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

Percentage Accurate: 45.4% → 81.7%

Time: 3.2s

Alternatives: 9

Speedup: 1.8×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 45.4% 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: 81.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := \left|\mathsf{max}\left(p, r\right)\right|\\ t_1 := \mathsf{max}\left(p, r\right) - \mathsf{min}\left(p, r\right)\\ t_2 := 4 \cdot {\left(\left|q\right|\right)}^{2}\\ t_3 := \left|\mathsf{min}\left(p, r\right)\right|\\ \mathbf{if}\;t\_2 \leq 10^{+144}:\\ \;\;\;\;0.5 \cdot \left(\left(t\_0 - \left(\mathsf{min}\left(p, r\right) - \mathsf{max}\left(p, r\right)\right)\right) + t\_3\right)\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+266}:\\ \;\;\;\;\left(\left(t\_3 + \sqrt{\mathsf{fma}\left(\left|q\right| \cdot 4, \left|q\right|, t\_1 \cdot t\_1\right)}\right) + t\_0\right) \cdot 0.5\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+307}:\\ \;\;\;\;\left(\mathsf{max}\left(p, r\right) - \left(\left(\mathsf{min}\left(p, r\right) - t\_0\right) - t\_3\right)\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0 + t\_3, 0.5, \left|q\right|\right)\\ \end{array} \]

FPCore

(FPCore (p r q)
 :precision binary64
 (let* ((t_0 (fabs (fmax p r)))
        (t_1 (- (fmax p r) (fmin p r)))
        (t_2 (* 4.0 (pow (fabs q) 2.0)))
        (t_3 (fabs (fmin p r))))
   (if (<= t_2 1e+144)
     (* 0.5 (+ (- t_0 (- (fmin p r) (fmax p r))) t_3))
     (if (<= t_2 5e+266)
       (*
        (+ (+ t_3 (sqrt (fma (* (fabs q) 4.0) (fabs q) (* t_1 t_1)))) t_0)
        0.5)
       (if (<= t_2 2e+307)
         (* (- (fmax p r) (- (- (fmin p r) t_0) t_3)) 0.5)
         (fma (+ t_0 t_3) 0.5 (fabs q)))))))

C

double code(double p, double r, double q) {
	double t_0 = fabs(fmax(p, r));
	double t_1 = fmax(p, r) - fmin(p, r);
	double t_2 = 4.0 * pow(fabs(q), 2.0);
	double t_3 = fabs(fmin(p, r));
	double tmp;
	if (t_2 <= 1e+144) {
		tmp = 0.5 * ((t_0 - (fmin(p, r) - fmax(p, r))) + t_3);
	} else if (t_2 <= 5e+266) {
		tmp = ((t_3 + sqrt(fma((fabs(q) * 4.0), fabs(q), (t_1 * t_1)))) + t_0) * 0.5;
	} else if (t_2 <= 2e+307) {
		tmp = (fmax(p, r) - ((fmin(p, r) - t_0) - t_3)) * 0.5;
	} else {
		tmp = fma((t_0 + t_3), 0.5, fabs(q));
	}
	return tmp;
}

Julia

function code(p, r, q)
	t_0 = abs(fmax(p, r))
	t_1 = Float64(fmax(p, r) - fmin(p, r))
	t_2 = Float64(4.0 * (abs(q) ^ 2.0))
	t_3 = abs(fmin(p, r))
	tmp = 0.0
	if (t_2 <= 1e+144)
		tmp = Float64(0.5 * Float64(Float64(t_0 - Float64(fmin(p, r) - fmax(p, r))) + t_3));
	elseif (t_2 <= 5e+266)
		tmp = Float64(Float64(Float64(t_3 + sqrt(fma(Float64(abs(q) * 4.0), abs(q), Float64(t_1 * t_1)))) + t_0) * 0.5);
	elseif (t_2 <= 2e+307)
		tmp = Float64(Float64(fmax(p, r) - Float64(Float64(fmin(p, r) - t_0) - t_3)) * 0.5);
	else
		tmp = fma(Float64(t_0 + t_3), 0.5, abs(q));
	end
	return tmp
end

