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

Percentage Accurate: 45.9% → 81.4%

Time: 3.1s

Alternatives: 12

Speedup: 2.0×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 45.9% accurate, 1.0× speedup

Math

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

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: 81.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} t_0 := \left|\mathsf{min}\left(p, r\right)\right|\\ t_1 := \left|\mathsf{max}\left(p, r\right)\right|\\ \mathbf{if}\;\left|q\right| \leq 7.2 \cdot 10^{+104}:\\ \;\;\;\;0.5 \cdot \left(\left(t\_1 - \left(\mathsf{min}\left(p, r\right) - \mathsf{max}\left(p, r\right)\right)\right) + t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_1 + t\_0, 0.5, \left|q\right|\right)\\ \end{array} \]

FPCore

(FPCore (p r q)
 :precision binary64
 (let* ((t_0 (fabs (fmin p r))) (t_1 (fabs (fmax p r))))
   (if (<= (fabs q) 7.2e+104)
     (* 0.5 (+ (- t_1 (- (fmin p r) (fmax p r))) t_0))
     (fma (+ t_1 t_0) 0.5 (fabs q)))))

C

double code(double p, double r, double q) {
	double t_0 = fabs(fmin(p, r));
	double t_1 = fabs(fmax(p, r));
	double tmp;
	if (fabs(q) <= 7.2e+104) {
		tmp = 0.5 * ((t_1 - (fmin(p, r) - fmax(p, r))) + t_0);
	} else {
		tmp = fma((t_1 + t_0), 0.5, fabs(q));
	}
	return tmp;
}

Julia

function code(p, r, q)
	t_0 = abs(fmin(p, r))
	t_1 = abs(fmax(p, r))
	tmp = 0.0
	if (abs(q) <= 7.2e+104)
		tmp = Float64(0.5 * Float64(Float64(t_1 - Float64(fmin(p, r) - fmax(p, r))) + t_0));
	else
		tmp = fma(Float64(t_1 + t_0), 0.5, abs(q));
	end
	return tmp
end

Wolfram

code[p_, r_, q_] := Block[{t$95$0 = N[Abs[N[Min[p, r], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[Max[p, r], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Abs[q], $MachinePrecision], 7.2e+104], N[(0.5 * N[(N[(t$95$1 - N[(N[Min[p, r], $MachinePrecision] - N[Max[p, r], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 + t$95$0), $MachinePrecision] * 0.5 + N[Abs[q], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if q < 7.20000000000000001e104

   1. Initial program 45.9%

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

   2. Taylor expanded in r around inf

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

   4. Applied rewrites 30.3%

      \[\leadsto \color{blue}{r \cdot \left(0.5 + 0.5 \cdot \frac{\left|p\right| + \left(\left|r\right| + -1 \cdot p\right)}{r}\right)} \]

   5. Taylor expanded in r around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-fma.f64 N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. lower-fabs.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-fabs.f64 N/A

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

      11. lower-*.f64 35.0

          \[\leadsto \mathsf{fma}\left(0.5, r, 0.5 \cdot \left(\left|p\right| + \left(\left|r\right| + -1 \cdot p\right)\right)\right) \]

   7. Applied rewrites 35.0%

      \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{r}, 0.5 \cdot \left(\left|p\right| + \left(\left|r\right| + -1 \cdot p\right)\right)\right) \]
   8. Step-by-step derivation

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lift-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. distribute-rgt-in N/A

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

      8. associate-+l+ N/A

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

      9. lower-fma.f64 N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. mul-1-neg N/A

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

      15. sub-flip-reverse N/A

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

      16. lower--.f64 N/A

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

      17. metadata-eval N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 N/A

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

      20. metadata-eval 35.4

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

   9. Applied rewrites 35.4%

      \[\leadsto \mathsf{fma}\left(\left|p\right|, 0.5, \mathsf{fma}\left(\left|r\right| - p, 0.5, r \cdot 0.5\right)\right) \]
   10. Step-by-step derivation

       1. metadata-eval N/A

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

       2. metadata-eval N/A

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

       3. metadata-eval N/A

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

       4. lift-fma.f64 N/A

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

       5. +-commutative N/A

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

       6. lift-fma.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. distribute-rgt-out N/A

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

       9. *-commutative N/A

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

       10. distribute-lft-out N/A

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

       11. lower-*.f64 N/A

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

       12. metadata-eval N/A

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

       13. lower-+.f64 N/A

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

       14. lift--.f64 N/A

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

       15. associate-+l- N/A

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

       16. lower--.f64 N/A

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

       17. lower--.f64 35.3

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

   11. Applied rewrites 35.3%

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

   if 7.20000000000000001e104 < q

   1. Initial program 45.9%

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

   2. Taylor expanded in q around inf

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

      1. metadata-eval N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-fabs.f64 N/A

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

      9. lower-fabs.f64 26.6

         \[\leadsto q \cdot \left(1 + 0.5 \cdot \frac{\left|p\right| + \left|r\right|}{q}\right) \]

   4. Applied rewrites 26.6%

      \[\leadsto \color{blue}{q \cdot \left(1 + 0.5 \cdot \frac{\left|p\right| + \left|r\right|}{q}\right)} \]
   5. Step-by-step derivation

      1. metadata-eval N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. *-commutative N/A

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

      7. *-rgt-identity N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. lower-fma.f64 N/A

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

   6. Applied rewrites 26.6%

      \[\leadsto \mathsf{fma}\left(\frac{\left|r\right| + \left|p\right|}{q}, \color{blue}{0.5 \cdot q}, q\right) \]
   7. Step-by-step derivation

      1. metadata-eval N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-fabs.f64 N/A

