The quadratic formula (r2)

Percentage Accurate: 52.5% → 89.3%

Time: 5.2s

Alternatives: 13

Speedup: 2.2×

Specification

Math

\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

FPCore

(FPCore (a b c)
  :precision binary64
  (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))

C

double code(double a, double b, double c) {
	return (-b - sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
}

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(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-b - sqrt(((b * b) - (4.0d0 * (a * c))))) / (2.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b - Math.sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
}

Python

def code(a, b, c):
	return (-b - math.sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) - sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c))))) / Float64(2.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b - sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) - N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Initial Program: 52.5% accurate, 1.0× speedup

Math

\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

FPCore

(FPCore (a b c)
  :precision binary64
  (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))

C

double code(double a, double b, double c) {
	return (-b - sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
}

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(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-b - sqrt(((b * b) - (4.0d0 * (a * c))))) / (2.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b - Math.sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
}

Python

def code(a, b, c):
	return (-b - math.sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) - sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c))))) / Float64(2.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b - sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) - N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Alternative 1: 89.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \frac{b + \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}}{-2 \cdot a}\\ t_1 := \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ t_2 := \mathsf{fma}\left(-0.5, \frac{b + \left|b\right|}{a}, \frac{c}{\left|b\right|}\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{-198}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;t\_1 \leq 10^{+290}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (a b c)
  :precision binary64
  (let* ((t_0 (/ (+ b (sqrt (fma -4.0 (* c a) (* b b)))) (* -2.0 a)))
       (t_1
        (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))
       (t_2 (fma -0.5 (/ (+ b (fabs b)) a) (/ c (fabs b)))))
  (if (<= t_1 (- INFINITY))
    t_2
    (if (<= t_1 -1e-198)
      t_0
      (if (<= t_1 0.0)
        (* -1.0 (/ c b))
        (if (<= t_1 1e+290) t_0 t_2))))))

C

double code(double a, double b, double c) {
	double t_0 = (b + sqrt(fma(-4.0, (c * a), (b * b)))) / (-2.0 * a);
	double t_1 = (-b - sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
	double t_2 = fma(-0.5, ((b + fabs(b)) / a), (c / fabs(b)));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_2;
	} else if (t_1 <= -1e-198) {
		tmp = t_0;
	} else if (t_1 <= 0.0) {
		tmp = -1.0 * (c / b);
	} else if (t_1 <= 1e+290) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(b + sqrt(fma(-4.0, Float64(c * a), Float64(b * b)))) / Float64(-2.0 * a))
	t_1 = Float64(Float64(Float64(-b) - sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c))))) / Float64(2.0 * a))
	t_2 = fma(-0.5, Float64(Float64(b + abs(b)) / a), Float64(c / abs(b)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = t_2;
	elseif (t_1 <= -1e-198)
		tmp = t_0;
	elseif (t_1 <= 0.0)
		tmp = Float64(-1.0 * Float64(c / b));
	elseif (t_1 <= 1e+290)
		tmp = t_0;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(b + N[Sqrt[N[(-4.0 * N[(c * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(-2.0 * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[((-b) - N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(-0.5 * N[(N[(b + N[Abs[b], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] + N[(c / N[Abs[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], t$95$2, If[LessEqual[t$95$1, -1e-198], t$95$0, If[LessEqual[t$95$1, 0.0], N[(-1.0 * N[(c / b), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+290], t$95$0, t$95$2]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 #s(literal 4 binary64) (*.f64 a c))))) (*.f64 #s(literal 2 binary64) a)) < -inf.0 or 1.0000000000000001e290 < (/.f64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 #s(literal 4 binary64) (*.f64 a c))))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 52.5%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

   2. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \sqrt{-4 \cdot \frac{c}{a}} + \frac{-1}{2} \cdot \frac{b}{a}} \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\sqrt{-4 \cdot \frac{c}{a}}}, \frac{-1}{2} \cdot \frac{b}{a}\right) \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \sqrt{-4 \cdot \frac{c}{a}}, \frac{-1}{2} \cdot \frac{b}{a}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \sqrt{-4 \cdot \frac{c}{a}}, \frac{-1}{2} \cdot \frac{b}{a}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \sqrt{-4 \cdot \frac{c}{a}}, \frac{-1}{2} \cdot \frac{b}{a}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \sqrt{-4 \cdot \frac{c}{a}}, \frac{-1}{2} \cdot \frac{b}{a}\right) \]

      6. lower-/.f64 19.6%

         \[\leadsto \mathsf{fma}\left(-0.5, \sqrt{-4 \cdot \frac{c}{a}}, -0.5 \cdot \frac{b}{a}\right) \]

   4. Applied rewrites 19.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \sqrt{-4 \cdot \frac{c}{a}}, -0.5 \cdot \frac{b}{a}\right)} \]

   5. Taylor expanded in c around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{b + \sqrt{{b}^{2}}}{a} + c \cdot \left(\frac{1}{\sqrt{{b}^{2}}} + \frac{a \cdot c}{{\left(\sqrt{{b}^{2}}\right)}^{3}}\right)} \]
   6. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\frac{b + \sqrt{{b}^{2}}}{a}}, c \cdot \left(\frac{1}{\sqrt{{b}^{2}}} + \frac{a \cdot c}{{\left(\sqrt{{b}^{2}}\right)}^{3}}\right)\right) \]

      2. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{b + \sqrt{{b}^{2}}}{\color{blue}{a}}, c \cdot \left(\frac{1}{\sqrt{{b}^{2}}} + \frac{a \cdot c}{{\left(\sqrt{{b}^{2}}\right)}^{3}}\right)\right) \]

      3. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{b + \sqrt{{b}^{2}}}{a}, c \cdot \left(\frac{1}{\sqrt{{b}^{2}}} + \frac{a \cdot c}{{\left(\sqrt{{b}^{2}}\right)}^{3}}\right)\right) \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{b + \sqrt{{b}^{2}}}{a}, c \cdot \left(\frac{1}{\sqrt{{b}^{2}}} + \frac{a \cdot c}{{\left(\sqrt{{b}^{2}}\right)}^{3}}\right)\right) \]

      5. lower-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{b + \sqrt{{b}^{2}}}{a}, c \cdot \left(\frac{1}{\sqrt{{b}^{2}}} + \frac{a \cdot c}{{\left(\sqrt{{b}^{2}}\right)}^{3}}\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{b + \sqrt{{b}^{2}}}{a}, c \cdot \left(\frac{1}{\sqrt{{b}^{2}}} + \frac{a \cdot c}{{\left(\sqrt{{b}^{2}}\right)}^{3}}\right)\right) \]

      7. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{b + \sqrt{{b}^{2}}}{a}, c \cdot \left(\frac{1}{\sqrt{{b}^{2}}} + \frac{a \cdot c}{{\left(\sqrt{{b}^{2}}\right)}^{3}}\right)\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{b + \sqrt{{b}^{2}}}{a}, c \cdot \left(\frac{1}{\sqrt{{b}^{2}}} + \frac{a \cdot c}{{\left(\sqrt{{b}^{2}}\right)}^{3}}\right)\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{b + \sqrt{{b}^{2}}}{a}, c \cdot \left(\frac{1}{\sqrt{{b}^{2}}} + \frac{a \cdot c}{{\left(\sqrt{{b}^{2}}\right)}^{3}}\right)\right) \]

      10. lower-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{b + \sqrt{{b}^{2}}}{a}, c \cdot \left(\frac{1}{\sqrt{{b}^{2}}} + \frac{a \cdot c}{{\left(\sqrt{{b}^{2}}\right)}^{3}}\right)\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{b + \sqrt{{b}^{2}}}{a}, c \cdot \left(\frac{1}{\sqrt{{b}^{2}}} + \frac{a \cdot c}{{\left(\sqrt{{b}^{2}}\right)}^{3}}\right)\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{b + \sqrt{{b}^{2}}}{a}, c \cdot \left(\frac{1}{\sqrt{{b}^{2}}} + \frac{a \cdot c}{{\left(\sqrt{{b}^{2}}\right)}^{3}}\right)\right) \]

   7. Applied rewrites 38.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{b + \sqrt{{b}^{2}}}{a}, c \cdot \left(\frac{1}{\sqrt{{b}^{2}}} + \frac{a \cdot c}{{\left(\sqrt{{b}^{2}}\right)}^{3}}\right)\right)} \]
   8. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{b + \sqrt{{b}^{2}}}{a} + \color{blue}{c \cdot \left(\frac{1}{\sqrt{{b}^{2}}} + \frac{a \cdot c}{{\left(\sqrt{{b}^{2}}\right)}^{3}}\right)} \]

      2. +-commutative N/A

         \[\leadsto c \cdot \left(\frac{1}{\sqrt{{b}^{2}}} + \frac{a \cdot c}{{\left(\sqrt{{b}^{2}}\right)}^{3}}\right) + \color{blue}{\frac{-1}{2} \cdot \frac{b + \sqrt{{b}^{2}}}{a}} \]

      3. lift-*.f64 N/A

         \[\leadsto c \cdot \left(\frac{1}{\sqrt{{b}^{2}}} + \frac{a \cdot c}{{\left(\sqrt{{b}^{2}}\right)}^{3}}\right) + \color{blue}{\frac{-1}{2}} \cdot \frac{b + \sqrt{{b}^{2}}}{a} \]

      4. *-commutative N/A

         \[\leadsto \left(\frac{1}{\sqrt{{b}^{2}}} + \frac{a \cdot c}{{\left(\sqrt{{b}^{2}}\right)}^{3}}\right) \cdot c + \color{blue}{\frac{-1}{2}} \cdot \frac{b + \sqrt{{b}^{2}}}{a} \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{\sqrt{{b}^{2}}} + \frac{a \cdot c}{{\left(\sqrt{{b}^{2}}\right)}^{3}}, \color{blue}{c}, \frac{-1}{2} \cdot \frac{b + \sqrt{{b}^{2}}}{a}\right) \]

   9. Applied rewrites 62.8%

      \[\leadsto \mathsf{fma}\left(\frac{\frac{c \cdot a}{b \cdot b} + 1}{\left|b\right|}, \color{blue}{c}, \frac{\left|b\right| + b}{a} \cdot -0.5\right) \]

   10. Taylor expanded in c around 0

       \[\leadsto \frac{-1}{2} \cdot \frac{b + \left|b\right|}{a} + \color{blue}{\frac{c}{\left|b\right|}} \]
   11. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{b + \left|b\right|}{\color{blue}{a}}, \frac{c}{\left|b\right|}\right) \]

       2. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{b + \left|b\right|}{a}, \frac{c}{\left|b\right|}\right) \]

       3. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{b + \left|b\right|}{a}, \frac{c}{\left|b\right|}\right) \]

       4. lower-fabs.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{b + \left|b\right|}{a}, \frac{c}{\left|b\right|}\right) \]

       5. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{b + \left|b\right|}{a}, \frac{c}{\left|b\right|}\right) \]

       6. lower-fabs.f64 68.4%

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{b + \left|b\right|}{a}, \frac{c}{\left|b\right|}\right) \]

   12. Applied rewrites 68.4%

       \[\leadsto \mathsf{fma}\left(-0.5, \color{blue}{\frac{b + \left|b\right|}{a}}, \frac{c}{\left|b\right|}\right) \]

   if -inf.0 < (/.f64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 #s(literal 4 binary64) (*.f64 a c))))) (*.f64 #s(literal 2 binary64) a)) < -9.9999999999999991e-199 or 0.0 < (/.f64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 #s(literal 4 binary64) (*.f64 a c))))) (*.f64 #s(literal 2 binary64) a)) < 1.0000000000000001e290

   1. Initial program 52.5%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}} \]

      2. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right)}{\mathsf{neg}\left(2 \cdot a\right)}} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right)}{\mathsf{neg}\left(2 \cdot a\right)}} \]

   3. Applied rewrites 52.5%

      \[\leadsto \color{blue}{\frac{b + \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}}{-2 \cdot a}} \]

   if -9.9999999999999991e-199 < (/.f64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 #s(literal 4 binary64) (*.f64 a c))))) (*.f64 #s(literal 2 binary64) a)) < 0.0

   1. Initial program 52.5%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

   2. Taylor expanded in b around -inf

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

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{c}{b}} \]

      2. lower-/.f64 34.4%

         \[\leadsto -1 \cdot \frac{c}{\color{blue}{b}} \]

   4. Applied rewrites 34.4%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 89.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(-0.5, \frac{b + \left|b\right|}{a}, \frac{c}{\left|b\right|}\right)\\ t_1 := \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ t_2 := \frac{-0.5}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+272}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{-198}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;t\_1 \leq 10^{+290}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (a b c)
  :precision binary64
  (let* ((t_0 (fma -0.5 (/ (+ b (fabs b)) a) (/ c (fabs b))))
       (t_1
        (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))
       (t_2 (* (/ -0.5 a) (+ b (sqrt (fma -4.0 (* c a) (* b b)))))))
  (if (<= t_1 -2e+272)
    t_0
    (if (<= t_1 -1e-198)
      t_2
      (if (<= t_1 0.0)
        (* -1.0 (/ c b))
        (if (<= t_1 1e+290) t_2 t_0))))))