Wolfram

code[p_, r_, q_] := Block[{t$95$0 = N[Abs[N[Max[p, r], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Max[p, r], $MachinePrecision] - N[Min[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[Abs[N[Min[p, r], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$2, 1e+144], N[(0.5 * N[(N[(t$95$0 - N[(N[Min[p, r], $MachinePrecision] - N[Max[p, r], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e+266], N[(N[(N[(t$95$3 + N[Sqrt[N[(N[(N[Abs[q], $MachinePrecision] * 4.0), $MachinePrecision] * N[Abs[q], $MachinePrecision] + N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[t$95$2, 2e+307], N[(N[(N[Max[p, r], $MachinePrecision] - N[(N[(N[Min[p, r], $MachinePrecision] - t$95$0), $MachinePrecision] - t$95$3), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(t$95$0 + t$95$3), $MachinePrecision] * 0.5 + N[Abs[q], $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 #s(literal 4 binary64) (pow.f64 q #s(literal 2 binary64))) < 1.00000000000000002e144

   1. Initial program 45.4%

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

   2. Taylor expanded in r around inf

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

   4. Applied rewrites 30.8%

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

   5. Taylor expanded in r around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-fma.f64 N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. lower-fabs.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-fabs.f64 N/A

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

      11. lower-*.f64 36.1%

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

   7. Applied rewrites 36.1%

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lift-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. distribute-rgt-in N/A

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

      8. associate-+l+ N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. metadata-eval N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

   9. Applied rewrites 36.4%

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

       1. metadata-eval N/A

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

       2. metadata-eval N/A

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

       3. metadata-eval N/A

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

       4. lift-fma.f64 N/A

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

       5. +-commutative N/A

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

       6. lift-fma.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. distribute-rgt-out N/A

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

       9. *-commutative N/A

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

       10. distribute-lft-out N/A

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

       11. lower-*.f64 N/A

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

       12. metadata-eval N/A

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

       13. lower-+.f64 N/A

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

       14. lift--.f64 N/A

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

       15. associate-+l- N/A

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

       16. lower--.f64 N/A

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

       17. lower--.f64 36.3%

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

   11. Applied rewrites 36.3%

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

   if 1.00000000000000002e144 < (*.f64 #s(literal 4 binary64) (pow.f64 q #s(literal 2 binary64))) < 4.9999999999999999e266

   1. Initial program 45.4%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
   2. 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 45.4%

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

   3. Applied rewrites 45.4%

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

   if 4.9999999999999999e266 < (*.f64 #s(literal 4 binary64) (pow.f64 q #s(literal 2 binary64))) < 1.99999999999999997e307

   1. Initial program 45.4%

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

   2. Taylor expanded in r around inf

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

   4. Applied rewrites 30.8%

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

   5. Taylor expanded in r around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-fma.f64 N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. lower-fabs.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-fabs.f64 N/A

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

      11. lower-*.f64 36.1%

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

   7. Applied rewrites 36.1%

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

      1. metadata-eval 36.1%

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

      2. metadata-eval 36.1%

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

   9. Applied rewrites 36.1%

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

   if 1.99999999999999997e307 < (*.f64 #s(literal 4 binary64) (pow.f64 q #s(literal 2 binary64)))

   1. Initial program 45.4%

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

   2. Taylor expanded in q around inf

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

      1. metadata-eval N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-fabs.f64 N/A

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

      9. lower-fabs.f64 26.6%

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

   4. Applied rewrites 26.6%

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

      1. metadata-eval N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. sum-to-mult-rev N/A

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

      9. lower-+.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. metadata-eval 29.2%

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

   6. Applied rewrites 29.2%

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

      1. metadata-eval N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-fabs.f64 N/A

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

      8. lift-fabs.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-fabs.f64 N/A

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

      11. lift-fabs.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. metadata-eval 29.2%

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

   8. Applied rewrites 29.2%

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

Alternative 2: 81.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := \left|\mathsf{max}\left(p, r\right)\right|\\ t_1 := \left|\mathsf{min}\left(p, r\right)\right|\\ \mathbf{if}\;4 \cdot {\left(\left|q\right|\right)}^{2} \leq 2 \cdot 10^{+95}:\\ \;\;\;\;0.5 \cdot \left(\left(t\_0 - \left(\mathsf{min}\left(p, r\right) - \mathsf{max}\left(p, r\right)\right)\right) + t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0 + t\_1, 0.5, \left|q\right|\right)\\ \end{array} \]