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

      9. lift-fabs.f64 N/A

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

      10. *-commutative N/A

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

      11. distribute-lft1-in N/A

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

      12. +-commutative N/A

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

   8. Applied rewrites 29.0%

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

Alternative 2: 81.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} t_0 := \left|\mathsf{min}\left(p, r\right)\right|\\ t_1 := \left|\mathsf{max}\left(p, r\right)\right|\\ \mathbf{if}\;\left|q\right| \leq 7.2 \cdot 10^{+104}:\\ \;\;\;\;0.5 \cdot \left(\left(t\_1 - \left(\mathsf{min}\left(p, r\right) - \mathsf{max}\left(p, r\right)\right)\right) + t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t\_1 + t\_0}{\left|q\right|}, 0.5 \cdot \left|q\right|, \left|q\right|\right)\\ \end{array} \]

FPCore

(FPCore (p r q)
 :precision binary64
 (let* ((t_0 (fabs (fmin p r))) (t_1 (fabs (fmax p r))))
   (if (<= (fabs q) 7.2e+104)
     (* 0.5 (+ (- t_1 (- (fmin p r) (fmax p r))) t_0))
     (fma (/ (+ t_1 t_0) (fabs q)) (* 0.5 (fabs q)) (fabs q)))))

C

double code(double p, double r, double q) {
	double t_0 = fabs(fmin(p, r));
	double t_1 = fabs(fmax(p, r));
	double tmp;
	if (fabs(q) <= 7.2e+104) {
		tmp = 0.5 * ((t_1 - (fmin(p, r) - fmax(p, r))) + t_0);
	} else {
		tmp = fma(((t_1 + t_0) / fabs(q)), (0.5 * fabs(q)), fabs(q));
	}
	return tmp;
}

Julia

function code(p, r, q)
	t_0 = abs(fmin(p, r))
	t_1 = abs(fmax(p, r))
	tmp = 0.0
	if (abs(q) <= 7.2e+104)
		tmp = Float64(0.5 * Float64(Float64(t_1 - Float64(fmin(p, r) - fmax(p, r))) + t_0));
	else
		tmp = fma(Float64(Float64(t_1 + t_0) / abs(q)), Float64(0.5 * abs(q)), abs(q));
	end
	return tmp
end

Wolfram

code[p_, r_, q_] := Block[{t$95$0 = N[Abs[N[Min[p, r], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[Max[p, r], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Abs[q], $MachinePrecision], 7.2e+104], N[(0.5 * N[(N[(t$95$1 - N[(N[Min[p, r], $MachinePrecision] - N[Max[p, r], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$1 + t$95$0), $MachinePrecision] / N[Abs[q], $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[Abs[q], $MachinePrecision]), $MachinePrecision] + N[Abs[q], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if q < 7.20000000000000001e104

   1. Initial program 45.9%

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

   2. Taylor expanded in r around inf

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

   4. Applied rewrites 30.3%

      \[\leadsto \color{blue}{r \cdot \left(0.5 + 0.5 \cdot \frac{\left|p\right| + \left(\left|r\right| + -1 \cdot p\right)}{r}\right)} \]

   5. Taylor expanded in r around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-fma.f64 N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. lower-fabs.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-fabs.f64 N/A

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

      11. lower-*.f64 35.0

          \[\leadsto \mathsf{fma}\left(0.5, r, 0.5 \cdot \left(\left|p\right| + \left(\left|r\right| + -1 \cdot p\right)\right)\right) \]

   7. Applied rewrites 35.0%

      \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{r}, 0.5 \cdot \left(\left|p\right| + \left(\left|r\right| + -1 \cdot p\right)\right)\right) \]
   8. Step-by-step derivation

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lift-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. distribute-rgt-in N/A

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

      8. associate-+l+ N/A

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

      9. lower-fma.f64 N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. mul-1-neg N/A

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

      15. sub-flip-reverse N/A

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

      16. lower--.f64 N/A

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

      17. metadata-eval N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 N/A

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

      20. metadata-eval 35.4

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

   9. Applied rewrites 35.4%

      \[\leadsto \mathsf{fma}\left(\left|p\right|, 0.5, \mathsf{fma}\left(\left|r\right| - p, 0.5, r \cdot 0.5\right)\right) \]
   10. Step-by-step derivation

       1. metadata-eval N/A

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

       2. metadata-eval N/A

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

       3. metadata-eval N/A

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

       4. lift-fma.f64 N/A

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

       5. +-commutative N/A

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

       6. lift-fma.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. distribute-rgt-out N/A

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

       9. *-commutative N/A

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

       10. distribute-lft-out N/A

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

       11. lower-*.f64 N/A

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

       12. metadata-eval N/A

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

       13. lower-+.f64 N/A

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

       14. lift--.f64 N/A

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

       15. associate-+l- N/A

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

       16. lower--.f64 N/A

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

       17. lower--.f64 35.3

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

   11. Applied rewrites 35.3%

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

   if 7.20000000000000001e104 < q

   1. Initial program 45.9%

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

   2. Taylor expanded in q around inf

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

      1. metadata-eval N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-fabs.f64 N/A

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

      9. lower-fabs.f64 26.6

         \[\leadsto q \cdot \left(1 + 0.5 \cdot \frac{\left|p\right| + \left|r\right|}{q}\right) \]

   4. Applied rewrites 26.6%

      \[\leadsto \color{blue}{q \cdot \left(1 + 0.5 \cdot \frac{\left|p\right| + \left|r\right|}{q}\right)} \]
   5. Step-by-step derivation

      1. metadata-eval N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. *-commutative N/A

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

      7. *-rgt-identity N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. lower-fma.f64 N/A

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

   6. Applied rewrites 26.6%

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

Alternative 3: 81.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} t_0 := \left|\mathsf{min}\left(p, r\right)\right|\\ t_1 := \left|\mathsf{max}\left(p, r\right)\right|\\ \mathbf{if}\;\left|q\right| \leq 7.2 \cdot 10^{+104}:\\ \;\;\;\;\left(t\_1 - \left(\left(\mathsf{min}\left(p, r\right) - t\_0\right) - \mathsf{max}\left(p, r\right)\right)\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_1 + t\_0, 0.5, \left|q\right|\right)\\ \end{array} \]

