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

Percentage Accurate: 24.4% → 58.6%

Time: 5.8s

Alternatives: 7

Speedup: 18.0×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 24.4% 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: 58.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \left|\mathsf{min}\left(p, r\right)\right|\\ t_1 := {\left(\left|q\right|\right)}^{2}\\ t_2 := \sqrt{{\left(\mathsf{min}\left(p, r\right) - \mathsf{max}\left(p, r\right)\right)}^{2}}\\ t_3 := \sqrt{\mathsf{fma}\left(\left|q\right| \cdot 4, \left|q\right|, \mathsf{min}\left(p, r\right) \cdot \mathsf{min}\left(p, r\right)\right)}\\ t_4 := 4 \cdot t\_1\\ t_5 := \left|\mathsf{max}\left(p, r\right)\right|\\ \mathbf{if}\;t\_4 \leq 5 \cdot 10^{-176}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{max}\left(p, r\right), \frac{\mathsf{min}\left(p, r\right)}{t\_0}, t\_5 - \left(t\_0 - t\_0\right)\right) \cdot 0.5\\ \mathbf{elif}\;t\_4 \leq 2 \cdot 10^{+34}:\\ \;\;\;\;\left(\frac{\mathsf{max}\left(p, r\right) \cdot \mathsf{min}\left(p, r\right)}{t\_3} - \left(\left(t\_3 - t\_0\right) - t\_5\right)\right) \cdot 0.5\\ \mathbf{elif}\;t\_4 \leq 10^{+71}:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{t\_1}{t\_2}, 0.5 \cdot \left(\left(t\_0 + t\_5\right) - t\_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-\left|q\right|\\ \end{array} \]

FPCore

(FPCore (p r q)
 :precision binary64
 (let* ((t_0 (fabs (fmin p r)))
        (t_1 (pow (fabs q) 2.0))
        (t_2 (sqrt (pow (- (fmin p r) (fmax p r)) 2.0)))
        (t_3 (sqrt (fma (* (fabs q) 4.0) (fabs q) (* (fmin p r) (fmin p r)))))
        (t_4 (* 4.0 t_1))
        (t_5 (fabs (fmax p r))))
   (if (<= t_4 5e-176)
     (* (fma (fmax p r) (/ (fmin p r) t_0) (- t_5 (- t_0 t_0))) 0.5)
     (if (<= t_4 2e+34)
       (* (- (/ (* (fmax p r) (fmin p r)) t_3) (- (- t_3 t_0) t_5)) 0.5)
       (if (<= t_4 1e+71)
         (fma -1.0 (/ t_1 t_2) (* 0.5 (- (+ t_0 t_5) t_2)))
         (- (fabs q)))))))

C

double code(double p, double r, double q) {
	double t_0 = fabs(fmin(p, r));
	double t_1 = pow(fabs(q), 2.0);
	double t_2 = sqrt(pow((fmin(p, r) - fmax(p, r)), 2.0));
	double t_3 = sqrt(fma((fabs(q) * 4.0), fabs(q), (fmin(p, r) * fmin(p, r))));
	double t_4 = 4.0 * t_1;
	double t_5 = fabs(fmax(p, r));
	double tmp;
	if (t_4 <= 5e-176) {
		tmp = fma(fmax(p, r), (fmin(p, r) / t_0), (t_5 - (t_0 - t_0))) * 0.5;
	} else if (t_4 <= 2e+34) {
		tmp = (((fmax(p, r) * fmin(p, r)) / t_3) - ((t_3 - t_0) - t_5)) * 0.5;
	} else if (t_4 <= 1e+71) {
		tmp = fma(-1.0, (t_1 / t_2), (0.5 * ((t_0 + t_5) - t_2)));
	} else {
		tmp = -fabs(q);
	}
	return tmp;
}