FPCore

(FPCore (p r q)
 :precision binary64
 (let* ((t_0 (fabs (fmax p r))) (t_1 (fabs (fmin p r))))
   (if (<= (* 4.0 (pow (fabs q) 2.0)) 2e+95)
     (* 0.5 (+ (- t_0 (- (fmin p r) (fmax p r))) t_1))
     (fma (+ t_0 t_1) 0.5 (fabs q)))))

C

double code(double p, double r, double q) {
	double t_0 = fabs(fmax(p, r));
	double t_1 = fabs(fmin(p, r));
	double tmp;
	if ((4.0 * pow(fabs(q), 2.0)) <= 2e+95) {
		tmp = 0.5 * ((t_0 - (fmin(p, r) - fmax(p, r))) + t_1);
	} else {
		tmp = fma((t_0 + t_1), 0.5, fabs(q));
	}
	return tmp;
}

Julia

function code(p, r, q)
	t_0 = abs(fmax(p, r))
	t_1 = abs(fmin(p, r))
	tmp = 0.0
	if (Float64(4.0 * (abs(q) ^ 2.0)) <= 2e+95)
		tmp = Float64(0.5 * Float64(Float64(t_0 - Float64(fmin(p, r) - fmax(p, r))) + t_1));
	else
		tmp = fma(Float64(t_0 + t_1), 0.5, abs(q));
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 4 binary64) (pow.f64 q #s(literal 2 binary64))) < 2.00000000000000004e95

   1. Initial program 45.4%

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

   2. Taylor expanded in r around inf

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

   4. Applied rewrites 30.8%

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

   5. Taylor expanded in r around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-fma.f64 N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. lower-fabs.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-fabs.f64 N/A

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

      11. lower-*.f64 36.1%

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

   7. Applied rewrites 36.1%

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lift-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. distribute-rgt-in N/A

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

      8. associate-+l+ N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. metadata-eval N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

   9. Applied rewrites 36.4%

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

       1. metadata-eval N/A

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

       2. metadata-eval N/A

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

       3. metadata-eval N/A

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

       4. lift-fma.f64 N/A

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

       5. +-commutative N/A

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

       6. lift-fma.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. distribute-rgt-out N/A

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

       9. *-commutative N/A

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

       10. distribute-lft-out N/A

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

       11. lower-*.f64 N/A

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

       12. metadata-eval N/A

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

       13. lower-+.f64 N/A

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

       14. lift--.f64 N/A

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

       15. associate-+l- N/A

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

       16. lower--.f64 N/A

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

       17. lower--.f64 36.3%

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

   11. Applied rewrites 36.3%

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

   if 2.00000000000000004e95 < (*.f64 #s(literal 4 binary64) (pow.f64 q #s(literal 2 binary64)))

   1. Initial program 45.4%

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

   2. Taylor expanded in q around inf

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

      1. metadata-eval N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-fabs.f64 N/A

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

      9. lower-fabs.f64 26.6%

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

   4. Applied rewrites 26.6%

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

      1. metadata-eval N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. sum-to-mult-rev N/A

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

      9. lower-+.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. metadata-eval 29.2%

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

   6. Applied rewrites 29.2%

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

      1. metadata-eval N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-fabs.f64 N/A

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

      8. lift-fabs.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-fabs.f64 N/A

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

      11. lift-fabs.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. metadata-eval 29.2%

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

   8. Applied rewrites 29.2%

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

Alternative 3: 81.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} t_0 := \left|\mathsf{max}\left(p, r\right)\right|\\ t_1 := \left|\mathsf{min}\left(p, r\right)\right|\\ \mathbf{if}\;\left|q\right| \leq 1.9 \cdot 10^{+48}:\\ \;\;\;\;\left(\mathsf{max}\left(p, r\right) - \left(\left(\mathsf{min}\left(p, r\right) - t\_0\right) - t\_1\right)\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0 + t\_1, 0.5, \left|q\right|\right)\\ \end{array} \]

FPCore

(FPCore (p r q)
 :precision binary64
 (let* ((t_0 (fabs (fmax p r))) (t_1 (fabs (fmin p r))))
   (if (<= (fabs q) 1.9e+48)
     (* (- (fmax p r) (- (- (fmin p r) t_0) t_1)) 0.5)
     (fma (+ t_0 t_1) 0.5 (fabs q)))))

C

double code(double p, double r, double q) {
	double t_0 = fabs(fmax(p, r));
	double t_1 = fabs(fmin(p, r));
	double tmp;
	if (fabs(q) <= 1.9e+48) {
		tmp = (fmax(p, r) - ((fmin(p, r) - t_0) - t_1)) * 0.5;
	} else {
		tmp = fma((t_0 + t_1), 0.5, fabs(q));
	}
	return tmp;
}

