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

Percentage Accurate: 23.8% → 60.0%

Time: 5.6s

Alternatives: 6

Speedup: 18.0×

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: 23.8% 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: 60.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_0 := \left|\mathsf{min}\left(p, r\right)\right|\\ \mathbf{if}\;\left|q\right| \leq 8.2 \cdot 10^{-41}:\\ \;\;\;\;-1 \cdot \left(\mathsf{min}\left(p, r\right) \cdot \left(-0.5 \cdot \frac{t\_0}{\mathsf{min}\left(p, r\right)} - 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{hypot}\left(\mathsf{max}\left(p, r\right) - \mathsf{min}\left(p, r\right), \left|q\right| \cdot 2\right) - t\_0\right) - \left|\mathsf{max}\left(p, r\right)\right|\right) \cdot -0.5\\ \end{array} \]

FPCore

(FPCore (p r q)
 :precision binary64
 (let* ((t_0 (fabs (fmin p r))))
   (if (<= (fabs q) 8.2e-41)
     (* -1.0 (* (fmin p r) (- (* -0.5 (/ t_0 (fmin p r))) 0.5)))
     (*
      (-
       (- (hypot (- (fmax p r) (fmin p r)) (* (fabs q) 2.0)) t_0)
       (fabs (fmax p r)))
      -0.5))))

C

double code(double p, double r, double q) {
	double t_0 = fabs(fmin(p, r));
	double tmp;
	if (fabs(q) <= 8.2e-41) {
		tmp = -1.0 * (fmin(p, r) * ((-0.5 * (t_0 / fmin(p, r))) - 0.5));
	} else {
		tmp = ((hypot((fmax(p, r) - fmin(p, r)), (fabs(q) * 2.0)) - t_0) - fabs(fmax(p, r))) * -0.5;
	}
	return tmp;
}

Java

public static double code(double p, double r, double q) {
	double t_0 = Math.abs(fmin(p, r));
	double tmp;
	if (Math.abs(q) <= 8.2e-41) {
		tmp = -1.0 * (fmin(p, r) * ((-0.5 * (t_0 / fmin(p, r))) - 0.5));
	} else {
		tmp = ((Math.hypot((fmax(p, r) - fmin(p, r)), (Math.abs(q) * 2.0)) - t_0) - Math.abs(fmax(p, r))) * -0.5;
	}
	return tmp;
}

Python

def code(p, r, q):
	t_0 = math.fabs(fmin(p, r))
	tmp = 0
	if math.fabs(q) <= 8.2e-41:
		tmp = -1.0 * (fmin(p, r) * ((-0.5 * (t_0 / fmin(p, r))) - 0.5))
	else:
		tmp = ((math.hypot((fmax(p, r) - fmin(p, r)), (math.fabs(q) * 2.0)) - t_0) - math.fabs(fmax(p, r))) * -0.5
	return tmp

Julia

function code(p, r, q)
	t_0 = abs(fmin(p, r))
	tmp = 0.0
	if (abs(q) <= 8.2e-41)
		tmp = Float64(-1.0 * Float64(fmin(p, r) * Float64(Float64(-0.5 * Float64(t_0 / fmin(p, r))) - 0.5)));
	else
		tmp = Float64(Float64(Float64(hypot(Float64(fmax(p, r) - fmin(p, r)), Float64(abs(q) * 2.0)) - t_0) - abs(fmax(p, r))) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(p, r, q)
	t_0 = abs(min(p, r));
	tmp = 0.0;
	if (abs(q) <= 8.2e-41)
		tmp = -1.0 * (min(p, r) * ((-0.5 * (t_0 / min(p, r))) - 0.5));
	else
		tmp = ((hypot((max(p, r) - min(p, r)), (abs(q) * 2.0)) - t_0) - abs(max(p, r))) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[p_, r_, q_] := Block[{t$95$0 = N[Abs[N[Min[p, r], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Abs[q], $MachinePrecision], 8.2e-41], N[(-1.0 * N[(N[Min[p, r], $MachinePrecision] * N[(N[(-0.5 * N[(t$95$0 / N[Min[p, r], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sqrt[N[(N[Max[p, r], $MachinePrecision] - N[Min[p, r], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(N[Abs[q], $MachinePrecision] * 2.0), $MachinePrecision] ^ 2], $MachinePrecision] - t$95$0), $MachinePrecision] - N[Abs[N[Max[p, r], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if q < 8.20000000000000028e-41

   1. Initial program 23.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. sub-negate-rev N/A

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

      5. distribute-lft-neg-out N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. lift-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 N/A

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

   3. Applied rewrites 21.4%

      \[\leadsto \color{blue}{\left(\left(\sqrt{\mathsf{fma}\left(q \cdot 4, q, \left(r - p\right) \cdot \left(r - p\right)\right)} - \left|p\right|\right) - \left|r\right|\right) \cdot -0.5} \]
   4. Step-by-step derivation

      1. lift-fabs.f64 N/A

         \[\leadsto \left(\left(\sqrt{\mathsf{fma}\left(q \cdot 4, q, \left(r - p\right) \cdot \left(r - p\right)\right)} - \left|p\right|\right) - \color{blue}{\left|r\right|}\right) \cdot \frac{-1}{2} \]

      2. neg-fabs N/A

         \[\leadsto \left(\left(\sqrt{\mathsf{fma}\left(q \cdot 4, q, \left(r - p\right) \cdot \left(r - p\right)\right)} - \left|p\right|\right) - \color{blue}{\left|\mathsf{neg}\left(r\right)\right|}\right) \cdot \frac{-1}{2} \]

      3. lift-neg.f64 N/A

         \[\leadsto \left(\left(\sqrt{\mathsf{fma}\left(q \cdot 4, q, \left(r - p\right) \cdot \left(r - p\right)\right)} - \left|p\right|\right) - \left|\color{blue}{-r}\right|\right) \cdot \frac{-1}{2} \]