Julia

function code(p, r, q)
	t_0 = abs(fmin(p, r))
	t_1 = abs(q) ^ 2.0
	t_2 = sqrt((Float64(fmin(p, r) - fmax(p, r)) ^ 2.0))
	t_3 = sqrt(fma(Float64(abs(q) * 4.0), abs(q), Float64(fmin(p, r) * fmin(p, r))))
	t_4 = Float64(4.0 * t_1)
	t_5 = abs(fmax(p, r))
	tmp = 0.0
	if (t_4 <= 5e-176)
		tmp = Float64(fma(fmax(p, r), Float64(fmin(p, r) / t_0), Float64(t_5 - Float64(t_0 - t_0))) * 0.5);
	elseif (t_4 <= 2e+34)
		tmp = Float64(Float64(Float64(Float64(fmax(p, r) * fmin(p, r)) / t_3) - Float64(Float64(t_3 - t_0) - t_5)) * 0.5);
	elseif (t_4 <= 1e+71)
		tmp = fma(-1.0, Float64(t_1 / t_2), Float64(0.5 * Float64(Float64(t_0 + t_5) - t_2)));
	else
		tmp = Float64(-abs(q));
	end
	return tmp
end

Wolfram

code[p_, r_, q_] := Block[{t$95$0 = N[Abs[N[Min[p, r], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Abs[q], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[Power[N[(N[Min[p, r], $MachinePrecision] - N[Max[p, r], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(N[(N[Abs[q], $MachinePrecision] * 4.0), $MachinePrecision] * N[Abs[q], $MachinePrecision] + N[(N[Min[p, r], $MachinePrecision] * N[Min[p, r], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(4.0 * t$95$1), $MachinePrecision]}, Block[{t$95$5 = N[Abs[N[Max[p, r], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$4, 5e-176], N[(N[(N[Max[p, r], $MachinePrecision] * N[(N[Min[p, r], $MachinePrecision] / t$95$0), $MachinePrecision] + N[(t$95$5 - N[(t$95$0 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[t$95$4, 2e+34], N[(N[(N[(N[(N[Max[p, r], $MachinePrecision] * N[Min[p, r], $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision] - N[(N[(t$95$3 - t$95$0), $MachinePrecision] - t$95$5), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[t$95$4, 1e+71], N[(-1.0 * N[(t$95$1 / t$95$2), $MachinePrecision] + N[(0.5 * N[(N[(t$95$0 + t$95$5), $MachinePrecision] - t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[Abs[q], $MachinePrecision])]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 #s(literal 4 binary64) (pow.f64 q #s(literal 2 binary64))) < 5e-176

   1. Initial program 24.4%

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

   2. Taylor expanded in r around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-fma.f64 N/A

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

   4. Applied rewrites 21.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \left(\left|p\right| + \left|r\right|\right) - \sqrt{\mathsf{fma}\left(4, {q}^{2}, {p}^{2}\right)}, 0.5 \cdot \frac{p \cdot r}{\sqrt{\mathsf{fma}\left(4, {q}^{2}, {p}^{2}\right)}}\right)} \]
   5. Step-by-step derivation

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. distribute-lft-out N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

   6. Applied rewrites 18.6%

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

   7. Taylor expanded in q around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-pow.f64 12.3

         \[\leadsto \mathsf{fma}\left(\frac{r}{\sqrt{{p}^{2}}}, p, \left(\left|r\right| - \sqrt{\mathsf{fma}\left(q \cdot q, 4, p \cdot p\right)}\right) + \left|p\right|\right) \cdot 0.5 \]

   9. Applied rewrites 12.3%

      \[\leadsto \mathsf{fma}\left(\frac{r}{\sqrt{{p}^{2}}}, p, \left(\left|r\right| - \sqrt{\mathsf{fma}\left(q \cdot q, 4, p \cdot p\right)}\right) + \left|p\right|\right) \cdot 0.5 \]

   10. Taylor expanded in q around 0

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

       1. lower-sqrt.f64 N/A

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

       2. lower-pow.f64 4.1

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

   12. Applied rewrites 4.1%

       \[\leadsto \mathsf{fma}\left(\frac{r}{\sqrt{{p}^{2}}}, p, \left(\left|r\right| - \sqrt{{p}^{2}}\right) + \left|p\right|\right) \cdot 0.5 \]
   13. Step-by-step derivation