Julia

function code(p, r, q)
	t_0 = abs(fmax(p, r))
	t_1 = abs(fmin(p, r))
	tmp = 0.0
	if (abs(q) <= 1.9e+48)
		tmp = Float64(Float64(fmax(p, r) - Float64(Float64(fmin(p, r) - t_0) - t_1)) * 0.5);
	else
		tmp = fma(Float64(t_0 + t_1), 0.5, abs(q));
	end
	return tmp
end

Wolfram

code[p_, r_, q_] := Block[{t$95$0 = N[Abs[N[Max[p, r], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[Min[p, r], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Abs[q], $MachinePrecision], 1.9e+48], N[(N[(N[Max[p, r], $MachinePrecision] - N[(N[(N[Min[p, r], $MachinePrecision] - t$95$0), $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(t$95$0 + t$95$1), $MachinePrecision] * 0.5 + N[Abs[q], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if q < 1.9e48

   1. Initial program 45.4%

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

   2. Taylor expanded in r around inf

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

   4. Applied rewrites 30.8%

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

   5. Taylor expanded in r around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-fma.f64 N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. lower-fabs.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-fabs.f64 N/A

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

      11. lower-*.f64 36.1%

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

   7. Applied rewrites 36.1%

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

      1. metadata-eval 36.1%

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

      2. metadata-eval 36.1%

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

   9. Applied rewrites 36.1%

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

   if 1.9e48 < q

   1. Initial program 45.4%

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

   2. Taylor expanded in q around inf

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

      1. metadata-eval N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-fabs.f64 N/A

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

      9. lower-fabs.f64 26.6%

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

   4. Applied rewrites 26.6%

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

      1. metadata-eval N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. sum-to-mult-rev N/A

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

      9. lower-+.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. metadata-eval 29.2%

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

   6. Applied rewrites 29.2%

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

      1. metadata-eval N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-fabs.f64 N/A

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

      8. lift-fabs.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-fabs.f64 N/A

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

      11. lift-fabs.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. metadata-eval 29.2%

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

   8. Applied rewrites 29.2%

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

Alternative 4: 63.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} t_0 := \left|\mathsf{min}\left(p, r\right)\right|\\ t_1 := \left|\mathsf{max}\left(p, r\right)\right|\\ \mathbf{if}\;\mathsf{max}\left(p, r\right) \leq -1.15 \cdot 10^{-238}:\\ \;\;\;\;0.5 \cdot \left(t\_0 - \left(\mathsf{min}\left(p, r\right) - t\_1\right)\right)\\ \mathbf{elif}\;\mathsf{max}\left(p, r\right) \leq 3.2 \cdot 10^{+166}:\\ \;\;\;\;\mathsf{fma}\left(t\_1 + t\_0, 0.5, \left|q\right|\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \mathsf{max}\left(p, r\right), 0.5 \cdot \left(t\_0 + t\_1\right)\right)\\ \end{array} \]

FPCore

(FPCore (p r q)
 :precision binary64
 (let* ((t_0 (fabs (fmin p r))) (t_1 (fabs (fmax p r))))
   (if (<= (fmax p r) -1.15e-238)
     (* 0.5 (- t_0 (- (fmin p r) t_1)))
     (if (<= (fmax p r) 3.2e+166)
       (fma (+ t_1 t_0) 0.5 (fabs q))
       (fma 0.5 (fmax p r) (* 0.5 (+ t_0 t_1)))))))

C

double code(double p, double r, double q) {
	double t_0 = fabs(fmin(p, r));
	double t_1 = fabs(fmax(p, r));
	double tmp;
	if (fmax(p, r) <= -1.15e-238) {
		tmp = 0.5 * (t_0 - (fmin(p, r) - t_1));
	} else if (fmax(p, r) <= 3.2e+166) {
		tmp = fma((t_1 + t_0), 0.5, fabs(q));
	} else {
		tmp = fma(0.5, fmax(p, r), (0.5 * (t_0 + t_1)));
	}
	return tmp;
}