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

         \[\leadsto \left(\left(\sqrt{\mathsf{fma}\left(q \cdot 4, q, \left(r - p\right) \cdot \left(r - p\right)\right)} - \left|p\right|\right) - \color{blue}{\sqrt{\left(-r\right) \cdot \left(-r\right)}}\right) \cdot \frac{-1}{2} \]

      5. sqrt-unprod N/A

         \[\leadsto \left(\left(\sqrt{\mathsf{fma}\left(q \cdot 4, q, \left(r - p\right) \cdot \left(r - p\right)\right)} - \left|p\right|\right) - \color{blue}{\sqrt{-r} \cdot \sqrt{-r}}\right) \cdot \frac{-1}{2} \]

      6. rem-square-sqrt 19.0%

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

      7. unpow1 N/A

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

      8. pow-to-exp N/A

         \[\leadsto \left(\left(\sqrt{\mathsf{fma}\left(q \cdot 4, q, \left(r - p\right) \cdot \left(r - p\right)\right)} - \left|p\right|\right) - \color{blue}{e^{\log \left(-r\right) \cdot 1}}\right) \cdot \frac{-1}{2} \]

      9. lower-unsound-exp.f64 N/A

         \[\leadsto \left(\left(\sqrt{\mathsf{fma}\left(q \cdot 4, q, \left(r - p\right) \cdot \left(r - p\right)\right)} - \left|p\right|\right) - \color{blue}{e^{\log \left(-r\right) \cdot 1}}\right) \cdot \frac{-1}{2} \]

      10. lower-unsound-*.f64 N/A

          \[\leadsto \left(\left(\sqrt{\mathsf{fma}\left(q \cdot 4, q, \left(r - p\right) \cdot \left(r - p\right)\right)} - \left|p\right|\right) - e^{\color{blue}{\log \left(-r\right) \cdot 1}}\right) \cdot \frac{-1}{2} \]

      11. lower-unsound-log.f64 8.9%

          \[\leadsto \left(\left(\sqrt{\mathsf{fma}\left(q \cdot 4, q, \left(r - p\right) \cdot \left(r - p\right)\right)} - \left|p\right|\right) - e^{\color{blue}{\log \left(-r\right)} \cdot 1}\right) \cdot -0.5 \]

   5. Applied rewrites 8.9%

      \[\leadsto \left(\left(\sqrt{\mathsf{fma}\left(q \cdot 4, q, \left(r - p\right) \cdot \left(r - p\right)\right)} - \left|p\right|\right) - \color{blue}{e^{\log \left(-r\right) \cdot 1}}\right) \cdot -0.5 \]

   6. Taylor expanded in p around -inf

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

   8. Applied rewrites 7.0%

      \[\leadsto \color{blue}{-1 \cdot \left(p \cdot \left(0.5 \cdot \frac{2 \cdot r - \left|p\right|}{p} - 0.5\right)\right)} \]

   9. Taylor expanded in r around 0

      \[\leadsto -1 \cdot \left(p \cdot \left(\frac{-1}{2} \cdot \frac{\left|p\right|}{p} - 0.5\right)\right) \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

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

       2. lower-/.f64 N/A

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

       3. lower-fabs.f64 17.7%

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

   11. Applied rewrites 17.7%

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

   if 8.20000000000000028e-41 < q

   1. Initial program 23.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. sub-negate-rev N/A

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

      5. distribute-lft-neg-out N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. lift-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 N/A

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

   3. Applied rewrites 21.4%

      \[\leadsto \color{blue}{\left(\left(\sqrt{\mathsf{fma}\left(q \cdot 4, q, \left(r - p\right) \cdot \left(r - p\right)\right)} - \left|p\right|\right) - \left|r\right|\right) \cdot -0.5} \]
   4. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\sqrt{\mathsf{fma}\left(q \cdot 4, q, \left(r - p\right) \cdot \left(r - p\right)\right)}} - \left|p\right|\right) - \left|r\right|\right) \cdot \frac{-1}{2} \]

      2. lift-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. metadata-eval N/A

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

      9. unswap-sqr N/A

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

      10. lower-hypot.f64 N/A

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

      11. lower-*.f64 47.3%

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

   5. Applied rewrites 47.3%

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

Alternative 2: 59.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_0 := \left|\mathsf{min}\left(p, r\right)\right|\\ \mathbf{if}\;\left|q\right| \leq 8.2 \cdot 10^{-41}:\\ \;\;\;\;-1 \cdot \left(\mathsf{min}\left(p, r\right) \cdot \left(-0.5 \cdot \frac{t\_0}{\mathsf{min}\left(p, r\right)} - 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{hypot}\left(\mathsf{max}\left(p, r\right) - \mathsf{min}\left(p, r\right), \left|q\right| \cdot 2\right) - t\_0\right) - \left(-\mathsf{max}\left(p, r\right)\right)\right) \cdot -0.5\\ \end{array} \]

FPCore

(FPCore (p r q)
 :precision binary64
 (let* ((t_0 (fabs (fmin p r))))
   (if (<= (fabs q) 8.2e-41)
     (* -1.0 (* (fmin p r) (- (* -0.5 (/ t_0 (fmin p r))) 0.5)))
     (*
      (-
       (- (hypot (- (fmax p r) (fmin p r)) (* (fabs q) 2.0)) t_0)
       (- (fmax p r)))
      -0.5))))

C

double code(double p, double r, double q) {
	double t_0 = fabs(fmin(p, r));
	double tmp;
	if (fabs(q) <= 8.2e-41) {
		tmp = -1.0 * (fmin(p, r) * ((-0.5 * (t_0 / fmin(p, r))) - 0.5));
	} else {
		tmp = ((hypot((fmax(p, r) - fmin(p, r)), (fabs(q) * 2.0)) - t_0) - -fmax(p, r)) * -0.5;
	}
	return tmp;
}

Java

public static double code(double p, double r, double q) {
	double t_0 = Math.abs(fmin(p, r));
	double tmp;
	if (Math.abs(q) <= 8.2e-41) {
		tmp = -1.0 * (fmin(p, r) * ((-0.5 * (t_0 / fmin(p, r))) - 0.5));
	} else {
		tmp = ((Math.hypot((fmax(p, r) - fmin(p, r)), (Math.abs(q) * 2.0)) - t_0) - -fmax(p, r)) * -0.5;
	}
	return tmp;
}