FPCore

(FPCore (p r q)
 :precision binary64
 (let* ((t_0 (fabs (fmin p r))) (t_1 (fabs (fmax p r))))
   (if (<= (fabs q) 7.2e+104)
     (* (- t_1 (- (- (fmin p r) t_0) (fmax p r))) 0.5)
     (fma (+ t_1 t_0) 0.5 (fabs q)))))

C

double code(double p, double r, double q) {
	double t_0 = fabs(fmin(p, r));
	double t_1 = fabs(fmax(p, r));
	double tmp;
	if (fabs(q) <= 7.2e+104) {
		tmp = (t_1 - ((fmin(p, r) - t_0) - fmax(p, r))) * 0.5;
	} else {
		tmp = fma((t_1 + t_0), 0.5, fabs(q));
	}
	return tmp;
}

Julia

function code(p, r, q)
	t_0 = abs(fmin(p, r))
	t_1 = abs(fmax(p, r))
	tmp = 0.0
	if (abs(q) <= 7.2e+104)
		tmp = Float64(Float64(t_1 - Float64(Float64(fmin(p, r) - t_0) - fmax(p, r))) * 0.5);
	else
		tmp = fma(Float64(t_1 + t_0), 0.5, abs(q));
	end
	return tmp
end

Wolfram

code[p_, r_, q_] := Block[{t$95$0 = N[Abs[N[Min[p, r], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[Max[p, r], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Abs[q], $MachinePrecision], 7.2e+104], N[(N[(t$95$1 - N[(N[(N[Min[p, r], $MachinePrecision] - t$95$0), $MachinePrecision] - N[Max[p, r], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(t$95$1 + t$95$0), $MachinePrecision] * 0.5 + N[Abs[q], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if q < 7.20000000000000001e104

   1. Initial program 45.9%

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

   2. Taylor expanded in r around inf

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

   4. Applied rewrites 30.3%

      \[\leadsto \color{blue}{r \cdot \left(0.5 + 0.5 \cdot \frac{\left|p\right| + \left(\left|r\right| + -1 \cdot p\right)}{r}\right)} \]

   5. Taylor expanded in r around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-fma.f64 N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. lower-fabs.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-fabs.f64 N/A

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

      11. lower-*.f64 35.0

          \[\leadsto \mathsf{fma}\left(0.5, r, 0.5 \cdot \left(\left|p\right| + \left(\left|r\right| + -1 \cdot p\right)\right)\right) \]

   7. Applied rewrites 35.0%

      \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{r}, 0.5 \cdot \left(\left|p\right| + \left(\left|r\right| + -1 \cdot p\right)\right)\right) \]
   8. Step-by-step derivation

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lift-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. distribute-rgt-in N/A

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

      8. associate-+l+ N/A

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

      9. lower-fma.f64 N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. mul-1-neg N/A

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

      15. sub-flip-reverse N/A

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

      16. lower--.f64 N/A

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

      17. metadata-eval N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 N/A

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

      20. metadata-eval 35.4

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

   9. Applied rewrites 35.4%

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

   11. Applied rewrites 35.1%

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

   if 7.20000000000000001e104 < q

   1. Initial program 45.9%

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

   2. Taylor expanded in q around inf

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

      1. metadata-eval N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-fabs.f64 N/A

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

      9. lower-fabs.f64 26.6

         \[\leadsto q \cdot \left(1 + 0.5 \cdot \frac{\left|p\right| + \left|r\right|}{q}\right) \]

   4. Applied rewrites 26.6%

      \[\leadsto \color{blue}{q \cdot \left(1 + 0.5 \cdot \frac{\left|p\right| + \left|r\right|}{q}\right)} \]
   5. Step-by-step derivation

      1. metadata-eval N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. *-commutative N/A

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

      7. *-rgt-identity N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. lower-fma.f64 N/A

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

   6. Applied rewrites 26.6%

      \[\leadsto \mathsf{fma}\left(\frac{\left|r\right| + \left|p\right|}{q}, \color{blue}{0.5 \cdot q}, q\right) \]
   7. Step-by-step derivation

      1. metadata-eval N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-fabs.f64 N/A

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

      9. lift-fabs.f64 N/A

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

      10. *-commutative N/A

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

      11. distribute-lft1-in N/A

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

      12. +-commutative N/A

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

   8. Applied rewrites 29.0%

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

Alternative 4: 81.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} t_0 := \left|\mathsf{max}\left(p, r\right)\right|\\ t_1 := \left|\mathsf{min}\left(p, r\right)\right|\\ \mathbf{if}\;\left|q\right| \leq 7.2 \cdot 10^{+104}:\\ \;\;\;\;\left(\mathsf{max}\left(p, r\right) - \left(\left(\mathsf{min}\left(p, r\right) - t\_0\right) - t\_1\right)\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0 + t\_1, 0.5, \left|q\right|\right)\\ \end{array} \]

FPCore

(FPCore (p r q)
 :precision binary64
 (let* ((t_0 (fabs (fmax p r))) (t_1 (fabs (fmin p r))))
   (if (<= (fabs q) 7.2e+104)
     (* (- (fmax p r) (- (- (fmin p r) t_0) t_1)) 0.5)
     (fma (+ t_0 t_1) 0.5 (fabs q)))))

C

double code(double p, double r, double q) {
	double t_0 = fabs(fmax(p, r));
	double t_1 = fabs(fmin(p, r));
	double tmp;
	if (fabs(q) <= 7.2e+104) {
		tmp = (fmax(p, r) - ((fmin(p, r) - t_0) - t_1)) * 0.5;
	} else {
		tmp = fma((t_0 + t_1), 0.5, fabs(q));
	}
	return tmp;
}

Julia

function code(p, r, q)
	t_0 = abs(fmax(p, r))
	t_1 = abs(fmin(p, r))
	tmp = 0.0
	if (abs(q) <= 7.2e+104)
		tmp = Float64(Float64(fmax(p, r) - Float64(Float64(fmin(p, r) - t_0) - t_1)) * 0.5);
	else
		tmp = fma(Float64(t_0 + t_1), 0.5, abs(q));
	end
	return tmp
end