Julia

function code(p, r, q)
	t_0 = abs(fmin(p, r))
	t_1 = abs(fmax(p, r))
	tmp = 0.0
	if (fmax(p, r) <= -1.15e-238)
		tmp = Float64(0.5 * Float64(t_0 - Float64(fmin(p, r) - t_1)));
	elseif (fmax(p, r) <= 3.2e+166)
		tmp = fma(Float64(t_1 + t_0), 0.5, abs(q));
	else
		tmp = fma(0.5, fmax(p, r), Float64(0.5 * Float64(t_0 + t_1)));
	end
	return tmp
end

Wolfram

code[p_, r_, q_] := Block[{t$95$0 = N[Abs[N[Min[p, r], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[Max[p, r], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Max[p, r], $MachinePrecision], -1.15e-238], N[(0.5 * N[(t$95$0 - N[(N[Min[p, r], $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Max[p, r], $MachinePrecision], 3.2e+166], N[(N[(t$95$1 + t$95$0), $MachinePrecision] * 0.5 + N[Abs[q], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Max[p, r], $MachinePrecision] + N[(0.5 * N[(t$95$0 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if r < -1.15000000000000002e-238

   1. Initial program 45.4%

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

   2. Taylor expanded in r around inf

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

   4. Applied rewrites 30.8%

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

   5. Taylor expanded in r around 0

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

      1. metadata-eval N/A

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

      2. lower-*.f64 N/A

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

      3. metadata-eval N/A

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

      4. lower-+.f64 N/A

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

      5. lower-fabs.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-fabs.f64 N/A

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

      8. lower-*.f64 25.1%

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

   7. Applied rewrites 25.1%

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

      1. lift-+.f64 N/A

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

      2. add-flip N/A

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

      3. lower--.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. sub-flip N/A

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

      8. sub-negate-rev N/A

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

      9. lower--.f64 25.1%

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

   9. Applied rewrites 25.1%

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

   if -1.15000000000000002e-238 < r < 3.19999999999999968e166

   1. Initial program 45.4%

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

   2. Taylor expanded in q around inf

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

      1. metadata-eval N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-fabs.f64 N/A

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

      9. lower-fabs.f64 26.6%

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

   4. Applied rewrites 26.6%

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

      1. metadata-eval N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. sum-to-mult-rev N/A

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

      9. lower-+.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. metadata-eval 29.2%

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

   6. Applied rewrites 29.2%

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

      1. metadata-eval N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-fabs.f64 N/A

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

      8. lift-fabs.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-fabs.f64 N/A

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

      11. lift-fabs.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. metadata-eval 29.2%

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

   8. Applied rewrites 29.2%

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

   if 3.19999999999999968e166 < r

   1. Initial program 45.4%

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

   2. Taylor expanded in r around inf

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

   4. Applied rewrites 30.8%

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

   5. Taylor expanded in r around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-fma.f64 N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. lower-fabs.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-fabs.f64 N/A

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

      11. lower-*.f64 36.1%

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

   7. Applied rewrites 36.1%

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

   8. Taylor expanded in p around 0

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

      1. lower-+.f64 N/A

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

      2. lower-fabs.f64 N/A

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

      3. lower-fabs.f64 24.9%

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

   10. Applied rewrites 24.9%

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

Alternative 5: 60.1% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (p r q)
 :precision binary64
 (let* ((t_0 (fabs (fmax p r))) (t_1 (fabs (fmin p r))))
   (if (<= (* 4.0 (pow (fabs q) 2.0)) 5.0)
     (* 0.5 (- t_1 (- (fmin p r) t_0)))
     (fma (+ t_0 t_1) 0.5 (fabs q)))))

C

double code(double p, double r, double q) {
	double t_0 = fabs(fmax(p, r));
	double t_1 = fabs(fmin(p, r));
	double tmp;
	if ((4.0 * pow(fabs(q), 2.0)) <= 5.0) {
		tmp = 0.5 * (t_1 - (fmin(p, r) - t_0));
	} else {
		tmp = fma((t_0 + t_1), 0.5, fabs(q));
	}
	return tmp;
}

Julia

function code(p, r, q)
	t_0 = abs(fmax(p, r))
	t_1 = abs(fmin(p, r))
	tmp = 0.0
	if (Float64(4.0 * (abs(q) ^ 2.0)) <= 5.0)
		tmp = Float64(0.5 * Float64(t_1 - Float64(fmin(p, r) - t_0)));
	else
		tmp = fma(Float64(t_0 + t_1), 0.5, abs(q));
	end
	return tmp
end

