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

Percentage Accurate: 45.7% → 80.5%

Time: 4.1s

Alternatives: 8

Speedup: 12.5×

Specification

\[\mathsf{TRUE}\left(\right)\]

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: 45.7% 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: 80.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} [p, r, q] = \mathsf{sort}([p, r, q])\\ \\ \begin{array}{l} t_0 := \left|p\right| + \left|r\right|\\ t_1 := \frac{1}{2} \cdot \left(t\_0 + \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right)\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{+149}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(t\_0 + \left(r - p\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: p, r, and q should be sorted in increasing order before calling this function.
(FPCore (p r q)
 :precision binary64
 (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+149) t_1 (* 0.5 (+ t_0 (- r p))))))

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+149) {
		tmp = t_1;
	} else {
		tmp = 0.5 * (t_0 + (r - p));
	}
	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+149) then
        tmp = t_1
    else
        tmp = 0.5d0 * (t_0 + (r - p))
    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+149) {
		tmp = t_1;
	} else {
		tmp = 0.5 * (t_0 + (r - p));
	}
	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+149:
		tmp = t_1
	else:
		tmp = 0.5 * (t_0 + (r - p))
	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+149)
		tmp = t_1;
	else
		tmp = Float64(0.5 * Float64(t_0 + Float64(r - p)));
	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+149)
		tmp = t_1;
	else
		tmp = 0.5 * (t_0 + (r - p));
	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+149], t$95$1, N[(0.5 * N[(t$95$0 + N[(r - p), $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))))))) < 4.9999999999999999e149

   1. Initial program 97.4%

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

   if 4.9999999999999999e149 < (*.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 7.4%

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

   3. Taylor expanded in r around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 36.0

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

   5. Applied rewrites 36.0%

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

   6. Taylor expanded in p around 0

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

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

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. unpow1 N/A

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

      5. metadata-eval N/A

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

      6. sqrt-pow1 N/A

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

      7. unpow2 N/A

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

      8. rem-sqrt-square-rev N/A

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

      9. fabs-mul N/A

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

      10. mul-1-neg N/A

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

      11. neg-fabs N/A

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

      12. rem-sqrt-square-rev N/A

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

      13. unpow2 N/A

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

      14. sqrt-pow1 N/A

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

      15. metadata-eval N/A

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

      16. unpow1 N/A

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

      17. lower--.f64 41.8

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

   8. Applied rewrites 41.8%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 41.8

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

   10. Applied rewrites 41.8%

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

Alternative 2: 71.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} [p, r, q] = \mathsf{sort}([p, r, q])\\ \\ \begin{array}{l} t_0 := \left|p\right| + \left|r\right|\\ \mathbf{if}\;4 \cdot {q}^{2} \leq 5 \cdot 10^{+250}:\\ \;\;\;\;0.5 \cdot \left(t\_0 + \left(r - p\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(t\_0 + \left(q + q\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: p, r, and q should be sorted in increasing order before calling this function.
(FPCore (p r q)
 :precision binary64
 (let* ((t_0 (+ (fabs p) (fabs r))))
   (if (<= (* 4.0 (pow q 2.0)) 5e+250)
     (* 0.5 (+ t_0 (- r p)))
     (* 0.5 (+ t_0 (+ q q))))))

C

assert(p < r && r < q);
double code(double p, double r, double q) {
	double t_0 = fabs(p) + fabs(r);
	double tmp;
	if ((4.0 * pow(q, 2.0)) <= 5e+250) {
		tmp = 0.5 * (t_0 + (r - p));
	} else {
		tmp = 0.5 * (t_0 + (q + q));
	}
	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) :: tmp
    t_0 = abs(p) + abs(r)
    if ((4.0d0 * (q ** 2.0d0)) <= 5d+250) then
        tmp = 0.5d0 * (t_0 + (r - p))
    else
        tmp = 0.5d0 * (t_0 + (q + q))
    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 tmp;
	if ((4.0 * Math.pow(q, 2.0)) <= 5e+250) {
		tmp = 0.5 * (t_0 + (r - p));
	} else {
		tmp = 0.5 * (t_0 + (q + q));
	}
	return tmp;
}

