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

Percentage Accurate: 44.9% → 81.4%

Time: 4.0s

Alternatives: 11

Speedup: 2.4×

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: 44.9% 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, 0.6× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(p, r\right) - \mathsf{min}\left(p, r\right)\\ t_1 := \left|\mathsf{min}\left(p, r\right)\right|\\ t_2 := \left|\mathsf{max}\left(p, r\right)\right|\\ t_3 := 4 \cdot {\left(\left|q\right|\right)}^{2}\\ t_4 := t\_1 + t\_2\\ t_5 := \mathsf{fma}\left(-0.5, \mathsf{min}\left(p, r\right), 0.5 \cdot \left(\mathsf{max}\left(p, r\right) + t\_4\right)\right)\\ \mathbf{if}\;t\_3 \leq 10^{+66}:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;t\_3 \leq 4 \cdot 10^{+174}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(\left|q\right| \cdot 4, \left|q\right|, t\_0 \cdot t\_0\right)} + \left(t\_2 + t\_1\right)\right) \cdot 0.5\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{+249}:\\ \;\;\;\;t\_5\\ \mathbf{else}:\\ \;\;\;\;\left|q\right| \cdot \left(1 + 0.5 \cdot \frac{t\_4}{\left|q\right|}\right)\\ \end{array} \]

FPCore

(FPCore (p r q)
  :precision binary64
  (let* ((t_0 (- (fmax p r) (fmin p r)))
       (t_1 (fabs (fmin p r)))
       (t_2 (fabs (fmax p r)))
       (t_3 (* 4.0 (pow (fabs q) 2.0)))
       (t_4 (+ t_1 t_2))
       (t_5 (fma -0.5 (fmin p r) (* 0.5 (+ (fmax p r) t_4)))))
  (if (<= t_3 1e+66)
    t_5
    (if (<= t_3 4e+174)
      (*
       (+
        (sqrt (fma (* (fabs q) 4.0) (fabs q) (* t_0 t_0)))
        (+ t_2 t_1))
       0.5)
      (if (<= t_3 2e+249)
        t_5
        (* (fabs q) (+ 1.0 (* 0.5 (/ t_4 (fabs q))))))))))

C

double code(double p, double r, double q) {
	double t_0 = fmax(p, r) - fmin(p, r);
	double t_1 = fabs(fmin(p, r));
	double t_2 = fabs(fmax(p, r));
	double t_3 = 4.0 * pow(fabs(q), 2.0);
	double t_4 = t_1 + t_2;
	double t_5 = fma(-0.5, fmin(p, r), (0.5 * (fmax(p, r) + t_4)));
	double tmp;
	if (t_3 <= 1e+66) {
		tmp = t_5;
	} else if (t_3 <= 4e+174) {
		tmp = (sqrt(fma((fabs(q) * 4.0), fabs(q), (t_0 * t_0))) + (t_2 + t_1)) * 0.5;
	} else if (t_3 <= 2e+249) {
		tmp = t_5;
	} else {
		tmp = fabs(q) * (1.0 + (0.5 * (t_4 / fabs(q))));
	}
	return tmp;
}

Julia

function code(p, r, q)
	t_0 = Float64(fmax(p, r) - fmin(p, r))
	t_1 = abs(fmin(p, r))
	t_2 = abs(fmax(p, r))
	t_3 = Float64(4.0 * (abs(q) ^ 2.0))
	t_4 = Float64(t_1 + t_2)
	t_5 = fma(-0.5, fmin(p, r), Float64(0.5 * Float64(fmax(p, r) + t_4)))
	tmp = 0.0
	if (t_3 <= 1e+66)
		tmp = t_5;
	elseif (t_3 <= 4e+174)
		tmp = Float64(Float64(sqrt(fma(Float64(abs(q) * 4.0), abs(q), Float64(t_0 * t_0))) + Float64(t_2 + t_1)) * 0.5);
	elseif (t_3 <= 2e+249)
		tmp = t_5;
	else
		tmp = Float64(abs(q) * Float64(1.0 + Float64(0.5 * Float64(t_4 / abs(q)))));
	end
	return tmp
end

Wolfram

code[p_, r_, q_] := Block[{t$95$0 = N[(N[Max[p, r], $MachinePrecision] - N[Min[p, r], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[Min[p, r], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Abs[N[Max[p, r], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(4.0 * N[Power[N[Abs[q], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$1 + t$95$2), $MachinePrecision]}, Block[{t$95$5 = N[(-0.5 * N[Min[p, r], $MachinePrecision] + N[(0.5 * N[(N[Max[p, r], $MachinePrecision] + t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, 1e+66], t$95$5, If[LessEqual[t$95$3, 4e+174], N[(N[(N[Sqrt[N[(N[(N[Abs[q], $MachinePrecision] * 4.0), $MachinePrecision] * N[Abs[q], $MachinePrecision] + N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(t$95$2 + t$95$1), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[t$95$3, 2e+249], t$95$5, N[(N[Abs[q], $MachinePrecision] * N[(1.0 + N[(0.5 * N[(t$95$4 / N[Abs[q], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal 4 binary64) (pow.f64 q #s(literal 2 binary64))) < 9.9999999999999995e65 or 4.0000000000000003e174 < (*.f64 #s(literal 4 binary64) (pow.f64 q #s(literal 2 binary64))) < 1.9999999999999998e249

   1. Initial program 44.9%

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

   2. Taylor expanded in p around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-fabs.f64 N/A

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

      9. lower-fabs.f64 30.5%

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

   4. Applied rewrites 30.5%

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

   5. Taylor expanded in p around 0

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

      1. metadata-eval N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. metadata-eval N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-fabs.f64 N/A

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

      8. lower-fabs.f64 35.5%

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

   7. Applied rewrites 35.5%

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

   if 9.9999999999999995e65 < (*.f64 #s(literal 4 binary64) (pow.f64 q #s(literal 2 binary64))) < 4.0000000000000003e174

   1. Initial program 44.9%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
   2. 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. lower-*.f64 44.9%

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

   3. Applied rewrites 44.9%

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

   if 1.9999999999999998e249 < (*.f64 #s(literal 4 binary64) (pow.f64 q #s(literal 2 binary64)))

   1. Initial program 44.9%

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

   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. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-fabs.f64 N/A