Wolfram

code[p_, r_, q_] := Block[{t$95$0 = N[Abs[N[Max[p, r], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[Min[p, r], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Abs[q], $MachinePrecision], 7.2e+104], N[(N[(N[Max[p, r], $MachinePrecision] - N[(N[(N[Min[p, r], $MachinePrecision] - t$95$0), $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(t$95$0 + t$95$1), $MachinePrecision] * 0.5 + N[Abs[q], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if q < 7.20000000000000001e104

   1. Initial program 45.9%

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

   2. Taylor expanded in r around inf

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

   4. Applied rewrites 30.3%

      \[\leadsto \color{blue}{r \cdot \left(0.5 + 0.5 \cdot \frac{\left|p\right| + \left(\left|r\right| + -1 \cdot p\right)}{r}\right)} \]

   5. Taylor expanded in r around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-fma.f64 N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. lower-fabs.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-fabs.f64 N/A

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

      11. lower-*.f64 35.0

          \[\leadsto \mathsf{fma}\left(0.5, r, 0.5 \cdot \left(\left|p\right| + \left(\left|r\right| + -1 \cdot p\right)\right)\right) \]

   7. Applied rewrites 35.0%

      \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{r}, 0.5 \cdot \left(\left|p\right| + \left(\left|r\right| + -1 \cdot p\right)\right)\right) \]
   8. Step-by-step derivation

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. distribute-lft-out N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

   9. Applied rewrites 35.0%

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

   if 7.20000000000000001e104 < q

   1. Initial program 45.9%

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

   2. Taylor expanded in q around inf

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

      1. metadata-eval N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-fabs.f64 N/A

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

      9. lower-fabs.f64 26.6

         \[\leadsto q \cdot \left(1 + 0.5 \cdot \frac{\left|p\right| + \left|r\right|}{q}\right) \]

   4. Applied rewrites 26.6%

      \[\leadsto \color{blue}{q \cdot \left(1 + 0.5 \cdot \frac{\left|p\right| + \left|r\right|}{q}\right)} \]
   5. Step-by-step derivation

      1. metadata-eval N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. *-commutative N/A

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

      7. *-rgt-identity N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. lower-fma.f64 N/A

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

   6. Applied rewrites 26.6%

      \[\leadsto \mathsf{fma}\left(\frac{\left|r\right| + \left|p\right|}{q}, \color{blue}{0.5 \cdot q}, q\right) \]
   7. Step-by-step derivation

      1. metadata-eval N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-fabs.f64 N/A

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

      9. lift-fabs.f64 N/A

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

      10. *-commutative N/A

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

      11. distribute-lft1-in N/A

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

      12. +-commutative N/A

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

   8. Applied rewrites 29.0%

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

Alternative 5: 65.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} t_0 := \left|\mathsf{min}\left(p, r\right)\right|\\ t_1 := \left|\mathsf{max}\left(p, r\right)\right|\\ \mathbf{if}\;\mathsf{max}\left(p, r\right) \leq -2.25 \cdot 10^{-192}:\\ \;\;\;\;0.5 \cdot \left(\left(t\_0 + t\_1\right) - \mathsf{min}\left(p, r\right)\right)\\ \mathbf{elif}\;\mathsf{max}\left(p, r\right) \leq 4 \cdot 10^{+101}:\\ \;\;\;\;\mathsf{fma}\left(t\_1 + t\_0, 0.5, \left|q\right|\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{max}\left(p, r\right) + t\_0\right) + t\_1\right) \cdot 0.5\\ \end{array} \]

FPCore

(FPCore (p r q)
 :precision binary64
 (let* ((t_0 (fabs (fmin p r))) (t_1 (fabs (fmax p r))))
   (if (<= (fmax p r) -2.25e-192)
     (* 0.5 (- (+ t_0 t_1) (fmin p r)))
     (if (<= (fmax p r) 4e+101)
       (fma (+ t_1 t_0) 0.5 (fabs q))
       (* (+ (+ (fmax p r) t_0) t_1) 0.5)))))

C

double code(double p, double r, double q) {
	double t_0 = fabs(fmin(p, r));
	double t_1 = fabs(fmax(p, r));
	double tmp;
	if (fmax(p, r) <= -2.25e-192) {
		tmp = 0.5 * ((t_0 + t_1) - fmin(p, r));
	} else if (fmax(p, r) <= 4e+101) {
		tmp = fma((t_1 + t_0), 0.5, fabs(q));
	} else {
		tmp = ((fmax(p, r) + t_0) + t_1) * 0.5;
	}
	return tmp;
}

Julia

function code(p, r, q)
	t_0 = abs(fmin(p, r))
	t_1 = abs(fmax(p, r))
	tmp = 0.0
	if (fmax(p, r) <= -2.25e-192)
		tmp = Float64(0.5 * Float64(Float64(t_0 + t_1) - fmin(p, r)));
	elseif (fmax(p, r) <= 4e+101)
		tmp = fma(Float64(t_1 + t_0), 0.5, abs(q));
	else
		tmp = Float64(Float64(Float64(fmax(p, r) + t_0) + t_1) * 0.5);
	end
	return tmp
end

Wolfram

code[p_, r_, q_] := Block[{t$95$0 = N[Abs[N[Min[p, r], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[Max[p, r], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Max[p, r], $MachinePrecision], -2.25e-192], N[(0.5 * N[(N[(t$95$0 + t$95$1), $MachinePrecision] - N[Min[p, r], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Max[p, r], $MachinePrecision], 4e+101], N[(N[(t$95$1 + t$95$0), $MachinePrecision] * 0.5 + N[Abs[q], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Max[p, r], $MachinePrecision] + t$95$0), $MachinePrecision] + t$95$1), $MachinePrecision] * 0.5), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if r < -2.25000000000000012e-192

   1. Initial program 45.9%

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

   2. Taylor expanded in r around inf

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

   4. Applied rewrites 30.3%

      \[\leadsto \color{blue}{r \cdot \left(0.5 + 0.5 \cdot \frac{\left|p\right| + \left(\left|r\right| + -1 \cdot p\right)}{r}\right)} \]