Python

def code(p, r, q):
	t_0 = math.fabs(fmin(p, r))
	tmp = 0
	if math.fabs(q) <= 8.2e-41:
		tmp = -1.0 * (fmin(p, r) * ((-0.5 * (t_0 / fmin(p, r))) - 0.5))
	else:
		tmp = ((math.hypot((fmax(p, r) - fmin(p, r)), (math.fabs(q) * 2.0)) - t_0) - -fmax(p, r)) * -0.5
	return tmp

Julia

function code(p, r, q)
	t_0 = abs(fmin(p, r))
	tmp = 0.0
	if (abs(q) <= 8.2e-41)
		tmp = Float64(-1.0 * Float64(fmin(p, r) * Float64(Float64(-0.5 * Float64(t_0 / fmin(p, r))) - 0.5)));
	else
		tmp = Float64(Float64(Float64(hypot(Float64(fmax(p, r) - fmin(p, r)), Float64(abs(q) * 2.0)) - t_0) - Float64(-fmax(p, r))) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(p, r, q)
	t_0 = abs(min(p, r));
	tmp = 0.0;
	if (abs(q) <= 8.2e-41)
		tmp = -1.0 * (min(p, r) * ((-0.5 * (t_0 / min(p, r))) - 0.5));
	else
		tmp = ((hypot((max(p, r) - min(p, r)), (abs(q) * 2.0)) - t_0) - -max(p, r)) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[p_, r_, q_] := Block[{t$95$0 = N[Abs[N[Min[p, r], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Abs[q], $MachinePrecision], 8.2e-41], N[(-1.0 * N[(N[Min[p, r], $MachinePrecision] * N[(N[(-0.5 * N[(t$95$0 / N[Min[p, r], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sqrt[N[(N[Max[p, r], $MachinePrecision] - N[Min[p, r], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(N[Abs[q], $MachinePrecision] * 2.0), $MachinePrecision] ^ 2], $MachinePrecision] - t$95$0), $MachinePrecision] - (-N[Max[p, r], $MachinePrecision])), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if q < 8.20000000000000028e-41

   1. Initial program 23.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. sub-negate-rev N/A

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

      5. distribute-lft-neg-out N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. lift-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 N/A

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

   3. Applied rewrites 21.4%

      \[\leadsto \color{blue}{\left(\left(\sqrt{\mathsf{fma}\left(q \cdot 4, q, \left(r - p\right) \cdot \left(r - p\right)\right)} - \left|p\right|\right) - \left|r\right|\right) \cdot -0.5} \]
   4. Step-by-step derivation

      1. lift-fabs.f64 N/A

         \[\leadsto \left(\left(\sqrt{\mathsf{fma}\left(q \cdot 4, q, \left(r - p\right) \cdot \left(r - p\right)\right)} - \left|p\right|\right) - \color{blue}{\left|r\right|}\right) \cdot \frac{-1}{2} \]

      2. neg-fabs N/A

         \[\leadsto \left(\left(\sqrt{\mathsf{fma}\left(q \cdot 4, q, \left(r - p\right) \cdot \left(r - p\right)\right)} - \left|p\right|\right) - \color{blue}{\left|\mathsf{neg}\left(r\right)\right|}\right) \cdot \frac{-1}{2} \]

      3. lift-neg.f64 N/A

         \[\leadsto \left(\left(\sqrt{\mathsf{fma}\left(q \cdot 4, q, \left(r - p\right) \cdot \left(r - p\right)\right)} - \left|p\right|\right) - \left|\color{blue}{-r}\right|\right) \cdot \frac{-1}{2} \]

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

         \[\leadsto \left(\left(\sqrt{\mathsf{fma}\left(q \cdot 4, q, \left(r - p\right) \cdot \left(r - p\right)\right)} - \left|p\right|\right) - \color{blue}{\sqrt{\left(-r\right) \cdot \left(-r\right)}}\right) \cdot \frac{-1}{2} \]

      5. sqrt-unprod N/A

         \[\leadsto \left(\left(\sqrt{\mathsf{fma}\left(q \cdot 4, q, \left(r - p\right) \cdot \left(r - p\right)\right)} - \left|p\right|\right) - \color{blue}{\sqrt{-r} \cdot \sqrt{-r}}\right) \cdot \frac{-1}{2} \]

      6. rem-square-sqrt 19.0%

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

      7. unpow1 N/A

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

      8. pow-to-exp N/A

         \[\leadsto \left(\left(\sqrt{\mathsf{fma}\left(q \cdot 4, q, \left(r - p\right) \cdot \left(r - p\right)\right)} - \left|p\right|\right) - \color{blue}{e^{\log \left(-r\right) \cdot 1}}\right) \cdot \frac{-1}{2} \]

      9. lower-unsound-exp.f64 N/A

         \[\leadsto \left(\left(\sqrt{\mathsf{fma}\left(q \cdot 4, q, \left(r - p\right) \cdot \left(r - p\right)\right)} - \left|p\right|\right) - \color{blue}{e^{\log \left(-r\right) \cdot 1}}\right) \cdot \frac{-1}{2} \]

      10. lower-unsound-*.f64 N/A

          \[\leadsto \left(\left(\sqrt{\mathsf{fma}\left(q \cdot 4, q, \left(r - p\right) \cdot \left(r - p\right)\right)} - \left|p\right|\right) - e^{\color{blue}{\log \left(-r\right) \cdot 1}}\right) \cdot \frac{-1}{2} \]

      11. lower-unsound-log.f64 8.9%

          \[\leadsto \left(\left(\sqrt{\mathsf{fma}\left(q \cdot 4, q, \left(r - p\right) \cdot \left(r - p\right)\right)} - \left|p\right|\right) - e^{\color{blue}{\log \left(-r\right)} \cdot 1}\right) \cdot -0.5 \]