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

      7. lower-fabs.f64 26.5%

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

   4. Applied rewrites 26.5%

      \[\leadsto \color{blue}{q \cdot \left(1 + 0.5 \cdot \frac{\left|p\right| + \left|r\right|}{q}\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 81.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := \left|\mathsf{min}\left(p, r\right)\right|\\ t_1 := \left|\mathsf{max}\left(p, r\right)\right|\\ t_2 := 4 \cdot {\left(\left|q\right|\right)}^{2}\\ t_3 := t\_0 + t\_1\\ t_4 := \mathsf{fma}\left(-0.5, \mathsf{min}\left(p, r\right), 0.5 \cdot \left(\mathsf{max}\left(p, r\right) + t\_3\right)\right)\\ \mathbf{if}\;t\_2 \leq 5 \cdot 10^{+67}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{+174}:\\ \;\;\;\;\left(\left(\left|q\right| + \left|q\right|\right) + \left(t\_1 + t\_0\right)\right) \cdot 0.5\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+249}:\\ \;\;\;\;t\_4\\ \mathbf{else}:\\ \;\;\;\;\left|q\right| \cdot \left(1 + 0.5 \cdot \frac{t\_3}{\left|q\right|}\right)\\ \end{array} \]

FPCore

(FPCore (p r q)
  :precision binary64
  (let* ((t_0 (fabs (fmin p r)))
       (t_1 (fabs (fmax p r)))
       (t_2 (* 4.0 (pow (fabs q) 2.0)))
       (t_3 (+ t_0 t_1))
       (t_4 (fma -0.5 (fmin p r) (* 0.5 (+ (fmax p r) t_3)))))
  (if (<= t_2 5e+67)
    t_4
    (if (<= t_2 4e+174)
      (* (+ (+ (fabs q) (fabs q)) (+ t_1 t_0)) 0.5)
      (if (<= t_2 2e+249)
        t_4
        (* (fabs q) (+ 1.0 (* 0.5 (/ t_3 (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 t_2 = 4.0 * pow(fabs(q), 2.0);
	double t_3 = t_0 + t_1;
	double t_4 = fma(-0.5, fmin(p, r), (0.5 * (fmax(p, r) + t_3)));
	double tmp;
	if (t_2 <= 5e+67) {
		tmp = t_4;
	} else if (t_2 <= 4e+174) {
		tmp = ((fabs(q) + fabs(q)) + (t_1 + t_0)) * 0.5;
	} else if (t_2 <= 2e+249) {
		tmp = t_4;
	} else {
		tmp = fabs(q) * (1.0 + (0.5 * (t_3 / fabs(q))));
	}
	return tmp;
}

Julia

function code(p, r, q)
	t_0 = abs(fmin(p, r))
	t_1 = abs(fmax(p, r))
	t_2 = Float64(4.0 * (abs(q) ^ 2.0))
	t_3 = Float64(t_0 + t_1)
	t_4 = fma(-0.5, fmin(p, r), Float64(0.5 * Float64(fmax(p, r) + t_3)))
	tmp = 0.0
	if (t_2 <= 5e+67)
		tmp = t_4;
	elseif (t_2 <= 4e+174)
		tmp = Float64(Float64(Float64(abs(q) + abs(q)) + Float64(t_1 + t_0)) * 0.5);
	elseif (t_2 <= 2e+249)
		tmp = t_4;
	else
		tmp = Float64(abs(q) * Float64(1.0 + Float64(0.5 * Float64(t_3 / 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]}, Block[{t$95$2 = N[(4.0 * N[Power[N[Abs[q], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$0 + t$95$1), $MachinePrecision]}, Block[{t$95$4 = N[(-0.5 * N[Min[p, r], $MachinePrecision] + N[(0.5 * N[(N[Max[p, r], $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 5e+67], t$95$4, If[LessEqual[t$95$2, 4e+174], N[(N[(N[(N[Abs[q], $MachinePrecision] + N[Abs[q], $MachinePrecision]), $MachinePrecision] + N[(t$95$1 + t$95$0), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[t$95$2, 2e+249], t$95$4, N[(N[Abs[q], $MachinePrecision] * N[(1.0 + N[(0.5 * N[(t$95$3 / N[Abs[q], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal 4 binary64) (pow.f64 q #s(literal 2 binary64))) < 4.9999999999999998e67 or 4.0000000000000003e174 < (*.f64 #s(literal 4 binary64) (pow.f64 q #s(literal 2 binary64))) < 1.9999999999999998e249

   1. Initial program 44.9%

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

   2. Taylor expanded in p around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-fabs.f64 N/A

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

      9. lower-fabs.f64 30.5%

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

   4. Applied rewrites 30.5%

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

   5. Taylor expanded in p around 0

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

      1. metadata-eval N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. metadata-eval N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-fabs.f64 N/A

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

      8. lower-fabs.f64 35.5%

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

   7. Applied rewrites 35.5%

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

   if 4.9999999999999998e67 < (*.f64 #s(literal 4 binary64) (pow.f64 q #s(literal 2 binary64))) < 4.0000000000000003e174

   1. Initial program 44.9%

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

   2. Taylor expanded in q around inf

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

      1. lower-*.f64 29.0%

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

   4. Applied rewrites 29.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. count-2-rev N/A

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

      10. lower-+.f64 N/A

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

      11. lower-+.f64 N/A

          \[\leadsto \left(\left(q + \color{blue}{q}\right) + \mathsf{Rewrite=>}\left(lift-+.f64, \left(\left|p\right| + \left|r\right|\right)\right)\right) \cdot \frac{1}{2} \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\left(q + \color{blue}{q}\right) + \mathsf{Rewrite=>}\left(+-commutative, \left(\left|r\right| + \left|p\right|\right)\right)\right) \cdot \frac{1}{2} \]

      13. lower-+.f64 N/A

          \[\leadsto \left(\left(q + \color{blue}{q}\right) + \mathsf{Rewrite=>}\left(lower-+.f64, \left(\left|r\right| + \left|p\right|\right)\right)\right) \cdot \frac{1}{2} \]

      14. lower-+.f64 N/A

          \[\leadsto \left(\left(q + \color{blue}{q}\right) + \left(\left|r\right| + \left|p\right|\right)\right) \cdot \mathsf{Rewrite=>}\left(metadata-eval, \frac{1}{2}\right) \]

   6. Applied rewrites 29.0%

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

   if 1.9999999999999998e249 < (*.f64 #s(literal 4 binary64) (pow.f64 q #s(literal 2 binary64)))

   1. Initial program 44.9%

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

   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. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-fabs.f64 N/A