Python

[p, r, q] = sort([p, r, q])
def code(p, r, q):
	t_0 = math.fabs(p) + math.fabs(r)
	tmp = 0
	if (4.0 * math.pow(q, 2.0)) <= 5e+250:
		tmp = 0.5 * (t_0 + (r - p))
	else:
		tmp = 0.5 * (t_0 + (q + q))
	return tmp

Julia

p, r, q = sort([p, r, q])
function code(p, r, q)
	t_0 = Float64(abs(p) + abs(r))
	tmp = 0.0
	if (Float64(4.0 * (q ^ 2.0)) <= 5e+250)
		tmp = Float64(0.5 * Float64(t_0 + Float64(r - p)));
	else
		tmp = Float64(0.5 * Float64(t_0 + Float64(q + q)));
	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);
	tmp = 0.0;
	if ((4.0 * (q ^ 2.0)) <= 5e+250)
		tmp = 0.5 * (t_0 + (r - p));
	else
		tmp = 0.5 * (t_0 + (q + q));
	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]}, If[LessEqual[N[(4.0 * N[Power[q, 2.0], $MachinePrecision]), $MachinePrecision], 5e+250], N[(0.5 * N[(t$95$0 + N[(r - p), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(t$95$0 + N[(q + q), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 4 binary64) (pow.f64 q #s(literal 2 binary64))) < 5.0000000000000002e250

   1. Initial program 51.7%

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

   3. Taylor expanded in r around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 39.7

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

   5. Applied rewrites 39.7%

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

   6. Taylor expanded in p around 0

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

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

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. unpow1 N/A

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

      5. metadata-eval N/A

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

      6. sqrt-pow1 N/A

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

      7. unpow2 N/A

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

      8. rem-sqrt-square-rev N/A

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

      9. fabs-mul N/A

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

      10. mul-1-neg N/A

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

      11. neg-fabs N/A

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

      12. rem-sqrt-square-rev N/A

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

      13. unpow2 N/A

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

      14. sqrt-pow1 N/A

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

      15. metadata-eval N/A

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

      16. unpow1 N/A

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

      17. lower--.f64 45.0

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

   8. Applied rewrites 45.0%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 45.0

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

   10. Applied rewrites 45.0%

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

   if 5.0000000000000002e250 < (*.f64 #s(literal 4 binary64) (pow.f64 q #s(literal 2 binary64)))

   1. Initial program 20.4%

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

   3. Taylor expanded in q around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 38.9

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

   5. Applied rewrites 38.9%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 38.9

         \[\leadsto \color{blue}{0.5} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + q \cdot 2\right) \]

   7. Applied rewrites 38.9%

      \[\leadsto \color{blue}{0.5} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + q \cdot 2\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 38.9

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

   9. Applied rewrites 38.9%

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

Alternative 3: 55.8% accurate, 8.9× speedup

Math

\[\begin{array}{l} [p, r, q] = \mathsf{sort}([p, r, q])\\ \\ \begin{array}{l} t_0 := \left|p\right| + \left|r\right|\\ \mathbf{if}\;r \leq 4 \cdot 10^{-222}:\\ \;\;\;\;0.5 \cdot \left(t\_0 + \left(-p\right)\right)\\ \mathbf{elif}\;r \leq 5.4 \cdot 10^{+68}:\\ \;\;\;\;\left(q \cdot 2 + r\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(t\_0 + r\right)\\ \end{array} \end{array} \]

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))))
   (if (<= r 4e-222)
     (* 0.5 (+ t_0 (- p)))
     (if (<= r 5.4e+68) (* (+ (* q 2.0) r) 0.5) (* 0.5 (+ t_0 r))))))