       1. lift-fma.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-*l/ N/A

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

       4. associate-/l* N/A

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

       5. lower-fma.f64 N/A

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

   14. Applied rewrites 31.0%

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

   if 5e-176 < (*.f64 #s(literal 4 binary64) (pow.f64 q #s(literal 2 binary64))) < 1.99999999999999989e34

   1. Initial program 24.4%

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

   2. Taylor expanded in r around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-fma.f64 N/A

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

   4. Applied rewrites 21.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \left(\left|p\right| + \left|r\right|\right) - \sqrt{\mathsf{fma}\left(4, {q}^{2}, {p}^{2}\right)}, 0.5 \cdot \frac{p \cdot r}{\sqrt{\mathsf{fma}\left(4, {q}^{2}, {p}^{2}\right)}}\right)} \]
   5. Step-by-step derivation

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. distribute-lft-out N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

   6. Applied rewrites 18.6%

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

      1. lift-fma.f64 N/A

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

      2. add-flip N/A

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

      3. lower--.f64 N/A

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

   8. Applied rewrites 26.9%

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

   if 1.99999999999999989e34 < (*.f64 #s(literal 4 binary64) (pow.f64 q #s(literal 2 binary64))) < 1e71

   1. Initial program 24.4%

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

   2. Taylor expanded in q around 0

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

      1. metadata-eval N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower--.f64 N/A

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

   4. Applied rewrites 18.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-1, \frac{{q}^{2}}{\sqrt{{\left(p - r\right)}^{2}}}, 0.5 \cdot \left(\left(\left|p\right| + \left|r\right|\right) - \sqrt{{\left(p - r\right)}^{2}}\right)\right)} \]

   if 1e71 < (*.f64 #s(literal 4 binary64) (pow.f64 q #s(literal 2 binary64)))

   1. Initial program 24.4%

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

   2. Taylor expanded in q around inf

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

      1. lower-*.f64 19.2

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

   4. Applied rewrites 19.2%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 19.2

         \[\leadsto -q \]

   6. Applied rewrites 19.2%

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

Alternative 2: 57.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := \left|\mathsf{min}\left(p, r\right)\right|\\ \mathbf{if}\;4 \cdot {\left(\left|q\right|\right)}^{2} \leq 5 \cdot 10^{-176}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{max}\left(p, r\right), \frac{\mathsf{min}\left(p, r\right)}{t\_0}, \left|\mathsf{max}\left(p, r\right)\right| - \left(t\_0 - t\_0\right)\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;-\left|q\right|\\ \end{array} \]

FPCore

(FPCore (p r q)
 :precision binary64
 (let* ((t_0 (fabs (fmin p r))))
   (if (<= (* 4.0 (pow (fabs q) 2.0)) 5e-176)
     (*
      (fma (fmax p r) (/ (fmin p r) t_0) (- (fabs (fmax p r)) (- t_0 t_0)))
      0.5)
     (- (fabs q)))))

C

double code(double p, double r, double q) {
	double t_0 = fabs(fmin(p, r));
	double tmp;
	if ((4.0 * pow(fabs(q), 2.0)) <= 5e-176) {
		tmp = fma(fmax(p, r), (fmin(p, r) / t_0), (fabs(fmax(p, r)) - (t_0 - t_0))) * 0.5;
	} else {
		tmp = -fabs(q);
	}
	return tmp;
}

Julia

function code(p, r, q)
	t_0 = abs(fmin(p, r))
	tmp = 0.0
	if (Float64(4.0 * (abs(q) ^ 2.0)) <= 5e-176)
		tmp = Float64(fma(fmax(p, r), Float64(fmin(p, r) / t_0), Float64(abs(fmax(p, r)) - Float64(t_0 - t_0))) * 0.5);
	else
		tmp = Float64(-abs(q));
	end
	return tmp
end

Wolfram

code[p_, r_, q_] := Block[{t$95$0 = N[Abs[N[Min[p, r], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(4.0 * N[Power[N[Abs[q], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 5e-176], N[(N[(N[Max[p, r], $MachinePrecision] * N[(N[Min[p, r], $MachinePrecision] / t$95$0), $MachinePrecision] + N[(N[Abs[N[Max[p, r], $MachinePrecision]], $MachinePrecision] - N[(t$95$0 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], (-N[Abs[q], $MachinePrecision])]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 4 binary64) (pow.f64 q #s(literal 2 binary64))) < 5e-176