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

      7. lower-fabs.f64 26.5%

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

   4. Applied rewrites 26.5%

      \[\leadsto \color{blue}{q \cdot \left(1 + 0.5 \cdot \frac{\left|p\right| + \left|r\right|}{q}\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 81.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} t_0 := \left|\mathsf{min}\left(p, r\right)\right|\\ t_1 := \left|\mathsf{max}\left(p, r\right)\right|\\ t_2 := \left(\left(\left|q\right| + \left|q\right|\right) + \left(t\_1 + t\_0\right)\right) \cdot 0.5\\ t_3 := \mathsf{fma}\left(-0.5, \mathsf{min}\left(p, r\right), 0.5 \cdot \left(\mathsf{max}\left(p, r\right) + \left(t\_0 + t\_1\right)\right)\right)\\ \mathbf{if}\;\left|q\right| \leq 4.9 \cdot 10^{+42}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;\left|q\right| \leq 8 \cdot 10^{+86}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\left|q\right| \leq 1.16 \cdot 10^{+127}:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (p r q)
  :precision binary64
  (let* ((t_0 (fabs (fmin p r)))
       (t_1 (fabs (fmax p r)))
       (t_2 (* (+ (+ (fabs q) (fabs q)) (+ t_1 t_0)) 0.5))
       (t_3 (fma -0.5 (fmin p r) (* 0.5 (+ (fmax p r) (+ t_0 t_1))))))
  (if (<= (fabs q) 4.9e+42)
    t_3
    (if (<= (fabs q) 8e+86)
      t_2
      (if (<= (fabs q) 1.16e+127) t_3 t_2)))))

C

double code(double p, double r, double q) {
	double t_0 = fabs(fmin(p, r));
	double t_1 = fabs(fmax(p, r));
	double t_2 = ((fabs(q) + fabs(q)) + (t_1 + t_0)) * 0.5;
	double t_3 = fma(-0.5, fmin(p, r), (0.5 * (fmax(p, r) + (t_0 + t_1))));
	double tmp;
	if (fabs(q) <= 4.9e+42) {
		tmp = t_3;
	} else if (fabs(q) <= 8e+86) {
		tmp = t_2;
	} else if (fabs(q) <= 1.16e+127) {
		tmp = t_3;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(p, r, q)
	t_0 = abs(fmin(p, r))
	t_1 = abs(fmax(p, r))
	t_2 = Float64(Float64(Float64(abs(q) + abs(q)) + Float64(t_1 + t_0)) * 0.5)
	t_3 = fma(-0.5, fmin(p, r), Float64(0.5 * Float64(fmax(p, r) + Float64(t_0 + t_1))))
	tmp = 0.0
	if (abs(q) <= 4.9e+42)
		tmp = t_3;
	elseif (abs(q) <= 8e+86)
		tmp = t_2;
	elseif (abs(q) <= 1.16e+127)
		tmp = t_3;
	else
		tmp = t_2;
	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]}, Block[{t$95$2 = N[(N[(N[(N[Abs[q], $MachinePrecision] + N[Abs[q], $MachinePrecision]), $MachinePrecision] + N[(t$95$1 + t$95$0), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$3 = N[(-0.5 * N[Min[p, r], $MachinePrecision] + N[(0.5 * N[(N[Max[p, r], $MachinePrecision] + N[(t$95$0 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[q], $MachinePrecision], 4.9e+42], t$95$3, If[LessEqual[N[Abs[q], $MachinePrecision], 8e+86], t$95$2, If[LessEqual[N[Abs[q], $MachinePrecision], 1.16e+127], t$95$3, t$95$2]]]]]]]

Derivation

1. Split input into 2 regimes

2. if q < 4.9000000000000002e42 or 8.0000000000000001e86 < q < 1.1599999999999999e127

   1. Initial program 44.9%

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

   2. Taylor expanded in p around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-fabs.f64 N/A

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

      9. lower-fabs.f64 30.5%

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

   4. Applied rewrites 30.5%

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

   5. Taylor expanded in p around 0

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

      1. metadata-eval N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. metadata-eval N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-fabs.f64 N/A

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

      8. lower-fabs.f64 35.5%

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

   7. Applied rewrites 35.5%

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

   if 4.9000000000000002e42 < q < 8.0000000000000001e86 or 1.1599999999999999e127 < q

   1. Initial program 44.9%

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

   2. Taylor expanded in q around inf

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

      1. lower-*.f64 29.0%

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

   4. Applied rewrites 29.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. count-2-rev N/A

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

      10. lower-+.f64 N/A

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

      11. lower-+.f64 N/A

          \[\leadsto \left(\left(q + \color{blue}{q}\right) + \mathsf{Rewrite=>}\left(lift-+.f64, \left(\left|p\right| + \left|r\right|\right)\right)\right) \cdot \frac{1}{2} \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\left(q + \color{blue}{q}\right) + \mathsf{Rewrite=>}\left(+-commutative, \left(\left|r\right| + \left|p\right|\right)\right)\right) \cdot \frac{1}{2} \]

      13. lower-+.f64 N/A

          \[\leadsto \left(\left(q + \color{blue}{q}\right) + \mathsf{Rewrite=>}\left(lower-+.f64, \left(\left|r\right| + \left|p\right|\right)\right)\right) \cdot \frac{1}{2} \]

      14. lower-+.f64 N/A

          \[\leadsto \left(\left(q + \color{blue}{q}\right) + \left(\left|r\right| + \left|p\right|\right)\right) \cdot \mathsf{Rewrite=>}\left(metadata-eval, \frac{1}{2}\right) \]

   6. Applied rewrites 29.0%

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

Alternative 4: 81.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} t_0 := \left|\mathsf{max}\left(p, r\right)\right|\\ t_1 := \left|\mathsf{min}\left(p, r\right)\right|\\ t_2 := \left(\left(\left|q\right| + \left|q\right|\right) + \left(t\_0 + t\_1\right)\right) \cdot 0.5\\ t_3 := -0.5 \cdot \left(\mathsf{min}\left(p, r\right) - \left(\left(\mathsf{max}\left(p, r\right) + t\_0\right) + t\_1\right)\right)\\ \mathbf{if}\;\left|q\right| \leq 4.9 \cdot 10^{+42}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;\left|q\right| \leq 8 \cdot 10^{+86}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\left|q\right| \leq 1.16 \cdot 10^{+127}:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (p r q)
  :precision binary64
  (let* ((t_0 (fabs (fmax p r)))
       (t_1 (fabs (fmin p r)))
       (t_2 (* (+ (+ (fabs q) (fabs q)) (+ t_0 t_1)) 0.5))
       (t_3 (* -0.5 (- (fmin p r) (+ (+ (fmax p r) t_0) t_1)))))
  (if (<= (fabs q) 4.9e+42)
    t_3
    (if (<= (fabs q) 8e+86)
      t_2
      (if (<= (fabs q) 1.16e+127) t_3 t_2)))))