C

assert(p < r && r < q);
double code(double p, double r, double q) {
	double t_0 = fabs(p) + fabs(r);
	double tmp;
	if (r <= 4e-222) {
		tmp = 0.5 * (t_0 + -p);
	} else if (r <= 5.4e+68) {
		tmp = ((q * 2.0) + r) * 0.5;
	} else {
		tmp = 0.5 * (t_0 + 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) :: tmp
    t_0 = abs(p) + abs(r)
    if (r <= 4d-222) then
        tmp = 0.5d0 * (t_0 + -p)
    else if (r <= 5.4d+68) then
        tmp = ((q * 2.0d0) + r) * 0.5d0
    else
        tmp = 0.5d0 * (t_0 + 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 tmp;
	if (r <= 4e-222) {
		tmp = 0.5 * (t_0 + -p);
	} else if (r <= 5.4e+68) {
		tmp = ((q * 2.0) + r) * 0.5;
	} else {
		tmp = 0.5 * (t_0 + r);
	}
	return tmp;
}

Python

[p, r, q] = sort([p, r, q])
def code(p, r, q):
	t_0 = math.fabs(p) + math.fabs(r)
	tmp = 0
	if r <= 4e-222:
		tmp = 0.5 * (t_0 + -p)
	elif r <= 5.4e+68:
		tmp = ((q * 2.0) + r) * 0.5
	else:
		tmp = 0.5 * (t_0 + r)
	return tmp

Julia

p, r, q = sort([p, r, q])
function code(p, r, q)
	t_0 = Float64(abs(p) + abs(r))
	tmp = 0.0
	if (r <= 4e-222)
		tmp = Float64(0.5 * Float64(t_0 + Float64(-p)));
	elseif (r <= 5.4e+68)
		tmp = Float64(Float64(Float64(q * 2.0) + r) * 0.5);
	else
		tmp = Float64(0.5 * Float64(t_0 + 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);
	tmp = 0.0;
	if (r <= 4e-222)
		tmp = 0.5 * (t_0 + -p);
	elseif (r <= 5.4e+68)
		tmp = ((q * 2.0) + r) * 0.5;
	else
		tmp = 0.5 * (t_0 + 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]}, If[LessEqual[r, 4e-222], N[(0.5 * N[(t$95$0 + (-p)), $MachinePrecision]), $MachinePrecision], If[LessEqual[r, 5.4e+68], N[(N[(N[(q * 2.0), $MachinePrecision] + r), $MachinePrecision] * 0.5), $MachinePrecision], N[(0.5 * N[(t$95$0 + r), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if r < 4.00000000000000019e-222

   1. Initial program 51.1%

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

   3. Taylor expanded in p around -inf

      \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \color{blue}{-1 \cdot p}\right) \]
   4. 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 25.9

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

   5. Applied rewrites 25.9%

      \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \color{blue}{\left(-p\right)}\right) \]
   6. 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 25.9

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

   7. Applied rewrites 25.9%

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

   if 4.00000000000000019e-222 < r < 5.39999999999999982e68

   1. Initial program 55.6%

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

   3. Taylor expanded in q around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 33.4

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

   5. Applied rewrites 33.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   7. Applied rewrites 28.7%

      \[\leadsto \color{blue}{\left(q \cdot 2 + \left(r + p\right)\right) \cdot 0.5} \]

   8. Taylor expanded in p around 0

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

   10. Applied rewrites 27.2%

       \[\leadsto \left(q \cdot 2 + \color{blue}{r}\right) \cdot 0.5 \]

   if 5.39999999999999982e68 < r

   1. Initial program 17.6%

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

   3. Taylor expanded in r around inf

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

   5. Applied rewrites 64.8%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 64.8

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

   7. Applied rewrites 64.8%

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

Alternative 4: 74.7% accurate, 10.0× speedup

Math

\[\begin{array}{l} [p, r, q] = \mathsf{sort}([p, r, q])\\ \\ \begin{array}{l} \mathbf{if}\;q \leq 2.5 \cdot 10^{+130}:\\ \;\;\;\;0.5 \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(r - p\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(q \cdot 2 + r\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