   5. Applied rewrites 8.9%

      \[\leadsto \left(\left(\sqrt{\mathsf{fma}\left(q \cdot 4, q, \left(r - p\right) \cdot \left(r - p\right)\right)} - \left|p\right|\right) - \color{blue}{e^{\log \left(-r\right) \cdot 1}}\right) \cdot -0.5 \]

   6. Taylor expanded in p around -inf

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

   8. Applied rewrites 7.0%

      \[\leadsto \color{blue}{-1 \cdot \left(p \cdot \left(0.5 \cdot \frac{2 \cdot r - \left|p\right|}{p} - 0.5\right)\right)} \]

   9. Taylor expanded in r around 0

      \[\leadsto -1 \cdot \left(p \cdot \left(\frac{-1}{2} \cdot \frac{\left|p\right|}{p} - 0.5\right)\right) \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

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

       2. lower-/.f64 N/A

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

       3. lower-fabs.f64 17.7%

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

   11. Applied rewrites 17.7%

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

   if 8.20000000000000028e-41 < q

   1. Initial program 23.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. sub-negate-rev N/A

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

      5. distribute-lft-neg-out N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. lift-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 N/A

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

   3. Applied rewrites 21.4%

      \[\leadsto \color{blue}{\left(\left(\sqrt{\mathsf{fma}\left(q \cdot 4, q, \left(r - p\right) \cdot \left(r - p\right)\right)} - \left|p\right|\right) - \left|r\right|\right) \cdot -0.5} \]
   4. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\sqrt{\mathsf{fma}\left(q \cdot 4, q, \left(r - p\right) \cdot \left(r - p\right)\right)}} - \left|p\right|\right) - \left|r\right|\right) \cdot \frac{-1}{2} \]

      2. lift-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. metadata-eval N/A

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

      9. unswap-sqr N/A

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

      10. lower-hypot.f64 N/A

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

      11. lower-*.f64 47.3%

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

   5. Applied rewrites 47.3%

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

      1. lift-fabs.f64 N/A

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

      2. neg-fabs N/A

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

      3. lift-neg.f64 N/A

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

      4. unpow1 N/A

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

      5. exp-to-pow N/A

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

      6. lift-log.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. exp-fabs N/A

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

      9. lift-*.f64 N/A

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

      10. lift-log.f64 N/A

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

      11. exp-to-pow N/A

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

      12. unpow1 41.2%

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

   7. Applied rewrites 41.2%

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

Alternative 3: 58.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|q\right| \leq 8.2 \cdot 10^{-41}:\\ \;\;\;\;-1 \cdot \left(\mathsf{min}\left(p, r\right) \cdot \left(-0.5 \cdot \frac{\left|\mathsf{min}\left(p, r\right)\right|}{\mathsf{min}\left(p, r\right)} - 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-\left|q\right|\\ \end{array} \]

FPCore

(FPCore (p r q)
 :precision binary64
 (if (<= (fabs q) 8.2e-41)
   (* -1.0 (* (fmin p r) (- (* -0.5 (/ (fabs (fmin p r)) (fmin p r))) 0.5)))
   (- (fabs q))))

C

double code(double p, double r, double q) {
	double tmp;
	if (fabs(q) <= 8.2e-41) {
		tmp = -1.0 * (fmin(p, r) * ((-0.5 * (fabs(fmin(p, r)) / fmin(p, r))) - 0.5));
	} else {
		tmp = -fabs(q);
	}
	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) <= 8.2d-41) then
        tmp = (-1.0d0) * (fmin(p, r) * (((-0.5d0) * (abs(fmin(p, r)) / fmin(p, r))) - 0.5d0))
    else
        tmp = -abs(q)
    end if
    code = tmp
end function

Java

public static double code(double p, double r, double q) {
	double tmp;
	if (Math.abs(q) <= 8.2e-41) {
		tmp = -1.0 * (fmin(p, r) * ((-0.5 * (Math.abs(fmin(p, r)) / fmin(p, r))) - 0.5));
	} else {
		tmp = -Math.abs(q);
	}
	return tmp;
}

Python

def code(p, r, q):
	tmp = 0
	if math.fabs(q) <= 8.2e-41:
		tmp = -1.0 * (fmin(p, r) * ((-0.5 * (math.fabs(fmin(p, r)) / fmin(p, r))) - 0.5))
	else:
		tmp = -math.fabs(q)
	return tmp

Julia

function code(p, r, q)
	tmp = 0.0
	if (abs(q) <= 8.2e-41)
		tmp = Float64(-1.0 * Float64(fmin(p, r) * Float64(Float64(-0.5 * Float64(abs(fmin(p, r)) / fmin(p, r))) - 0.5)));
	else
		tmp = Float64(-abs(q));
	end
	return tmp
end

MATLAB

function tmp_2 = code(p, r, q)
	tmp = 0.0;
	if (abs(q) <= 8.2e-41)
		tmp = -1.0 * (min(p, r) * ((-0.5 * (abs(min(p, r)) / min(p, r))) - 0.5));
	else
		tmp = -abs(q);
	end
	tmp_2 = tmp;
end

Wolfram

code[p_, r_, q_] := If[LessEqual[N[Abs[q], $MachinePrecision], 8.2e-41], N[(-1.0 * N[(N[Min[p, r], $MachinePrecision] * N[(N[(-0.5 * N[(N[Abs[N[Min[p, r], $MachinePrecision]], $MachinePrecision] / N[Min[p, r], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[Abs[q], $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if q < 8.20000000000000028e-41

   1. Initial program 23.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. sub-negate-rev N/A

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

      5. distribute-lft-neg-out N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. lift-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 N/A

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

   3. Applied rewrites 21.4%

      \[\leadsto \color{blue}{\left(\left(\sqrt{\mathsf{fma}\left(q \cdot 4, q, \left(r - p\right) \cdot \left(r - p\right)\right)} - \left|p\right|\right) - \left|r\right|\right) \cdot -0.5} \]
   4. Step-by-step derivation

      1. lift-fabs.f64 N/A

         \[\leadsto \left(\left(\sqrt{\mathsf{fma}\left(q \cdot 4, q, \left(r - p\right) \cdot \left(r - p\right)\right)} - \left|p\right|\right) - \color{blue}{\left|r\right|}\right) \cdot \frac{-1}{2} \]