C

double code(double a, double b, double c) {
	double t_0 = fma(-0.5, ((b + fabs(b)) / a), (c / fabs(b)));
	double t_1 = (-b - sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
	double t_2 = (-0.5 / a) * (b + sqrt(fma(-4.0, (c * a), (b * b))));
	double tmp;
	if (t_1 <= -2e+272) {
		tmp = t_0;
	} else if (t_1 <= -1e-198) {
		tmp = t_2;
	} else if (t_1 <= 0.0) {
		tmp = -1.0 * (c / b);
	} else if (t_1 <= 1e+290) {
		tmp = t_2;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(-0.5, Float64(Float64(b + abs(b)) / a), Float64(c / abs(b)))
	t_1 = Float64(Float64(Float64(-b) - sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c))))) / Float64(2.0 * a))
	t_2 = Float64(Float64(-0.5 / a) * Float64(b + sqrt(fma(-4.0, Float64(c * a), Float64(b * b)))))
	tmp = 0.0
	if (t_1 <= -2e+272)
		tmp = t_0;
	elseif (t_1 <= -1e-198)
		tmp = t_2;
	elseif (t_1 <= 0.0)
		tmp = Float64(-1.0 * Float64(c / b));
	elseif (t_1 <= 1e+290)
		tmp = t_2;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(-0.5 * N[(N[(b + N[Abs[b], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] + N[(c / N[Abs[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[((-b) - N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(-0.5 / a), $MachinePrecision] * N[(b + N[Sqrt[N[(-4.0 * N[(c * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+272], t$95$0, If[LessEqual[t$95$1, -1e-198], t$95$2, If[LessEqual[t$95$1, 0.0], N[(-1.0 * N[(c / b), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+290], t$95$2, t$95$0]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 #s(literal 4 binary64) (*.f64 a c))))) (*.f64 #s(literal 2 binary64) a)) < -2.0000000000000001e272 or 1.0000000000000001e290 < (/.f64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 #s(literal 4 binary64) (*.f64 a c))))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 52.5%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

   2. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \sqrt{-4 \cdot \frac{c}{a}} + \frac{-1}{2} \cdot \frac{b}{a}} \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\sqrt{-4 \cdot \frac{c}{a}}}, \frac{-1}{2} \cdot \frac{b}{a}\right) \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \sqrt{-4 \cdot \frac{c}{a}}, \frac{-1}{2} \cdot \frac{b}{a}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \sqrt{-4 \cdot \frac{c}{a}}, \frac{-1}{2} \cdot \frac{b}{a}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \sqrt{-4 \cdot \frac{c}{a}}, \frac{-1}{2} \cdot \frac{b}{a}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \sqrt{-4 \cdot \frac{c}{a}}, \frac{-1}{2} \cdot \frac{b}{a}\right) \]

      6. lower-/.f64 19.6%

         \[\leadsto \mathsf{fma}\left(-0.5, \sqrt{-4 \cdot \frac{c}{a}}, -0.5 \cdot \frac{b}{a}\right) \]

   4. Applied rewrites 19.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \sqrt{-4 \cdot \frac{c}{a}}, -0.5 \cdot \frac{b}{a}\right)} \]

   5. Taylor expanded in c around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{b + \sqrt{{b}^{2}}}{a} + c \cdot \left(\frac{1}{\sqrt{{b}^{2}}} + \frac{a \cdot c}{{\left(\sqrt{{b}^{2}}\right)}^{3}}\right)} \]
   6. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\frac{b + \sqrt{{b}^{2}}}{a}}, c \cdot \left(\frac{1}{\sqrt{{b}^{2}}} + \frac{a \cdot c}{{\left(\sqrt{{b}^{2}}\right)}^{3}}\right)\right) \]

      2. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{b + \sqrt{{b}^{2}}}{\color{blue}{a}}, c \cdot \left(\frac{1}{\sqrt{{b}^{2}}} + \frac{a \cdot c}{{\left(\sqrt{{b}^{2}}\right)}^{3}}\right)\right) \]

      3. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{b + \sqrt{{b}^{2}}}{a}, c \cdot \left(\frac{1}{\sqrt{{b}^{2}}} + \frac{a \cdot c}{{\left(\sqrt{{b}^{2}}\right)}^{3}}\right)\right) \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{b + \sqrt{{b}^{2}}}{a}, c \cdot \left(\frac{1}{\sqrt{{b}^{2}}} + \frac{a \cdot c}{{\left(\sqrt{{b}^{2}}\right)}^{3}}\right)\right) \]

      5. lower-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{b + \sqrt{{b}^{2}}}{a}, c \cdot \left(\frac{1}{\sqrt{{b}^{2}}} + \frac{a \cdot c}{{\left(\sqrt{{b}^{2}}\right)}^{3}}\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{b + \sqrt{{b}^{2}}}{a}, c \cdot \left(\frac{1}{\sqrt{{b}^{2}}} + \frac{a \cdot c}{{\left(\sqrt{{b}^{2}}\right)}^{3}}\right)\right) \]

      7. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{b + \sqrt{{b}^{2}}}{a}, c \cdot \left(\frac{1}{\sqrt{{b}^{2}}} + \frac{a \cdot c}{{\left(\sqrt{{b}^{2}}\right)}^{3}}\right)\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{b + \sqrt{{b}^{2}}}{a}, c \cdot \left(\frac{1}{\sqrt{{b}^{2}}} + \frac{a \cdot c}{{\left(\sqrt{{b}^{2}}\right)}^{3}}\right)\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{b + \sqrt{{b}^{2}}}{a}, c \cdot \left(\frac{1}{\sqrt{{b}^{2}}} + \frac{a \cdot c}{{\left(\sqrt{{b}^{2}}\right)}^{3}}\right)\right) \]

      10. lower-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{b + \sqrt{{b}^{2}}}{a}, c \cdot \left(\frac{1}{\sqrt{{b}^{2}}} + \frac{a \cdot c}{{\left(\sqrt{{b}^{2}}\right)}^{3}}\right)\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{b + \sqrt{{b}^{2}}}{a}, c \cdot \left(\frac{1}{\sqrt{{b}^{2}}} + \frac{a \cdot c}{{\left(\sqrt{{b}^{2}}\right)}^{3}}\right)\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{b + \sqrt{{b}^{2}}}{a}, c \cdot \left(\frac{1}{\sqrt{{b}^{2}}} + \frac{a \cdot c}{{\left(\sqrt{{b}^{2}}\right)}^{3}}\right)\right) \]

   7. Applied rewrites 38.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{b + \sqrt{{b}^{2}}}{a}, c \cdot \left(\frac{1}{\sqrt{{b}^{2}}} + \frac{a \cdot c}{{\left(\sqrt{{b}^{2}}\right)}^{3}}\right)\right)} \]
   8. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{b + \sqrt{{b}^{2}}}{a} + \color{blue}{c \cdot \left(\frac{1}{\sqrt{{b}^{2}}} + \frac{a \cdot c}{{\left(\sqrt{{b}^{2}}\right)}^{3}}\right)} \]

      2. +-commutative N/A

         \[\leadsto c \cdot \left(\frac{1}{\sqrt{{b}^{2}}} + \frac{a \cdot c}{{\left(\sqrt{{b}^{2}}\right)}^{3}}\right) + \color{blue}{\frac{-1}{2} \cdot \frac{b + \sqrt{{b}^{2}}}{a}} \]

      3. lift-*.f64 N/A

         \[\leadsto c \cdot \left(\frac{1}{\sqrt{{b}^{2}}} + \frac{a \cdot c}{{\left(\sqrt{{b}^{2}}\right)}^{3}}\right) + \color{blue}{\frac{-1}{2}} \cdot \frac{b + \sqrt{{b}^{2}}}{a} \]

      4. *-commutative N/A

         \[\leadsto \left(\frac{1}{\sqrt{{b}^{2}}} + \frac{a \cdot c}{{\left(\sqrt{{b}^{2}}\right)}^{3}}\right) \cdot c + \color{blue}{\frac{-1}{2}} \cdot \frac{b + \sqrt{{b}^{2}}}{a} \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{\sqrt{{b}^{2}}} + \frac{a \cdot c}{{\left(\sqrt{{b}^{2}}\right)}^{3}}, \color{blue}{c}, \frac{-1}{2} \cdot \frac{b + \sqrt{{b}^{2}}}{a}\right) \]

   9. Applied rewrites 62.8%

      \[\leadsto \mathsf{fma}\left(\frac{\frac{c \cdot a}{b \cdot b} + 1}{\left|b\right|}, \color{blue}{c}, \frac{\left|b\right| + b}{a} \cdot -0.5\right) \]

   10. Taylor expanded in c around 0

       \[\leadsto \frac{-1}{2} \cdot \frac{b + \left|b\right|}{a} + \color{blue}{\frac{c}{\left|b\right|}} \]
   11. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{b + \left|b\right|}{\color{blue}{a}}, \frac{c}{\left|b\right|}\right) \]

       2. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{b + \left|b\right|}{a}, \frac{c}{\left|b\right|}\right) \]

       3. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{b + \left|b\right|}{a}, \frac{c}{\left|b\right|}\right) \]

       4. lower-fabs.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{b + \left|b\right|}{a}, \frac{c}{\left|b\right|}\right) \]

       5. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{b + \left|b\right|}{a}, \frac{c}{\left|b\right|}\right) \]

       6. lower-fabs.f64 68.4%

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{b + \left|b\right|}{a}, \frac{c}{\left|b\right|}\right) \]

   12. Applied rewrites 68.4%

       \[\leadsto \mathsf{fma}\left(-0.5, \color{blue}{\frac{b + \left|b\right|}{a}}, \frac{c}{\left|b\right|}\right) \]

   if -2.0000000000000001e272 < (/.f64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 #s(literal 4 binary64) (*.f64 a c))))) (*.f64 #s(literal 2 binary64) a)) < -9.9999999999999991e-199 or 0.0 < (/.f64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 #s(literal 4 binary64) (*.f64 a c))))) (*.f64 #s(literal 2 binary64) a)) < 1.0000000000000001e290

   1. Initial program 52.5%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}} \]

      2. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right)}{\mathsf{neg}\left(2 \cdot a\right)}} \]

      3. mult-flip N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(2 \cdot a\right)}} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{\mathsf{neg}\left(2 \cdot a\right)} \cdot \left(\mathsf{neg}\left(\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right)\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\mathsf{neg}\left(2 \cdot a\right)} \cdot \left(\mathsf{neg}\left(\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right)\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{1}{\mathsf{neg}\left(\color{blue}{2 \cdot a}\right)} \cdot \left(\mathsf{neg}\left(\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right)\right) \]

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

         \[\leadsto \frac{1}{\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot a}} \cdot \left(\mathsf{neg}\left(\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right)\right) \]

      8. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{\mathsf{neg}\left(2\right)}}{a}} \cdot \left(\mathsf{neg}\left(\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \frac{\frac{1}{\color{blue}{-2}}}{a} \cdot \left(\mathsf{neg}\left(\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \frac{\color{blue}{\frac{-1}{2}}}{a} \cdot \left(\mathsf{neg}\left(\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\frac{1}{2}\right)}}{a} \cdot \left(\mathsf{neg}\left(\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right)\right) \]

      12. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{a}} \cdot \left(\mathsf{neg}\left(\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \frac{\color{blue}{\frac{-1}{2}}}{a} \cdot \left(\mathsf{neg}\left(\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right)\right) \]

      14. lift--.f64 N/A

          \[\leadsto \frac{\frac{-1}{2}}{a} \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}\right)\right) \]

      15. sub-negate-rev N/A

          \[\leadsto \frac{\frac{-1}{2}}{a} \cdot \color{blue}{\left(\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - \left(-b\right)\right)} \]

      16. sub-flip N/A

          \[\leadsto \frac{\frac{-1}{2}}{a} \cdot \color{blue}{\left(\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} + \left(\mathsf{neg}\left(\left(-b\right)\right)\right)\right)} \]

      17. +-commutative N/A

          \[\leadsto \frac{\frac{-1}{2}}{a} \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(-b\right)\right)\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)} \]

      18. lower-+.f64 N/A

          \[\leadsto \frac{\frac{-1}{2}}{a} \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(-b\right)\right)\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)} \]

   3. Applied rewrites 52.4%

      \[\leadsto \color{blue}{\frac{-0.5}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}\right)} \]

   if -9.9999999999999991e-199 < (/.f64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 #s(literal 4 binary64) (*.f64 a c))))) (*.f64 #s(literal 2 binary64) a)) < 0.0