   1. Initial program 24.4%

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

   2. Taylor expanded in r around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-fma.f64 N/A

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

   4. Applied rewrites 21.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \left(\left|p\right| + \left|r\right|\right) - \sqrt{\mathsf{fma}\left(4, {q}^{2}, {p}^{2}\right)}, 0.5 \cdot \frac{p \cdot r}{\sqrt{\mathsf{fma}\left(4, {q}^{2}, {p}^{2}\right)}}\right)} \]
   5. Step-by-step derivation

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. distribute-lft-out N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

   6. Applied rewrites 18.6%

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

   7. Taylor expanded in q around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-pow.f64 12.3

         \[\leadsto \mathsf{fma}\left(\frac{r}{\sqrt{{p}^{2}}}, p, \left(\left|r\right| - \sqrt{\mathsf{fma}\left(q \cdot q, 4, p \cdot p\right)}\right) + \left|p\right|\right) \cdot 0.5 \]

   9. Applied rewrites 12.3%

      \[\leadsto \mathsf{fma}\left(\frac{r}{\sqrt{{p}^{2}}}, p, \left(\left|r\right| - \sqrt{\mathsf{fma}\left(q \cdot q, 4, p \cdot p\right)}\right) + \left|p\right|\right) \cdot 0.5 \]

   10. Taylor expanded in q around 0

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

       1. lower-sqrt.f64 N/A

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

       2. lower-pow.f64 4.1

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

   12. Applied rewrites 4.1%

       \[\leadsto \mathsf{fma}\left(\frac{r}{\sqrt{{p}^{2}}}, p, \left(\left|r\right| - \sqrt{{p}^{2}}\right) + \left|p\right|\right) \cdot 0.5 \]
   13. Step-by-step derivation

       1. lift-fma.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-*l/ N/A

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

       4. associate-/l* N/A

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

       5. lower-fma.f64 N/A

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

   14. Applied rewrites 31.0%

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

   if 5e-176 < (*.f64 #s(literal 4 binary64) (pow.f64 q #s(literal 2 binary64)))

   1. Initial program 24.4%

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

   2. Taylor expanded in q around inf

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

      1. lower-*.f64 19.2

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

   4. Applied rewrites 19.2%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 19.2

         \[\leadsto -q \]

   6. Applied rewrites 19.2%

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

Alternative 3: 57.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;4 \cdot {\left(\left|q\right|\right)}^{2} \leq 2 \cdot 10^{-140}:\\ \;\;\;\;\left(0.5 - \frac{\left|\mathsf{min}\left(p, r\right)\right| - \left(\mathsf{max}\left(p, r\right) - \left|\mathsf{max}\left(p, r\right)\right|\right)}{\mathsf{min}\left(p, r\right)} \cdot -0.5\right) \cdot \mathsf{min}\left(p, r\right)\\ \mathbf{else}:\\ \;\;\;\;-\left|q\right|\\ \end{array} \]

FPCore

(FPCore (p r q)
 :precision binary64
 (if (<= (* 4.0 (pow (fabs q) 2.0)) 2e-140)
   (*
    (-
     0.5
     (*
      (/ (- (fabs (fmin p r)) (- (fmax p r) (fabs (fmax p r)))) (fmin p r))
      -0.5))
    (fmin p r))
   (- (fabs q))))

C

double code(double p, double r, double q) {
	double tmp;
	if ((4.0 * pow(fabs(q), 2.0)) <= 2e-140) {
		tmp = (0.5 - (((fabs(fmin(p, r)) - (fmax(p, r) - fabs(fmax(p, r)))) / fmin(p, r)) * -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 ((4.0d0 * (abs(q) ** 2.0d0)) <= 2d-140) then
        tmp = (0.5d0 - (((abs(fmin(p, r)) - (fmax(p, r) - abs(fmax(p, r)))) / fmin(p, r)) * (-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 ((4.0 * Math.pow(Math.abs(q), 2.0)) <= 2e-140) {
		tmp = (0.5 - (((Math.abs(fmin(p, r)) - (fmax(p, r) - Math.abs(fmax(p, r)))) / fmin(p, r)) * -0.5)) * fmin(p, r);
	} else {
		tmp = -Math.abs(q);
	}
	return tmp;
}