NOTE: p, r, and q should be sorted in increasing order before calling this function.
(FPCore (p r q)
 :precision binary64
 (if (<= q 2.5e+130)
   (* 0.5 (+ (+ (fabs p) (fabs r)) (- r p)))
   (* (+ (* q 2.0) r) 0.5)))

C

assert(p < r && r < q);
double code(double p, double r, double q) {
	double tmp;
	if (q <= 2.5e+130) {
		tmp = 0.5 * ((fabs(p) + fabs(r)) + (r - p));
	} else {
		tmp = ((q * 2.0) + r) * 0.5;
	}
	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 <= 2.5d+130) then
        tmp = 0.5d0 * ((abs(p) + abs(r)) + (r - p))
    else
        tmp = ((q * 2.0d0) + r) * 0.5d0
    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 <= 2.5e+130) {
		tmp = 0.5 * ((Math.abs(p) + Math.abs(r)) + (r - p));
	} else {
		tmp = ((q * 2.0) + r) * 0.5;
	}
	return tmp;
}

Python

[p, r, q] = sort([p, r, q])
def code(p, r, q):
	tmp = 0
	if q <= 2.5e+130:
		tmp = 0.5 * ((math.fabs(p) + math.fabs(r)) + (r - p))
	else:
		tmp = ((q * 2.0) + r) * 0.5
	return tmp

Julia

p, r, q = sort([p, r, q])
function code(p, r, q)
	tmp = 0.0
	if (q <= 2.5e+130)
		tmp = Float64(0.5 * Float64(Float64(abs(p) + abs(r)) + Float64(r - p)));
	else
		tmp = Float64(Float64(Float64(q * 2.0) + r) * 0.5);
	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 <= 2.5e+130)
		tmp = 0.5 * ((abs(p) + abs(r)) + (r - p));
	else
		tmp = ((q * 2.0) + r) * 0.5;
	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, 2.5e+130], N[(0.5 * N[(N[(N[Abs[p], $MachinePrecision] + N[Abs[r], $MachinePrecision]), $MachinePrecision] + N[(r - p), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(q * 2.0), $MachinePrecision] + r), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if q < 2.4999999999999998e130

   1. Initial program 46.5%

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

   3. Taylor expanded in r around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 35.9

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

   5. Applied rewrites 35.9%

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

   6. Taylor expanded in p around 0

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

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

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. unpow1 N/A

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

      5. metadata-eval N/A

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

      6. sqrt-pow1 N/A

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

      7. unpow2 N/A

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

      8. rem-sqrt-square-rev N/A

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

      9. fabs-mul N/A

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

      10. mul-1-neg N/A

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

      11. neg-fabs N/A

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

      12. rem-sqrt-square-rev N/A

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

      13. unpow2 N/A

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

      14. sqrt-pow1 N/A

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

      15. metadata-eval N/A

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

      16. unpow1 N/A

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

      17. lower--.f64 40.3

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

   8. Applied rewrites 40.3%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 40.3

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

   10. Applied rewrites 40.3%

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

   if 2.4999999999999998e130 < q

   1. Initial program 17.2%

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

   3. Taylor expanded in q around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 73.7

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

   5. Applied rewrites 73.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   7. Applied rewrites 69.1%

      \[\leadsto \color{blue}{\left(q \cdot 2 + \left(r + p\right)\right) \cdot 0.5} \]

   8. Taylor expanded in p around 0

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

   10. Applied rewrites 68.8%

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

Alternative 5: 49.7% accurate, 12.5× speedup

Math

\[\begin{array}{l} [p, r, q] = \mathsf{sort}([p, r, q])\\ \\ \begin{array}{l} \mathbf{if}\;q \leq 3 \cdot 10^{+85}:\\ \;\;\;\;0.5 \cdot \left(\left(\left|p\right| + r\right) + r\right)\\ \mathbf{else}:\\ \;\;\;\;\left(q \cdot 2 + r\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