   5. Taylor expanded in r around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-fma.f64 N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. lower-fabs.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-fabs.f64 N/A

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

      11. lower-*.f64 35.0

          \[\leadsto \mathsf{fma}\left(0.5, r, 0.5 \cdot \left(\left|p\right| + \left(\left|r\right| + -1 \cdot p\right)\right)\right) \]

   7. Applied rewrites 35.0%

      \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{r}, 0.5 \cdot \left(\left|p\right| + \left(\left|r\right| + -1 \cdot p\right)\right)\right) \]
   8. Step-by-step derivation

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lift-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-+.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. mul-1-neg N/A

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

      13. sub-flip-reverse N/A

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

      14. associate-+l- N/A

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

      15. lower--.f64 N/A

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

      16. lower--.f64 N/A

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

      17. metadata-eval N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 N/A

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

      20. metadata-eval 35.1

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

   9. Applied rewrites 35.1%

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

   10. Taylor expanded in r around 0

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

       1. metadata-eval N/A

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

       2. lower-*.f64 N/A

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

       3. metadata-eval N/A

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

       4. lower--.f64 N/A

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

       5. lower-+.f64 N/A

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

       6. lower-fabs.f64 N/A

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

       7. lower-fabs.f64 24.4

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

   12. Applied rewrites 24.4%

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

   if -2.25000000000000012e-192 < r < 3.9999999999999999e101

   1. Initial program 45.9%

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

   2. Taylor expanded in q around inf

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

      1. metadata-eval N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-fabs.f64 N/A

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

      9. lower-fabs.f64 26.6

         \[\leadsto q \cdot \left(1 + 0.5 \cdot \frac{\left|p\right| + \left|r\right|}{q}\right) \]

   4. Applied rewrites 26.6%

      \[\leadsto \color{blue}{q \cdot \left(1 + 0.5 \cdot \frac{\left|p\right| + \left|r\right|}{q}\right)} \]
   5. Step-by-step derivation

      1. metadata-eval N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. *-commutative N/A

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

      7. *-rgt-identity N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. lower-fma.f64 N/A

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

   6. Applied rewrites 26.6%

      \[\leadsto \mathsf{fma}\left(\frac{\left|r\right| + \left|p\right|}{q}, \color{blue}{0.5 \cdot q}, q\right) \]
   7. Step-by-step derivation

      1. metadata-eval N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-fabs.f64 N/A

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

      9. lift-fabs.f64 N/A

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

      10. *-commutative N/A

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

      11. distribute-lft1-in N/A

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

      12. +-commutative N/A

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

   8. Applied rewrites 29.0%

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

   if 3.9999999999999999e101 < r

   1. Initial program 45.9%

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

   2. Taylor expanded in r around inf

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

   4. Applied rewrites 30.3%

      \[\leadsto \color{blue}{r \cdot \left(0.5 + 0.5 \cdot \frac{\left|p\right| + \left(\left|r\right| + -1 \cdot p\right)}{r}\right)} \]

   5. Taylor expanded in r around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-fma.f64 N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. lower-fabs.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-fabs.f64 N/A

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

      11. lower-*.f64 35.0

          \[\leadsto \mathsf{fma}\left(0.5, r, 0.5 \cdot \left(\left|p\right| + \left(\left|r\right| + -1 \cdot p\right)\right)\right) \]

   7. Applied rewrites 35.0%

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

   8. Taylor expanded in p around 0

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

      1. lower-fabs.f64 24.5

         \[\leadsto \mathsf{fma}\left(0.5, r, 0.5 \cdot \left(\left|p\right| + \left|r\right|\right)\right) \]

   10. Applied rewrites 24.5%

       \[\leadsto \mathsf{fma}\left(0.5, r, 0.5 \cdot \left(\left|p\right| + \left|r\right|\right)\right) \]
   11. Step-by-step derivation

       1. metadata-eval 24.5

          \[\leadsto \mathsf{fma}\left(0.5, r, 0.5 \cdot \left(\left|p\right| + \left|r\right|\right)\right) \]

       2. metadata-eval 24.5

          \[\leadsto \mathsf{fma}\left(0.5, r, 0.5 \cdot \left(\left|p\right| + \left|r\right|\right)\right) \]

       3. metadata-eval N/A

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

       4. metadata-eval N/A

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

   12. Applied rewrites 24.5%

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

Alternative 6: 60.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} t_0 := \left|\mathsf{min}\left(p, r\right)\right|\\ t_1 := \left|\mathsf{max}\left(p, r\right)\right|\\ \mathbf{if}\;\mathsf{min}\left(p, r\right) \leq -5.8 \cdot 10^{+44}:\\ \;\;\;\;0.5 \cdot \left(\left(t\_0 + t\_1\right) - \mathsf{min}\left(p, r\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_1 + t\_0, 0.5, \left|q\right|\right)\\ \end{array} \]

FPCore

(FPCore (p r q)
 :precision binary64
 (let* ((t_0 (fabs (fmin p r))) (t_1 (fabs (fmax p r))))
   (if (<= (fmin p r) -5.8e+44)
     (* 0.5 (- (+ t_0 t_1) (fmin p r)))
     (fma (+ t_1 t_0) 0.5 (fabs q)))))

C

double code(double p, double r, double q) {
	double t_0 = fabs(fmin(p, r));
	double t_1 = fabs(fmax(p, r));
	double tmp;
	if (fmin(p, r) <= -5.8e+44) {
		tmp = 0.5 * ((t_0 + t_1) - fmin(p, r));
	} else {
		tmp = fma((t_1 + t_0), 0.5, fabs(q));
	}
	return tmp;
}

Julia

function code(p, r, q)
	t_0 = abs(fmin(p, r))
	t_1 = abs(fmax(p, r))
	tmp = 0.0
	if (fmin(p, r) <= -5.8e+44)
		tmp = Float64(0.5 * Float64(Float64(t_0 + t_1) - fmin(p, r)));
	else
		tmp = fma(Float64(t_1 + t_0), 0.5, abs(q));
	end
	return tmp
end

