The quadratic formula (r2)

Percentage Accurate: 52.6% → 85.1%

Time: 4.1s

Alternatives: 12

Speedup: 2.6×

Specification

Math

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

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: 52.6% accurate, 1.0× speedup

Math

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

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: 85.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.8 \cdot 10^{-17}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \leq 3.2 \cdot 10^{+89}:\\ \;\;\;\;\frac{b}{a} \cdot -0.5 - \frac{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}}{a + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} + \frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.8e-17)
   (- (/ c b))
   (if (<= b 3.2e+89)
     (- (* (/ b a) -0.5) (/ (sqrt (fma (* -4.0 a) c (* b b))) (+ a a)))
     (+ (/ c b) (/ (- b) a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.8e-17) {
		tmp = -(c / b);
	} else if (b <= 3.2e+89) {
		tmp = ((b / a) * -0.5) - (sqrt(fma((-4.0 * a), c, (b * b))) / (a + a));
	} else {
		tmp = (c / b) + (-b / a);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.8e-17)
		tmp = Float64(-Float64(c / b));
	elseif (b <= 3.2e+89)
		tmp = Float64(Float64(Float64(b / a) * -0.5) - Float64(sqrt(fma(Float64(-4.0 * a), c, Float64(b * b))) / Float64(a + a)));
	else
		tmp = Float64(Float64(c / b) + Float64(Float64(-b) / a));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.8e-17], (-N[(c / b), $MachinePrecision]), If[LessEqual[b, 3.2e+89], N[(N[(N[(b / a), $MachinePrecision] * -0.5), $MachinePrecision] - N[(N[Sqrt[N[(N[(-4.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] + N[((-b) / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.7999999999999999e-17

   1. Initial program 15.6%

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

   2. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 89.6

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

   4. Applied rewrites 89.6%

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

   if -2.7999999999999999e-17 < b < 3.19999999999999987e89

   1. Initial program 76.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. div-sub N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. count-2-rev N/A

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

      13. lower-+.f64 N/A

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

      14. lower-/.f64 N/A

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

   3. Applied rewrites 76.9%

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

   4. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{b}{a}} - \frac{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}}{a + a} \]
   5. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-/.f64 76.9

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

   6. Applied rewrites 76.9%

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

   if 3.19999999999999987e89 < b

   1. Initial program 55.2%

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

   2. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. mul-1-neg N/A

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

      6. lift-neg.f64 N/A

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

      7. lower-/.f64 95.3

         \[\leadsto \frac{c}{b} + \frac{-b}{\color{blue}{a}} \]

   4. Applied rewrites 95.3%

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

Alternative 2: 85.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.8 \cdot 10^{-17}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \leq 3.2 \cdot 10^{+89}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}}{a + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} + \frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.8e-17)
   (- (/ c b))
   (if (<= b 3.2e+89)
     (/ (- (- b) (sqrt (fma (* -4.0 a) c (* b b)))) (+ a a))
     (+ (/ c b) (/ (- b) a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.8e-17) {
		tmp = -(c / b);
	} else if (b <= 3.2e+89) {
		tmp = (-b - sqrt(fma((-4.0 * a), c, (b * b)))) / (a + a);
	} else {
		tmp = (c / b) + (-b / a);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.8e-17)
		tmp = Float64(-Float64(c / b));
	elseif (b <= 3.2e+89)
		tmp = Float64(Float64(Float64(-b) - sqrt(fma(Float64(-4.0 * a), c, Float64(b * b)))) / Float64(a + a));
	else
		tmp = Float64(Float64(c / b) + Float64(Float64(-b) / a));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.8e-17], (-N[(c / b), $MachinePrecision]), If[LessEqual[b, 3.2e+89], N[(N[((-b) - N[Sqrt[N[(N[(-4.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] + N[((-b) / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.7999999999999999e-17

   1. Initial program 15.6%

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

   2. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 89.6

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

   4. Applied rewrites 89.6%

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

   if -2.7999999999999999e-17 < b < 3.19999999999999987e89

   1. Initial program 76.9%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 76.9

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

      14. lift-*.f64 N/A

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

      15. count-2-rev N/A

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

      16. lower-+.f64 76.9

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

   3. Applied rewrites 76.9%

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

   if 3.19999999999999987e89 < b

   1. Initial program 55.2%

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

   2. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. mul-1-neg N/A

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

      6. lift-neg.f64 N/A

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

      7. lower-/.f64 95.3

         \[\leadsto \frac{c}{b} + \frac{-b}{\color{blue}{a}} \]

