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

Percentage Accurate: 23.6% → 67.2%

Time: 5.0s

Alternatives: 8

Speedup: 83.3×

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: 23.6% 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: 67.2% accurate, 5.9× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;q\_m \leq 3.5 \cdot 10^{-64}:\\ \;\;\;\;\frac{1}{2} \cdot \left(\left(p + \left|p\right|\right) + \mathsf{fma}\left(-1, r, \left|r\right|\right)\right)\\ \mathbf{elif}\;q\_m \leq 3.2 \cdot 10^{+66}:\\ \;\;\;\;\mathsf{fma}\left(\left|p\right| + p, 0.5, \frac{\left(-q\_m\right) \cdot q\_m}{r}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left|r\right| + \left|p\right|\right) - \left(q\_m + q\_m\right)\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

q_m = (fabs.f64 q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
(FPCore (p r q_m)
 :precision binary64
 (if (<= q_m 3.5e-64)
   (* (/ 1.0 2.0) (+ (+ p (fabs p)) (fma -1.0 r (fabs r))))
   (if (<= q_m 3.2e+66)
     (fma (+ (fabs p) p) 0.5 (/ (* (- q_m) q_m) r))
     (* (- (+ (fabs r) (fabs p)) (+ q_m q_m)) 0.5))))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double tmp;
	if (q_m <= 3.5e-64) {
		tmp = (1.0 / 2.0) * ((p + fabs(p)) + fma(-1.0, r, fabs(r)));
	} else if (q_m <= 3.2e+66) {
		tmp = fma((fabs(p) + p), 0.5, ((-q_m * q_m) / r));
	} else {
		tmp = ((fabs(r) + fabs(p)) - (q_m + q_m)) * 0.5;
	}
	return tmp;
}

Julia

q_m = abs(q)
p, r, q_m = sort([p, r, q_m])
function code(p, r, q_m)
	tmp = 0.0
	if (q_m <= 3.5e-64)
		tmp = Float64(Float64(1.0 / 2.0) * Float64(Float64(p + abs(p)) + fma(-1.0, r, abs(r))));
	elseif (q_m <= 3.2e+66)
		tmp = fma(Float64(abs(p) + p), 0.5, Float64(Float64(Float64(-q_m) * q_m) / r));
	else
		tmp = Float64(Float64(Float64(abs(r) + abs(p)) - Float64(q_m + q_m)) * 0.5);
	end
	return tmp
end

Wolfram

q_m = N[Abs[q], $MachinePrecision]
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
code[p_, r_, q$95$m_] := If[LessEqual[q$95$m, 3.5e-64], N[(N[(1.0 / 2.0), $MachinePrecision] * N[(N[(p + N[Abs[p], $MachinePrecision]), $MachinePrecision] + N[(-1.0 * r + N[Abs[r], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[q$95$m, 3.2e+66], N[(N[(N[Abs[p], $MachinePrecision] + p), $MachinePrecision] * 0.5 + N[(N[((-q$95$m) * q$95$m), $MachinePrecision] / r), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Abs[r], $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision] - N[(q$95$m + q$95$m), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if q < 3.5000000000000003e-64

   1. Initial program 23.3%

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

   3. Taylor expanded in r around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. div-add-rev N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lift-fabs.f64 N/A

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

      9. lift-fabs.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 12.8

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

   5. Applied rewrites 12.8%

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

   6. Taylor expanded in p around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. +-commutative N/A

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

      6. div-add N/A

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

      7. lift-fabs.f64 N/A

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

      8. lift-fabs.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-/.f64 5.0

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

   8. Applied rewrites 5.0%

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

   9. Taylor expanded in r around 0

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

       1. associate-+r+ N/A

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

       2. lower-+.f64 N/A

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

       3. lower-+.f64 N/A

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

       4. lift-fabs.f64 N/A

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

       5. mul-1-neg N/A

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

       6. +-commutative N/A

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

       7. mul-1-neg N/A

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

       8. lower-fma.f64 N/A

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

       9. lift-fabs.f64 28.2

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

   11. Applied rewrites 28.2%

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

   if 3.5000000000000003e-64 < q < 3.2e66

   1. Initial program 8.9%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) - \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
   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 6.5