Wolfram

code[p_, r_, q_] := Block[{t$95$0 = N[Abs[N[Min[p, r], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[Max[p, r], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Min[p, r], $MachinePrecision], -5.8e+44], N[(0.5 * N[(N[(t$95$0 + t$95$1), $MachinePrecision] - N[Min[p, r], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 + t$95$0), $MachinePrecision] * 0.5 + N[Abs[q], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if p < -5.8000000000000004e44

   1. Initial program 45.9%

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

   2. Taylor expanded in r around inf

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

   4. Applied rewrites 30.3%

      \[\leadsto \color{blue}{r \cdot \left(0.5 + 0.5 \cdot \frac{\left|p\right| + \left(\left|r\right| + -1 \cdot p\right)}{r}\right)} \]

   5. Taylor expanded in r around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-fma.f64 N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. lower-fabs.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-fabs.f64 N/A

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

      11. lower-*.f64 35.0

          \[\leadsto \mathsf{fma}\left(0.5, r, 0.5 \cdot \left(\left|p\right| + \left(\left|r\right| + -1 \cdot p\right)\right)\right) \]

   7. Applied rewrites 35.0%

      \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{r}, 0.5 \cdot \left(\left|p\right| + \left(\left|r\right| + -1 \cdot p\right)\right)\right) \]
   8. Step-by-step derivation

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lift-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-+.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. mul-1-neg N/A

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

      13. sub-flip-reverse N/A

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

      14. associate-+l- N/A

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

      15. lower--.f64 N/A

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

      16. lower--.f64 N/A

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

      17. metadata-eval N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 N/A

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

      20. metadata-eval 35.1

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

   9. Applied rewrites 35.1%

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

   10. Taylor expanded in r around 0

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

       1. metadata-eval N/A

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

       2. lower-*.f64 N/A

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

       3. metadata-eval N/A

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

       4. lower--.f64 N/A

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

       5. lower-+.f64 N/A

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

       6. lower-fabs.f64 N/A

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

       7. lower-fabs.f64 24.4

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

   12. Applied rewrites 24.4%

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

   if -5.8000000000000004e44 < p

   1. Initial program 45.9%

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

   2. Taylor expanded in q around inf

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

      1. metadata-eval N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-fabs.f64 N/A

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

      9. lower-fabs.f64 26.6

         \[\leadsto q \cdot \left(1 + 0.5 \cdot \frac{\left|p\right| + \left|r\right|}{q}\right) \]

   4. Applied rewrites 26.6%

      \[\leadsto \color{blue}{q \cdot \left(1 + 0.5 \cdot \frac{\left|p\right| + \left|r\right|}{q}\right)} \]
   5. Step-by-step derivation

      1. metadata-eval N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. *-commutative N/A

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

      7. *-rgt-identity N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. lower-fma.f64 N/A

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

   6. Applied rewrites 26.6%

      \[\leadsto \mathsf{fma}\left(\frac{\left|r\right| + \left|p\right|}{q}, \color{blue}{0.5 \cdot q}, q\right) \]
   7. Step-by-step derivation

      1. metadata-eval N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-fabs.f64 N/A

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

      9. lift-fabs.f64 N/A

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

      10. *-commutative N/A

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

      11. distribute-lft1-in N/A

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

      12. +-commutative N/A

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

   8. Applied rewrites 29.0%

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

Alternative 7: 45.2% accurate, 4.6× speedup

Math

\[\mathsf{fma}\left(\left|r\right| + \left|p\right|, 0.5, \left|q\right|\right) \]

FPCore

(FPCore (p r q) :precision binary64 (fma (+ (fabs r) (fabs p)) 0.5 (fabs q)))

C

double code(double p, double r, double q) {
	return fma((fabs(r) + fabs(p)), 0.5, fabs(q));
}

Julia

function code(p, r, q)
	return fma(Float64(abs(r) + abs(p)), 0.5, abs(q))
end

Wolfram

code[p_, r_, q_] := N[(N[(N[Abs[r], $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision] * 0.5 + N[Abs[q], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 45.9%

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

2. Taylor expanded in q around inf

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

   1. metadata-eval N/A

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

   2. lower-*.f64 N/A

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

   3. lower-+.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. metadata-eval N/A

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

   6. lower-/.f64 N/A

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

   7. lower-+.f64 N/A

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

   8. lower-fabs.f64 N/A

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

   9. lower-fabs.f64 26.6

      \[\leadsto q \cdot \left(1 + 0.5 \cdot \frac{\left|p\right| + \left|r\right|}{q}\right) \]

4. Applied rewrites 26.6%

   \[\leadsto \color{blue}{q \cdot \left(1 + 0.5 \cdot \frac{\left|p\right| + \left|r\right|}{q}\right)} \]
5. Step-by-step derivation

   1. metadata-eval N/A

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

   2. lift-*.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. +-commutative N/A

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

   5. distribute-lft-in N/A

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

   6. *-commutative N/A

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

   7. *-rgt-identity N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. associate-*l* N/A

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

   11. lower-fma.f64 N/A

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

6. Applied rewrites 26.6%

   \[\leadsto \mathsf{fma}\left(\frac{\left|r\right| + \left|p\right|}{q}, \color{blue}{0.5 \cdot q}, q\right) \]
7. Step-by-step derivation

   1. metadata-eval N/A

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

   2. lift-fma.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. lift-/.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. +-commutative N/A

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

   8. lift-fabs.f64 N/A

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

   9. lift-fabs.f64 N/A

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

   10. *-commutative N/A

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

   11. distribute-lft1-in N/A

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

   12. +-commutative N/A

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

8. Applied rewrites 29.0%

   \[\leadsto \mathsf{fma}\left(\left|r\right| + \left|p\right|, \color{blue}{0.5}, q\right) \]
9. Add Preprocessing

Alternative 8: 38.7% accurate, 4.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|q\right| \leq 2.75 \cdot 10^{-31}:\\ \;\;\;\;0.5 \cdot \left(\left|p\right| + \left|r\right|\right)\\ \mathbf{else}:\\ \;\;\;\;\left|q\right| \cdot 1\\ \end{array} \]