      2. neg-fabs N/A

         \[\leadsto \left(\left(\sqrt{\mathsf{fma}\left(q \cdot 4, q, \left(r - p\right) \cdot \left(r - p\right)\right)} - \left|p\right|\right) - \color{blue}{\left|\mathsf{neg}\left(r\right)\right|}\right) \cdot \frac{-1}{2} \]

      3. lift-neg.f64 N/A

         \[\leadsto \left(\left(\sqrt{\mathsf{fma}\left(q \cdot 4, q, \left(r - p\right) \cdot \left(r - p\right)\right)} - \left|p\right|\right) - \left|\color{blue}{-r}\right|\right) \cdot \frac{-1}{2} \]

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

         \[\leadsto \left(\left(\sqrt{\mathsf{fma}\left(q \cdot 4, q, \left(r - p\right) \cdot \left(r - p\right)\right)} - \left|p\right|\right) - \color{blue}{\sqrt{\left(-r\right) \cdot \left(-r\right)}}\right) \cdot \frac{-1}{2} \]

      5. sqrt-unprod N/A

         \[\leadsto \left(\left(\sqrt{\mathsf{fma}\left(q \cdot 4, q, \left(r - p\right) \cdot \left(r - p\right)\right)} - \left|p\right|\right) - \color{blue}{\sqrt{-r} \cdot \sqrt{-r}}\right) \cdot \frac{-1}{2} \]

      6. rem-square-sqrt 19.0%

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

      7. unpow1 N/A

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

      8. pow-to-exp N/A

         \[\leadsto \left(\left(\sqrt{\mathsf{fma}\left(q \cdot 4, q, \left(r - p\right) \cdot \left(r - p\right)\right)} - \left|p\right|\right) - \color{blue}{e^{\log \left(-r\right) \cdot 1}}\right) \cdot \frac{-1}{2} \]

      9. lower-unsound-exp.f64 N/A

         \[\leadsto \left(\left(\sqrt{\mathsf{fma}\left(q \cdot 4, q, \left(r - p\right) \cdot \left(r - p\right)\right)} - \left|p\right|\right) - \color{blue}{e^{\log \left(-r\right) \cdot 1}}\right) \cdot \frac{-1}{2} \]

      10. lower-unsound-*.f64 N/A

          \[\leadsto \left(\left(\sqrt{\mathsf{fma}\left(q \cdot 4, q, \left(r - p\right) \cdot \left(r - p\right)\right)} - \left|p\right|\right) - e^{\color{blue}{\log \left(-r\right) \cdot 1}}\right) \cdot \frac{-1}{2} \]

      11. lower-unsound-log.f64 8.9%

          \[\leadsto \left(\left(\sqrt{\mathsf{fma}\left(q \cdot 4, q, \left(r - p\right) \cdot \left(r - p\right)\right)} - \left|p\right|\right) - e^{\color{blue}{\log \left(-r\right)} \cdot 1}\right) \cdot -0.5 \]

   5. Applied rewrites 8.9%

      \[\leadsto \left(\left(\sqrt{\mathsf{fma}\left(q \cdot 4, q, \left(r - p\right) \cdot \left(r - p\right)\right)} - \left|p\right|\right) - \color{blue}{e^{\log \left(-r\right) \cdot 1}}\right) \cdot -0.5 \]

   6. Taylor expanded in p around -inf

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

   8. Applied rewrites 7.0%

      \[\leadsto \color{blue}{-1 \cdot \left(p \cdot \left(0.5 \cdot \frac{2 \cdot r - \left|p\right|}{p} - 0.5\right)\right)} \]

   9. Taylor expanded in r around 0

      \[\leadsto -1 \cdot \left(p \cdot \left(\frac{-1}{2} \cdot \frac{\left|p\right|}{p} - 0.5\right)\right) \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

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

       2. lower-/.f64 N/A

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

       3. lower-fabs.f64 17.7%

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

   11. Applied rewrites 17.7%

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

   if 8.20000000000000028e-41 < q

   1. Initial program 23.8%

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

   2. Taylor expanded in q around inf

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

      1. lower-*.f64 19.6%

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

   4. Applied rewrites 19.6%

      \[\leadsto \color{blue}{-1 \cdot q} \]
   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 19.6%

         \[\leadsto -q \]

   6. Applied rewrites 19.6%

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

Alternative 4: 41.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} t_0 := 2 \cdot \mathsf{max}\left(p, r\right)\\ \mathbf{if}\;\left|q\right| \leq 1.65 \cdot 10^{-151}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, t\_0 - \left|\mathsf{min}\left(p, r\right)\right|, 0.5 \cdot \mathsf{min}\left(p, r\right)\right)\\ \mathbf{elif}\;\left|q\right| \leq 4.5 \cdot 10^{-58}:\\ \;\;\;\;t\_0 \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;-\left|q\right|\\ \end{array} \]

FPCore

(FPCore (p r q)
 :precision binary64
 (let* ((t_0 (* 2.0 (fmax p r))))
   (if (<= (fabs q) 1.65e-151)
     (fma -0.5 (- t_0 (fabs (fmin p r))) (* 0.5 (fmin p r)))
     (if (<= (fabs q) 4.5e-58) (* t_0 -0.5) (- (fabs q))))))

C

double code(double p, double r, double q) {
	double t_0 = 2.0 * fmax(p, r);
	double tmp;
	if (fabs(q) <= 1.65e-151) {
		tmp = fma(-0.5, (t_0 - fabs(fmin(p, r))), (0.5 * fmin(p, r)));
	} else if (fabs(q) <= 4.5e-58) {
		tmp = t_0 * -0.5;
	} else {
		tmp = -fabs(q);
	}
	return tmp;
}

Julia

function code(p, r, q)
	t_0 = Float64(2.0 * fmax(p, r))
	tmp = 0.0
	if (abs(q) <= 1.65e-151)
		tmp = fma(-0.5, Float64(t_0 - abs(fmin(p, r))), Float64(0.5 * fmin(p, r)));
	elseif (abs(q) <= 4.5e-58)
		tmp = Float64(t_0 * -0.5);
	else
		tmp = Float64(-abs(q));
	end
	return tmp
end