   4. Applied rewrites 95.3%

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

Alternative 3: 85.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.8 \cdot 10^{-17}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \leq 3.2 \cdot 10^{+89}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} + \frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.8e-17)
   (- (/ c b))
   (if (<= b 3.2e+89)
     (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a))
     (+ (/ c b) (/ (- b) a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.8e-17) {
		tmp = -(c / b);
	} else if (b <= 3.2e+89) {
		tmp = (-b - sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
	} else {
		tmp = (c / b) + (-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 <= (-2.8d-17)) then
        tmp = -(c / b)
    else if (b <= 3.2d+89) then
        tmp = (-b - sqrt(((b * b) - (4.0d0 * (a * c))))) / (2.0d0 * a)
    else
        tmp = (c / b) + (-b / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.8e-17) {
		tmp = -(c / b);
	} else if (b <= 3.2e+89) {
		tmp = (-b - Math.sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
	} else {
		tmp = (c / b) + (-b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.8e-17:
		tmp = -(c / b)
	elif b <= 3.2e+89:
		tmp = (-b - math.sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a)
	else:
		tmp = (c / b) + (-b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.8e-17)
		tmp = Float64(-Float64(c / b));
	elseif (b <= 3.2e+89)
		tmp = Float64(Float64(Float64(-b) - sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c))))) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(c / b) + Float64(Float64(-b) / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.8e-17)
		tmp = -(c / b);
	elseif (b <= 3.2e+89)
		tmp = (-b - sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
	else
		tmp = (c / b) + (-b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.8e-17], (-N[(c / b), $MachinePrecision]), If[LessEqual[b, 3.2e+89], 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], N[(N[(c / b), $MachinePrecision] + N[((-b) / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.7999999999999999e-17

   1. Initial program 15.6%

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

   2. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 89.6

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

   4. Applied rewrites 89.6%

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

   if -2.7999999999999999e-17 < b < 3.19999999999999987e89

   1. Initial program 76.9%

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

   if 3.19999999999999987e89 < b

   1. Initial program 55.2%

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

   2. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. mul-1-neg N/A

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

      6. lift-neg.f64 N/A

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

      7. lower-/.f64 95.3

         \[\leadsto \frac{c}{b} + \frac{-b}{\color{blue}{a}} \]

   4. Applied rewrites 95.3%

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

Alternative 4: 80.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.9 \cdot 10^{-91}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \leq 1.5 \cdot 10^{-98}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\left(a \cdot -4\right) \cdot c}}{a + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} + \frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.9e-91)
   (- (/ c b))
   (if (<= b 1.5e-98)
     (/ (- (- b) (sqrt (* (* a -4.0) c))) (+ a a))
     (+ (/ c b) (/ (- b) a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.9e-91) {
		tmp = -(c / b);
	} else if (b <= 1.5e-98) {
		tmp = (-b - sqrt(((a * -4.0) * c))) / (a + a);
	} else {
		tmp = (c / b) + (-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 <= (-1.9d-91)) then
        tmp = -(c / b)
    else if (b <= 1.5d-98) then
        tmp = (-b - sqrt(((a * (-4.0d0)) * c))) / (a + a)
    else
        tmp = (c / b) + (-b / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.9e-91) {
		tmp = -(c / b);
	} else if (b <= 1.5e-98) {
		tmp = (-b - Math.sqrt(((a * -4.0) * c))) / (a + a);
	} else {
		tmp = (c / b) + (-b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.9e-91:
		tmp = -(c / b)
	elif b <= 1.5e-98:
		tmp = (-b - math.sqrt(((a * -4.0) * c))) / (a + a)
	else:
		tmp = (c / b) + (-b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.9e-91)
		tmp = Float64(-Float64(c / b));
	elseif (b <= 1.5e-98)
		tmp = Float64(Float64(Float64(-b) - sqrt(Float64(Float64(a * -4.0) * c))) / Float64(a + a));
	else
		tmp = Float64(Float64(c / b) + Float64(Float64(-b) / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.9e-91)
		tmp = -(c / b);
	elseif (b <= 1.5e-98)
		tmp = (-b - sqrt(((a * -4.0) * c))) / (a + a);
	else
		tmp = (c / b) + (-b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.9e-91], (-N[(c / b), $MachinePrecision]), If[LessEqual[b, 1.5e-98], N[(N[((-b) - N[Sqrt[N[(N[(a * -4.0), $MachinePrecision] * c), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] + N[((-b) / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.89999999999999989e-91