Python

def code(p, r, q):
	tmp = 0
	if (4.0 * math.pow(math.fabs(q), 2.0)) <= 2e-140:
		tmp = (0.5 - (((math.fabs(fmin(p, r)) - (fmax(p, r) - math.fabs(fmax(p, r)))) / fmin(p, r)) * -0.5)) * fmin(p, r)
	else:
		tmp = -math.fabs(q)
	return tmp

Julia

function code(p, r, q)
	tmp = 0.0
	if (Float64(4.0 * (abs(q) ^ 2.0)) <= 2e-140)
		tmp = Float64(Float64(0.5 - Float64(Float64(Float64(abs(fmin(p, r)) - Float64(fmax(p, r) - abs(fmax(p, r)))) / fmin(p, r)) * -0.5)) * fmin(p, r));
	else
		tmp = Float64(-abs(q));
	end
	return tmp
end

MATLAB

function tmp_2 = code(p, r, q)
	tmp = 0.0;
	if ((4.0 * (abs(q) ^ 2.0)) <= 2e-140)
		tmp = (0.5 - (((abs(min(p, r)) - (max(p, r) - abs(max(p, r)))) / min(p, r)) * -0.5)) * min(p, r);
	else
		tmp = -abs(q);
	end
	tmp_2 = tmp;
end

Wolfram

code[p_, r_, q_] := If[LessEqual[N[(4.0 * N[Power[N[Abs[q], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 2e-140], N[(N[(0.5 - N[(N[(N[(N[Abs[N[Min[p, r], $MachinePrecision]], $MachinePrecision] - N[(N[Max[p, r], $MachinePrecision] - N[Abs[N[Max[p, r], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Min[p, r], $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] * N[Min[p, r], $MachinePrecision]), $MachinePrecision], (-N[Abs[q], $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 4 binary64) (pow.f64 q #s(literal 2 binary64))) < 2e-140

   1. Initial program 24.4%

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

   2. Taylor expanded in p around -inf

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

      1. metadata-eval N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

   4. Applied rewrites 8.3%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower--.f64 17.3

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

   6. Applied rewrites 17.3%

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

      1. metadata-eval N/A

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

      2. lift-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

   8. Applied rewrites 17.3%

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

   if 2e-140 < (*.f64 #s(literal 4 binary64) (pow.f64 q #s(literal 2 binary64)))

   1. Initial program 24.4%

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

   2. Taylor expanded in q around inf

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

      1. lower-*.f64 19.2

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

   4. Applied rewrites 19.2%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 19.2

         \[\leadsto -q \]

   6. Applied rewrites 19.2%

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

Alternative 4: 44.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|q\right| \leq 3.3 \cdot 10^{-88}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \mathsf{min}\left(p, r\right), 0.5 \cdot \left(\left(\left|\mathsf{min}\left(p, r\right)\right| + \left|\mathsf{max}\left(p, r\right)\right|\right) - \mathsf{max}\left(p, r\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-\left|q\right|\\ \end{array} \]

FPCore

(FPCore (p r q)
 :precision binary64
 (if (<= (fabs q) 3.3e-88)
   (fma
    0.5
    (fmin p r)
    (* 0.5 (- (+ (fabs (fmin p r)) (fabs (fmax p r))) (fmax p r))))
   (- (fabs q))))

C

double code(double p, double r, double q) {
	double tmp;
	if (fabs(q) <= 3.3e-88) {
		tmp = fma(0.5, fmin(p, r), (0.5 * ((fabs(fmin(p, r)) + fabs(fmax(p, r))) - fmax(p, r))));
	} else {
		tmp = -fabs(q);
	}
	return tmp;
}

Julia

function code(p, r, q)
	tmp = 0.0
	if (abs(q) <= 3.3e-88)
		tmp = fma(0.5, fmin(p, r), Float64(0.5 * Float64(Float64(abs(fmin(p, r)) + abs(fmax(p, r))) - fmax(p, r))));
	else
		tmp = Float64(-abs(q));
	end
	return tmp
end