Wolfram

code[p_, r_, q_] := Block[{t$95$0 = N[Abs[N[Max[p, r], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[Min[p, r], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(4.0 * N[Power[N[Abs[q], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 5.0], N[(0.5 * N[(t$95$1 - N[(N[Min[p, r], $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 + t$95$1), $MachinePrecision] * 0.5 + N[Abs[q], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 4 binary64) (pow.f64 q #s(literal 2 binary64))) < 5

   1. Initial program 45.4%

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

   2. Taylor expanded in r around inf

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

   4. Applied rewrites 30.8%

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

   5. Taylor expanded in r around 0

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

      1. metadata-eval N/A

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

      2. lower-*.f64 N/A

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

      3. metadata-eval N/A

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

      4. lower-+.f64 N/A

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

      5. lower-fabs.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-fabs.f64 N/A

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

      8. lower-*.f64 25.1%

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

   7. Applied rewrites 25.1%

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

      1. lift-+.f64 N/A

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

      2. add-flip N/A

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

      3. lower--.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. sub-flip N/A

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

      8. sub-negate-rev N/A

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

      9. lower--.f64 25.1%

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

   9. Applied rewrites 25.1%

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

   if 5 < (*.f64 #s(literal 4 binary64) (pow.f64 q #s(literal 2 binary64)))

   1. Initial program 45.4%

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

   2. Taylor expanded in q around inf

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

      1. metadata-eval N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-fabs.f64 N/A

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

      9. lower-fabs.f64 26.6%

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

   4. Applied rewrites 26.6%

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

      1. metadata-eval N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. sum-to-mult-rev N/A

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

      9. lower-+.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. metadata-eval 29.2%

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

   6. Applied rewrites 29.2%

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

      1. metadata-eval N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-fabs.f64 N/A

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

      8. lift-fabs.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-fabs.f64 N/A

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

      11. lift-fabs.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. metadata-eval 29.2%

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

   8. Applied rewrites 29.2%

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

Alternative 6: 45.2% accurate, 4.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 45.4%

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

2. Taylor expanded in q around inf

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

   1. metadata-eval N/A

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

   2. lower-*.f64 N/A

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

   3. lower-+.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. metadata-eval N/A

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

   6. lower-/.f64 N/A

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

   7. lower-+.f64 N/A

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

   8. lower-fabs.f64 N/A

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

   9. lower-fabs.f64 26.6%

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

4. Applied rewrites 26.6%

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

   1. metadata-eval N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-+.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. associate-*r/ N/A

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

   8. sum-to-mult-rev N/A

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

   9. lower-+.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lift-+.f64 N/A

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

   13. +-commutative N/A

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

   14. lower-+.f64 N/A

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

   15. metadata-eval 29.2%

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

6. Applied rewrites 29.2%

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

   1. metadata-eval N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. +-commutative N/A

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

   7. lift-fabs.f64 N/A

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

   8. lift-fabs.f64 N/A

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

   9. lower-fma.f64 N/A

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

   10. lift-fabs.f64 N/A

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

   11. lift-fabs.f64 N/A

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

   12. +-commutative N/A

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

   13. lift-+.f64 N/A

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

   14. metadata-eval 29.2%

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

8. Applied rewrites 29.2%

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

Alternative 7: 39.2% accurate, 4.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|q\right| \leq 1.55 \cdot 10^{-34}:\\ \;\;\;\;0.5 \cdot \left(\left|p\right| + \left|r\right|\right)\\ \mathbf{else}:\\ \;\;\;\;\left|q\right|\\ \end{array} \]

FPCore

(FPCore (p r q)
 :precision binary64
 (if (<= (fabs q) 1.55e-34) (* 0.5 (+ (fabs p) (fabs r))) (fabs q)))

C

double code(double p, double r, double q) {
	double tmp;
	if (fabs(q) <= 1.55e-34) {
		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) <= 1.55d-34) 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) <= 1.55e-34) {
		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) <= 1.55e-34:
		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) <= 1.55e-34)
		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) <= 1.55e-34)
		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], 1.55e-34], 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 < 1.5499999999999999e-34

   1. Initial program 45.4%

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

   2. Taylor expanded in q around inf

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

      1. metadata-eval N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-fabs.f64 N/A