FPCore

(FPCore (p r q)
 :precision binary64
 (if (<= (fabs q) 2.75e-31) (* 0.5 (+ (fabs p) (fabs r))) (* (fabs q) 1.0)))

C

double code(double p, double r, double q) {
	double tmp;
	if (fabs(q) <= 2.75e-31) {
		tmp = 0.5 * (fabs(p) + fabs(r));
	} else {
		tmp = fabs(q) * 1.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(p, r, q)
use fmin_fmax_functions
    real(8), intent (in) :: p
    real(8), intent (in) :: r
    real(8), intent (in) :: q
    real(8) :: tmp
    if (abs(q) <= 2.75d-31) then
        tmp = 0.5d0 * (abs(p) + abs(r))
    else
        tmp = abs(q) * 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double p, double r, double q) {
	double tmp;
	if (Math.abs(q) <= 2.75e-31) {
		tmp = 0.5 * (Math.abs(p) + Math.abs(r));
	} else {
		tmp = Math.abs(q) * 1.0;
	}
	return tmp;
}

Python

def code(p, r, q):
	tmp = 0
	if math.fabs(q) <= 2.75e-31:
		tmp = 0.5 * (math.fabs(p) + math.fabs(r))
	else:
		tmp = math.fabs(q) * 1.0
	return tmp

Julia

function code(p, r, q)
	tmp = 0.0
	if (abs(q) <= 2.75e-31)
		tmp = Float64(0.5 * Float64(abs(p) + abs(r)));
	else
		tmp = Float64(abs(q) * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(p, r, q)
	tmp = 0.0;
	if (abs(q) <= 2.75e-31)
		tmp = 0.5 * (abs(p) + abs(r));
	else
		tmp = abs(q) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[p_, r_, q_] := If[LessEqual[N[Abs[q], $MachinePrecision], 2.75e-31], N[(0.5 * N[(N[Abs[p], $MachinePrecision] + N[Abs[r], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Abs[q], $MachinePrecision] * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if q < 2.74999999999999979e-31

   1. Initial program 45.9%

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

   2. Taylor expanded in q around inf

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

      1. metadata-eval N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-fabs.f64 N/A

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

      9. lower-fabs.f64 26.6

         \[\leadsto q \cdot \left(1 + 0.5 \cdot \frac{\left|p\right| + \left|r\right|}{q}\right) \]

   4. Applied rewrites 26.6%

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

   5. Taylor expanded in q around 0

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

      1. metadata-eval N/A

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

      2. lower-*.f64 N/A

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

      3. metadata-eval N/A

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

      4. lower-+.f64 N/A

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

      5. lower-fabs.f64 N/A

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

      6. lower-fabs.f64 14.4

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

   7. Applied rewrites 14.4%

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

   if 2.74999999999999979e-31 < q

   1. Initial program 45.9%

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

   2. Taylor expanded in q around inf

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

      1. metadata-eval N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-fabs.f64 N/A

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

      9. lower-fabs.f64 26.6

         \[\leadsto q \cdot \left(1 + 0.5 \cdot \frac{\left|p\right| + \left|r\right|}{q}\right) \]

   4. Applied rewrites 26.6%

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

   5. Taylor expanded in q around inf

      \[\leadsto q \cdot 1 \]
   6. Step-by-step derivation

   7. Applied rewrites 18.2%

      \[\leadsto q \cdot 1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 36.6% accurate, 4.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|q\right| \leq 3.5 \cdot 10^{-175}:\\ \;\;\;\;0.5 \cdot \mathsf{max}\left(p, r\right)\\ \mathbf{else}:\\ \;\;\;\;\left|q\right| \cdot 1\\ \end{array} \]

FPCore

(FPCore (p r q)
 :precision binary64
 (if (<= (fabs q) 3.5e-175) (* 0.5 (fmax p r)) (* (fabs q) 1.0)))

C

double code(double p, double r, double q) {
	double tmp;
	if (fabs(q) <= 3.5e-175) {
		tmp = 0.5 * fmax(p, r);
	} else {
		tmp = fabs(q) * 1.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(p, r, q)
use fmin_fmax_functions
    real(8), intent (in) :: p
    real(8), intent (in) :: r
    real(8), intent (in) :: q
    real(8) :: tmp
    if (abs(q) <= 3.5d-175) then
        tmp = 0.5d0 * fmax(p, r)
    else
        tmp = abs(q) * 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double p, double r, double q) {
	double tmp;
	if (Math.abs(q) <= 3.5e-175) {
		tmp = 0.5 * fmax(p, r);
	} else {
		tmp = Math.abs(q) * 1.0;
	}
	return tmp;
}

Python

def code(p, r, q):
	tmp = 0
	if math.fabs(q) <= 3.5e-175:
		tmp = 0.5 * fmax(p, r)
	else:
		tmp = math.fabs(q) * 1.0
	return tmp

Julia

function code(p, r, q)
	tmp = 0.0
	if (abs(q) <= 3.5e-175)
		tmp = Float64(0.5 * fmax(p, r));
	else
		tmp = Float64(abs(q) * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(p, r, q)
	tmp = 0.0;
	if (abs(q) <= 3.5e-175)
		tmp = 0.5 * max(p, r);
	else
		tmp = abs(q) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[p_, r_, q_] := If[LessEqual[N[Abs[q], $MachinePrecision], 3.5e-175], N[(0.5 * N[Max[p, r], $MachinePrecision]), $MachinePrecision], N[(N[Abs[q], $MachinePrecision] * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if q < 3.49999999999999999e-175

   1. Initial program 45.9%

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

   2. Taylor expanded in r around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot r} \]
   3. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{1}{2} \cdot r \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{r} \]

      3. metadata-eval 5.2

         \[\leadsto 0.5 \cdot r \]

   4. Applied rewrites 5.2%

      \[\leadsto \color{blue}{0.5 \cdot r} \]

   if 3.49999999999999999e-175 < q

   1. Initial program 45.9%

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

   2. Taylor expanded in q around inf

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

      1. metadata-eval N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-fabs.f64 N/A