C

double code(double p, double r, double q) {
	double t_0 = fabs(fmax(p, r));
	double t_1 = fabs(fmin(p, r));
	double t_2 = ((fabs(q) + fabs(q)) + (t_0 + t_1)) * 0.5;
	double t_3 = -0.5 * (fmin(p, r) - ((fmax(p, r) + t_0) + t_1));
	double tmp;
	if (fabs(q) <= 4.9e+42) {
		tmp = t_3;
	} else if (fabs(q) <= 8e+86) {
		tmp = t_2;
	} else if (fabs(q) <= 1.16e+127) {
		tmp = t_3;
	} else {
		tmp = t_2;
	}
	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) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = abs(fmax(p, r))
    t_1 = abs(fmin(p, r))
    t_2 = ((abs(q) + abs(q)) + (t_0 + t_1)) * 0.5d0
    t_3 = (-0.5d0) * (fmin(p, r) - ((fmax(p, r) + t_0) + t_1))
    if (abs(q) <= 4.9d+42) then
        tmp = t_3
    else if (abs(q) <= 8d+86) then
        tmp = t_2
    else if (abs(q) <= 1.16d+127) then
        tmp = t_3
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double p, double r, double q) {
	double t_0 = Math.abs(fmax(p, r));
	double t_1 = Math.abs(fmin(p, r));
	double t_2 = ((Math.abs(q) + Math.abs(q)) + (t_0 + t_1)) * 0.5;
	double t_3 = -0.5 * (fmin(p, r) - ((fmax(p, r) + t_0) + t_1));
	double tmp;
	if (Math.abs(q) <= 4.9e+42) {
		tmp = t_3;
	} else if (Math.abs(q) <= 8e+86) {
		tmp = t_2;
	} else if (Math.abs(q) <= 1.16e+127) {
		tmp = t_3;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(p, r, q):
	t_0 = math.fabs(fmax(p, r))
	t_1 = math.fabs(fmin(p, r))
	t_2 = ((math.fabs(q) + math.fabs(q)) + (t_0 + t_1)) * 0.5
	t_3 = -0.5 * (fmin(p, r) - ((fmax(p, r) + t_0) + t_1))
	tmp = 0
	if math.fabs(q) <= 4.9e+42:
		tmp = t_3
	elif math.fabs(q) <= 8e+86:
		tmp = t_2
	elif math.fabs(q) <= 1.16e+127:
		tmp = t_3
	else:
		tmp = t_2
	return tmp

Julia

function code(p, r, q)
	t_0 = abs(fmax(p, r))
	t_1 = abs(fmin(p, r))
	t_2 = Float64(Float64(Float64(abs(q) + abs(q)) + Float64(t_0 + t_1)) * 0.5)
	t_3 = Float64(-0.5 * Float64(fmin(p, r) - Float64(Float64(fmax(p, r) + t_0) + t_1)))
	tmp = 0.0
	if (abs(q) <= 4.9e+42)
		tmp = t_3;
	elseif (abs(q) <= 8e+86)
		tmp = t_2;
	elseif (abs(q) <= 1.16e+127)
		tmp = t_3;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(p, r, q)
	t_0 = abs(max(p, r));
	t_1 = abs(min(p, r));
	t_2 = ((abs(q) + abs(q)) + (t_0 + t_1)) * 0.5;
	t_3 = -0.5 * (min(p, r) - ((max(p, r) + t_0) + t_1));
	tmp = 0.0;
	if (abs(q) <= 4.9e+42)
		tmp = t_3;
	elseif (abs(q) <= 8e+86)
		tmp = t_2;
	elseif (abs(q) <= 1.16e+127)
		tmp = t_3;
	else
		tmp = t_2;
	end
	tmp_2 = 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]}, Block[{t$95$2 = N[(N[(N[(N[Abs[q], $MachinePrecision] + N[Abs[q], $MachinePrecision]), $MachinePrecision] + N[(t$95$0 + t$95$1), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$3 = N[(-0.5 * N[(N[Min[p, r], $MachinePrecision] - N[(N[(N[Max[p, r], $MachinePrecision] + t$95$0), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[q], $MachinePrecision], 4.9e+42], t$95$3, If[LessEqual[N[Abs[q], $MachinePrecision], 8e+86], t$95$2, If[LessEqual[N[Abs[q], $MachinePrecision], 1.16e+127], t$95$3, t$95$2]]]]]]]

Derivation

1. Split input into 2 regimes

2. if q < 4.9000000000000002e42 or 8.0000000000000001e86 < q < 1.1599999999999999e127

   1. Initial program 44.9%

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

   2. Taylor expanded in p around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-fabs.f64 N/A

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

      9. lower-fabs.f64 30.5%

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

   4. Applied rewrites 30.5%

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

   5. Taylor expanded in p around 0

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

      1. metadata-eval N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. metadata-eval N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-fabs.f64 N/A

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

      8. lower-fabs.f64 35.5%

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

   7. Applied rewrites 35.5%

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

      1. metadata-eval N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

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

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. distribute-lft-out-- N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 35.5%

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

      10. lift-+.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. associate-+r+ N/A

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

      14. lower-+.f64 N/A

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

      15. lower-+.f64 35.5%

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

   9. Applied rewrites 35.5%

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

   if 4.9000000000000002e42 < q < 8.0000000000000001e86 or 1.1599999999999999e127 < q

   1. Initial program 44.9%

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

   2. Taylor expanded in q around inf

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

      1. lower-*.f64 29.0%

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

   4. Applied rewrites 29.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. count-2-rev N/A

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

      10. lower-+.f64 N/A

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

      11. lower-+.f64 N/A

          \[\leadsto \left(\left(q + \color{blue}{q}\right) + \mathsf{Rewrite=>}\left(lift-+.f64, \left(\left|p\right| + \left|r\right|\right)\right)\right) \cdot \frac{1}{2} \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\left(q + \color{blue}{q}\right) + \mathsf{Rewrite=>}\left(+-commutative, \left(\left|r\right| + \left|p\right|\right)\right)\right) \cdot \frac{1}{2} \]

      13. lower-+.f64 N/A

          \[\leadsto \left(\left(q + \color{blue}{q}\right) + \mathsf{Rewrite=>}\left(lower-+.f64, \left(\left|r\right| + \left|p\right|\right)\right)\right) \cdot \frac{1}{2} \]