   1. Initial program 19.4%

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

   2. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 84.4

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

   4. Applied rewrites 84.4%

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

   if -1.89999999999999989e-91 < b < 1.5e-98

   1. Initial program 75.0%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 75.0

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

      14. lift-*.f64 N/A

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

      15. count-2-rev N/A

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

      16. lower-+.f64 75.0

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

   3. Applied rewrites 75.0%

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

   4. Taylor expanded in a around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 70.8

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

   6. Applied rewrites 70.8%

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

   if 1.5e-98 < b

   1. Initial program 69.9%

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

   2. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. mul-1-neg N/A

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

      6. lift-neg.f64 N/A

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

      7. lower-/.f64 83.2

         \[\leadsto \frac{c}{b} + \frac{-b}{\color{blue}{a}} \]

   4. Applied rewrites 83.2%

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

Alternative 5: 79.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.8 \cdot 10^{-17}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \leq 6.2 \cdot 10^{-109}:\\ \;\;\;\;\frac{-\sqrt{\left(a \cdot -4\right) \cdot c}}{a + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} + \frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.8e-17)
   (- (/ c b))
   (if (<= b 6.2e-109)
     (/ (- (sqrt (* (* a -4.0) c))) (+ a a))
     (+ (/ c b) (/ (- b) a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.8e-17) {
		tmp = -(c / b);
	} else if (b <= 6.2e-109) {
		tmp = -sqrt(((a * -4.0) * c)) / (a + a);
	} else {
		tmp = (c / b) + (-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 <= (-2.8d-17)) then
        tmp = -(c / b)
    else if (b <= 6.2d-109) then
        tmp = -sqrt(((a * (-4.0d0)) * c)) / (a + a)
    else
        tmp = (c / b) + (-b / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.8e-17) {
		tmp = -(c / b);
	} else if (b <= 6.2e-109) {
		tmp = -Math.sqrt(((a * -4.0) * c)) / (a + a);
	} else {
		tmp = (c / b) + (-b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.8e-17:
		tmp = -(c / b)
	elif b <= 6.2e-109:
		tmp = -math.sqrt(((a * -4.0) * c)) / (a + a)
	else:
		tmp = (c / b) + (-b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.8e-17)
		tmp = Float64(-Float64(c / b));
	elseif (b <= 6.2e-109)
		tmp = Float64(Float64(-sqrt(Float64(Float64(a * -4.0) * c))) / Float64(a + a));
	else
		tmp = Float64(Float64(c / b) + Float64(Float64(-b) / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.8e-17)
		tmp = -(c / b);
	elseif (b <= 6.2e-109)
		tmp = -sqrt(((a * -4.0) * c)) / (a + a);
	else
		tmp = (c / b) + (-b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.8e-17], (-N[(c / b), $MachinePrecision]), If[LessEqual[b, 6.2e-109], N[((-N[Sqrt[N[(N[(a * -4.0), $MachinePrecision] * c), $MachinePrecision]], $MachinePrecision]) / N[(a + a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] + N[((-b) / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.7999999999999999e-17

   1. Initial program 15.6%

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

   2. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 89.6

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

   4. Applied rewrites 89.6%

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

   if -2.7999999999999999e-17 < b < 6.1999999999999999e-109

   1. Initial program 69.2%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 69.3

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

      14. lift-*.f64 N/A

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

      15. count-2-rev N/A

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

      16. lower-+.f64 69.3

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

   3. Applied rewrites 69.3%

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

   4. Taylor expanded in a around inf

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

      1. pow2 N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. metadata-eval N/A

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

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

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

      6. pow2 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. sqrt-unprod N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 65.0

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

   6. Applied rewrites 65.0%

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

   if 6.1999999999999999e-109 < b

   1. Initial program 70.0%

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

   2. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. mul-1-neg N/A

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

      6. lift-neg.f64 N/A

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

      7. lower-/.f64 82.5

         \[\leadsto \frac{c}{b} + \frac{-b}{\color{blue}{a}} \]

