
        tmp = r
    end if
    code = tmp
end function

Java

assert p < r && r < q;
public static double code(double p, double r, double q) {
	double tmp;
	if (r <= 3.5e+52) {
		tmp = q;
	} else {
		tmp = r;
	}
	return tmp;
}

Python

[p, r, q] = sort([p, r, q])
def code(p, r, q):
	tmp = 0
	if r <= 3.5e+52:
		tmp = q
	else:
		tmp = r
	return tmp

Julia

p, r, q = sort([p, r, q])
function code(p, r, q)
	tmp = 0.0
	if (r <= 3.5e+52)
		tmp = q;
	else
		tmp = r;
	end
	return tmp
end

MATLAB

p, r, q = num2cell(sort([p, r, q])){:}
function tmp_2 = code(p, r, q)
	tmp = 0.0;
	if (r <= 3.5e+52)
		tmp = q;
	else
		tmp = r;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: p, r, and q should be sorted in increasing order before calling this function.
code[p_, r_, q_] := If[LessEqual[r, 3.5e+52], q, r]

Derivation

1. Split input into 2 regimes

2. if r < 3.5e52

   1. Initial program 55.2%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]

   2. Taylor expanded in q around inf

      \[\leadsto \color{blue}{q} \]
   3. Step-by-step derivation

   4. Applied rewrites 22.6%

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

   if 3.5e52 < r

   1. Initial program 29.9%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]

   2. Taylor expanded in p around inf

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

   4. Applied rewrites 15.1%

      \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \color{blue}{p}\right) \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + p\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2}} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + p\right) \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\left|p\right| + \left|r\right|\right) + p\right) \cdot \frac{1}{2}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left|p\right| + \left|r\right|\right) + p\right) \cdot \frac{1}{2}} \]

   6. Applied rewrites 14.8%

      \[\leadsto \color{blue}{\left(p + \left(r + p\right)\right) \cdot 0.5} \]

   7. Taylor expanded in p around -inf

      \[\leadsto \color{blue}{r} \]
   8. Step-by-step derivation

   9. Applied rewrites 73.6%

      \[\leadsto \color{blue}{r} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 35.5% accurate, 56.9× speedup

Math

\[\begin{array}{l} [p, r, q] = \mathsf{sort}([p, r, q])\\ \\ r \end{array} \]

FPCore

NOTE: p, r, and q should be sorted in increasing order before calling this function.
(FPCore (p r q) :precision binary64 r)

C

assert(p < r && r < q);
double code(double p, double r, double q) {
	return r;
}

Fortran

NOTE: p, r, and q should be sorted in increasing order before calling this function.
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(p, r, q)
use fmin_fmax_functions
    real(8), intent (in) :: p
    real(8), intent (in) :: r
    real(8), intent (in) :: q
    code = r
end function

Java

assert p < r && r < q;
public static double code(double p, double r, double q) {
	return r;
}

Python

[p, r, q] = sort([p, r, q])
def code(p, r, q):
	return r

Julia

p, r, q = sort([p, r, q])
function code(p, r, q)
	return r
end

MATLAB

p, r, q = num2cell(sort([p, r, q])){:}
function tmp = code(p, r, q)
	tmp = r;
end

Wolfram

NOTE: p, r, and q should be sorted in increasing order before calling this function.
code[p_, r_, q_] := r

Derivation

1. Initial program 45.8%

   \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]

2. Taylor expanded in p around inf

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

4. Applied rewrites 8.6%

   \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \color{blue}{p}\right) \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + p\right)} \]

   2. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{2}} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + p\right) \]

   3. *-commutative N/A

      \[\leadsto \color{blue}{\left(\left(\left|p\right| + \left|r\right|\right) + p\right) \cdot \frac{1}{2}} \]

   4. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\left(\left|p\right| + \left|r\right|\right) + p\right) \cdot \frac{1}{2}} \]

6. Applied rewrites 7.7%

   \[\leadsto \color{blue}{\left(p + \left(r + p\right)\right) \cdot 0.5} \]

7. Taylor expanded in p around -inf

   \[\leadsto \color{blue}{r} \]
8. Step-by-step derivation

9. Applied rewrites 35.5%

   \[\leadsto \color{blue}{r} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025110 
(FPCore (p r q)
  :name "1/2(abs(p)+abs(r) + sqrt((p-r)^2 + 4q^2))"
  :precision binary64
  (* (/ 1.0 2.0) (+ (+ (fabs p) (fabs r)) (sqrt (+ (pow (- p r) 2.0) (* 4.0 (pow q 2.0)))))))