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

   5. Applied rewrites 6.5%

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

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

      5. lift-+.f64 N/A

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

      6. lift-fabs.f64 N/A

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

      7. lift-fabs.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-fabs.f64 N/A

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

      10. lift-fabs.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. metadata-eval 6.5

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

   7. Applied rewrites 6.5%

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

      1. lift-fabs.f64 N/A

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

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

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

      3. unpow2 N/A

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

      4. pow-to-exp N/A

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

      5. exp-sqrt-rev N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-log.f64 5.2

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

   9. Applied rewrites 5.2%

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

   10. Taylor expanded in r around inf

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

       1. *-commutative N/A

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

       2. metadata-eval N/A

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

       3. exp-sqrt N/A

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

       4. pow-to-exp N/A

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

       5. unpow2 N/A

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

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

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

       7. +-commutative N/A

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

       8. +-commutative N/A

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

       9. *-commutative N/A

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

       10. lower-fma.f64 N/A

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

   12. Applied rewrites 25.5%

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

   if 3.2e66 < q

   1. Initial program 14.8%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) - \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
   2. 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 60.2

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

   5. Applied rewrites 60.2%

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

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

      5. lift-+.f64 N/A

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

      6. lift-fabs.f64 N/A

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

      7. lift-fabs.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-fabs.f64 N/A

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

      10. lift-fabs.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. metadata-eval 60.2

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

   7. Applied rewrites 60.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 60.2

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

   9. Applied rewrites 60.2%

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

4. Final simplification 35.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;q \leq 3.5 \cdot 10^{-64}:\\ \;\;\;\;\frac{1}{2} \cdot \left(\left(p + \left|p\right|\right) + \mathsf{fma}\left(-1, r, \left|r\right|\right)\right)\\ \mathbf{elif}\;q \leq 3.2 \cdot 10^{+66}:\\ \;\;\;\;\mathsf{fma}\left(\left|p\right| + p, 0.5, \frac{\left(-q\right) \cdot q}{r}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left|r\right| + \left|p\right|\right) - \left(q + q\right)\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 58.3% accurate, 6.9× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;q\_m \leq 3.2 \cdot 10^{+66}:\\ \;\;\;\;\mathsf{fma}\left(\left|p\right| + p, 0.5, \frac{\left(-q\_m\right) \cdot q\_m}{r}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left|r\right| + \left|p\right|\right) - \left(q\_m + q\_m\right)\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

q_m = (fabs.f64 q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
(FPCore (p r q_m)
 :precision binary64
 (if (<= q_m 3.2e+66)
   (fma (+ (fabs p) p) 0.5 (/ (* (- q_m) q_m) r))
   (* (- (+ (fabs r) (fabs p)) (+ q_m q_m)) 0.5)))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double tmp;
	if (q_m <= 3.2e+66) {
		tmp = fma((fabs(p) + p), 0.5, ((-q_m * q_m) / r));
	} else {
		tmp = ((fabs(r) + fabs(p)) - (q_m + q_m)) * 0.5;
	}
	return tmp;
}

Julia

q_m = abs(q)
p, r, q_m = sort([p, r, q_m])
function code(p, r, q_m)
	tmp = 0.0
	if (q_m <= 3.2e+66)
		tmp = fma(Float64(abs(p) + p), 0.5, Float64(Float64(Float64(-q_m) * q_m) / r));
	else
		tmp = Float64(Float64(Float64(abs(r) + abs(p)) - Float64(q_m + q_m)) * 0.5);
	end
	return tmp
end

Wolfram

q_m = N[Abs[q], $MachinePrecision]
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
code[p_, r_, q$95$m_] := If[LessEqual[q$95$m, 3.2e+66], N[(N[(N[Abs[p], $MachinePrecision] + p), $MachinePrecision] * 0.5 + N[(N[((-q$95$m) * q$95$m), $MachinePrecision] / r), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Abs[r], $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision] - N[(q$95$m + q$95$m), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if q < 3.2e66