   1. Initial program 52.5%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

   2. Taylor expanded in b around -inf

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

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{c}{b}} \]

      2. lower-/.f64 34.4%

         \[\leadsto -1 \cdot \frac{c}{\color{blue}{b}} \]

   4. Applied rewrites 34.4%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 80.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;b \leq -6.5 \cdot 10^{-96}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 1.8 \cdot 10^{-134}:\\ \;\;\;\;\frac{b + \sqrt{-4 \cdot \left(a \cdot c\right)}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{b + \left|b\right|}{a}, \frac{c}{\left|b\right|}\right)\\ \end{array} \]

FPCore

(FPCore (a b c)
  :precision binary64
  (if (<= b -6.5e-96)
  (* -1.0 (/ c b))
  (if (<= b 1.8e-134)
    (/ (+ b (sqrt (* -4.0 (* a c)))) (* -2.0 a))
    (fma -0.5 (/ (+ b (fabs b)) a) (/ c (fabs b))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -6.5e-96) {
		tmp = -1.0 * (c / b);
	} else if (b <= 1.8e-134) {
		tmp = (b + sqrt((-4.0 * (a * c)))) / (-2.0 * a);
	} else {
		tmp = fma(-0.5, ((b + fabs(b)) / a), (c / fabs(b)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -6.5e-96)
		tmp = Float64(-1.0 * Float64(c / b));
	elseif (b <= 1.8e-134)
		tmp = Float64(Float64(b + sqrt(Float64(-4.0 * Float64(a * c)))) / Float64(-2.0 * a));
	else
		tmp = fma(-0.5, Float64(Float64(b + abs(b)) / a), Float64(c / abs(b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -6.5e-96], N[(-1.0 * N[(c / b), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.8e-134], N[(N[(b + N[Sqrt[N[(-4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(-2.0 * a), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(N[(b + N[Abs[b], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] + N[(c / N[Abs[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -6.5e-96

   1. Initial program 52.5%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

   2. Taylor expanded in b around -inf

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

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{c}{b}} \]

      2. lower-/.f64 34.4%

         \[\leadsto -1 \cdot \frac{c}{\color{blue}{b}} \]

   4. Applied rewrites 34.4%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]

   if -6.5e-96 < b < 1.7999999999999999e-134

   1. Initial program 52.5%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}} \]

      2. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right)}{\mathsf{neg}\left(2 \cdot a\right)}} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right)}{\mathsf{neg}\left(2 \cdot a\right)}} \]

   3. Applied rewrites 52.5%

      \[\leadsto \color{blue}{\frac{b + \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}}{-2 \cdot a}} \]

   4. Taylor expanded in b around 0

      \[\leadsto \frac{\color{blue}{b + \sqrt{-4 \cdot \left(a \cdot c\right)}}}{-2 \cdot a} \]
   5. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{b + \color{blue}{\sqrt{-4 \cdot \left(a \cdot c\right)}}}{-2 \cdot a} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \frac{b + \sqrt{-4 \cdot \left(a \cdot c\right)}}{-2 \cdot a} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{b + \sqrt{-4 \cdot \left(a \cdot c\right)}}{-2 \cdot a} \]

      4. lower-*.f64 34.1%

         \[\leadsto \frac{b + \sqrt{-4 \cdot \left(a \cdot c\right)}}{-2 \cdot a} \]

   6. Applied rewrites 34.1%

      \[\leadsto \frac{\color{blue}{b + \sqrt{-4 \cdot \left(a \cdot c\right)}}}{-2 \cdot a} \]

   if 1.7999999999999999e-134 < b

   1. Initial program 52.5%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

   2. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \sqrt{-4 \cdot \frac{c}{a}} + \frac{-1}{2} \cdot \frac{b}{a}} \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\sqrt{-4 \cdot \frac{c}{a}}}, \frac{-1}{2} \cdot \frac{b}{a}\right) \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \sqrt{-4 \cdot \frac{c}{a}}, \frac{-1}{2} \cdot \frac{b}{a}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \sqrt{-4 \cdot \frac{c}{a}}, \frac{-1}{2} \cdot \frac{b}{a}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \sqrt{-4 \cdot \frac{c}{a}}, \frac{-1}{2} \cdot \frac{b}{a}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \sqrt{-4 \cdot \frac{c}{a}}, \frac{-1}{2} \cdot \frac{b}{a}\right) \]

      6. lower-/.f64 19.6%

         \[\leadsto \mathsf{fma}\left(-0.5, \sqrt{-4 \cdot \frac{c}{a}}, -0.5 \cdot \frac{b}{a}\right) \]

   4. Applied rewrites 19.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \sqrt{-4 \cdot \frac{c}{a}}, -0.5 \cdot \frac{b}{a}\right)} \]

   5. Taylor expanded in c around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{b + \sqrt{{b}^{2}}}{a} + c \cdot \left(\frac{1}{\sqrt{{b}^{2}}} + \frac{a \cdot c}{{\left(\sqrt{{b}^{2}}\right)}^{3}}\right)} \]
   6. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\frac{b + \sqrt{{b}^{2}}}{a}}, c \cdot \left(\frac{1}{\sqrt{{b}^{2}}} + \frac{a \cdot c}{{\left(\sqrt{{b}^{2}}\right)}^{3}}\right)\right) \]

      2. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{b + \sqrt{{b}^{2}}}{\color{blue}{a}}, c \cdot \left(\frac{1}{\sqrt{{b}^{2}}} + \frac{a \cdot c}{{\left(\sqrt{{b}^{2}}\right)}^{3}}\right)\right) \]

      3. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{b + \sqrt{{b}^{2}}}{a}, c \cdot \left(\frac{1}{\sqrt{{b}^{2}}} + \frac{a \cdot c}{{\left(\sqrt{{b}^{2}}\right)}^{3}}\right)\right) \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{b + \sqrt{{b}^{2}}}{a}, c \cdot \left(\frac{1}{\sqrt{{b}^{2}}} + \frac{a \cdot c}{{\left(\sqrt{{b}^{2}}\right)}^{3}}\right)\right) \]

      5. lower-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{b + \sqrt{{b}^{2}}}{a}, c \cdot \left(\frac{1}{\sqrt{{b}^{2}}} + \frac{a \cdot c}{{\left(\sqrt{{b}^{2}}\right)}^{3}}\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{b + \sqrt{{b}^{2}}}{a}, c \cdot \left(\frac{1}{\sqrt{{b}^{2}}} + \frac{a \cdot c}{{\left(\sqrt{{b}^{2}}\right)}^{3}}\right)\right) \]

      7. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{b + \sqrt{{b}^{2}}}{a}, c \cdot \left(\frac{1}{\sqrt{{b}^{2}}} + \frac{a \cdot c}{{\left(\sqrt{{b}^{2}}\right)}^{3}}\right)\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{b + \sqrt{{b}^{2}}}{a}, c \cdot \left(\frac{1}{\sqrt{{b}^{2}}} + \frac{a \cdot c}{{\left(\sqrt{{b}^{2}}\right)}^{3}}\right)\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{b + \sqrt{{b}^{2}}}{a}, c \cdot \left(\frac{1}{\sqrt{{b}^{2}}} + \frac{a \cdot c}{{\left(\sqrt{{b}^{2}}\right)}^{3}}\right)\right) \]

      10. lower-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{b + \sqrt{{b}^{2}}}{a}, c \cdot \left(\frac{1}{\sqrt{{b}^{2}}} + \frac{a \cdot c}{{\left(\sqrt{{b}^{2}}\right)}^{3}}\right)\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{b + \sqrt{{b}^{2}}}{a}, c \cdot \left(\frac{1}{\sqrt{{b}^{2}}} + \frac{a \cdot c}{{\left(\sqrt{{b}^{2}}\right)}^{3}}\right)\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{b + \sqrt{{b}^{2}}}{a}, c \cdot \left(\frac{1}{\sqrt{{b}^{2}}} + \frac{a \cdot c}{{\left(\sqrt{{b}^{2}}\right)}^{3}}\right)\right) \]

   7. Applied rewrites 38.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{b + \sqrt{{b}^{2}}}{a}, c \cdot \left(\frac{1}{\sqrt{{b}^{2}}} + \frac{a \cdot c}{{\left(\sqrt{{b}^{2}}\right)}^{3}}\right)\right)} \]
   8. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{b + \sqrt{{b}^{2}}}{a} + \color{blue}{c \cdot \left(\frac{1}{\sqrt{{b}^{2}}} + \frac{a \cdot c}{{\left(\sqrt{{b}^{2}}\right)}^{3}}\right)} \]

      2. +-commutative N/A

         \[\leadsto c \cdot \left(\frac{1}{\sqrt{{b}^{2}}} + \frac{a \cdot c}{{\left(\sqrt{{b}^{2}}\right)}^{3}}\right) + \color{blue}{\frac{-1}{2} \cdot \frac{b + \sqrt{{b}^{2}}}{a}} \]

      3. lift-*.f64 N/A

         \[\leadsto c \cdot \left(\frac{1}{\sqrt{{b}^{2}}} + \frac{a \cdot c}{{\left(\sqrt{{b}^{2}}\right)}^{3}}\right) + \color{blue}{\frac{-1}{2}} \cdot \frac{b + \sqrt{{b}^{2}}}{a} \]

      4. *-commutative N/A

         \[\leadsto \left(\frac{1}{\sqrt{{b}^{2}}} + \frac{a \cdot c}{{\left(\sqrt{{b}^{2}}\right)}^{3}}\right) \cdot c + \color{blue}{\frac{-1}{2}} \cdot \frac{b + \sqrt{{b}^{2}}}{a} \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{\sqrt{{b}^{2}}} + \frac{a \cdot c}{{\left(\sqrt{{b}^{2}}\right)}^{3}}, \color{blue}{c}, \frac{-1}{2} \cdot \frac{b + \sqrt{{b}^{2}}}{a}\right) \]

   9. Applied rewrites 62.8%

      \[\leadsto \mathsf{fma}\left(\frac{\frac{c \cdot a}{b \cdot b} + 1}{\left|b\right|}, \color{blue}{c}, \frac{\left|b\right| + b}{a} \cdot -0.5\right) \]

   10. Taylor expanded in c around 0

       \[\leadsto \frac{-1}{2} \cdot \frac{b + \left|b\right|}{a} + \color{blue}{\frac{c}{\left|b\right|}} \]
   11. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{b + \left|b\right|}{\color{blue}{a}}, \frac{c}{\left|b\right|}\right) \]

       2. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{b + \left|b\right|}{a}, \frac{c}{\left|b\right|}\right) \]

       3. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{b + \left|b\right|}{a}, \frac{c}{\left|b\right|}\right) \]

       4. lower-fabs.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{b + \left|b\right|}{a}, \frac{c}{\left|b\right|}\right) \]

       5. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{b + \left|b\right|}{a}, \frac{c}{\left|b\right|}\right) \]

       6. lower-fabs.f64 68.4%

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{b + \left|b\right|}{a}, \frac{c}{\left|b\right|}\right) \]

   12. Applied rewrites 68.4%

       \[\leadsto \mathsf{fma}\left(-0.5, \color{blue}{\frac{b + \left|b\right|}{a}}, \frac{c}{\left|b\right|}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 80.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;b \leq -6.5 \cdot 10^{-96}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 1.8 \cdot 10^{-134}:\\ \;\;\;\;-0.5 \cdot \frac{\sqrt{\left|-4 \cdot \left(c \cdot a\right)\right|}}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{b + \left|b\right|}{a}, \frac{c}{\left|b\right|}\right)\\ \end{array} \]

FPCore

(FPCore (a b c)
  :precision binary64
  (if (<= b -6.5e-96)
  (* -1.0 (/ c b))
  (if (<= b 1.8e-134)
    (* -0.5 (/ (sqrt (fabs (* -4.0 (* c a)))) a))
    (fma -0.5 (/ (+ b (fabs b)) a) (/ c (fabs b))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -6.5e-96) {
		tmp = -1.0 * (c / b);
	} else if (b <= 1.8e-134) {
		tmp = -0.5 * (sqrt(fabs((-4.0 * (c * a)))) / a);
	} else {
		tmp = fma(-0.5, ((b + fabs(b)) / a), (c / fabs(b)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -6.5e-96)
		tmp = Float64(-1.0 * Float64(c / b));
	elseif (b <= 1.8e-134)
		tmp = Float64(-0.5 * Float64(sqrt(abs(Float64(-4.0 * Float64(c * a)))) / a));
	else
		tmp = fma(-0.5, Float64(Float64(b + abs(b)) / a), Float64(c / abs(b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -6.5e-96], N[(-1.0 * N[(c / b), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.8e-134], N[(-0.5 * N[(N[Sqrt[N[Abs[N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(N[(b + N[Abs[b], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] + N[(c / N[Abs[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -6.5e-96