Wolfram

code[p_, r_, q_] := Block[{t$95$0 = N[(2.0 * N[Max[p, r], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[q], $MachinePrecision], 1.65e-151], N[(-0.5 * N[(t$95$0 - N[Abs[N[Min[p, r], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[Min[p, r], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[q], $MachinePrecision], 4.5e-58], N[(t$95$0 * -0.5), $MachinePrecision], (-N[Abs[q], $MachinePrecision])]]]

Derivation

1. Split input into 3 regimes

2. if q < 1.6499999999999999e-151

   1. Initial program 23.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. sub-negate-rev N/A

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

      5. distribute-lft-neg-out N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. lift-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 N/A

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

   3. Applied rewrites 21.4%

      \[\leadsto \color{blue}{\left(\left(\sqrt{\mathsf{fma}\left(q \cdot 4, q, \left(r - p\right) \cdot \left(r - p\right)\right)} - \left|p\right|\right) - \left|r\right|\right) \cdot -0.5} \]
   4. Step-by-step derivation

      1. lift-fabs.f64 N/A

         \[\leadsto \left(\left(\sqrt{\mathsf{fma}\left(q \cdot 4, q, \left(r - p\right) \cdot \left(r - p\right)\right)} - \left|p\right|\right) - \color{blue}{\left|r\right|}\right) \cdot \frac{-1}{2} \]

      2. neg-fabs N/A

         \[\leadsto \left(\left(\sqrt{\mathsf{fma}\left(q \cdot 4, q, \left(r - p\right) \cdot \left(r - p\right)\right)} - \left|p\right|\right) - \color{blue}{\left|\mathsf{neg}\left(r\right)\right|}\right) \cdot \frac{-1}{2} \]

      3. lift-neg.f64 N/A

         \[\leadsto \left(\left(\sqrt{\mathsf{fma}\left(q \cdot 4, q, \left(r - p\right) \cdot \left(r - p\right)\right)} - \left|p\right|\right) - \left|\color{blue}{-r}\right|\right) \cdot \frac{-1}{2} \]

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

         \[\leadsto \left(\left(\sqrt{\mathsf{fma}\left(q \cdot 4, q, \left(r - p\right) \cdot \left(r - p\right)\right)} - \left|p\right|\right) - \color{blue}{\sqrt{\left(-r\right) \cdot \left(-r\right)}}\right) \cdot \frac{-1}{2} \]

      5. sqrt-unprod N/A

         \[\leadsto \left(\left(\sqrt{\mathsf{fma}\left(q \cdot 4, q, \left(r - p\right) \cdot \left(r - p\right)\right)} - \left|p\right|\right) - \color{blue}{\sqrt{-r} \cdot \sqrt{-r}}\right) \cdot \frac{-1}{2} \]

      6. rem-square-sqrt 19.0%

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

      7. unpow1 N/A

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

      8. pow-to-exp N/A

         \[\leadsto \left(\left(\sqrt{\mathsf{fma}\left(q \cdot 4, q, \left(r - p\right) \cdot \left(r - p\right)\right)} - \left|p\right|\right) - \color{blue}{e^{\log \left(-r\right) \cdot 1}}\right) \cdot \frac{-1}{2} \]

      9. lower-unsound-exp.f64 N/A

         \[\leadsto \left(\left(\sqrt{\mathsf{fma}\left(q \cdot 4, q, \left(r - p\right) \cdot \left(r - p\right)\right)} - \left|p\right|\right) - \color{blue}{e^{\log \left(-r\right) \cdot 1}}\right) \cdot \frac{-1}{2} \]

      10. lower-unsound-*.f64 N/A

          \[\leadsto \left(\left(\sqrt{\mathsf{fma}\left(q \cdot 4, q, \left(r - p\right) \cdot \left(r - p\right)\right)} - \left|p\right|\right) - e^{\color{blue}{\log \left(-r\right) \cdot 1}}\right) \cdot \frac{-1}{2} \]

      11. lower-unsound-log.f64 8.9%

          \[\leadsto \left(\left(\sqrt{\mathsf{fma}\left(q \cdot 4, q, \left(r - p\right) \cdot \left(r - p\right)\right)} - \left|p\right|\right) - e^{\color{blue}{\log \left(-r\right)} \cdot 1}\right) \cdot -0.5 \]

   5. Applied rewrites 8.9%

      \[\leadsto \left(\left(\sqrt{\mathsf{fma}\left(q \cdot 4, q, \left(r - p\right) \cdot \left(r - p\right)\right)} - \left|p\right|\right) - \color{blue}{e^{\log \left(-r\right) \cdot 1}}\right) \cdot -0.5 \]

   6. Taylor expanded in p around -inf

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

   8. Applied rewrites 7.0%

      \[\leadsto \color{blue}{-1 \cdot \left(p \cdot \left(0.5 \cdot \frac{2 \cdot r - \left|p\right|}{p} - 0.5\right)\right)} \]

   9. Taylor expanded in p around 0

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

       1. metadata-eval N/A

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

       2. lower-fma.f64 N/A

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

       3. lower--.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. lower-fabs.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. metadata-eval 7.1%

          \[\leadsto \mathsf{fma}\left(-0.5, 2 \cdot r - \left|p\right|, 0.5 \cdot p\right) \]

   11. Applied rewrites 7.1%

       \[\leadsto \mathsf{fma}\left(-0.5, \color{blue}{2 \cdot r - \left|p\right|}, 0.5 \cdot p\right) \]

   if 1.6499999999999999e-151 < q < 4.5000000000000003e-58

   1. Initial program 23.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. sub-negate-rev N/A

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

      5. distribute-lft-neg-out N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. lift-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 N/A

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

   3. Applied rewrites 21.4%

      \[\leadsto \color{blue}{\left(\left(\sqrt{\mathsf{fma}\left(q \cdot 4, q, \left(r - p\right) \cdot \left(r - p\right)\right)} - \left|p\right|\right) - \left|r\right|\right) \cdot -0.5} \]
   4. Step-by-step derivation