      14. lower-+.f64 N/A

          \[\leadsto \left(\left(q + \color{blue}{q}\right) + \left(\left|r\right| + \left|p\right|\right)\right) \cdot \mathsf{Rewrite=>}\left(metadata-eval, \frac{1}{2}\right) \]

   6. Applied rewrites 29.0%

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

Alternative 5: 80.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|q\right| \leq 1.16 \cdot 10^{+127}:\\ \;\;\;\;-0.5 \cdot \left(\mathsf{min}\left(p, r\right) - \left(\left(\mathsf{max}\left(p, r\right) + \left|\mathsf{max}\left(p, r\right)\right|\right) + \left|\mathsf{min}\left(p, r\right)\right|\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left|q\right|\\ \end{array} \]

FPCore

(FPCore (p r q)
  :precision binary64
  (if (<= (fabs q) 1.16e+127)
  (*
   -0.5
   (-
    (fmin p r)
    (+ (+ (fmax p r) (fabs (fmax p r))) (fabs (fmin p r)))))
  (fabs q)))

C

double code(double p, double r, double q) {
	double tmp;
	if (fabs(q) <= 1.16e+127) {
		tmp = -0.5 * (fmin(p, r) - ((fmax(p, r) + fabs(fmax(p, r))) + fabs(fmin(p, r))));
	} 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) <= 1.16d+127) then
        tmp = (-0.5d0) * (fmin(p, r) - ((fmax(p, r) + abs(fmax(p, r))) + abs(fmin(p, r))))
    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) <= 1.16e+127) {
		tmp = -0.5 * (fmin(p, r) - ((fmax(p, r) + Math.abs(fmax(p, r))) + Math.abs(fmin(p, r))));
	} else {
		tmp = Math.abs(q);
	}
	return tmp;
}

Python

def code(p, r, q):
	tmp = 0
	if math.fabs(q) <= 1.16e+127:
		tmp = -0.5 * (fmin(p, r) - ((fmax(p, r) + math.fabs(fmax(p, r))) + math.fabs(fmin(p, r))))
	else:
		tmp = math.fabs(q)
	return tmp

Julia

function code(p, r, q)
	tmp = 0.0
	if (abs(q) <= 1.16e+127)
		tmp = Float64(-0.5 * Float64(fmin(p, r) - Float64(Float64(fmax(p, r) + abs(fmax(p, r))) + abs(fmin(p, r)))));
	else
		tmp = abs(q);
	end
	return tmp
end

MATLAB

function tmp_2 = code(p, r, q)
	tmp = 0.0;
	if (abs(q) <= 1.16e+127)
		tmp = -0.5 * (min(p, r) - ((max(p, r) + abs(max(p, r))) + abs(min(p, r))));
	else
		tmp = abs(q);
	end
	tmp_2 = tmp;
end

Wolfram

code[p_, r_, q_] := If[LessEqual[N[Abs[q], $MachinePrecision], 1.16e+127], N[(-0.5 * N[(N[Min[p, r], $MachinePrecision] - N[(N[(N[Max[p, r], $MachinePrecision] + N[Abs[N[Max[p, r], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[Abs[N[Min[p, r], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Abs[q], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if q < 1.1599999999999999e127

   1. Initial program 44.9%

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

   2. Taylor expanded in p around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-fabs.f64 N/A

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

      9. lower-fabs.f64 30.5%

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

   4. Applied rewrites 30.5%

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

   5. Taylor expanded in p around 0

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

      1. metadata-eval N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. metadata-eval N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-fabs.f64 N/A

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

      8. lower-fabs.f64 35.5%

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

   7. Applied rewrites 35.5%

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

      1. metadata-eval N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

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

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. distribute-lft-out-- N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 35.5%

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

      10. lift-+.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. associate-+r+ N/A

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

      14. lower-+.f64 N/A

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

      15. lower-+.f64 35.5%

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

   9. Applied rewrites 35.5%

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

   if 1.1599999999999999e127 < q

   1. Initial program 44.9%

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

   2. Taylor expanded in p around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-fabs.f64 N/A

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

      9. lower-fabs.f64 30.5%

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

   4. Applied rewrites 30.5%

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

   5. Taylor expanded in p around 0

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

      1. metadata-eval N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. metadata-eval N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-fabs.f64 N/A

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

      8. lower-fabs.f64 35.5%

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

   7. Applied rewrites 35.5%

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

      1. metadata-eval N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

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

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. distribute-lft-out-- N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 35.5%

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

      10. lift-+.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. associate-+r+ N/A

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

      14. lower-+.f64 N/A

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

      15. lower-+.f64 35.5%

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

   9. Applied rewrites 35.5%

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

   10. Taylor expanded in q around inf

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

   12. Applied rewrites 18.3%

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

Alternative 6: 62.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} t_0 := \left|\mathsf{min}\left(p, r\right)\right| + \left|\mathsf{max}\left(p, r\right)\right|\\ \mathbf{if}\;\mathsf{min}\left(p, r\right) \leq -1.6 \cdot 10^{+92}:\\ \;\;\;\;-0.5 \cdot \left(\mathsf{min}\left(p, r\right) - t\_0\right)\\ \mathbf{elif}\;\mathsf{min}\left(p, r\right) \leq 1.05 \cdot 10^{-195}:\\ \;\;\;\;\left|q\right|\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\mathsf{max}\left(p, r\right) + t\_0\right)\\ \end{array} \]

FPCore

(FPCore (p r q)
  :precision binary64
  (let* ((t_0 (+ (fabs (fmin p r)) (fabs (fmax p r)))))
  (if (<= (fmin p r) -1.6e+92)
    (* -0.5 (- (fmin p r) t_0))
    (if (<= (fmin p r) 1.05e-195)
      (fabs q)
      (* 0.5 (+ (fmax p r) t_0))))))

C

double code(double p, double r, double q) {
	double t_0 = fabs(fmin(p, r)) + fabs(fmax(p, r));
	double tmp;
	if (fmin(p, r) <= -1.6e+92) {
		tmp = -0.5 * (fmin(p, r) - t_0);
	} else if (fmin(p, r) <= 1.05e-195) {
		tmp = fabs(q);
	} else {
		tmp = 0.5 * (fmax(p, r) + t_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) :: t_0
    real(8) :: tmp
    t_0 = abs(fmin(p, r)) + abs(fmax(p, r))
    if (fmin(p, r) <= (-1.6d+92)) then
        tmp = (-0.5d0) * (fmin(p, r) - t_0)
    else if (fmin(p, r) <= 1.05d-195) then
        tmp = abs(q)
    else
        tmp = 0.5d0 * (fmax(p, r) + t_0)
    end if
    code = tmp
end function