   1. Initial program 52.5%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

   2. Taylor expanded in b around -inf

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

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{c}{b}} \]

      2. lower-/.f64 34.4%

         \[\leadsto -1 \cdot \frac{c}{\color{blue}{b}} \]

   4. Applied rewrites 34.4%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]

   if -6.5e-96 < b < 1.7999999999999999e-134

   1. Initial program 52.5%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

   2. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{\sqrt{\mathsf{neg}\left(4 \cdot \left(a \cdot c\right)\right)}}{a}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \color{blue}{\frac{\sqrt{\mathsf{neg}\left(4 \cdot \left(a \cdot c\right)\right)}}{a}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{\sqrt{\mathsf{neg}\left(4 \cdot \left(a \cdot c\right)\right)}}{\color{blue}{a}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{\sqrt{\mathsf{neg}\left(4 \cdot \left(a \cdot c\right)\right)}}{a} \]

      4. lower-neg.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{\sqrt{-4 \cdot \left(a \cdot c\right)}}{a} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{\sqrt{-4 \cdot \left(a \cdot c\right)}}{a} \]

      6. lower-*.f64 28.9%

         \[\leadsto -0.5 \cdot \frac{\sqrt{-4 \cdot \left(a \cdot c\right)}}{a} \]

   4. Applied rewrites 28.9%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{\sqrt{-4 \cdot \left(a \cdot c\right)}}{a}} \]
   5. Step-by-step derivation

      1. lift-neg.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{\sqrt{\mathsf{neg}\left(4 \cdot \left(a \cdot c\right)\right)}}{a} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{\sqrt{\mathsf{neg}\left(4 \cdot \left(a \cdot c\right)\right)}}{a} \]

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

         \[\leadsto \frac{-1}{2} \cdot \frac{\sqrt{\left(\mathsf{neg}\left(4\right)\right) \cdot \left(a \cdot c\right)}}{a} \]

      4. metadata-eval N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{\sqrt{-4 \cdot \left(a \cdot c\right)}}{a} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{\sqrt{-4 \cdot \left(a \cdot c\right)}}{a} \]

      6. *-commutative N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{\sqrt{-4 \cdot \left(c \cdot a\right)}}{a} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{\sqrt{-4 \cdot \left(c \cdot a\right)}}{a} \]

      8. rem-square-sqrt N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{\sqrt{\sqrt{-4 \cdot \left(c \cdot a\right)} \cdot \sqrt{-4 \cdot \left(c \cdot a\right)}}}{a} \]

      9. metadata-eval N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{\sqrt{\sqrt{\left(\mathsf{neg}\left(4\right)\right) \cdot \left(c \cdot a\right)} \cdot \sqrt{-4 \cdot \left(c \cdot a\right)}}}{a} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{-1}{2} \cdot \frac{\sqrt{\sqrt{\left(\mathsf{neg}\left(4\right)\right) \cdot \left(c \cdot a\right)} \cdot \sqrt{-4 \cdot \left(c \cdot a\right)}}}{a} \]

      11. *-commutative N/A

          \[\leadsto \frac{-1}{2} \cdot \frac{\sqrt{\sqrt{\left(\mathsf{neg}\left(4\right)\right) \cdot \left(a \cdot c\right)} \cdot \sqrt{-4 \cdot \left(c \cdot a\right)}}}{a} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{-1}{2} \cdot \frac{\sqrt{\sqrt{\left(\mathsf{neg}\left(4\right)\right) \cdot \left(a \cdot c\right)} \cdot \sqrt{-4 \cdot \left(c \cdot a\right)}}}{a} \]

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

          \[\leadsto \frac{-1}{2} \cdot \frac{\sqrt{\sqrt{\mathsf{neg}\left(4 \cdot \left(a \cdot c\right)\right)} \cdot \sqrt{-4 \cdot \left(c \cdot a\right)}}}{a} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{-1}{2} \cdot \frac{\sqrt{\sqrt{\mathsf{neg}\left(4 \cdot \left(a \cdot c\right)\right)} \cdot \sqrt{-4 \cdot \left(c \cdot a\right)}}}{a} \]

      15. lift-neg.f64 N/A

          \[\leadsto \frac{-1}{2} \cdot \frac{\sqrt{\sqrt{-4 \cdot \left(a \cdot c\right)} \cdot \sqrt{-4 \cdot \left(c \cdot a\right)}}}{a} \]

      16. lift-sqrt.f64 N/A

          \[\leadsto \frac{-1}{2} \cdot \frac{\sqrt{\sqrt{-4 \cdot \left(a \cdot c\right)} \cdot \sqrt{-4 \cdot \left(c \cdot a\right)}}}{a} \]

      17. metadata-eval N/A

          \[\leadsto \frac{-1}{2} \cdot \frac{\sqrt{\sqrt{-4 \cdot \left(a \cdot c\right)} \cdot \sqrt{\left(\mathsf{neg}\left(4\right)\right) \cdot \left(c \cdot a\right)}}}{a} \]

      18. lift-*.f64 N/A

          \[\leadsto \frac{-1}{2} \cdot \frac{\sqrt{\sqrt{-4 \cdot \left(a \cdot c\right)} \cdot \sqrt{\left(\mathsf{neg}\left(4\right)\right) \cdot \left(c \cdot a\right)}}}{a} \]

      19. *-commutative N/A

          \[\leadsto \frac{-1}{2} \cdot \frac{\sqrt{\sqrt{-4 \cdot \left(a \cdot c\right)} \cdot \sqrt{\left(\mathsf{neg}\left(4\right)\right) \cdot \left(a \cdot c\right)}}}{a} \]

      20. lift-*.f64 N/A

          \[\leadsto \frac{-1}{2} \cdot \frac{\sqrt{\sqrt{-4 \cdot \left(a \cdot c\right)} \cdot \sqrt{\left(\mathsf{neg}\left(4\right)\right) \cdot \left(a \cdot c\right)}}}{a} \]

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

          \[\leadsto \frac{-1}{2} \cdot \frac{\sqrt{\sqrt{-4 \cdot \left(a \cdot c\right)} \cdot \sqrt{\mathsf{neg}\left(4 \cdot \left(a \cdot c\right)\right)}}}{a} \]

      22. lift-*.f64 N/A

          \[\leadsto \frac{-1}{2} \cdot \frac{\sqrt{\sqrt{-4 \cdot \left(a \cdot c\right)} \cdot \sqrt{\mathsf{neg}\left(4 \cdot \left(a \cdot c\right)\right)}}}{a} \]

      23. lift-neg.f64 N/A

          \[\leadsto \frac{-1}{2} \cdot \frac{\sqrt{\sqrt{-4 \cdot \left(a \cdot c\right)} \cdot \sqrt{-4 \cdot \left(a \cdot c\right)}}}{a} \]

      24. lift-sqrt.f64 N/A

          \[\leadsto \frac{-1}{2} \cdot \frac{\sqrt{\sqrt{-4 \cdot \left(a \cdot c\right)} \cdot \sqrt{-4 \cdot \left(a \cdot c\right)}}}{a} \]

      25. sqr-abs-rev N/A

          \[\leadsto \frac{-1}{2} \cdot \frac{\sqrt{\left|\sqrt{-4 \cdot \left(a \cdot c\right)}\right| \cdot \left|\sqrt{-4 \cdot \left(a \cdot c\right)}\right|}}{a} \]

   6. Applied rewrites 30.0%

      \[\leadsto -0.5 \cdot \frac{\sqrt{\left|-4 \cdot \left(c \cdot a\right)\right|}}{a} \]

   if 1.7999999999999999e-134 < b

   1. Initial program 52.5%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

   2. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \sqrt{-4 \cdot \frac{c}{a}} + \frac{-1}{2} \cdot \frac{b}{a}} \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\sqrt{-4 \cdot \frac{c}{a}}}, \frac{-1}{2} \cdot \frac{b}{a}\right) \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \sqrt{-4 \cdot \frac{c}{a}}, \frac{-1}{2} \cdot \frac{b}{a}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \sqrt{-4 \cdot \frac{c}{a}}, \frac{-1}{2} \cdot \frac{b}{a}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \sqrt{-4 \cdot \frac{c}{a}}, \frac{-1}{2} \cdot \frac{b}{a}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \sqrt{-4 \cdot \frac{c}{a}}, \frac{-1}{2} \cdot \frac{b}{a}\right) \]

      6. lower-/.f64 19.6%

         \[\leadsto \mathsf{fma}\left(-0.5, \sqrt{-4 \cdot \frac{c}{a}}, -0.5 \cdot \frac{b}{a}\right) \]

   4. Applied rewrites 19.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \sqrt{-4 \cdot \frac{c}{a}}, -0.5 \cdot \frac{b}{a}\right)} \]

   5. Taylor expanded in c around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{b + \sqrt{{b}^{2}}}{a} + c \cdot \left(\frac{1}{\sqrt{{b}^{2}}} + \frac{a \cdot c}{{\left(\sqrt{{b}^{2}}\right)}^{3}}\right)} \]
   6. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\frac{b + \sqrt{{b}^{2}}}{a}}, c \cdot \left(\frac{1}{\sqrt{{b}^{2}}} + \frac{a \cdot c}{{\left(\sqrt{{b}^{2}}\right)}^{3}}\right)\right) \]

      2. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{b + \sqrt{{b}^{2}}}{\color{blue}{a}}, c \cdot \left(\frac{1}{\sqrt{{b}^{2}}} + \frac{a \cdot c}{{\left(\sqrt{{b}^{2}}\right)}^{3}}\right)\right) \]

      3. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{b + \sqrt{{b}^{2}}}{a}, c \cdot \left(\frac{1}{\sqrt{{b}^{2}}} + \frac{a \cdot c}{{\left(\sqrt{{b}^{2}}\right)}^{3}}\right)\right) \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{b + \sqrt{{b}^{2}}}{a}, c \cdot \left(\frac{1}{\sqrt{{b}^{2}}} + \frac{a \cdot c}{{\left(\sqrt{{b}^{2}}\right)}^{3}}\right)\right) \]

      5. lower-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{b + \sqrt{{b}^{2}}}{a}, c \cdot \left(\frac{1}{\sqrt{{b}^{2}}} + \frac{a \cdot c}{{\left(\sqrt{{b}^{2}}\right)}^{3}}\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{b + \sqrt{{b}^{2}}}{a}, c \cdot \left(\frac{1}{\sqrt{{b}^{2}}} + \frac{a \cdot c}{{\left(\sqrt{{b}^{2}}\right)}^{3}}\right)\right) \]

      7. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{b + \sqrt{{b}^{2}}}{a}, c \cdot \left(\frac{1}{\sqrt{{b}^{2}}} + \frac{a \cdot c}{{\left(\sqrt{{b}^{2}}\right)}^{3}}\right)\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{b + \sqrt{{b}^{2}}}{a}, c \cdot \left(\frac{1}{\sqrt{{b}^{2}}} + \frac{a \cdot c}{{\left(\sqrt{{b}^{2}}\right)}^{3}}\right)\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{b + \sqrt{{b}^{2}}}{a}, c \cdot \left(\frac{1}{\sqrt{{b}^{2}}} + \frac{a \cdot c}{{\left(\sqrt{{b}^{2}}\right)}^{3}}\right)\right) \]

      10. lower-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{b + \sqrt{{b}^{2}}}{a}, c \cdot \left(\frac{1}{\sqrt{{b}^{2}}} + \frac{a \cdot c}{{\left(\sqrt{{b}^{2}}\right)}^{3}}\right)\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{b + \sqrt{{b}^{2}}}{a}, c \cdot \left(\frac{1}{\sqrt{{b}^{2}}} + \frac{a \cdot c}{{\left(\sqrt{{b}^{2}}\right)}^{3}}\right)\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{b + \sqrt{{b}^{2}}}{a}, c \cdot \left(\frac{1}{\sqrt{{b}^{2}}} + \frac{a \cdot c}{{\left(\sqrt{{b}^{2}}\right)}^{3}}\right)\right) \]

   7. Applied rewrites 38.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{b + \sqrt{{b}^{2}}}{a}, c \cdot \left(\frac{1}{\sqrt{{b}^{2}}} + \frac{a \cdot c}{{\left(\sqrt{{b}^{2}}\right)}^{3}}\right)\right)} \]
   8. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{b + \sqrt{{b}^{2}}}{a} + \color{blue}{c \cdot \left(\frac{1}{\sqrt{{b}^{2}}} + \frac{a \cdot c}{{\left(\sqrt{{b}^{2}}\right)}^{3}}\right)} \]

      2. +-commutative N/A

         \[\leadsto c \cdot \left(\frac{1}{\sqrt{{b}^{2}}} + \frac{a \cdot c}{{\left(\sqrt{{b}^{2}}\right)}^{3}}\right) + \color{blue}{\frac{-1}{2} \cdot \frac{b + \sqrt{{b}^{2}}}{a}} \]