Wolfram

code[p_, r_, q_] := If[LessEqual[N[Abs[q], $MachinePrecision], 3.3e-88], N[(0.5 * N[Min[p, r], $MachinePrecision] + N[(0.5 * N[(N[(N[Abs[N[Min[p, r], $MachinePrecision]], $MachinePrecision] + N[Abs[N[Max[p, r], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[Max[p, r], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[Abs[q], $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if q < 3.29999999999999994e-88

   1. Initial program 24.4%

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

   2. Taylor expanded in p around -inf

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

      1. metadata-eval N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

   4. Applied rewrites 8.3%

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

   5. Taylor expanded in p around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-fma.f64 N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. lower--.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-fabs.f64 N/A

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

      10. lower-fabs.f64 8.3

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

   7. Applied rewrites 8.3%

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

   if 3.29999999999999994e-88 < q

   1. Initial program 24.4%

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

   2. Taylor expanded in q around inf

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

      1. lower-*.f64 19.2

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

   4. Applied rewrites 19.2%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 19.2

         \[\leadsto -q \]

   6. Applied rewrites 19.2%

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

Alternative 5: 42.3% accurate, 2.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|q\right| \leq 2.8 \cdot 10^{-88}:\\ \;\;\;\;0.5 \cdot \left(\left(\left|\mathsf{min}\left(p, r\right)\right| + \left|\mathsf{max}\left(p, r\right)\right|\right) - \mathsf{max}\left(p, r\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-\left|q\right|\\ \end{array} \]

FPCore

(FPCore (p r q)
 :precision binary64
 (if (<= (fabs q) 2.8e-88)
   (* 0.5 (- (+ (fabs (fmin p r)) (fabs (fmax p r))) (fmax p r)))
   (- (fabs q))))

C

double code(double p, double r, double q) {
	double tmp;
	if (fabs(q) <= 2.8e-88) {
		tmp = 0.5 * ((fabs(fmin(p, r)) + fabs(fmax(p, r))) - fmax(p, r));
	} else {
		tmp = -fabs(q);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(p, r, q)
use fmin_fmax_functions
    real(8), intent (in) :: p
    real(8), intent (in) :: r
    real(8), intent (in) :: q
    real(8) :: tmp
    if (abs(q) <= 2.8d-88) then
        tmp = 0.5d0 * ((abs(fmin(p, r)) + abs(fmax(p, r))) - fmax(p, r))
    else
        tmp = -abs(q)
    end if
    code = tmp
end function

Java

public static double code(double p, double r, double q) {
	double tmp;
	if (Math.abs(q) <= 2.8e-88) {
		tmp = 0.5 * ((Math.abs(fmin(p, r)) + Math.abs(fmax(p, r))) - fmax(p, r));
	} else {
		tmp = -Math.abs(q);
	}
	return tmp;
}

Python

def code(p, r, q):
	tmp = 0
	if math.fabs(q) <= 2.8e-88:
		tmp = 0.5 * ((math.fabs(fmin(p, r)) + math.fabs(fmax(p, r))) - fmax(p, r))
	else:
		tmp = -math.fabs(q)
	return tmp

Julia

function code(p, r, q)
	tmp = 0.0
	if (abs(q) <= 2.8e-88)
		tmp = Float64(0.5 * Float64(Float64(abs(fmin(p, r)) + abs(fmax(p, r))) - fmax(p, r)));
	else
		tmp = Float64(-abs(q));
	end
	return tmp
end

MATLAB

function tmp_2 = code(p, r, q)
	tmp = 0.0;
	if (abs(q) <= 2.8e-88)
		tmp = 0.5 * ((abs(min(p, r)) + abs(max(p, r))) - max(p, r));
	else
		tmp = -abs(q);
	end
	tmp_2 = tmp;
end

Wolfram

code[p_, r_, q_] := If[LessEqual[N[Abs[q], $MachinePrecision], 2.8e-88], N[(0.5 * N[(N[(N[Abs[N[Min[p, r], $MachinePrecision]], $MachinePrecision] + N[Abs[N[Max[p, r], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[Max[p, r], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[Abs[q], $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if q < 2.79999999999999976e-88