   1. Initial program 21.0%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) - \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
   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 5.2

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

   5. Applied rewrites 5.2%

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

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

      5. lift-+.f64 N/A

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

      6. lift-fabs.f64 N/A

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

      7. lift-fabs.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-fabs.f64 N/A

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

      10. lift-fabs.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. metadata-eval 5.2

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

   7. Applied rewrites 5.2%

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

      1. lift-fabs.f64 N/A

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

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

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

      3. unpow2 N/A

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

      4. pow-to-exp N/A

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

      5. exp-sqrt-rev N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-log.f64 3.9

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

   9. Applied rewrites 3.9%

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

   10. Taylor expanded in r around inf

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

       1. *-commutative N/A

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

       2. metadata-eval N/A

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

       3. exp-sqrt N/A

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

       4. pow-to-exp N/A

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

       5. unpow2 N/A

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

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

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

       7. +-commutative N/A

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

       8. +-commutative N/A

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

       9. *-commutative N/A

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

       10. lower-fma.f64 N/A

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

   12. Applied rewrites 23.6%

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

   if 3.2e66 < q

   1. Initial program 14.8%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) - \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
   2. 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 60.2

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

   5. Applied rewrites 60.2%

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

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

      5. lift-+.f64 N/A

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

      6. lift-fabs.f64 N/A

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

      7. lift-fabs.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-fabs.f64 N/A

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

      10. lift-fabs.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. metadata-eval 60.2

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

   7. Applied rewrites 60.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 60.2

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

   9. Applied rewrites 60.2%

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

4. Final simplification 32.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;q \leq 3.2 \cdot 10^{+66}:\\ \;\;\;\;\mathsf{fma}\left(\left|p\right| + p, 0.5, \frac{\left(-q\right) \cdot q}{r}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left|r\right| + \left|p\right|\right) - \left(q + q\right)\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 57.5% accurate, 10.0× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;q\_m \leq 2.2 \cdot 10^{-31}:\\ \;\;\;\;0.5 \cdot \left(p + \left(\left(\left|r\right| - r\right) + \left|p\right|\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left|p\right| - q\_m \cdot 2\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

q_m = (fabs.f64 q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
(FPCore (p r q_m)
 :precision binary64
 (if (<= q_m 2.2e-31)
   (* 0.5 (+ p (+ (- (fabs r) r) (fabs p))))
   (* (- (fabs p) (* q_m 2.0)) 0.5)))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double tmp;
	if (q_m <= 2.2e-31) {
		tmp = 0.5 * (p + ((fabs(r) - r) + fabs(p)));
	} else {
		tmp = (fabs(p) - (q_m * 2.0)) * 0.5;
	}
	return tmp;
}

Fortran

q_m =     private
NOTE: p, r, and q_m 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_m)
use fmin_fmax_functions
    real(8), intent (in) :: p
    real(8), intent (in) :: r
    real(8), intent (in) :: q_m
    real(8) :: tmp
    if (q_m <= 2.2d-31) then
        tmp = 0.5d0 * (p + ((abs(r) - r) + abs(p)))
    else
        tmp = (abs(p) - (q_m * 2.0d0)) * 0.5d0
    end if
    code = tmp
end function

Java

q_m = Math.abs(q);
assert p < r && r < q_m;
public static double code(double p, double r, double q_m) {
	double tmp;
	if (q_m <= 2.2e-31) {
		tmp = 0.5 * (p + ((Math.abs(r) - r) + Math.abs(p)));
	} else {
		tmp = (Math.abs(p) - (q_m * 2.0)) * 0.5;
	}
	return tmp;
}

Python

q_m = math.fabs(q)
[p, r, q_m] = sort([p, r, q_m])
def code(p, r, q_m):
	tmp = 0
	if q_m <= 2.2e-31:
		tmp = 0.5 * (p + ((math.fabs(r) - r) + math.fabs(p)))
	else:
		tmp = (math.fabs(p) - (q_m * 2.0)) * 0.5
	return tmp