Java

public static double code(double p, double r, double q) {
	double t_0 = Math.abs(fmin(p, r)) + Math.abs(fmax(p, r));
	double tmp;
	if (fmin(p, r) <= -1.6e+92) {
		tmp = -0.5 * (fmin(p, r) - t_0);
	} else if (fmin(p, r) <= 1.05e-195) {
		tmp = Math.abs(q);
	} else {
		tmp = 0.5 * (fmax(p, r) + t_0);
	}
	return tmp;
}

Python

def code(p, r, q):
	t_0 = math.fabs(fmin(p, r)) + math.fabs(fmax(p, r))
	tmp = 0
	if fmin(p, r) <= -1.6e+92:
		tmp = -0.5 * (fmin(p, r) - t_0)
	elif fmin(p, r) <= 1.05e-195:
		tmp = math.fabs(q)
	else:
		tmp = 0.5 * (fmax(p, r) + t_0)
	return tmp

Julia

function code(p, r, q)
	t_0 = Float64(abs(fmin(p, r)) + abs(fmax(p, r)))
	tmp = 0.0
	if (fmin(p, r) <= -1.6e+92)
		tmp = Float64(-0.5 * Float64(fmin(p, r) - t_0));
	elseif (fmin(p, r) <= 1.05e-195)
		tmp = abs(q);
	else
		tmp = Float64(0.5 * Float64(fmax(p, r) + t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(p, r, q)
	t_0 = abs(min(p, r)) + abs(max(p, r));
	tmp = 0.0;
	if (min(p, r) <= -1.6e+92)
		tmp = -0.5 * (min(p, r) - t_0);
	elseif (min(p, r) <= 1.05e-195)
		tmp = abs(q);
	else
		tmp = 0.5 * (max(p, r) + t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[p_, r_, q_] := Block[{t$95$0 = N[(N[Abs[N[Min[p, r], $MachinePrecision]], $MachinePrecision] + N[Abs[N[Max[p, r], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Min[p, r], $MachinePrecision], -1.6e+92], N[(-0.5 * N[(N[Min[p, r], $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Min[p, r], $MachinePrecision], 1.05e-195], N[Abs[q], $MachinePrecision], N[(0.5 * N[(N[Max[p, r], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if p < -1.6000000000000001e92

   1. Initial program 44.9%

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

   2. Taylor expanded in p around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-fabs.f64 N/A

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

      9. lower-fabs.f64 30.5%

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

   4. Applied rewrites 30.5%

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

   5. Taylor expanded in p around 0

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

      1. metadata-eval N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. metadata-eval N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-fabs.f64 N/A

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

      8. lower-fabs.f64 35.5%

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

   7. Applied rewrites 35.5%

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

      1. metadata-eval N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

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

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. distribute-lft-out-- N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 35.5%

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

      10. lift-+.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. associate-+r+ N/A

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

      14. lower-+.f64 N/A

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

      15. lower-+.f64 35.5%

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

   9. Applied rewrites 35.5%

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

   10. Taylor expanded in r around 0

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

       1. lower--.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. lower-fabs.f64 N/A

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

       4. lower-fabs.f64 24.8%

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

   12. Applied rewrites 24.8%

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

   if -1.6000000000000001e92 < p < 1.05e-195

   1. Initial program 44.9%

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

   2. Taylor expanded in p around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-fabs.f64 N/A

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

      9. lower-fabs.f64 30.5%

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

   4. Applied rewrites 30.5%

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

   5. Taylor expanded in p around 0

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

      1. metadata-eval N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. metadata-eval N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-fabs.f64 N/A

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

      8. lower-fabs.f64 35.5%

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

   7. Applied rewrites 35.5%

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

      1. metadata-eval N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

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

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. distribute-lft-out-- N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 35.5%

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

      10. lift-+.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. associate-+r+ N/A

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

      14. lower-+.f64 N/A

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

      15. lower-+.f64 35.5%

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

   9. Applied rewrites 35.5%

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

   10. Taylor expanded in q around inf

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

   12. Applied rewrites 18.3%

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

   if 1.05e-195 < p

   1. Initial program 44.9%

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

   2. Taylor expanded in p around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-fabs.f64 N/A

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

      9. lower-fabs.f64 30.5%

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

   4. Applied rewrites 30.5%

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

   5. Taylor expanded in p around 0

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

      1. metadata-eval N/A

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

      2. lower-*.f64 N/A

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

      3. metadata-eval N/A

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

      4. lower-+.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-fabs.f64 N/A

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

      7. lower-fabs.f64 24.5%

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

   7. Applied rewrites 24.5%

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

Alternative 7: 56.2% accurate, 2.4× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|q\right| \leq 1.95 \cdot 10^{-19}:\\ \;\;\;\;-0.5 \cdot \left(\mathsf{min}\left(p, r\right) - \left(\left|\mathsf{min}\left(p, r\right)\right| + \left|\mathsf{max}\left(p, r\right)\right|\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left|q\right|\\ \end{array} \]

FPCore

(FPCore (p r q)
  :precision binary64
  (if (<= (fabs q) 1.95e-19)
  (* -0.5 (- (fmin p r) (+ (fabs (fmin p r)) (fabs (fmax p r)))))
  (fabs q)))

C

double code(double p, double r, double q) {
	double tmp;
	if (fabs(q) <= 1.95e-19) {
		tmp = -0.5 * (fmin(p, r) - (fabs(fmin(p, r)) + fabs(fmax(p, r))));
	} 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) <= 1.95d-19) then
        tmp = (-0.5d0) * (fmin(p, r) - (abs(fmin(p, r)) + abs(fmax(p, r))))
    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) <= 1.95e-19) {
		tmp = -0.5 * (fmin(p, r) - (Math.abs(fmin(p, r)) + Math.abs(fmax(p, r))));
	} else {
		tmp = Math.abs(q);
	}
	return tmp;
}

Python

def code(p, r, q):
	tmp = 0
	if math.fabs(q) <= 1.95e-19:
		tmp = -0.5 * (fmin(p, r) - (math.fabs(fmin(p, r)) + math.fabs(fmax(p, r))))
	else:
		tmp = math.fabs(q)
	return tmp

Julia

function code(p, r, q)
	tmp = 0.0
	if (abs(q) <= 1.95e-19)
		tmp = Float64(-0.5 * Float64(fmin(p, r) - Float64(abs(fmin(p, r)) + abs(fmax(p, r)))));
	else
		tmp = abs(q);
	end
	return tmp
end