NOTE: p, r, and q should be sorted in increasing order before calling this function.
(FPCore (p r q)
 :precision binary64
 (if (<= q 3e+85) (* 0.5 (+ (+ (fabs p) r) r)) (* (+ (* q 2.0) r) 0.5)))

C

assert(p < r && r < q);
double code(double p, double r, double q) {
	double tmp;
	if (q <= 3e+85) {
		tmp = 0.5 * ((fabs(p) + r) + r);
	} else {
		tmp = ((q * 2.0) + r) * 0.5;
	}
	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 <= 3d+85) then
        tmp = 0.5d0 * ((abs(p) + r) + r)
    else
        tmp = ((q * 2.0d0) + r) * 0.5d0
    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 <= 3e+85) {
		tmp = 0.5 * ((Math.abs(p) + r) + r);
	} else {
		tmp = ((q * 2.0) + r) * 0.5;
	}
	return tmp;
}

Python

[p, r, q] = sort([p, r, q])
def code(p, r, q):
	tmp = 0
	if q <= 3e+85:
		tmp = 0.5 * ((math.fabs(p) + r) + r)
	else:
		tmp = ((q * 2.0) + r) * 0.5
	return tmp

Julia

p, r, q = sort([p, r, q])
function code(p, r, q)
	tmp = 0.0
	if (q <= 3e+85)
		tmp = Float64(0.5 * Float64(Float64(abs(p) + r) + r));
	else
		tmp = Float64(Float64(Float64(q * 2.0) + r) * 0.5);
	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 <= 3e+85)
		tmp = 0.5 * ((abs(p) + r) + r);
	else
		tmp = ((q * 2.0) + r) * 0.5;
	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, 3e+85], N[(0.5 * N[(N[(N[Abs[p], $MachinePrecision] + r), $MachinePrecision] + r), $MachinePrecision]), $MachinePrecision], N[(N[(N[(q * 2.0), $MachinePrecision] + r), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if q < 3e85

   1. Initial program 45.6%

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

   3. Taylor expanded in r around inf

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

   5. Applied rewrites 29.3%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 29.3

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

   7. Applied rewrites 29.3%

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

   8. Taylor expanded in r around 0

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

      1. rem-sqrt-square-rev N/A

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

      2. unpow2 N/A

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

      3. sqrt-pow1 N/A

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

      4. metadata-eval N/A

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

      5. unpow1 28.9

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

   10. Applied rewrites 28.9%

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

   if 3e85 < q

   1. Initial program 28.9%

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

   3. Taylor expanded in q around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 68.3

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

   5. Applied rewrites 68.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   7. Applied rewrites 62.4%

      \[\leadsto \color{blue}{\left(q \cdot 2 + \left(r + p\right)\right) \cdot 0.5} \]

   8. Taylor expanded in p around 0

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

   10. Applied rewrites 62.2%

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

Alternative 6: 49.3% accurate, 12.5× speedup

Math

\[\begin{array}{l} [p, r, q] = \mathsf{sort}([p, r, q])\\ \\ \begin{array}{l} \mathbf{if}\;q \leq 3 \cdot 10^{+85}:\\ \;\;\;\;0.5 \cdot \left(\left(\left|p\right| + r\right) + r\right)\\ \mathbf{else}:\\ \;\;\;\;q\\ \end{array} \end{array} \]

FPCore

NOTE: p, r, and q should be sorted in increasing order before calling this function.
(FPCore (p r q)
 :precision binary64
 (if (<= q 3e+85) (* 0.5 (+ (+ (fabs p) r) r)) q))

C

assert(p < r && r < q);
double code(double p, double r, double q) {
	double tmp;
	if (q <= 3e+85) {
		tmp = 0.5 * ((fabs(p) + r) + r);
	} else {
		tmp = q;
	}
	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 <= 3d+85) then
        tmp = 0.5d0 * ((abs(p) + r) + r)
    else
        tmp = q
    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 <= 3e+85) {
		tmp = 0.5 * ((Math.abs(p) + r) + r);
	} else {
		tmp = q;
	}
	return tmp;
}