Julia

q_m = abs(q)
p, r, q_m = sort([p, r, q_m])
function code(p, r, q_m)
	tmp = 0.0
	if (q_m <= 2.2e-31)
		tmp = Float64(0.5 * Float64(p + Float64(Float64(abs(r) - r) + abs(p))));
	else
		tmp = Float64(Float64(abs(p) - Float64(q_m * 2.0)) * 0.5);
	end
	return tmp
end

MATLAB

q_m = abs(q);
p, r, q_m = num2cell(sort([p, r, q_m])){:}
function tmp_2 = code(p, r, q_m)
	tmp = 0.0;
	if (q_m <= 2.2e-31)
		tmp = 0.5 * (p + ((abs(r) - r) + abs(p)));
	else
		tmp = (abs(p) - (q_m * 2.0)) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

q_m = N[Abs[q], $MachinePrecision]
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
code[p_, r_, q$95$m_] := If[LessEqual[q$95$m, 2.2e-31], N[(0.5 * N[(p + N[(N[(N[Abs[r], $MachinePrecision] - r), $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Abs[p], $MachinePrecision] - N[(q$95$m * 2.0), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if q < 2.2000000000000001e-31

   1. Initial program 23.1%

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

   3. Taylor expanded in p around -inf

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

      1. metadata-eval N/A

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

      2. associate-*r* N/A

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

      3. mul-1-neg N/A

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower--.f64 N/A

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

   5. Applied rewrites 21.3%

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

   6. Taylor expanded in p around 0

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

      1. distribute-lft-out N/A

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

      2. lower-*.f64 N/A

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

      3. associate-+r- N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lift-fabs.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-fabs.f64 21.3

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

   8. Applied rewrites 21.3%

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

   if 2.2000000000000001e-31 < q

   1. Initial program 11.9%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) - \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
   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 45.9

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

   5. Applied rewrites 45.9%

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

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

      5. lift-+.f64 N/A

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

      6. lift-fabs.f64 N/A

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

      7. lift-fabs.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-fabs.f64 N/A

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

      10. lift-fabs.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. metadata-eval 45.9

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

   7. Applied rewrites 45.9%

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

      1. lift-fabs.f64 N/A

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

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

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

      3. unpow2 N/A

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

      4. pow-to-exp N/A

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

      5. exp-sqrt-rev N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-log.f64 24.3

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

   9. Applied rewrites 24.3%

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

   10. Taylor expanded in r around 0

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

       1. exp-sqrt N/A

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

       2. pow-to-exp N/A

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

       3. unpow2 N/A

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

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

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

       5. lift-fabs.f64 46.9

          \[\leadsto \left(\left|p\right| - q \cdot 2\right) \cdot 0.5 \]

   12. Applied rewrites 46.9%

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

Alternative 4: 57.2% accurate, 11.4× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;q\_m \leq 2.2 \cdot 10^{-31}:\\ \;\;\;\;\left(\left|p\right| + p\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(\left|p\right| - q\_m \cdot 2\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

q_m = (fabs.f64 q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
(FPCore (p r q_m)
 :precision binary64
 (if (<= q_m 2.2e-31) (* (+ (fabs p) p) 0.5) (* (- (fabs p) (* q_m 2.0)) 0.5)))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double tmp;
	if (q_m <= 2.2e-31) {
		tmp = (fabs(p) + p) * 0.5;
	} else {
		tmp = (fabs(p) - (q_m * 2.0)) * 0.5;
	}
	return tmp;
}

Fortran

q_m =     private
NOTE: p, r, and q_m 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_m)
use fmin_fmax_functions
    real(8), intent (in) :: p
    real(8), intent (in) :: r
    real(8), intent (in) :: q_m
    real(8) :: tmp
    if (q_m <= 2.2d-31) then
        tmp = (abs(p) + p) * 0.5d0
    else
        tmp = (abs(p) - (q_m * 2.0d0)) * 0.5d0
    end if
    code = tmp
end function