      1. lift-fabs.f64 N/A

         \[\leadsto \left(\left(\sqrt{\mathsf{fma}\left(q \cdot 4, q, \left(r - p\right) \cdot \left(r - p\right)\right)} - \left|p\right|\right) - \color{blue}{\left|r\right|}\right) \cdot \frac{-1}{2} \]

      2. neg-fabs N/A

         \[\leadsto \left(\left(\sqrt{\mathsf{fma}\left(q \cdot 4, q, \left(r - p\right) \cdot \left(r - p\right)\right)} - \left|p\right|\right) - \color{blue}{\left|\mathsf{neg}\left(r\right)\right|}\right) \cdot \frac{-1}{2} \]

      3. lift-neg.f64 N/A

         \[\leadsto \left(\left(\sqrt{\mathsf{fma}\left(q \cdot 4, q, \left(r - p\right) \cdot \left(r - p\right)\right)} - \left|p\right|\right) - \left|\color{blue}{-r}\right|\right) \cdot \frac{-1}{2} \]

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

         \[\leadsto \left(\left(\sqrt{\mathsf{fma}\left(q \cdot 4, q, \left(r - p\right) \cdot \left(r - p\right)\right)} - \left|p\right|\right) - \color{blue}{\sqrt{\left(-r\right) \cdot \left(-r\right)}}\right) \cdot \frac{-1}{2} \]

      5. sqrt-unprod N/A

         \[\leadsto \left(\left(\sqrt{\mathsf{fma}\left(q \cdot 4, q, \left(r - p\right) \cdot \left(r - p\right)\right)} - \left|p\right|\right) - \color{blue}{\sqrt{-r} \cdot \sqrt{-r}}\right) \cdot \frac{-1}{2} \]

      6. rem-square-sqrt 19.0%

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

      7. unpow1 N/A

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

      8. pow-to-exp N/A

         \[\leadsto \left(\left(\sqrt{\mathsf{fma}\left(q \cdot 4, q, \left(r - p\right) \cdot \left(r - p\right)\right)} - \left|p\right|\right) - \color{blue}{e^{\log \left(-r\right) \cdot 1}}\right) \cdot \frac{-1}{2} \]

      9. lower-unsound-exp.f64 N/A

         \[\leadsto \left(\left(\sqrt{\mathsf{fma}\left(q \cdot 4, q, \left(r - p\right) \cdot \left(r - p\right)\right)} - \left|p\right|\right) - \color{blue}{e^{\log \left(-r\right) \cdot 1}}\right) \cdot \frac{-1}{2} \]

      10. lower-unsound-*.f64 N/A

          \[\leadsto \left(\left(\sqrt{\mathsf{fma}\left(q \cdot 4, q, \left(r - p\right) \cdot \left(r - p\right)\right)} - \left|p\right|\right) - e^{\color{blue}{\log \left(-r\right) \cdot 1}}\right) \cdot \frac{-1}{2} \]

      11. lower-unsound-log.f64 8.9%

          \[\leadsto \left(\left(\sqrt{\mathsf{fma}\left(q \cdot 4, q, \left(r - p\right) \cdot \left(r - p\right)\right)} - \left|p\right|\right) - e^{\color{blue}{\log \left(-r\right)} \cdot 1}\right) \cdot -0.5 \]

   5. Applied rewrites 8.9%

      \[\leadsto \left(\left(\sqrt{\mathsf{fma}\left(q \cdot 4, q, \left(r - p\right) \cdot \left(r - p\right)\right)} - \left|p\right|\right) - \color{blue}{e^{\log \left(-r\right) \cdot 1}}\right) \cdot -0.5 \]

   6. Taylor expanded in r around inf

      \[\leadsto \color{blue}{\left(2 \cdot r\right)} \cdot -0.5 \]
   7. Step-by-step derivation

      1. lower-*.f64 9.2%

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

   8. Applied rewrites 9.2%

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

   if 4.5000000000000003e-58 < q

   1. Initial program 23.8%

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

   2. Taylor expanded in q around inf

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

      1. lower-*.f64 19.6%

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

   4. Applied rewrites 19.6%

      \[\leadsto \color{blue}{-1 \cdot q} \]
   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 19.6%

         \[\leadsto -q \]

   6. Applied rewrites 19.6%

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

Alternative 5: 41.2% accurate, 3.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|q\right| \leq 4.5 \cdot 10^{-58}:\\ \;\;\;\;\left(2 \cdot \mathsf{max}\left(p, r\right)\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;-\left|q\right|\\ \end{array} \]

FPCore

(FPCore (p r q)
 :precision binary64
 (if (<= (fabs q) 4.5e-58) (* (* 2.0 (fmax p r)) -0.5) (- (fabs q))))

C

double code(double p, double r, double q) {
	double tmp;
	if (fabs(q) <= 4.5e-58) {
		tmp = (2.0 * fmax(p, r)) * -0.5;
	} else {
		tmp = -fabs(q);
	}
	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) <= 4.5d-58) then
        tmp = (2.0d0 * fmax(p, r)) * (-0.5d0)
    else
        tmp = -abs(q)
    end if
    code = tmp
end function

Java

public static double code(double p, double r, double q) {
	double tmp;
	if (Math.abs(q) <= 4.5e-58) {
		tmp = (2.0 * fmax(p, r)) * -0.5;
	} else {
		tmp = -Math.abs(q);
	}
	return tmp;
}

Python

def code(p, r, q):
	tmp = 0
	if math.fabs(q) <= 4.5e-58:
		tmp = (2.0 * fmax(p, r)) * -0.5
	else:
		tmp = -math.fabs(q)
	return tmp

Julia

function code(p, r, q)
	tmp = 0.0
	if (abs(q) <= 4.5e-58)
		tmp = Float64(Float64(2.0 * fmax(p, r)) * -0.5);
	else
		tmp = Float64(-abs(q));
	end
	return tmp
end