      3. lift-*.f64 N/A

         \[\leadsto c \cdot \left(\frac{1}{\sqrt{{b}^{2}}} + \frac{a \cdot c}{{\left(\sqrt{{b}^{2}}\right)}^{3}}\right) + \color{blue}{\frac{-1}{2}} \cdot \frac{b + \sqrt{{b}^{2}}}{a} \]

      4. *-commutative N/A

         \[\leadsto \left(\frac{1}{\sqrt{{b}^{2}}} + \frac{a \cdot c}{{\left(\sqrt{{b}^{2}}\right)}^{3}}\right) \cdot c + \color{blue}{\frac{-1}{2}} \cdot \frac{b + \sqrt{{b}^{2}}}{a} \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{\sqrt{{b}^{2}}} + \frac{a \cdot c}{{\left(\sqrt{{b}^{2}}\right)}^{3}}, \color{blue}{c}, \frac{-1}{2} \cdot \frac{b + \sqrt{{b}^{2}}}{a}\right) \]

   9. Applied rewrites 62.8%

      \[\leadsto \mathsf{fma}\left(\frac{\frac{c \cdot a}{b \cdot b} + 1}{\left|b\right|}, \color{blue}{c}, \frac{\left|b\right| + b}{a} \cdot -0.5\right) \]

   10. Taylor expanded in c around 0

       \[\leadsto \frac{-1}{2} \cdot \frac{b + \left|b\right|}{a} + \color{blue}{\frac{c}{\left|b\right|}} \]
   11. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{b + \left|b\right|}{\color{blue}{a}}, \frac{c}{\left|b\right|}\right) \]

       2. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{b + \left|b\right|}{a}, \frac{c}{\left|b\right|}\right) \]

       3. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{b + \left|b\right|}{a}, \frac{c}{\left|b\right|}\right) \]

       4. lower-fabs.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{b + \left|b\right|}{a}, \frac{c}{\left|b\right|}\right) \]

       5. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{b + \left|b\right|}{a}, \frac{c}{\left|b\right|}\right) \]

       6. lower-fabs.f64 68.4%

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{b + \left|b\right|}{a}, \frac{c}{\left|b\right|}\right) \]

   12. Applied rewrites 68.4%

       \[\leadsto \mathsf{fma}\left(-0.5, \color{blue}{\frac{b + \left|b\right|}{a}}, \frac{c}{\left|b\right|}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 80.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;b \leq -6.5 \cdot 10^{-96}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 1.8 \cdot 10^{-134}:\\ \;\;\;\;-0.5 \cdot \frac{\sqrt{\left|-4 \cdot \left(c \cdot a\right)\right|}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

FPCore

(FPCore (a b c)
  :precision binary64
  (if (<= b -6.5e-96)
  (* -1.0 (/ c b))
  (if (<= b 1.8e-134)
    (* -0.5 (/ (sqrt (fabs (* -4.0 (* c a)))) a))
    (/ (- b) a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -6.5e-96) {
		tmp = -1.0 * (c / b);
	} else if (b <= 1.8e-134) {
		tmp = -0.5 * (sqrt(fabs((-4.0 * (c * a)))) / a);
	} else {
		tmp = -b / a;
	}
	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(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-6.5d-96)) then
        tmp = (-1.0d0) * (c / b)
    else if (b <= 1.8d-134) then
        tmp = (-0.5d0) * (sqrt(abs(((-4.0d0) * (c * a)))) / a)
    else
        tmp = -b / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -6.5e-96) {
		tmp = -1.0 * (c / b);
	} else if (b <= 1.8e-134) {
		tmp = -0.5 * (Math.sqrt(Math.abs((-4.0 * (c * a)))) / a);
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -6.5e-96:
		tmp = -1.0 * (c / b)
	elif b <= 1.8e-134:
		tmp = -0.5 * (math.sqrt(math.fabs((-4.0 * (c * a)))) / a)
	else:
		tmp = -b / a
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -6.5e-96)
		tmp = Float64(-1.0 * Float64(c / b));
	elseif (b <= 1.8e-134)
		tmp = Float64(-0.5 * Float64(sqrt(abs(Float64(-4.0 * Float64(c * a)))) / a));
	else
		tmp = Float64(Float64(-b) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -6.5e-96)
		tmp = -1.0 * (c / b);
	elseif (b <= 1.8e-134)
		tmp = -0.5 * (sqrt(abs((-4.0 * (c * a)))) / a);
	else
		tmp = -b / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -6.5e-96], N[(-1.0 * N[(c / b), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.8e-134], N[(-0.5 * N[(N[Sqrt[N[Abs[N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[((-b) / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -6.5e-96

   1. Initial program 52.5%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

   2. Taylor expanded in b around -inf

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

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{c}{b}} \]

      2. lower-/.f64 34.4%

         \[\leadsto -1 \cdot \frac{c}{\color{blue}{b}} \]

   4. Applied rewrites 34.4%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]

   if -6.5e-96 < b < 1.7999999999999999e-134

   1. Initial program 52.5%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

   2. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{\sqrt{\mathsf{neg}\left(4 \cdot \left(a \cdot c\right)\right)}}{a}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \color{blue}{\frac{\sqrt{\mathsf{neg}\left(4 \cdot \left(a \cdot c\right)\right)}}{a}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{\sqrt{\mathsf{neg}\left(4 \cdot \left(a \cdot c\right)\right)}}{\color{blue}{a}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{\sqrt{\mathsf{neg}\left(4 \cdot \left(a \cdot c\right)\right)}}{a} \]

      4. lower-neg.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{\sqrt{-4 \cdot \left(a \cdot c\right)}}{a} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{\sqrt{-4 \cdot \left(a \cdot c\right)}}{a} \]

      6. lower-*.f64 28.9%

         \[\leadsto -0.5 \cdot \frac{\sqrt{-4 \cdot \left(a \cdot c\right)}}{a} \]

   4. Applied rewrites 28.9%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{\sqrt{-4 \cdot \left(a \cdot c\right)}}{a}} \]
   5. Step-by-step derivation

      1. lift-neg.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{\sqrt{\mathsf{neg}\left(4 \cdot \left(a \cdot c\right)\right)}}{a} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{\sqrt{\mathsf{neg}\left(4 \cdot \left(a \cdot c\right)\right)}}{a} \]

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

         \[\leadsto \frac{-1}{2} \cdot \frac{\sqrt{\left(\mathsf{neg}\left(4\right)\right) \cdot \left(a \cdot c\right)}}{a} \]

      4. metadata-eval N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{\sqrt{-4 \cdot \left(a \cdot c\right)}}{a} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{\sqrt{-4 \cdot \left(a \cdot c\right)}}{a} \]

      6. *-commutative N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{\sqrt{-4 \cdot \left(c \cdot a\right)}}{a} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{\sqrt{-4 \cdot \left(c \cdot a\right)}}{a} \]

      8. rem-square-sqrt N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{\sqrt{\sqrt{-4 \cdot \left(c \cdot a\right)} \cdot \sqrt{-4 \cdot \left(c \cdot a\right)}}}{a} \]

      9. metadata-eval N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{\sqrt{\sqrt{\left(\mathsf{neg}\left(4\right)\right) \cdot \left(c \cdot a\right)} \cdot \sqrt{-4 \cdot \left(c \cdot a\right)}}}{a} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{-1}{2} \cdot \frac{\sqrt{\sqrt{\left(\mathsf{neg}\left(4\right)\right) \cdot \left(c \cdot a\right)} \cdot \sqrt{-4 \cdot \left(c \cdot a\right)}}}{a} \]

      11. *-commutative N/A

          \[\leadsto \frac{-1}{2} \cdot \frac{\sqrt{\sqrt{\left(\mathsf{neg}\left(4\right)\right) \cdot \left(a \cdot c\right)} \cdot \sqrt{-4 \cdot \left(c \cdot a\right)}}}{a} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{-1}{2} \cdot \frac{\sqrt{\sqrt{\left(\mathsf{neg}\left(4\right)\right) \cdot \left(a \cdot c\right)} \cdot \sqrt{-4 \cdot \left(c \cdot a\right)}}}{a} \]

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

          \[\leadsto \frac{-1}{2} \cdot \frac{\sqrt{\sqrt{\mathsf{neg}\left(4 \cdot \left(a \cdot c\right)\right)} \cdot \sqrt{-4 \cdot \left(c \cdot a\right)}}}{a} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{-1}{2} \cdot \frac{\sqrt{\sqrt{\mathsf{neg}\left(4 \cdot \left(a \cdot c\right)\right)} \cdot \sqrt{-4 \cdot \left(c \cdot a\right)}}}{a} \]

      15. lift-neg.f64 N/A

          \[\leadsto \frac{-1}{2} \cdot \frac{\sqrt{\sqrt{-4 \cdot \left(a \cdot c\right)} \cdot \sqrt{-4 \cdot \left(c \cdot a\right)}}}{a} \]

      16. lift-sqrt.f64 N/A

          \[\leadsto \frac{-1}{2} \cdot \frac{\sqrt{\sqrt{-4 \cdot \left(a \cdot c\right)} \cdot \sqrt{-4 \cdot \left(c \cdot a\right)}}}{a} \]

      17. metadata-eval N/A

          \[\leadsto \frac{-1}{2} \cdot \frac{\sqrt{\sqrt{-4 \cdot \left(a \cdot c\right)} \cdot \sqrt{\left(\mathsf{neg}\left(4\right)\right) \cdot \left(c \cdot a\right)}}}{a} \]

      18. lift-*.f64 N/A

          \[\leadsto \frac{-1}{2} \cdot \frac{\sqrt{\sqrt{-4 \cdot \left(a \cdot c\right)} \cdot \sqrt{\left(\mathsf{neg}\left(4\right)\right) \cdot \left(c \cdot a\right)}}}{a} \]

      19. *-commutative N/A

          \[\leadsto \frac{-1}{2} \cdot \frac{\sqrt{\sqrt{-4 \cdot \left(a \cdot c\right)} \cdot \sqrt{\left(\mathsf{neg}\left(4\right)\right) \cdot \left(a \cdot c\right)}}}{a} \]

      20. lift-*.f64 N/A

          \[\leadsto \frac{-1}{2} \cdot \frac{\sqrt{\sqrt{-4 \cdot \left(a \cdot c\right)} \cdot \sqrt{\left(\mathsf{neg}\left(4\right)\right) \cdot \left(a \cdot c\right)}}}{a} \]

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

          \[\leadsto \frac{-1}{2} \cdot \frac{\sqrt{\sqrt{-4 \cdot \left(a \cdot c\right)} \cdot \sqrt{\mathsf{neg}\left(4 \cdot \left(a \cdot c\right)\right)}}}{a} \]

      22. lift-*.f64 N/A

          \[\leadsto \frac{-1}{2} \cdot \frac{\sqrt{\sqrt{-4 \cdot \left(a \cdot c\right)} \cdot \sqrt{\mathsf{neg}\left(4 \cdot \left(a \cdot c\right)\right)}}}{a} \]

      23. lift-neg.f64 N/A

          \[\leadsto \frac{-1}{2} \cdot \frac{\sqrt{\sqrt{-4 \cdot \left(a \cdot c\right)} \cdot \sqrt{-4 \cdot \left(a \cdot c\right)}}}{a} \]

      24. lift-sqrt.f64 N/A

          \[\leadsto \frac{-1}{2} \cdot \frac{\sqrt{\sqrt{-4 \cdot \left(a \cdot c\right)} \cdot \sqrt{-4 \cdot \left(a \cdot c\right)}}}{a} \]

      25. sqr-abs-rev N/A

          \[\leadsto \frac{-1}{2} \cdot \frac{\sqrt{\left|\sqrt{-4 \cdot \left(a \cdot c\right)}\right| \cdot \left|\sqrt{-4 \cdot \left(a \cdot c\right)}\right|}}{a} \]

   6. Applied rewrites 30.0%

      \[\leadsto -0.5 \cdot \frac{\sqrt{\left|-4 \cdot \left(c \cdot a\right)\right|}}{a} \]

   if 1.7999999999999999e-134 < b

   1. Initial program 52.5%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

   2. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{b}{a}} \]

      2. lower-/.f64 36.4%

         \[\leadsto -1 \cdot \frac{b}{\color{blue}{a}} \]

   4. Applied rewrites 36.4%

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

      1. lift-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{b}{a}} \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{b}{a}\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \mathsf{neg}\left(\frac{b}{a}\right) \]

      4. distribute-neg-frac N/A

         \[\leadsto \frac{\mathsf{neg}\left(b\right)}{\color{blue}{a}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(b\right)}{\color{blue}{a}} \]

      6. lower-neg.f64 36.4%

         \[\leadsto \frac{-b}{a} \]