Java

q_m = Math.abs(q);
assert p < r && r < q_m;
public static double code(double p, double r, double q_m) {
	double tmp;
	if (q_m <= 2.2e-31) {
		tmp = (Math.abs(p) + p) * 0.5;
	} else {
		tmp = (Math.abs(p) - (q_m * 2.0)) * 0.5;
	}
	return tmp;
}

Python

q_m = math.fabs(q)
[p, r, q_m] = sort([p, r, q_m])
def code(p, r, q_m):
	tmp = 0
	if q_m <= 2.2e-31:
		tmp = (math.fabs(p) + p) * 0.5
	else:
		tmp = (math.fabs(p) - (q_m * 2.0)) * 0.5
	return tmp

Julia

q_m = abs(q)
p, r, q_m = sort([p, r, q_m])
function code(p, r, q_m)
	tmp = 0.0
	if (q_m <= 2.2e-31)
		tmp = Float64(Float64(abs(p) + p) * 0.5);
	else
		tmp = Float64(Float64(abs(p) - Float64(q_m * 2.0)) * 0.5);
	end
	return tmp
end

MATLAB

q_m = abs(q);
p, r, q_m = num2cell(sort([p, r, q_m])){:}
function tmp_2 = code(p, r, q_m)
	tmp = 0.0;
	if (q_m <= 2.2e-31)
		tmp = (abs(p) + p) * 0.5;
	else
		tmp = (abs(p) - (q_m * 2.0)) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

q_m = N[Abs[q], $MachinePrecision]
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
code[p_, r_, q$95$m_] := If[LessEqual[q$95$m, 2.2e-31], N[(N[(N[Abs[p], $MachinePrecision] + p), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[Abs[p], $MachinePrecision] - N[(q$95$m * 2.0), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if q < 2.2000000000000001e-31

   1. Initial program 23.1%

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

   3. Taylor expanded in 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 5.6

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

   5. Applied rewrites 5.6%

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

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

      5. lift-+.f64 N/A

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

      6. lift-fabs.f64 N/A

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

      7. lift-fabs.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-fabs.f64 N/A

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

      10. lift-fabs.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. metadata-eval 5.6

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

   7. Applied rewrites 5.6%

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

      1. lift-fabs.f64 N/A

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

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

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

      3. unpow2 N/A

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

      4. pow-to-exp N/A

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

      5. exp-sqrt-rev N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-log.f64 4.3

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

   9. Applied rewrites 4.3%

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

   10. Taylor expanded in r around inf

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

       1. exp-sqrt N/A

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

       2. pow-to-exp N/A

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

       3. unpow2 N/A

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

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

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

       5. +-commutative N/A

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

       6. mul-1-neg N/A

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

       7. lower--.f64 N/A

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

       8. lift-fabs.f64 N/A

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

       9. lift-neg.f64 21.1

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

   12. Applied rewrites 21.1%

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

   if 2.2000000000000001e-31 < q

   1. Initial program 11.9%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) - \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
   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 45.9

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

   5. Applied rewrites 45.9%

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

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

      5. lift-+.f64 N/A

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

      6. lift-fabs.f64 N/A

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

      7. lift-fabs.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-fabs.f64 N/A

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

      10. lift-fabs.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. metadata-eval 45.9

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

   7. Applied rewrites 45.9%

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

      1. lift-fabs.f64 N/A

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

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

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

      3. unpow2 N/A

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

      4. pow-to-exp N/A

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

      5. exp-sqrt-rev N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-log.f64 24.3

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

   9. Applied rewrites 24.3%

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

   10. Taylor expanded in r around 0

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

       1. exp-sqrt N/A

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

       2. pow-to-exp N/A

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

       3. unpow2 N/A

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

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

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

       5. lift-fabs.f64 46.9

          \[\leadsto \left(\left|p\right| - q \cdot 2\right) \cdot 0.5 \]