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

      9. lower-fabs.f64 26.6%

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

   4. Applied rewrites 26.6%

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

   5. Taylor expanded in q around 0

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

      1. metadata-eval N/A

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

      2. lower-*.f64 N/A

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

      3. metadata-eval N/A

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

      4. lower-+.f64 N/A

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

      5. lower-fabs.f64 N/A

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

      6. lower-fabs.f64 14.4%

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

   7. Applied rewrites 14.4%

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

   if 1.5499999999999999e-34 < q

   1. Initial program 45.4%

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

   2. Taylor expanded in r around inf

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

   4. Applied rewrites 30.8%

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

   5. Taylor expanded in r around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-fma.f64 N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. lower-fabs.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-fabs.f64 N/A

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

      11. lower-*.f64 36.1%

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

   7. Applied rewrites 36.1%

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lift-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. distribute-rgt-in N/A

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

      8. associate-+l+ N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. metadata-eval N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

   9. Applied rewrites 36.4%

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

   10. Taylor expanded in q around inf

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

   12. Applied rewrites 18.4%

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

Alternative 8: 36.9% accurate, 4.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|q\right| \leq 2.35 \cdot 10^{-131}:\\ \;\;\;\;-0.5 \cdot \mathsf{min}\left(p, r\right)\\ \mathbf{else}:\\ \;\;\;\;\left|q\right|\\ \end{array} \]

FPCore

(FPCore (p r q)
 :precision binary64
 (if (<= (fabs q) 2.35e-131) (* -0.5 (fmin p r)) (fabs q)))

C

double code(double p, double r, double q) {
	double tmp;
	if (fabs(q) <= 2.35e-131) {
		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) <= 2.35d-131) 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) <= 2.35e-131) {
		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) <= 2.35e-131:
		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) <= 2.35e-131)
		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) <= 2.35e-131)
		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], 2.35e-131], N[(-0.5 * N[Min[p, r], $MachinePrecision]), $MachinePrecision], N[Abs[q], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if q < 2.3499999999999998e-131

   1. Initial program 45.4%

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

   2. Taylor expanded in p around -inf

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

      1. lower-*.f64 5.4%

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

   4. Applied rewrites 5.4%

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

   if 2.3499999999999998e-131 < q

   1. Initial program 45.4%

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

   2. Taylor expanded in r around inf

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

   4. Applied rewrites 30.8%

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

   5. Taylor expanded in r around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-fma.f64 N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. lower-fabs.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-fabs.f64 N/A

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

      11. lower-*.f64 36.1%

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

   7. Applied rewrites 36.1%

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lift-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. distribute-rgt-in N/A

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

      8. associate-+l+ N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. metadata-eval N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

   9. Applied rewrites 36.4%

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

   10. Taylor expanded in q around inf

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

   12. Applied rewrites 18.4%

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

Alternative 9: 35.2% accurate, 26.0× speedup

Math

\[\left|q\right| \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(p, r, q)
use fmin_fmax_functions
    real(8), intent (in) :: p
    real(8), intent (in) :: r
    real(8), intent (in) :: q
    code = abs(q)
end function

Java

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

Python

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

Julia

function code(p, r, q)
	return 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 45.4%

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

2. Taylor expanded in r around inf

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

   1. metadata-eval N/A

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

   2. metadata-eval N/A

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

   3. lower-*.f64 N/A

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

   4. lower-+.f64 N/A

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

   5. metadata-eval N/A

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

   6. lower-*.f64 N/A

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

   7. metadata-eval N/A

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

   8. lower-/.f64 N/A

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

4. Applied rewrites 30.8%

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

5. Taylor expanded in r around 0

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

   1. metadata-eval N/A

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

   2. metadata-eval N/A

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

   3. lower-fma.f64 N/A

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

   4. metadata-eval N/A

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

   5. lower-*.f64 N/A

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

   6. metadata-eval N/A

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

   7. lower-+.f64 N/A

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

   8. lower-fabs.f64 N/A

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

   9. lower-+.f64 N/A

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

   10. lower-fabs.f64 N/A

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

   11. lower-*.f64 36.1%

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

7. Applied rewrites 36.1%

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

   1. metadata-eval N/A

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

   2. metadata-eval N/A

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

   3. lift-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. distribute-rgt-in N/A

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

   8. associate-+l+ N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. metadata-eval N/A

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

   12. *-commutative N/A

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

   13. lower-fma.f64 N/A

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

9. Applied rewrites 36.4%

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

10. Taylor expanded in q around inf

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

12. Applied rewrites 18.4%

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

Reproduce

herbie shell --seed 2025187 
(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)))))))