MATLAB

function tmp_2 = code(p, r, q)
	tmp = 0.0;
	if (abs(q) <= 1.95e-19)
		tmp = -0.5 * (min(p, r) - (abs(min(p, r)) + abs(max(p, r))));
	else
		tmp = abs(q);
	end
	tmp_2 = tmp;
end

Wolfram

code[p_, r_, q_] := If[LessEqual[N[Abs[q], $MachinePrecision], 1.95e-19], N[(-0.5 * N[(N[Min[p, r], $MachinePrecision] - N[(N[Abs[N[Min[p, r], $MachinePrecision]], $MachinePrecision] + N[Abs[N[Max[p, r], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Abs[q], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if q < 1.95e-19

   1. Initial program 44.9%

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

   2. Taylor expanded in p around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-fabs.f64 N/A

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

      9. lower-fabs.f64 30.5%

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

   4. Applied rewrites 30.5%

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

   5. Taylor expanded in p around 0

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

      1. metadata-eval N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. metadata-eval N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-fabs.f64 N/A

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

      8. lower-fabs.f64 35.5%

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

   7. Applied rewrites 35.5%

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

      1. metadata-eval N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

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

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. distribute-lft-out-- N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 35.5%

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

      10. lift-+.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. associate-+r+ N/A

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

      14. lower-+.f64 N/A

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

      15. lower-+.f64 35.5%

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

   9. Applied rewrites 35.5%

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

   10. Taylor expanded in r around 0

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

       1. lower--.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. lower-fabs.f64 N/A

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

       4. lower-fabs.f64 24.8%

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

   12. Applied rewrites 24.8%

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

   if 1.95e-19 < q

   1. Initial program 44.9%

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

   2. Taylor expanded in p around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-fabs.f64 N/A

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

      9. lower-fabs.f64 30.5%

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

   4. Applied rewrites 30.5%

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

   5. Taylor expanded in p around 0

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

      1. metadata-eval N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. metadata-eval N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-fabs.f64 N/A

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

      8. lower-fabs.f64 35.5%

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

   7. Applied rewrites 35.5%

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

      1. metadata-eval N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

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

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. distribute-lft-out-- N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 35.5%

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

      10. lift-+.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. associate-+r+ N/A

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

      14. lower-+.f64 N/A

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

      15. lower-+.f64 35.5%

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

   9. Applied rewrites 35.5%

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

   10. Taylor expanded in q around inf

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

   12. Applied rewrites 18.3%

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

Alternative 8: 39.0% accurate, 4.4× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|q\right| \leq 1.05 \cdot 10^{-60}:\\ \;\;\;\;0.5 \cdot \left(\left|p\right| + \left|r\right|\right)\\ \mathbf{else}:\\ \;\;\;\;\left|q\right|\\ \end{array} \]

FPCore

(FPCore (p r q)
  :precision binary64
  (if (<= (fabs q) 1.05e-60) (* 0.5 (+ (fabs p) (fabs r))) (fabs q)))

C

double code(double p, double r, double q) {
	double tmp;
	if (fabs(q) <= 1.05e-60) {
		tmp = 0.5 * (fabs(p) + fabs(r));
	} 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) <= 1.05d-60) then
        tmp = 0.5d0 * (abs(p) + abs(r))
    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) <= 1.05e-60) {
		tmp = 0.5 * (Math.abs(p) + Math.abs(r));
	} else {
		tmp = Math.abs(q);
	}
	return tmp;
}

Python

def code(p, r, q):
	tmp = 0
	if math.fabs(q) <= 1.05e-60:
		tmp = 0.5 * (math.fabs(p) + math.fabs(r))
	else:
		tmp = math.fabs(q)
	return tmp

Julia

function code(p, r, q)
	tmp = 0.0
	if (abs(q) <= 1.05e-60)
		tmp = Float64(0.5 * Float64(abs(p) + abs(r)));
	else
		tmp = abs(q);
	end
	return tmp
end

MATLAB

function tmp_2 = code(p, r, q)
	tmp = 0.0;
	if (abs(q) <= 1.05e-60)
		tmp = 0.5 * (abs(p) + abs(r));
	else
		tmp = abs(q);
	end
	tmp_2 = tmp;
end

Wolfram

code[p_, r_, q_] := If[LessEqual[N[Abs[q], $MachinePrecision], 1.05e-60], N[(0.5 * N[(N[Abs[p], $MachinePrecision] + N[Abs[r], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Abs[q], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if q < 1.05e-60

   1. Initial program 44.9%

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

   2. Taylor expanded in p around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-fabs.f64 N/A

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

      9. lower-fabs.f64 30.5%

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

   4. Applied rewrites 30.5%

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

   5. Taylor expanded in p around 0

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

      1. metadata-eval N/A

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

      2. lower-*.f64 N/A

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

      3. metadata-eval N/A

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

      4. lower-+.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-fabs.f64 N/A

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

      7. lower-fabs.f64 24.5%

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

   7. Applied rewrites 24.5%

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

   8. Taylor expanded in r around 0

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

      1. metadata-eval N/A

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

      2. lower-*.f64 N/A

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

      3. metadata-eval N/A

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

      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%

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

   10. Applied rewrites 14.4%

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

   if 1.05e-60 < q

   1. Initial program 44.9%

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

   2. Taylor expanded in p around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-fabs.f64 N/A

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

      9. lower-fabs.f64 30.5%

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

   4. Applied rewrites 30.5%

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

   5. Taylor expanded in p around 0

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

      1. metadata-eval N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. metadata-eval N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-fabs.f64 N/A

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

      8. lower-fabs.f64 35.5%

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

   7. Applied rewrites 35.5%

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

      1. metadata-eval N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

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

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. distribute-lft-out-- N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 35.5%

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

      10. lift-+.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. associate-+r+ N/A

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

      14. lower-+.f64 N/A

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

      15. lower-+.f64 35.5%

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

   9. Applied rewrites 35.5%

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

   10. Taylor expanded in q around inf

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

   12. Applied rewrites 18.3%

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

Alternative 9: 36.6% accurate, 5.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|q\right| \leq 5.6 \cdot 10^{-151}:\\ \;\;\;\;0.5 \cdot \mathsf{max}\left(p, r\right)\\ \mathbf{else}:\\ \;\;\;\;\left|q\right|\\ \end{array} \]

FPCore

(FPCore (p r q)
  :precision binary64
  (if (<= (fabs q) 5.6e-151) (* 0.5 (fmax p r)) (fabs q)))