   12. Applied rewrites 46.9%

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

4. Final simplification 29.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;q \leq 2.2 \cdot 10^{-31}:\\ \;\;\;\;\left(\left|p\right| + p\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(\left|p\right| - q \cdot 2\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 56.4% accurate, 12.5× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;q\_m \leq 1.15 \cdot 10^{+18}:\\ \;\;\;\;\left(\left|p\right| + p\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(r - q\_m \cdot 2\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

q_m = (fabs.f64 q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
(FPCore (p r q_m)
 :precision binary64
 (if (<= q_m 1.15e+18) (* (+ (fabs p) p) 0.5) (* (- r (* q_m 2.0)) 0.5)))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double tmp;
	if (q_m <= 1.15e+18) {
		tmp = (fabs(p) + p) * 0.5;
	} else {
		tmp = (r - (q_m * 2.0)) * 0.5;
	}
	return tmp;
}

Fortran

q_m =     private
NOTE: p, r, and q_m 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_m)
use fmin_fmax_functions
    real(8), intent (in) :: p
    real(8), intent (in) :: r
    real(8), intent (in) :: q_m
    real(8) :: tmp
    if (q_m <= 1.15d+18) then
        tmp = (abs(p) + p) * 0.5d0
    else
        tmp = (r - (q_m * 2.0d0)) * 0.5d0
    end if
    code = tmp
end function

Java

q_m = Math.abs(q);
assert p < r && r < q_m;
public static double code(double p, double r, double q_m) {
	double tmp;
	if (q_m <= 1.15e+18) {
		tmp = (Math.abs(p) + p) * 0.5;
	} else {
		tmp = (r - (q_m * 2.0)) * 0.5;
	}
	return tmp;
}

Python

q_m = math.fabs(q)
[p, r, q_m] = sort([p, r, q_m])
def code(p, r, q_m):
	tmp = 0
	if q_m <= 1.15e+18:
		tmp = (math.fabs(p) + p) * 0.5
	else:
		tmp = (r - (q_m * 2.0)) * 0.5
	return tmp

Julia

q_m = abs(q)
p, r, q_m = sort([p, r, q_m])
function code(p, r, q_m)
	tmp = 0.0
	if (q_m <= 1.15e+18)
		tmp = Float64(Float64(abs(p) + p) * 0.5);
	else
		tmp = Float64(Float64(r - Float64(q_m * 2.0)) * 0.5);
	end
	return tmp
end

MATLAB

q_m = abs(q);
p, r, q_m = num2cell(sort([p, r, q_m])){:}
function tmp_2 = code(p, r, q_m)
	tmp = 0.0;
	if (q_m <= 1.15e+18)
		tmp = (abs(p) + p) * 0.5;
	else
		tmp = (r - (q_m * 2.0)) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

q_m = N[Abs[q], $MachinePrecision]
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
code[p_, r_, q$95$m_] := If[LessEqual[q$95$m, 1.15e+18], N[(N[(N[Abs[p], $MachinePrecision] + p), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(r - N[(q$95$m * 2.0), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if q < 1.15e18

   1. Initial program 22.2%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) - \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
   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 5.5

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

   5. Applied rewrites 5.5%

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

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

      5. lift-+.f64 N/A

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

      6. lift-fabs.f64 N/A

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

      7. lift-fabs.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-fabs.f64 N/A

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

      10. lift-fabs.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. metadata-eval 5.5

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

   7. Applied rewrites 5.5%

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

      1. lift-fabs.f64 N/A

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

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

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

      3. unpow2 N/A

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

      4. pow-to-exp N/A

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

      5. exp-sqrt-rev N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-log.f64 4.2

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

   9. Applied rewrites 4.2%

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

   10. Taylor expanded in r around inf

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

       1. exp-sqrt N/A

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

       2. pow-to-exp N/A

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

       3. unpow2 N/A

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

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

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

       5. +-commutative N/A

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

       6. mul-1-neg N/A

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

       7. lower--.f64 N/A

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

       8. lift-fabs.f64 N/A

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

       9. lift-neg.f64 20.8

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

   12. Applied rewrites 20.8%

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

   if 1.15e18 < q

   1. Initial program 12.9%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) - \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
   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 50.5