Python

[p, r, q] = sort([p, r, q])
def code(p, r, q):
	tmp = 0
	if q <= 3e+85:
		tmp = 0.5 * ((math.fabs(p) + r) + r)
	else:
		tmp = q
	return tmp

Julia

p, r, q = sort([p, r, q])
function code(p, r, q)
	tmp = 0.0
	if (q <= 3e+85)
		tmp = Float64(0.5 * Float64(Float64(abs(p) + r) + r));
	else
		tmp = q;
	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 <= 3e+85)
		tmp = 0.5 * ((abs(p) + r) + r);
	else
		tmp = q;
	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, 3e+85], N[(0.5 * N[(N[(N[Abs[p], $MachinePrecision] + r), $MachinePrecision] + r), $MachinePrecision]), $MachinePrecision], q]

Derivation

1. Split input into 2 regimes

2. if q < 3e85

   1. Initial program 45.6%

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

   3. Taylor expanded in r around inf

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

   5. Applied rewrites 29.3%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 29.3

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

   7. Applied rewrites 29.3%

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

   8. Taylor expanded in r around 0

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

      1. rem-sqrt-square-rev N/A

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

      2. unpow2 N/A

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

      3. sqrt-pow1 N/A

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

      4. metadata-eval N/A

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

      5. unpow1 28.9

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

   10. Applied rewrites 28.9%

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

   if 3e85 < q

   1. Initial program 28.9%

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

   3. Taylor expanded in q around inf

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

   5. Applied rewrites 61.6%

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

Alternative 7: 40.5% accurate, 35.6× speedup

Math

\[\begin{array}{l} [p, r, q] = \mathsf{sort}([p, r, q])\\ \\ \begin{array}{l} \mathbf{if}\;r \leq 5.4 \cdot 10^{+68}:\\ \;\;\;\;q\\ \mathbf{else}:\\ \;\;\;\;r\\ \end{array} \end{array} \]

FPCore

NOTE: p, r, and q should be sorted in increasing order before calling this function.
(FPCore (p r q) :precision binary64 (if (<= r 5.4e+68) q r))

C

assert(p < r && r < q);
double code(double p, double r, double q) {
	double tmp;
	if (r <= 5.4e+68) {
		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 <= 5.4d+68) 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 <= 5.4e+68) {
		tmp = q;
	} else {
		tmp = r;
	}
	return tmp;
}

Python

[p, r, q] = sort([p, r, q])
def code(p, r, q):
	tmp = 0
	if r <= 5.4e+68:
		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 <= 5.4e+68)
		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 <= 5.4e+68)
		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, 5.4e+68], q, r]

Derivation

1. Split input into 2 regimes

2. if r < 5.39999999999999982e68

   1. Initial program 52.2%

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

   3. Taylor expanded in q around inf

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

   5. Applied rewrites 18.4%

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

   if 5.39999999999999982e68 < r

   1. Initial program 17.6%

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

   3. Taylor expanded in p around 0

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

      1. metadata-eval N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 18.2%

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

   6. Taylor expanded in r around inf

      \[\leadsto r \]
   7. Step-by-step derivation

   8. Applied rewrites 62.3%

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

Alternative 8: 17.4% accurate, 250.0× speedup

Math

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

FPCore

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

C

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

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 = q
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 42.2%

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

3. Taylor expanded in q around inf

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

5. Applied rewrites 16.4%

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

Reproduce

herbie shell --seed 2025050 
(FPCore (p r q)
  :name "1/2(abs(p)+abs(r) + sqrt((p-r)^2 + 4q^2))"
  :precision binary64
  :pre (TRUE)
  (* (/ 1.0 2.0) (+ (+ (fabs p) (fabs r)) (sqrt (+ (pow (- p r) 2.0) (* 4.0 (pow q 2.0)))))))