   1. Initial program 24.4%

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

   2. Taylor expanded in p around -inf

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

      1. metadata-eval N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

   4. Applied rewrites 8.3%

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

   5. Taylor expanded in p around 0

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

      1. metadata-eval N/A

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

      2. lower-*.f64 N/A

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

      3. metadata-eval N/A

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

      4. lower--.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-fabs.f64 N/A

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

      7. lower-fabs.f64 7.0

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

   7. Applied rewrites 7.0%

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

   if 2.79999999999999976e-88 < q

   1. Initial program 24.4%

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

   2. Taylor expanded in q around inf

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

      1. lower-*.f64 19.2

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

   4. Applied rewrites 19.2%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 19.2

         \[\leadsto -q \]

   6. Applied rewrites 19.2%

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

Alternative 6: 37.8% accurate, 3.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|q\right| \leq 2.5 \cdot 10^{-88}:\\ \;\;\;\;r \cdot \left(0.5 \cdot \frac{p}{r}\right)\\ \mathbf{else}:\\ \;\;\;\;-\left|q\right|\\ \end{array} \]

FPCore

(FPCore (p r q)
 :precision binary64
 (if (<= (fabs q) 2.5e-88) (* r (* 0.5 (/ p r))) (- (fabs q))))

C

double code(double p, double r, double q) {
	double tmp;
	if (fabs(q) <= 2.5e-88) {
		tmp = r * (0.5 * (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) <= 2.5d-88) then
        tmp = r * (0.5d0 * (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) <= 2.5e-88) {
		tmp = r * (0.5 * (p / r));
	} else {
		tmp = -Math.abs(q);
	}
	return tmp;
}

Python

def code(p, r, q):
	tmp = 0
	if math.fabs(q) <= 2.5e-88:
		tmp = r * (0.5 * (p / r))
	else:
		tmp = -math.fabs(q)
	return tmp

Julia

function code(p, r, q)
	tmp = 0.0
	if (abs(q) <= 2.5e-88)
		tmp = Float64(r * Float64(0.5 * Float64(p / r)));
	else
		tmp = Float64(-abs(q));
	end
	return tmp
end

MATLAB

function tmp_2 = code(p, r, q)
	tmp = 0.0;
	if (abs(q) <= 2.5e-88)
		tmp = r * (0.5 * (p / r));
	else
		tmp = -abs(q);
	end
	tmp_2 = tmp;
end

Wolfram

code[p_, r_, q_] := If[LessEqual[N[Abs[q], $MachinePrecision], 2.5e-88], N[(r * N[(0.5 * N[(p / r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[Abs[q], $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if q < 2.50000000000000004e-88

   1. Initial program 24.4%

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

   2. Taylor expanded in r around inf

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

   4. Applied rewrites 8.2%

      \[\leadsto \color{blue}{r \cdot \left(\mathsf{fma}\left(-1, \frac{{q}^{2}}{{r}^{2}}, 0.5 \cdot \frac{\left(\left|p\right| + \left|r\right|\right) - -1 \cdot p}{r}\right) - 0.5\right)} \]

   5. Taylor expanded in p around inf

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

      1. metadata-eval N/A

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

      2. lower-*.f64 N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 6.7

         \[\leadsto r \cdot \left(0.5 \cdot \frac{p}{r}\right) \]

   7. Applied rewrites 6.7%

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

   if 2.50000000000000004e-88 < q

   1. Initial program 24.4%

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

   2. Taylor expanded in q around inf

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

      1. lower-*.f64 19.2

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

   4. Applied rewrites 19.2%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 19.2

         \[\leadsto -q \]

   6. Applied rewrites 19.2%

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

Alternative 7: 36.9% accurate, 18.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 24.4%

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

2. Taylor expanded in q around inf

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

   1. lower-*.f64 19.2

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

4. Applied rewrites 19.2%

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

   1. lift-*.f64 N/A

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

   2. mul-1-neg N/A

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

   3. lower-neg.f64 19.2

      \[\leadsto -q \]

6. Applied rewrites 19.2%

   \[\leadsto -q \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2025173 
(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)))))))