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

   5. Applied rewrites 50.5%

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

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

      5. lift-+.f64 N/A

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

      6. lift-fabs.f64 N/A

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

      7. lift-fabs.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-fabs.f64 N/A

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

      10. lift-fabs.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. metadata-eval 50.5

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

   7. Applied rewrites 50.5%

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

      1. lift-fabs.f64 N/A

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

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

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

      3. unpow2 N/A

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

      4. pow-to-exp N/A

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

      5. exp-sqrt-rev N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-log.f64 26.9

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

   9. Applied rewrites 26.9%

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

   10. Taylor expanded in r around inf

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

       1. exp-sqrt 51.2

          \[\leadsto \left(r - q \cdot 2\right) \cdot 0.5 \]

       2. pow-to-exp 51.2

          \[\leadsto \left(r - q \cdot 2\right) \cdot 0.5 \]

       3. unpow2 51.2

          \[\leadsto \left(r - q \cdot 2\right) \cdot 0.5 \]

       4. rem-sqrt-square-rev 51.2

          \[\leadsto \left(r - q \cdot 2\right) \cdot 0.5 \]

   12. Applied rewrites 51.2%

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

4. Final simplification 29.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;q \leq 1.15 \cdot 10^{+18}:\\ \;\;\;\;\left(\left|p\right| + p\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(r - q \cdot 2\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 56.9% accurate, 14.7× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;q\_m \leq 800000000000:\\ \;\;\;\;\left(\left|p\right| + p\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;-q\_m\\ \end{array} \end{array} \]

FPCore

q_m = (fabs.f64 q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
(FPCore (p r q_m)
 :precision binary64
 (if (<= q_m 800000000000.0) (* (+ (fabs p) p) 0.5) (- q_m)))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double tmp;
	if (q_m <= 800000000000.0) {
		tmp = (fabs(p) + p) * 0.5;
	} else {
		tmp = -q_m;
	}
	return tmp;
}

Fortran

q_m =     private
NOTE: p, r, and q_m 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_m)
use fmin_fmax_functions
    real(8), intent (in) :: p
    real(8), intent (in) :: r
    real(8), intent (in) :: q_m
    real(8) :: tmp
    if (q_m <= 800000000000.0d0) then
        tmp = (abs(p) + p) * 0.5d0
    else
        tmp = -q_m
    end if
    code = tmp
end function

Java

q_m = Math.abs(q);
assert p < r && r < q_m;
public static double code(double p, double r, double q_m) {
	double tmp;
	if (q_m <= 800000000000.0) {
		tmp = (Math.abs(p) + p) * 0.5;
	} else {
		tmp = -q_m;
	}
	return tmp;
}

Python

q_m = math.fabs(q)
[p, r, q_m] = sort([p, r, q_m])
def code(p, r, q_m):
	tmp = 0
	if q_m <= 800000000000.0:
		tmp = (math.fabs(p) + p) * 0.5
	else:
		tmp = -q_m
	return tmp

Julia

q_m = abs(q)
p, r, q_m = sort([p, r, q_m])
function code(p, r, q_m)
	tmp = 0.0
	if (q_m <= 800000000000.0)
		tmp = Float64(Float64(abs(p) + p) * 0.5);
	else
		tmp = Float64(-q_m);
	end
	return tmp
end

MATLAB

q_m = abs(q);
p, r, q_m = num2cell(sort([p, r, q_m])){:}
function tmp_2 = code(p, r, q_m)
	tmp = 0.0;
	if (q_m <= 800000000000.0)
		tmp = (abs(p) + p) * 0.5;
	else
		tmp = -q_m;
	end
	tmp_2 = tmp;
end

Wolfram

q_m = N[Abs[q], $MachinePrecision]
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
code[p_, r_, q$95$m_] := If[LessEqual[q$95$m, 800000000000.0], N[(N[(N[Abs[p], $MachinePrecision] + p), $MachinePrecision] * 0.5), $MachinePrecision], (-q$95$m)]