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

      9. lower-fabs.f64 26.6

         \[\leadsto q \cdot \left(1 + 0.5 \cdot \frac{\left|p\right| + \left|r\right|}{q}\right) \]

   4. Applied rewrites 26.6%

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

   5. Taylor expanded in q around inf

      \[\leadsto q \cdot 1 \]
   6. Step-by-step derivation

   7. Applied rewrites 18.2%

      \[\leadsto q \cdot 1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 18.2% accurate, 4.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\mathsf{max}\left(p, r\right) \leq 1.75 \cdot 10^{-37}:\\ \;\;\;\;-0.5 \cdot \mathsf{min}\left(p, r\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{max}\left(p, r\right)\\ \end{array} \]

FPCore

(FPCore (p r q)
 :precision binary64
 (if (<= (fmax p r) 1.75e-37) (* -0.5 (fmin p r)) (* 0.5 (fmax p r))))

C

double code(double p, double r, double q) {
	double tmp;
	if (fmax(p, r) <= 1.75e-37) {
		tmp = -0.5 * fmin(p, r);
	} else {
		tmp = 0.5 * fmax(p, r);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(p, r, q)
use fmin_fmax_functions
    real(8), intent (in) :: p
    real(8), intent (in) :: r
    real(8), intent (in) :: q
    real(8) :: tmp
    if (fmax(p, r) <= 1.75d-37) then
        tmp = (-0.5d0) * fmin(p, r)
    else
        tmp = 0.5d0 * fmax(p, r)
    end if
    code = tmp
end function

Java

public static double code(double p, double r, double q) {
	double tmp;
	if (fmax(p, r) <= 1.75e-37) {
		tmp = -0.5 * fmin(p, r);
	} else {
		tmp = 0.5 * fmax(p, r);
	}
	return tmp;
}

Python

def code(p, r, q):
	tmp = 0
	if fmax(p, r) <= 1.75e-37:
		tmp = -0.5 * fmin(p, r)
	else:
		tmp = 0.5 * fmax(p, r)
	return tmp

Julia

function code(p, r, q)
	tmp = 0.0
	if (fmax(p, r) <= 1.75e-37)
		tmp = Float64(-0.5 * fmin(p, r));
	else
		tmp = Float64(0.5 * fmax(p, r));
	end
	return tmp
end

MATLAB

function tmp_2 = code(p, r, q)
	tmp = 0.0;
	if (max(p, r) <= 1.75e-37)
		tmp = -0.5 * min(p, r);
	else
		tmp = 0.5 * max(p, r);
	end
	tmp_2 = tmp;
end

Wolfram

code[p_, r_, q_] := If[LessEqual[N[Max[p, r], $MachinePrecision], 1.75e-37], N[(-0.5 * N[Min[p, r], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Max[p, r], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 1.7500000000000001e-37

   1. Initial program 45.9%

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

   2. Taylor expanded in p around -inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot p} \]
   3. Step-by-step derivation

      1. lower-*.f64 5.3

         \[\leadsto -0.5 \cdot \color{blue}{p} \]

   4. Applied rewrites 5.3%

      \[\leadsto \color{blue}{-0.5 \cdot p} \]

   if 1.7500000000000001e-37 < r

   1. Initial program 45.9%

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

   2. Taylor expanded in r around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot r} \]
   3. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{1}{2} \cdot r \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{r} \]

      3. metadata-eval 5.2

         \[\leadsto 0.5 \cdot r \]

   4. Applied rewrites 5.2%

      \[\leadsto \color{blue}{0.5 \cdot r} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 13.1% accurate, 8.0× speedup

Math

\[-0.5 \cdot \mathsf{min}\left(p, r\right) \]

FPCore

(FPCore (p r q) :precision binary64 (* -0.5 (fmin p r)))

C

double code(double p, double r, double q) {
	return -0.5 * fmin(p, r);
}

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 = (-0.5d0) * fmin(p, r)
end function

Java

public static double code(double p, double r, double q) {
	return -0.5 * fmin(p, r);
}

Python

def code(p, r, q):
	return -0.5 * fmin(p, r)

Julia

function code(p, r, q)
	return Float64(-0.5 * fmin(p, r))
end

MATLAB

function tmp = code(p, r, q)
	tmp = -0.5 * min(p, r);
end

Wolfram

code[p_, r_, q_] := N[(-0.5 * N[Min[p, r], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 45.9%

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

2. Taylor expanded in p around -inf

   \[\leadsto \color{blue}{\frac{-1}{2} \cdot p} \]
3. Step-by-step derivation

   1. lower-*.f64 5.3

      \[\leadsto -0.5 \cdot \color{blue}{p} \]

4. Applied rewrites 5.3%

   \[\leadsto \color{blue}{-0.5 \cdot p} \]
5. Add Preprocessing

Alternative 12: 8.7% accurate, 28.9× speedup

Math

\[-q \]

FPCore

(FPCore (p r q) :precision binary64 (- q))

C

double code(double p, double r, double q) {
	return -q;
}

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 = -q
end function

Java

public static double code(double p, double r, double q) {
	return -q;
}

Python

def code(p, r, q):
	return -q

Julia

function code(p, r, q)
	return Float64(-q)
end

MATLAB

function tmp = code(p, r, q)
	tmp = -q;
end

Wolfram

code[p_, r_, q_] := (-q)

Derivation

1. Initial program 45.9%

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

2. Taylor expanded in q around -inf

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

   1. lower-*.f64 18.2

      \[\leadsto -1 \cdot \color{blue}{q} \]

4. Applied rewrites 18.2%

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

   1. lift-*.f64 N/A

      \[\leadsto -1 \cdot \color{blue}{q} \]

   2. mul-1-neg N/A

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

   3. lower-neg.f64 18.2

      \[\leadsto -q \]

6. Applied rewrites 18.2%

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

Reproduce

herbie shell --seed 2025178 
(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)))))))