C

double code(double p, double r, double q) {
	double tmp;
	if (fabs(q) <= 5.6e-151) {
		tmp = 0.5 * fmax(p, r);
	} 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) <= 5.6d-151) then
        tmp = 0.5d0 * fmax(p, r)
    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) <= 5.6e-151) {
		tmp = 0.5 * fmax(p, r);
	} else {
		tmp = Math.abs(q);
	}
	return tmp;
}

Python

def code(p, r, q):
	tmp = 0
	if math.fabs(q) <= 5.6e-151:
		tmp = 0.5 * fmax(p, r)
	else:
		tmp = math.fabs(q)
	return tmp

Julia

function code(p, r, q)
	tmp = 0.0
	if (abs(q) <= 5.6e-151)
		tmp = Float64(0.5 * fmax(p, r));
	else
		tmp = abs(q);
	end
	return tmp
end

MATLAB

function tmp_2 = code(p, r, q)
	tmp = 0.0;
	if (abs(q) <= 5.6e-151)
		tmp = 0.5 * max(p, r);
	else
		tmp = abs(q);
	end
	tmp_2 = tmp;
end

Wolfram

code[p_, r_, q_] := If[LessEqual[N[Abs[q], $MachinePrecision], 5.6e-151], N[(0.5 * N[Max[p, r], $MachinePrecision]), $MachinePrecision], N[Abs[q], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if q < 5.6000000000000002e-151

   1. Initial program 44.9%

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

   2. Taylor expanded in r around inf

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

      1. lower-*.f64 5.2%

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

   4. Applied rewrites 5.2%

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

   if 5.6000000000000002e-151 < q

   1. Initial program 44.9%

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

   2. Taylor expanded in p around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-fabs.f64 N/A

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

      9. lower-fabs.f64 30.5%

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

   4. Applied rewrites 30.5%

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

   5. Taylor expanded in p around 0

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

      1. metadata-eval N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. metadata-eval N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-fabs.f64 N/A

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

      8. lower-fabs.f64 35.5%

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

   7. Applied rewrites 35.5%

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

      1. metadata-eval N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

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

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. distribute-lft-out-- N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 35.5%

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

      10. lift-+.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. associate-+r+ N/A

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

      14. lower-+.f64 N/A

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

      15. lower-+.f64 35.5%

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

   9. Applied rewrites 35.5%

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

   10. Taylor expanded in q around inf

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

   12. Applied rewrites 18.3%

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

Alternative 10: 36.5% accurate, 5.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|q\right| \leq 5.6 \cdot 10^{-151}:\\ \;\;\;\;-0.5 \cdot \mathsf{min}\left(p, r\right)\\ \mathbf{else}:\\ \;\;\;\;\left|q\right|\\ \end{array} \]

FPCore

(FPCore (p r q)
  :precision binary64
  (if (<= (fabs q) 5.6e-151) (* -0.5 (fmin p r)) (fabs q)))

C

double code(double p, double r, double q) {
	double tmp;
	if (fabs(q) <= 5.6e-151) {
		tmp = -0.5 * fmin(p, r);
	} 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) <= 5.6d-151) then
        tmp = (-0.5d0) * fmin(p, r)
    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) <= 5.6e-151) {
		tmp = -0.5 * fmin(p, r);
	} else {
		tmp = Math.abs(q);
	}
	return tmp;
}

Python

def code(p, r, q):
	tmp = 0
	if math.fabs(q) <= 5.6e-151:
		tmp = -0.5 * fmin(p, r)
	else:
		tmp = math.fabs(q)
	return tmp

Julia

function code(p, r, q)
	tmp = 0.0
	if (abs(q) <= 5.6e-151)
		tmp = Float64(-0.5 * fmin(p, r));
	else
		tmp = abs(q);
	end
	return tmp
end

MATLAB

function tmp_2 = code(p, r, q)
	tmp = 0.0;
	if (abs(q) <= 5.6e-151)
		tmp = -0.5 * min(p, r);
	else
		tmp = abs(q);
	end
	tmp_2 = tmp;
end

Wolfram

code[p_, r_, q_] := If[LessEqual[N[Abs[q], $MachinePrecision], 5.6e-151], N[(-0.5 * N[Min[p, r], $MachinePrecision]), $MachinePrecision], N[Abs[q], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if q < 5.6000000000000002e-151

   1. Initial program 44.9%

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

   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 5.6000000000000002e-151 < q

   1. Initial program 44.9%

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

   2. Taylor expanded in p around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-fabs.f64 N/A

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

      9. lower-fabs.f64 30.5%

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

   4. Applied rewrites 30.5%

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

   5. Taylor expanded in p around 0

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

      1. metadata-eval N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. metadata-eval N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-fabs.f64 N/A

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

      8. lower-fabs.f64 35.5%

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

   7. Applied rewrites 35.5%

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

      1. metadata-eval N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

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

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. distribute-lft-out-- N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 35.5%

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

      10. lift-+.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. associate-+r+ N/A

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

      14. lower-+.f64 N/A

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

      15. lower-+.f64 35.5%

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

   9. Applied rewrites 35.5%

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

   10. Taylor expanded in q around inf

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

   12. Applied rewrites 18.3%

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

Alternative 11: 35.1% accurate, 29.6× 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 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 44.9%

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

2. Taylor expanded in p around -inf

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-+.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-+.f64 N/A

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

   7. lower-+.f64 N/A

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

   8. lower-fabs.f64 N/A

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

   9. lower-fabs.f64 30.5%

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

4. Applied rewrites 30.5%

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

5. Taylor expanded in p around 0

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

   1. metadata-eval N/A

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

   2. lower-fma.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. metadata-eval N/A

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

   5. lower-+.f64 N/A

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

   6. lower-+.f64 N/A

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

   7. lower-fabs.f64 N/A

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

   8. lower-fabs.f64 35.5%

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

7. Applied rewrites 35.5%

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

   1. metadata-eval N/A

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

   2. lift-fma.f64 N/A

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

   3. lift-*.f64 N/A

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

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

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

   5. metadata-eval N/A

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

   6. metadata-eval N/A

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

   7. distribute-lft-out-- N/A

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

   8. lower-*.f64 N/A

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

   9. lower--.f64 35.5%

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

   10. lift-+.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. associate-+r+ N/A

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

   14. lower-+.f64 N/A

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

   15. lower-+.f64 35.5%

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

9. Applied rewrites 35.5%

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

10. Taylor expanded in q around inf

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

12. Applied rewrites 18.3%

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

Reproduce

herbie shell --seed 2025210 
(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)))))))