MATLAB

function tmp_2 = code(p, r, q)
	tmp = 0.0;
	if (abs(q) <= 4.5e-58)
		tmp = (2.0 * max(p, r)) * -0.5;
	else
		tmp = -abs(q);
	end
	tmp_2 = tmp;
end

Wolfram

code[p_, r_, q_] := If[LessEqual[N[Abs[q], $MachinePrecision], 4.5e-58], N[(N[(2.0 * N[Max[p, r], $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision], (-N[Abs[q], $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if q < 4.5000000000000003e-58

   1. Initial program 23.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. sub-negate-rev N/A

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

      5. distribute-lft-neg-out N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. lift-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 N/A

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

   3. Applied rewrites 21.4%

      \[\leadsto \color{blue}{\left(\left(\sqrt{\mathsf{fma}\left(q \cdot 4, q, \left(r - p\right) \cdot \left(r - p\right)\right)} - \left|p\right|\right) - \left|r\right|\right) \cdot -0.5} \]
   4. Step-by-step derivation

      1. lift-fabs.f64 N/A

         \[\leadsto \left(\left(\sqrt{\mathsf{fma}\left(q \cdot 4, q, \left(r - p\right) \cdot \left(r - p\right)\right)} - \left|p\right|\right) - \color{blue}{\left|r\right|}\right) \cdot \frac{-1}{2} \]

      2. neg-fabs N/A

         \[\leadsto \left(\left(\sqrt{\mathsf{fma}\left(q \cdot 4, q, \left(r - p\right) \cdot \left(r - p\right)\right)} - \left|p\right|\right) - \color{blue}{\left|\mathsf{neg}\left(r\right)\right|}\right) \cdot \frac{-1}{2} \]

      3. lift-neg.f64 N/A

         \[\leadsto \left(\left(\sqrt{\mathsf{fma}\left(q \cdot 4, q, \left(r - p\right) \cdot \left(r - p\right)\right)} - \left|p\right|\right) - \left|\color{blue}{-r}\right|\right) \cdot \frac{-1}{2} \]

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

         \[\leadsto \left(\left(\sqrt{\mathsf{fma}\left(q \cdot 4, q, \left(r - p\right) \cdot \left(r - p\right)\right)} - \left|p\right|\right) - \color{blue}{\sqrt{\left(-r\right) \cdot \left(-r\right)}}\right) \cdot \frac{-1}{2} \]

      5. sqrt-unprod N/A

         \[\leadsto \left(\left(\sqrt{\mathsf{fma}\left(q \cdot 4, q, \left(r - p\right) \cdot \left(r - p\right)\right)} - \left|p\right|\right) - \color{blue}{\sqrt{-r} \cdot \sqrt{-r}}\right) \cdot \frac{-1}{2} \]

      6. rem-square-sqrt 19.0%

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

      7. unpow1 N/A

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

      8. pow-to-exp N/A

         \[\leadsto \left(\left(\sqrt{\mathsf{fma}\left(q \cdot 4, q, \left(r - p\right) \cdot \left(r - p\right)\right)} - \left|p\right|\right) - \color{blue}{e^{\log \left(-r\right) \cdot 1}}\right) \cdot \frac{-1}{2} \]

      9. lower-unsound-exp.f64 N/A

         \[\leadsto \left(\left(\sqrt{\mathsf{fma}\left(q \cdot 4, q, \left(r - p\right) \cdot \left(r - p\right)\right)} - \left|p\right|\right) - \color{blue}{e^{\log \left(-r\right) \cdot 1}}\right) \cdot \frac{-1}{2} \]

      10. lower-unsound-*.f64 N/A

          \[\leadsto \left(\left(\sqrt{\mathsf{fma}\left(q \cdot 4, q, \left(r - p\right) \cdot \left(r - p\right)\right)} - \left|p\right|\right) - e^{\color{blue}{\log \left(-r\right) \cdot 1}}\right) \cdot \frac{-1}{2} \]

      11. lower-unsound-log.f64 8.9%

          \[\leadsto \left(\left(\sqrt{\mathsf{fma}\left(q \cdot 4, q, \left(r - p\right) \cdot \left(r - p\right)\right)} - \left|p\right|\right) - e^{\color{blue}{\log \left(-r\right)} \cdot 1}\right) \cdot -0.5 \]

   5. Applied rewrites 8.9%

      \[\leadsto \left(\left(\sqrt{\mathsf{fma}\left(q \cdot 4, q, \left(r - p\right) \cdot \left(r - p\right)\right)} - \left|p\right|\right) - \color{blue}{e^{\log \left(-r\right) \cdot 1}}\right) \cdot -0.5 \]

   6. Taylor expanded in r around inf

      \[\leadsto \color{blue}{\left(2 \cdot r\right)} \cdot -0.5 \]
   7. Step-by-step derivation

      1. lower-*.f64 9.2%

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

   8. Applied rewrites 9.2%

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

   if 4.5000000000000003e-58 < q

   1. Initial program 23.8%

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

   2. Taylor expanded in q around inf

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

      1. lower-*.f64 19.6%

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

   4. Applied rewrites 19.6%

      \[\leadsto \color{blue}{-1 \cdot q} \]
   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 19.6%

         \[\leadsto -q \]

   6. Applied rewrites 19.6%

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

Alternative 6: 35.6% accurate, 18.0× speedup

Math

\[-\left|q\right| \]

FPCore

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

C

double code(double p, double r, double q) {
	return -fabs(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 = -abs(q)
end function

Java

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

Python

def code(p, r, q):
	return -math.fabs(q)

Julia

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

MATLAB

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

Wolfram

code[p_, r_, q_] := (-N[Abs[q], $MachinePrecision])

Derivation

1. Initial program 23.8%

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

2. Taylor expanded in q around inf

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

   1. lower-*.f64 19.6%

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

4. Applied rewrites 19.6%

   \[\leadsto \color{blue}{-1 \cdot q} \]
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 19.6%

      \[\leadsto -q \]

6. Applied rewrites 19.6%

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

Reproduce

herbie shell --seed 2025188 
(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)))))))