   6. Applied rewrites 36.4%

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

Alternative 6: 80.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;b \leq -6.5 \cdot 10^{-96}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 1.8 \cdot 10^{-134}:\\ \;\;\;\;\frac{\sqrt{-4 \cdot \left(c \cdot a\right)} \cdot -0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

FPCore

(FPCore (a b c)
  :precision binary64
  (if (<= b -6.5e-96)
  (* -1.0 (/ c b))
  (if (<= b 1.8e-134)
    (/ (* (sqrt (* -4.0 (* c a))) -0.5) a)
    (/ (- b) a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -6.5e-96) {
		tmp = -1.0 * (c / b);
	} else if (b <= 1.8e-134) {
		tmp = (sqrt((-4.0 * (c * a))) * -0.5) / a;
	} else {
		tmp = -b / a;
	}
	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(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-6.5d-96)) then
        tmp = (-1.0d0) * (c / b)
    else if (b <= 1.8d-134) then
        tmp = (sqrt(((-4.0d0) * (c * a))) * (-0.5d0)) / a
    else
        tmp = -b / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -6.5e-96) {
		tmp = -1.0 * (c / b);
	} else if (b <= 1.8e-134) {
		tmp = (Math.sqrt((-4.0 * (c * a))) * -0.5) / a;
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -6.5e-96:
		tmp = -1.0 * (c / b)
	elif b <= 1.8e-134:
		tmp = (math.sqrt((-4.0 * (c * a))) * -0.5) / a
	else:
		tmp = -b / a
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -6.5e-96)
		tmp = Float64(-1.0 * Float64(c / b));
	elseif (b <= 1.8e-134)
		tmp = Float64(Float64(sqrt(Float64(-4.0 * Float64(c * a))) * -0.5) / a);
	else
		tmp = Float64(Float64(-b) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -6.5e-96)
		tmp = -1.0 * (c / b);
	elseif (b <= 1.8e-134)
		tmp = (sqrt((-4.0 * (c * a))) * -0.5) / a;
	else
		tmp = -b / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -6.5e-96], N[(-1.0 * N[(c / b), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.8e-134], N[(N[(N[Sqrt[N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * -0.5), $MachinePrecision] / a), $MachinePrecision], N[((-b) / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -6.5e-96

   1. Initial program 52.5%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

   2. Taylor expanded in b around -inf

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

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{c}{b}} \]

      2. lower-/.f64 34.4%

         \[\leadsto -1 \cdot \frac{c}{\color{blue}{b}} \]

   4. Applied rewrites 34.4%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]

   if -6.5e-96 < b < 1.7999999999999999e-134

   1. Initial program 52.5%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

   2. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{\sqrt{\mathsf{neg}\left(4 \cdot \left(a \cdot c\right)\right)}}{a}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \color{blue}{\frac{\sqrt{\mathsf{neg}\left(4 \cdot \left(a \cdot c\right)\right)}}{a}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{\sqrt{\mathsf{neg}\left(4 \cdot \left(a \cdot c\right)\right)}}{\color{blue}{a}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{\sqrt{\mathsf{neg}\left(4 \cdot \left(a \cdot c\right)\right)}}{a} \]

      4. lower-neg.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{\sqrt{-4 \cdot \left(a \cdot c\right)}}{a} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{\sqrt{-4 \cdot \left(a \cdot c\right)}}{a} \]

      6. lower-*.f64 28.9%

         \[\leadsto -0.5 \cdot \frac{\sqrt{-4 \cdot \left(a \cdot c\right)}}{a} \]

   4. Applied rewrites 28.9%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{\sqrt{-4 \cdot \left(a \cdot c\right)}}{a}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \color{blue}{\frac{\sqrt{-4 \cdot \left(a \cdot c\right)}}{a}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{\sqrt{-4 \cdot \left(a \cdot c\right)}}{\color{blue}{a}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\frac{-1}{2} \cdot \sqrt{-4 \cdot \left(a \cdot c\right)}}{\color{blue}{a}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{2} \cdot \sqrt{-4 \cdot \left(a \cdot c\right)}}{\color{blue}{a}} \]

      5. *-commutative N/A

         \[\leadsto \frac{\sqrt{-4 \cdot \left(a \cdot c\right)} \cdot \frac{-1}{2}}{a} \]

      6. lower-*.f64 28.9%

         \[\leadsto \frac{\sqrt{-4 \cdot \left(a \cdot c\right)} \cdot -0.5}{a} \]

      7. lift-neg.f64 N/A

         \[\leadsto \frac{\sqrt{\mathsf{neg}\left(4 \cdot \left(a \cdot c\right)\right)} \cdot \frac{-1}{2}}{a} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\mathsf{neg}\left(4 \cdot \left(a \cdot c\right)\right)} \cdot \frac{-1}{2}}{a} \]

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

         \[\leadsto \frac{\sqrt{\left(\mathsf{neg}\left(4\right)\right) \cdot \left(a \cdot c\right)} \cdot \frac{-1}{2}}{a} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\sqrt{-4 \cdot \left(a \cdot c\right)} \cdot \frac{-1}{2}}{a} \]

      11. lower-*.f64 28.9%

          \[\leadsto \frac{\sqrt{-4 \cdot \left(a \cdot c\right)} \cdot -0.5}{a} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{-4 \cdot \left(a \cdot c\right)} \cdot \frac{-1}{2}}{a} \]

      13. *-commutative N/A

          \[\leadsto \frac{\sqrt{-4 \cdot \left(c \cdot a\right)} \cdot \frac{-1}{2}}{a} \]

      14. lift-*.f64 28.9%

          \[\leadsto \frac{\sqrt{-4 \cdot \left(c \cdot a\right)} \cdot -0.5}{a} \]

   6. Applied rewrites 28.9%

      \[\leadsto \frac{\sqrt{-4 \cdot \left(c \cdot a\right)} \cdot -0.5}{\color{blue}{a}} \]

   if 1.7999999999999999e-134 < b

   1. Initial program 52.5%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

   2. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{b}{a}} \]

      2. lower-/.f64 36.4%

         \[\leadsto -1 \cdot \frac{b}{\color{blue}{a}} \]

   4. Applied rewrites 36.4%

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

      1. lift-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{b}{a}} \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{b}{a}\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \mathsf{neg}\left(\frac{b}{a}\right) \]

      4. distribute-neg-frac N/A

         \[\leadsto \frac{\mathsf{neg}\left(b\right)}{\color{blue}{a}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(b\right)}{\color{blue}{a}} \]

      6. lower-neg.f64 36.4%

         \[\leadsto \frac{-b}{a} \]

   6. Applied rewrites 36.4%

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

Alternative 7: 80.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;b \leq -6.5 \cdot 10^{-96}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 1.8 \cdot 10^{-134}:\\ \;\;\;\;\sqrt{-4 \cdot \left(c \cdot a\right)} \cdot \frac{-0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

FPCore

(FPCore (a b c)
  :precision binary64
  (if (<= b -6.5e-96)
  (* -1.0 (/ c b))
  (if (<= b 1.8e-134)
    (* (sqrt (* -4.0 (* c a))) (/ -0.5 a))
    (/ (- b) a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -6.5e-96) {
		tmp = -1.0 * (c / b);
	} else if (b <= 1.8e-134) {
		tmp = sqrt((-4.0 * (c * a))) * (-0.5 / a);
	} else {
		tmp = -b / a;
	}
	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(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-6.5d-96)) then
        tmp = (-1.0d0) * (c / b)
    else if (b <= 1.8d-134) then
        tmp = sqrt(((-4.0d0) * (c * a))) * ((-0.5d0) / a)
    else
        tmp = -b / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -6.5e-96) {
		tmp = -1.0 * (c / b);
	} else if (b <= 1.8e-134) {
		tmp = Math.sqrt((-4.0 * (c * a))) * (-0.5 / a);
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -6.5e-96:
		tmp = -1.0 * (c / b)
	elif b <= 1.8e-134:
		tmp = math.sqrt((-4.0 * (c * a))) * (-0.5 / a)
	else:
		tmp = -b / a
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -6.5e-96)
		tmp = Float64(-1.0 * Float64(c / b));
	elseif (b <= 1.8e-134)
		tmp = Float64(sqrt(Float64(-4.0 * Float64(c * a))) * Float64(-0.5 / a));
	else
		tmp = Float64(Float64(-b) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -6.5e-96)
		tmp = -1.0 * (c / b);
	elseif (b <= 1.8e-134)
		tmp = sqrt((-4.0 * (c * a))) * (-0.5 / a);
	else
		tmp = -b / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -6.5e-96], N[(-1.0 * N[(c / b), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.8e-134], N[(N[Sqrt[N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-0.5 / a), $MachinePrecision]), $MachinePrecision], N[((-b) / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -6.5e-96

   1. Initial program 52.5%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

   2. Taylor expanded in b around -inf

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

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{c}{b}} \]

      2. lower-/.f64 34.4%

         \[\leadsto -1 \cdot \frac{c}{\color{blue}{b}} \]

   4. Applied rewrites 34.4%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]

   if -6.5e-96 < b < 1.7999999999999999e-134

   1. Initial program 52.5%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

   2. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{\sqrt{\mathsf{neg}\left(4 \cdot \left(a \cdot c\right)\right)}}{a}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \color{blue}{\frac{\sqrt{\mathsf{neg}\left(4 \cdot \left(a \cdot c\right)\right)}}{a}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{\sqrt{\mathsf{neg}\left(4 \cdot \left(a \cdot c\right)\right)}}{\color{blue}{a}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{\sqrt{\mathsf{neg}\left(4 \cdot \left(a \cdot c\right)\right)}}{a} \]

      4. lower-neg.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{\sqrt{-4 \cdot \left(a \cdot c\right)}}{a} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{\sqrt{-4 \cdot \left(a \cdot c\right)}}{a} \]

      6. lower-*.f64 28.9%

         \[\leadsto -0.5 \cdot \frac{\sqrt{-4 \cdot \left(a \cdot c\right)}}{a} \]

   4. Applied rewrites 28.9%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{\sqrt{-4 \cdot \left(a \cdot c\right)}}{a}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \color{blue}{\frac{\sqrt{-4 \cdot \left(a \cdot c\right)}}{a}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\sqrt{-4 \cdot \left(a \cdot c\right)}}{a} \cdot \color{blue}{\frac{-1}{2}} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\sqrt{-4 \cdot \left(a \cdot c\right)}}{a} \cdot \frac{-1}{2} \]

      4. mult-flip N/A

         \[\leadsto \left(\sqrt{-4 \cdot \left(a \cdot c\right)} \cdot \frac{1}{a}\right) \cdot \frac{-1}{2} \]

      5. associate-*l* N/A

         \[\leadsto \sqrt{-4 \cdot \left(a \cdot c\right)} \cdot \color{blue}{\left(\frac{1}{a} \cdot \frac{-1}{2}\right)} \]

      6. metadata-eval N/A

         \[\leadsto \sqrt{-4 \cdot \left(a \cdot c\right)} \cdot \left(\frac{1}{a} \cdot \frac{1}{\color{blue}{-2}}\right) \]

      7. frac-times N/A

         \[\leadsto \sqrt{-4 \cdot \left(a \cdot c\right)} \cdot \frac{1 \cdot 1}{\color{blue}{a \cdot -2}} \]

      8. metadata-eval N/A

         \[\leadsto \sqrt{-4 \cdot \left(a \cdot c\right)} \cdot \frac{1}{\color{blue}{a} \cdot -2} \]

      9. *-commutative N/A

         \[\leadsto \sqrt{-4 \cdot \left(a \cdot c\right)} \cdot \frac{1}{-2 \cdot \color{blue}{a}} \]

      10. lift-*.f64 N/A

          \[\leadsto \sqrt{-4 \cdot \left(a \cdot c\right)} \cdot \frac{1}{-2 \cdot \color{blue}{a}} \]

      11. lower-*.f64 N/A

          \[\leadsto \sqrt{-4 \cdot \left(a \cdot c\right)} \cdot \color{blue}{\frac{1}{-2 \cdot a}} \]

      12. lift-neg.f64 N/A

          \[\leadsto \sqrt{\mathsf{neg}\left(4 \cdot \left(a \cdot c\right)\right)} \cdot \frac{1}{-2 \cdot a} \]

      13. lift-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{neg}\left(4 \cdot \left(a \cdot c\right)\right)} \cdot \frac{1}{-2 \cdot a} \]

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

          \[\leadsto \sqrt{\left(\mathsf{neg}\left(4\right)\right) \cdot \left(a \cdot c\right)} \cdot \frac{1}{-2 \cdot a} \]

      15. metadata-eval N/A

          \[\leadsto \sqrt{-4 \cdot \left(a \cdot c\right)} \cdot \frac{1}{-2 \cdot a} \]

      16. lower-*.f64 N/A

          \[\leadsto \sqrt{-4 \cdot \left(a \cdot c\right)} \cdot \frac{1}{-2 \cdot a} \]