Derivation

1. Split input into 2 regimes

2. if q < 8e11

   1. Initial program 22.2%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) - \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
   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 5.5

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

   5. Applied rewrites 5.5%

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

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

      5. lift-+.f64 N/A

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

      6. lift-fabs.f64 N/A

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

      7. lift-fabs.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-fabs.f64 N/A

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

      10. lift-fabs.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. metadata-eval 5.5

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

   7. Applied rewrites 5.5%

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

      1. lift-fabs.f64 N/A

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

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

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

      3. unpow2 N/A

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

      4. pow-to-exp N/A

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

      5. exp-sqrt-rev N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-log.f64 4.2

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

   9. Applied rewrites 4.2%

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

   10. Taylor expanded in r around inf

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

       1. exp-sqrt N/A

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

       2. pow-to-exp N/A

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

       3. unpow2 N/A

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

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

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

       5. +-commutative N/A

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

       6. mul-1-neg N/A

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

       7. lower--.f64 N/A

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

       8. lift-fabs.f64 N/A

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

       9. lift-neg.f64 20.8

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

   12. Applied rewrites 20.8%

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

   if 8e11 < q

   1. Initial program 12.9%

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

   3. Taylor expanded in q around inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(q\right) \]

      2. lower-neg.f64 51.3

         \[\leadsto -q \]

   5. Applied rewrites 51.3%

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

4. Final simplification 29.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;q \leq 800000000000:\\ \;\;\;\;\left(\left|p\right| + p\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;-q\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 35.0% accurate, 83.3× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ -q\_m \end{array} \]

FPCore

q_m = (fabs.f64 q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
(FPCore (p r q_m) :precision binary64 (- q_m))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	return -q_m;
}

Fortran

q_m =     private
NOTE: p, r, and q_m 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_m)
use fmin_fmax_functions
    real(8), intent (in) :: p
    real(8), intent (in) :: r
    real(8), intent (in) :: q_m
    code = -q_m
end function

Java

q_m = Math.abs(q);
assert p < r && r < q_m;
public static double code(double p, double r, double q_m) {
	return -q_m;
}

Python

q_m = math.fabs(q)
[p, r, q_m] = sort([p, r, q_m])
def code(p, r, q_m):
	return -q_m

Julia

q_m = abs(q)
p, r, q_m = sort([p, r, q_m])
function code(p, r, q_m)
	return Float64(-q_m)
end

MATLAB

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

Wolfram

q_m = N[Abs[q], $MachinePrecision]
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
code[p_, r_, q$95$m_] := (-q$95$m)

Derivation

1. Initial program 19.5%

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

3. Taylor expanded in q around inf

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

   1. mul-1-neg N/A

      \[\leadsto \mathsf{neg}\left(q\right) \]

   2. lower-neg.f64 19.3

      \[\leadsto -q \]

5. Applied rewrites 19.3%

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

Alternative 8: 3.3% accurate, 250.0× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ q\_m \end{array} \]

FPCore

q_m = (fabs.f64 q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
(FPCore (p r q_m) :precision binary64 q_m)

C

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

Fortran

q_m =     private
NOTE: p, r, and q_m 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_m)
use fmin_fmax_functions
    real(8), intent (in) :: p
    real(8), intent (in) :: r
    real(8), intent (in) :: q_m
    code = q_m
end function

Java

q_m = Math.abs(q);
assert p < r && r < q_m;
public static double code(double p, double r, double q_m) {
	return q_m;
}

Python

q_m = math.fabs(q)
[p, r, q_m] = sort([p, r, q_m])
def code(p, r, q_m):
	return q_m

Julia

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

MATLAB

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

Wolfram

q_m = N[Abs[q], $MachinePrecision]
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
code[p_, r_, q$95$m_] := q$95$m

Derivation

1. Initial program 19.5%

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

3. Taylor expanded in q around -inf

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

5. Applied rewrites 18.4%

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

Reproduce

herbie shell --seed 2025082 
(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)))))))