      17. lift-*.f64 N/A

          \[\leadsto \sqrt{-4 \cdot \left(a \cdot c\right)} \cdot \frac{1}{-2 \cdot a} \]

      18. *-commutative N/A

          \[\leadsto \sqrt{-4 \cdot \left(c \cdot a\right)} \cdot \frac{1}{-2 \cdot a} \]

      19. lift-*.f64 N/A

          \[\leadsto \sqrt{-4 \cdot \left(c \cdot a\right)} \cdot \frac{1}{-2 \cdot a} \]

      20. lift-*.f64 N/A

          \[\leadsto \sqrt{-4 \cdot \left(c \cdot a\right)} \cdot \frac{1}{-2 \cdot \color{blue}{a}} \]

      21. associate-/r* N/A

          \[\leadsto \sqrt{-4 \cdot \left(c \cdot a\right)} \cdot \frac{\frac{1}{-2}}{\color{blue}{a}} \]

      22. metadata-eval N/A

          \[\leadsto \sqrt{-4 \cdot \left(c \cdot a\right)} \cdot \frac{\frac{-1}{2}}{a} \]

      23. lower-/.f64 28.9%

          \[\leadsto \sqrt{-4 \cdot \left(c \cdot a\right)} \cdot \frac{-0.5}{\color{blue}{a}} \]

   6. Applied rewrites 28.9%

      \[\leadsto \sqrt{-4 \cdot \left(c \cdot a\right)} \cdot \color{blue}{\frac{-0.5}{a}} \]

   if 1.7999999999999999e-134 < b

   1. Initial program 52.5%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

   2. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{b}{a}} \]

      2. lower-/.f64 36.4%

         \[\leadsto -1 \cdot \frac{b}{\color{blue}{a}} \]

   4. Applied rewrites 36.4%

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

      1. lift-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{b}{a}} \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{b}{a}\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \mathsf{neg}\left(\frac{b}{a}\right) \]

      4. distribute-neg-frac N/A

         \[\leadsto \frac{\mathsf{neg}\left(b\right)}{\color{blue}{a}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(b\right)}{\color{blue}{a}} \]

      6. lower-neg.f64 36.4%

         \[\leadsto \frac{-b}{a} \]

   6. Applied rewrites 36.4%

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

Alternative 8: 80.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;b \leq -6.5 \cdot 10^{-96}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 1.8 \cdot 10^{-134}:\\ \;\;\;\;0.5 \cdot \left(c \cdot \sqrt{\frac{-4}{a \cdot c}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

FPCore

(FPCore (a b c)
  :precision binary64
  (if (<= b -6.5e-96)
  (* -1.0 (/ c b))
  (if (<= b 1.8e-134)
    (* 0.5 (* c (sqrt (/ -4.0 (* a c)))))
    (/ (- b) a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -6.5e-96) {
		tmp = -1.0 * (c / b);
	} else if (b <= 1.8e-134) {
		tmp = 0.5 * (c * sqrt((-4.0 / (a * c))));
	} else {
		tmp = -b / a;
	}
	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(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-6.5d-96)) then
        tmp = (-1.0d0) * (c / b)
    else if (b <= 1.8d-134) then
        tmp = 0.5d0 * (c * sqrt(((-4.0d0) / (a * c))))
    else
        tmp = -b / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -6.5e-96) {
		tmp = -1.0 * (c / b);
	} else if (b <= 1.8e-134) {
		tmp = 0.5 * (c * Math.sqrt((-4.0 / (a * c))));
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -6.5e-96:
		tmp = -1.0 * (c / b)
	elif b <= 1.8e-134:
		tmp = 0.5 * (c * math.sqrt((-4.0 / (a * c))))
	else:
		tmp = -b / a
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -6.5e-96)
		tmp = Float64(-1.0 * Float64(c / b));
	elseif (b <= 1.8e-134)
		tmp = Float64(0.5 * Float64(c * sqrt(Float64(-4.0 / Float64(a * c)))));
	else
		tmp = Float64(Float64(-b) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -6.5e-96)
		tmp = -1.0 * (c / b);
	elseif (b <= 1.8e-134)
		tmp = 0.5 * (c * sqrt((-4.0 / (a * c))));
	else
		tmp = -b / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -6.5e-96], N[(-1.0 * N[(c / b), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.8e-134], N[(0.5 * N[(c * N[Sqrt[N[(-4.0 / N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-b) / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -6.5e-96

   1. Initial program 52.5%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

   2. Taylor expanded in b around -inf

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

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{c}{b}} \]

      2. lower-/.f64 34.4%

         \[\leadsto -1 \cdot \frac{c}{\color{blue}{b}} \]

   4. Applied rewrites 34.4%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]

   if -6.5e-96 < b < 1.7999999999999999e-134

   1. Initial program 52.5%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

   2. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \sqrt{-4 \cdot \frac{c}{a}} + \frac{-1}{2} \cdot \frac{b}{a}} \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\sqrt{-4 \cdot \frac{c}{a}}}, \frac{-1}{2} \cdot \frac{b}{a}\right) \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \sqrt{-4 \cdot \frac{c}{a}}, \frac{-1}{2} \cdot \frac{b}{a}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \sqrt{-4 \cdot \frac{c}{a}}, \frac{-1}{2} \cdot \frac{b}{a}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \sqrt{-4 \cdot \frac{c}{a}}, \frac{-1}{2} \cdot \frac{b}{a}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \sqrt{-4 \cdot \frac{c}{a}}, \frac{-1}{2} \cdot \frac{b}{a}\right) \]

      6. lower-/.f64 19.6%

         \[\leadsto \mathsf{fma}\left(-0.5, \sqrt{-4 \cdot \frac{c}{a}}, -0.5 \cdot \frac{b}{a}\right) \]

   4. Applied rewrites 19.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \sqrt{-4 \cdot \frac{c}{a}}, -0.5 \cdot \frac{b}{a}\right)} \]

   5. Taylor expanded in c around -inf

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(c \cdot \sqrt{\frac{-4}{a \cdot c}}\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(c \cdot \color{blue}{\sqrt{\frac{-4}{a \cdot c}}}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(c \cdot \sqrt{\frac{-4}{a \cdot c}}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(c \cdot \sqrt{\frac{-4}{a \cdot c}}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(c \cdot \sqrt{\frac{-4}{a \cdot c}}\right) \]

      5. lower-*.f64 27.6%

         \[\leadsto 0.5 \cdot \left(c \cdot \sqrt{\frac{-4}{a \cdot c}}\right) \]

   7. Applied rewrites 27.6%

      \[\leadsto 0.5 \cdot \color{blue}{\left(c \cdot \sqrt{\frac{-4}{a \cdot c}}\right)} \]

   if 1.7999999999999999e-134 < b

   1. Initial program 52.5%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

   2. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{b}{a}} \]

      2. lower-/.f64 36.4%

         \[\leadsto -1 \cdot \frac{b}{\color{blue}{a}} \]

   4. Applied rewrites 36.4%

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

      1. lift-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{b}{a}} \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{b}{a}\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \mathsf{neg}\left(\frac{b}{a}\right) \]

      4. distribute-neg-frac N/A

         \[\leadsto \frac{\mathsf{neg}\left(b\right)}{\color{blue}{a}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(b\right)}{\color{blue}{a}} \]

      6. lower-neg.f64 36.4%

         \[\leadsto \frac{-b}{a} \]

   6. Applied rewrites 36.4%

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

Alternative 9: 72.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;b \leq -3.8 \cdot 10^{-172}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 1.8 \cdot 10^{-134}:\\ \;\;\;\;0.5 \cdot \sqrt{-4 \cdot \frac{c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

FPCore

(FPCore (a b c)
  :precision binary64
  (if (<= b -3.8e-172)
  (* -1.0 (/ c b))
  (if (<= b 1.8e-134) (* 0.5 (sqrt (* -4.0 (/ c a)))) (/ (- b) a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.8e-172) {
		tmp = -1.0 * (c / b);
	} else if (b <= 1.8e-134) {
		tmp = 0.5 * sqrt((-4.0 * (c / a)));
	} else {
		tmp = -b / a;
	}
	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(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-3.8d-172)) then
        tmp = (-1.0d0) * (c / b)
    else if (b <= 1.8d-134) then
        tmp = 0.5d0 * sqrt(((-4.0d0) * (c / a)))
    else
        tmp = -b / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.8e-172) {
		tmp = -1.0 * (c / b);
	} else if (b <= 1.8e-134) {
		tmp = 0.5 * Math.sqrt((-4.0 * (c / a)));
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -3.8e-172:
		tmp = -1.0 * (c / b)
	elif b <= 1.8e-134:
		tmp = 0.5 * math.sqrt((-4.0 * (c / a)))
	else:
		tmp = -b / a
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.8e-172)
		tmp = Float64(-1.0 * Float64(c / b));
	elseif (b <= 1.8e-134)
		tmp = Float64(0.5 * sqrt(Float64(-4.0 * Float64(c / a))));
	else
		tmp = Float64(Float64(-b) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -3.8e-172)
		tmp = -1.0 * (c / b);
	elseif (b <= 1.8e-134)
		tmp = 0.5 * sqrt((-4.0 * (c / a)));
	else
		tmp = -b / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.8e-172], N[(-1.0 * N[(c / b), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.8e-134], N[(0.5 * N[Sqrt[N[(-4.0 * N[(c / a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[((-b) / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.7999999999999999e-172

   1. Initial program 52.5%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

   2. Taylor expanded in b around -inf

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

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{c}{b}} \]

      2. lower-/.f64 34.4%

         \[\leadsto -1 \cdot \frac{c}{\color{blue}{b}} \]

   4. Applied rewrites 34.4%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]

   if -3.7999999999999999e-172 < b < 1.7999999999999999e-134

   1. Initial program 52.5%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

   2. Taylor expanded in a around -inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \sqrt{-4 \cdot \frac{c}{a}}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\sqrt{-4 \cdot \frac{c}{a}}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{-4 \cdot \frac{c}{a}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{-4 \cdot \frac{c}{a}} \]

      4. lower-/.f64 16.8%

         \[\leadsto 0.5 \cdot \sqrt{-4 \cdot \frac{c}{a}} \]

   4. Applied rewrites 16.8%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{-4 \cdot \frac{c}{a}}} \]

   if 1.7999999999999999e-134 < b

   1. Initial program 52.5%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

   2. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{b}{a}} \]

      2. lower-/.f64 36.4%

         \[\leadsto -1 \cdot \frac{b}{\color{blue}{a}} \]

   4. Applied rewrites 36.4%

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

      1. lift-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{b}{a}} \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{b}{a}\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \mathsf{neg}\left(\frac{b}{a}\right) \]

      4. distribute-neg-frac N/A

         \[\leadsto \frac{\mathsf{neg}\left(b\right)}{\color{blue}{a}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(b\right)}{\color{blue}{a}} \]

      6. lower-neg.f64 36.4%

         \[\leadsto \frac{-b}{a} \]

   6. Applied rewrites 36.4%

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

Alternative 10: 72.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;b \leq -6.6 \cdot 10^{-124}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{-135}:\\ \;\;\;\;-0.5 \cdot \sqrt{c \cdot \frac{-4}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

FPCore

(FPCore (a b c)
  :precision binary64
  (if (<= b -6.6e-124)
  (* -1.0 (/ c b))
  (if (<= b 1.35e-135) (* -0.5 (sqrt (* c (/ -4.0 a)))) (/ (- b) a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -6.6e-124) {
		tmp = -1.0 * (c / b);
	} else if (b <= 1.35e-135) {
		tmp = -0.5 * sqrt((c * (-4.0 / a)));
	} else {
		tmp = -b / a;
	}
	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(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-6.6d-124)) then
        tmp = (-1.0d0) * (c / b)
    else if (b <= 1.35d-135) then
        tmp = (-0.5d0) * sqrt((c * ((-4.0d0) / a)))
    else
        tmp = -b / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -6.6e-124) {
		tmp = -1.0 * (c / b);
	} else if (b <= 1.35e-135) {
		tmp = -0.5 * Math.sqrt((c * (-4.0 / a)));
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -6.6e-124:
		tmp = -1.0 * (c / b)
	elif b <= 1.35e-135:
		tmp = -0.5 * math.sqrt((c * (-4.0 / a)))
	else:
		tmp = -b / a
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -6.6e-124)
		tmp = Float64(-1.0 * Float64(c / b));
	elseif (b <= 1.35e-135)
		tmp = Float64(-0.5 * sqrt(Float64(c * Float64(-4.0 / a))));
	else
		tmp = Float64(Float64(-b) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -6.6e-124)
		tmp = -1.0 * (c / b);
	elseif (b <= 1.35e-135)
		tmp = -0.5 * sqrt((c * (-4.0 / a)));
	else
		tmp = -b / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -6.6e-124], N[(-1.0 * N[(c / b), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.35e-135], N[(-0.5 * N[Sqrt[N[(c * N[(-4.0 / a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[((-b) / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -6.5999999999999997e-124

   1. Initial program 52.5%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

   2. Taylor expanded in b around -inf

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

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{c}{b}} \]

      2. lower-/.f64 34.4%

         \[\leadsto -1 \cdot \frac{c}{\color{blue}{b}} \]

   4. Applied rewrites 34.4%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]

   if -6.5999999999999997e-124 < b < 1.35e-135

   1. Initial program 52.5%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

   2. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \sqrt{-4 \cdot \frac{c}{a}}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \color{blue}{\sqrt{-4 \cdot \frac{c}{a}}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \sqrt{-4 \cdot \frac{c}{a}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \sqrt{-4 \cdot \frac{c}{a}} \]

      4. lower-/.f64 16.8%

         \[\leadsto -0.5 \cdot \sqrt{-4 \cdot \frac{c}{a}} \]

   4. Applied rewrites 16.8%

      \[\leadsto \color{blue}{-0.5 \cdot \sqrt{-4 \cdot \frac{c}{a}}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \sqrt{-4 \cdot \frac{c}{a}} \]

      2. *-commutative N/A

         \[\leadsto \frac{-1}{2} \cdot \sqrt{\frac{c}{a} \cdot -4} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \sqrt{\frac{c}{a} \cdot -4} \]

      4. associate-*l/ N/A

         \[\leadsto \frac{-1}{2} \cdot \sqrt{\frac{c \cdot -4}{a}} \]

      5. associate-/l* N/A

         \[\leadsto \frac{-1}{2} \cdot \sqrt{c \cdot \frac{-4}{a}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \sqrt{c \cdot \frac{-4}{a}} \]

      7. lower-/.f64 16.8%

         \[\leadsto -0.5 \cdot \sqrt{c \cdot \frac{-4}{a}} \]

   6. Applied rewrites 16.8%

      \[\leadsto -0.5 \cdot \sqrt{c \cdot \frac{-4}{a}} \]

   if 1.35e-135 < b

   1. Initial program 52.5%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

   2. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{b}{a}} \]

      2. lower-/.f64 36.4%

         \[\leadsto -1 \cdot \frac{b}{\color{blue}{a}} \]

   4. Applied rewrites 36.4%

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

      1. lift-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{b}{a}} \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{b}{a}\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \mathsf{neg}\left(\frac{b}{a}\right) \]

      4. distribute-neg-frac N/A

         \[\leadsto \frac{\mathsf{neg}\left(b\right)}{\color{blue}{a}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(b\right)}{\color{blue}{a}} \]

      6. lower-neg.f64 36.4%

         \[\leadsto \frac{-b}{a} \]

   6. Applied rewrites 36.4%

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

Alternative 11: 72.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;b \leq -6.6 \cdot 10^{-124}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{-135}:\\ \;\;\;\;-0.5 \cdot \sqrt{-4 \cdot \frac{c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

FPCore

(FPCore (a b c)
  :precision binary64
  (if (<= b -6.6e-124)
  (* -1.0 (/ c b))
  (if (<= b 1.35e-135) (* -0.5 (sqrt (* -4.0 (/ c a)))) (/ (- b) a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -6.6e-124) {
		tmp = -1.0 * (c / b);
	} else if (b <= 1.35e-135) {
		tmp = -0.5 * sqrt((-4.0 * (c / a)));
	} else {
		tmp = -b / a;
	}
	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(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-6.6d-124)) then
        tmp = (-1.0d0) * (c / b)
    else if (b <= 1.35d-135) then
        tmp = (-0.5d0) * sqrt(((-4.0d0) * (c / a)))
    else
        tmp = -b / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -6.6e-124) {
		tmp = -1.0 * (c / b);
	} else if (b <= 1.35e-135) {
		tmp = -0.5 * Math.sqrt((-4.0 * (c / a)));
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -6.6e-124:
		tmp = -1.0 * (c / b)
	elif b <= 1.35e-135:
		tmp = -0.5 * math.sqrt((-4.0 * (c / a)))
	else:
		tmp = -b / a
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -6.6e-124)
		tmp = Float64(-1.0 * Float64(c / b));
	elseif (b <= 1.35e-135)
		tmp = Float64(-0.5 * sqrt(Float64(-4.0 * Float64(c / a))));
	else
		tmp = Float64(Float64(-b) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -6.6e-124)
		tmp = -1.0 * (c / b);
	elseif (b <= 1.35e-135)
		tmp = -0.5 * sqrt((-4.0 * (c / a)));
	else
		tmp = -b / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -6.6e-124], N[(-1.0 * N[(c / b), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.35e-135], N[(-0.5 * N[Sqrt[N[(-4.0 * N[(c / a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[((-b) / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -6.5999999999999997e-124

   1. Initial program 52.5%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

   2. Taylor expanded in b around -inf

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

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{c}{b}} \]

      2. lower-/.f64 34.4%

         \[\leadsto -1 \cdot \frac{c}{\color{blue}{b}} \]

   4. Applied rewrites 34.4%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]

   if -6.5999999999999997e-124 < b < 1.35e-135

   1. Initial program 52.5%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

   2. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \sqrt{-4 \cdot \frac{c}{a}}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \color{blue}{\sqrt{-4 \cdot \frac{c}{a}}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \sqrt{-4 \cdot \frac{c}{a}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \sqrt{-4 \cdot \frac{c}{a}} \]

      4. lower-/.f64 16.8%

         \[\leadsto -0.5 \cdot \sqrt{-4 \cdot \frac{c}{a}} \]

   4. Applied rewrites 16.8%

      \[\leadsto \color{blue}{-0.5 \cdot \sqrt{-4 \cdot \frac{c}{a}}} \]

   if 1.35e-135 < b

   1. Initial program 52.5%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

   2. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{b}{a}} \]

      2. lower-/.f64 36.4%

         \[\leadsto -1 \cdot \frac{b}{\color{blue}{a}} \]

   4. Applied rewrites 36.4%

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

      1. lift-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{b}{a}} \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{b}{a}\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \mathsf{neg}\left(\frac{b}{a}\right) \]

      4. distribute-neg-frac N/A

         \[\leadsto \frac{\mathsf{neg}\left(b\right)}{\color{blue}{a}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(b\right)}{\color{blue}{a}} \]

      6. lower-neg.f64 36.4%

         \[\leadsto \frac{-b}{a} \]

   6. Applied rewrites 36.4%

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

Alternative 12: 68.3% accurate, 2.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-308}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

FPCore

(FPCore (a b c)
  :precision binary64
  (if (<= b -5e-308) (* -1.0 (/ c b)) (/ (- b) a)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-308) {
		tmp = -1.0 * (c / b);
	} else {
		tmp = -b / a;
	}
	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(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-5d-308)) then
        tmp = (-1.0d0) * (c / b)
    else
        tmp = -b / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-308) {
		tmp = -1.0 * (c / b);
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e-308:
		tmp = -1.0 * (c / b)
	else:
		tmp = -b / a
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e-308)
		tmp = Float64(-1.0 * Float64(c / b));
	else
		tmp = Float64(Float64(-b) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5e-308)
		tmp = -1.0 * (c / b);
	else
		tmp = -b / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-308], N[(-1.0 * N[(c / b), $MachinePrecision]), $MachinePrecision], N[((-b) / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -4.9999999999999995e-308

   1. Initial program 52.5%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

   2. Taylor expanded in b around -inf

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

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{c}{b}} \]

      2. lower-/.f64 34.4%

         \[\leadsto -1 \cdot \frac{c}{\color{blue}{b}} \]

   4. Applied rewrites 34.4%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]

   if -4.9999999999999995e-308 < b

   1. Initial program 52.5%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

   2. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{b}{a}} \]

      2. lower-/.f64 36.4%

         \[\leadsto -1 \cdot \frac{b}{\color{blue}{a}} \]

   4. Applied rewrites 36.4%

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

      1. lift-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{b}{a}} \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{b}{a}\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \mathsf{neg}\left(\frac{b}{a}\right) \]

      4. distribute-neg-frac N/A

         \[\leadsto \frac{\mathsf{neg}\left(b\right)}{\color{blue}{a}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(b\right)}{\color{blue}{a}} \]

      6. lower-neg.f64 36.4%

         \[\leadsto \frac{-b}{a} \]

   6. Applied rewrites 36.4%

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

Alternative 13: 36.4% accurate, 4.3× speedup

Math

\[\frac{-b}{a} \]

FPCore

(FPCore (a b c)
  :precision binary64
  (/ (- b) a))

C

double code(double a, double b, double c) {
	return -b / a;
}

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(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = -b / a
end function

Java

public static double code(double a, double b, double c) {
	return -b / a;
}

Python

def code(a, b, c):
	return -b / a

Julia

function code(a, b, c)
	return Float64(Float64(-b) / a)
end

MATLAB

function tmp = code(a, b, c)
	tmp = -b / a;
end

Wolfram

code[a_, b_, c_] := N[((-b) / a), $MachinePrecision]

Derivation

1. Initial program 52.5%

   \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

2. Taylor expanded in b around inf

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

   1. lower-*.f64 N/A

      \[\leadsto -1 \cdot \color{blue}{\frac{b}{a}} \]

   2. lower-/.f64 36.4%

      \[\leadsto -1 \cdot \frac{b}{\color{blue}{a}} \]

4. Applied rewrites 36.4%

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

   1. lift-*.f64 N/A

      \[\leadsto -1 \cdot \color{blue}{\frac{b}{a}} \]

   2. mul-1-neg N/A

      \[\leadsto \mathsf{neg}\left(\frac{b}{a}\right) \]

   3. lift-/.f64 N/A

      \[\leadsto \mathsf{neg}\left(\frac{b}{a}\right) \]

   4. distribute-neg-frac N/A

      \[\leadsto \frac{\mathsf{neg}\left(b\right)}{\color{blue}{a}} \]

   5. lower-/.f64 N/A

      \[\leadsto \frac{\mathsf{neg}\left(b\right)}{\color{blue}{a}} \]

   6. lower-neg.f64 36.4%

      \[\leadsto \frac{-b}{a} \]

6. Applied rewrites 36.4%

   \[\leadsto \frac{-b}{\color{blue}{a}} \]
7. Add Preprocessing

Developer Target 1: 70.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\\ \mathbf{if}\;b < 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + t\_0}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - t\_0}{2 \cdot a}\\ \end{array} \]

FPCore

(FPCore (a b c)
  :precision binary64
  (let* ((t_0 (sqrt (- (* b b) (* 4.0 (* a c))))))
  (if (< b 0.0)
    (/ c (* a (/ (+ (- b) t_0) (* 2.0 a))))
    (/ (- (- b) t_0) (* 2.0 a)))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((b * b) - (4.0 * (a * c))));
	double tmp;
	if (b < 0.0) {
		tmp = c / (a * ((-b + t_0) / (2.0 * a)));
	} else {
		tmp = (-b - t_0) / (2.0 * a);
	}
	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(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(((b * b) - (4.0d0 * (a * c))))
    if (b < 0.0d0) then
        tmp = c / (a * ((-b + t_0) / (2.0d0 * a)))
    else
        tmp = (-b - t_0) / (2.0d0 * a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.sqrt(((b * b) - (4.0 * (a * c))));
	double tmp;
	if (b < 0.0) {
		tmp = c / (a * ((-b + t_0) / (2.0 * a)));
	} else {
		tmp = (-b - t_0) / (2.0 * a);
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = math.sqrt(((b * b) - (4.0 * (a * c))))
	tmp = 0
	if b < 0.0:
		tmp = c / (a * ((-b + t_0) / (2.0 * a)))
	else:
		tmp = (-b - t_0) / (2.0 * a)
	return tmp

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c))))
	tmp = 0.0
	if (b < 0.0)
		tmp = Float64(c / Float64(a * Float64(Float64(Float64(-b) + t_0) / Float64(2.0 * a))));
	else
		tmp = Float64(Float64(Float64(-b) - t_0) / Float64(2.0 * a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = sqrt(((b * b) - (4.0 * (a * c))));
	tmp = 0.0;
	if (b < 0.0)
		tmp = c / (a * ((-b + t_0) / (2.0 * a)));
	else
		tmp = (-b - t_0) / (2.0 * a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[Less[b, 0.0], N[(c / N[(a * N[(N[((-b) + t$95$0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-b) - t$95$0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]]]

Reproduce

herbie shell --seed 2025253 
(FPCore (a b c)
  :name "The quadratic formula (r2)"
  :precision binary64

  :alt
  (! :herbie-platform c (let ((d (sqrt (- (* b b) (* 4 (* a c)))))) (let ((r1 (/ (+ (- b) d) (* 2 a)))) (let ((r2 (/ (- (- b) d) (* 2 a)))) (if (< b 0) (/ c (* a r1)) r2)))))

  (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))