   4. Applied rewrites 82.5%

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

Alternative 6: 79.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.8 \cdot 10^{-17}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \leq 6.2 \cdot 10^{-109}:\\ \;\;\;\;\sqrt{\frac{-1}{c \cdot a}} \cdot c\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} + \frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.8e-17)
   (- (/ c b))
   (if (<= b 6.2e-109) (* (sqrt (/ -1.0 (* c a))) c) (+ (/ c b) (/ (- b) a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.8e-17) {
		tmp = -(c / b);
	} else if (b <= 6.2e-109) {
		tmp = sqrt((-1.0 / (c * a))) * c;
	} else {
		tmp = (c / b) + (-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 <= (-2.8d-17)) then
        tmp = -(c / b)
    else if (b <= 6.2d-109) then
        tmp = sqrt(((-1.0d0) / (c * a))) * c
    else
        tmp = (c / b) + (-b / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.8e-17) {
		tmp = -(c / b);
	} else if (b <= 6.2e-109) {
		tmp = Math.sqrt((-1.0 / (c * a))) * c;
	} else {
		tmp = (c / b) + (-b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.8e-17:
		tmp = -(c / b)
	elif b <= 6.2e-109:
		tmp = math.sqrt((-1.0 / (c * a))) * c
	else:
		tmp = (c / b) + (-b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.8e-17)
		tmp = Float64(-Float64(c / b));
	elseif (b <= 6.2e-109)
		tmp = Float64(sqrt(Float64(-1.0 / Float64(c * a))) * c);
	else
		tmp = Float64(Float64(c / b) + Float64(Float64(-b) / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.8e-17)
		tmp = -(c / b);
	elseif (b <= 6.2e-109)
		tmp = sqrt((-1.0 / (c * a))) * c;
	else
		tmp = (c / b) + (-b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.8e-17], (-N[(c / b), $MachinePrecision]), If[LessEqual[b, 6.2e-109], N[(N[Sqrt[N[(-1.0 / N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision], N[(N[(c / b), $MachinePrecision] + N[((-b) / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.7999999999999999e-17

   1. Initial program 15.6%

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

   2. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 89.6

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

   4. Applied rewrites 89.6%

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

   if -2.7999999999999999e-17 < b < 6.1999999999999999e-109

   1. Initial program 69.2%

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

   2. Taylor expanded in a around -inf

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. sqrt-unprod N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 31.2

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

   4. Applied rewrites 31.2%

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

   5. Taylor expanded in c around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. sqrt-unprod N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 64.2

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

   7. Applied rewrites 64.2%

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

   8. Taylor expanded in a around inf

      \[\leadsto \left(\sqrt{\frac{1}{a \cdot c}} \cdot \sqrt{-1}\right) \cdot c \]
   9. Step-by-step derivation

      1. *-commutative N/A

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

      2. sqrt-prod N/A

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

      3. lower-sqrt.f64 N/A

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

      4. associate-*l/ N/A

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 64.3

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

   10. Applied rewrites 64.3%

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

   if 6.1999999999999999e-109 < b

   1. Initial program 70.0%

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

   2. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. mul-1-neg N/A

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

      6. lift-neg.f64 N/A

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

      7. lower-/.f64 82.5

         \[\leadsto \frac{c}{b} + \frac{-b}{\color{blue}{a}} \]

   4. Applied rewrites 82.5%

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

Alternative 7: 71.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.3 \cdot 10^{-158}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \leq 1.85 \cdot 10^{-196}:\\ \;\;\;\;-\sqrt{-\frac{c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} + \frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.3e-158)
   (- (/ c b))
   (if (<= b 1.85e-196) (- (sqrt (- (/ c a)))) (+ (/ c b) (/ (- b) a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.3e-158) {
		tmp = -(c / b);
	} else if (b <= 1.85e-196) {
		tmp = -sqrt(-(c / a));
	} else {
		tmp = (c / b) + (-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 <= (-2.3d-158)) then
        tmp = -(c / b)
    else if (b <= 1.85d-196) then
        tmp = -sqrt(-(c / a))
    else
        tmp = (c / b) + (-b / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.3e-158) {
		tmp = -(c / b);
	} else if (b <= 1.85e-196) {
		tmp = -Math.sqrt(-(c / a));
	} else {
		tmp = (c / b) + (-b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.3e-158:
		tmp = -(c / b)
	elif b <= 1.85e-196:
		tmp = -math.sqrt(-(c / a))
	else:
		tmp = (c / b) + (-b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.3e-158)
		tmp = Float64(-Float64(c / b));
	elseif (b <= 1.85e-196)
		tmp = Float64(-sqrt(Float64(-Float64(c / a))));
	else
		tmp = Float64(Float64(c / b) + Float64(Float64(-b) / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.3e-158)
		tmp = -(c / b);
	elseif (b <= 1.85e-196)
		tmp = -sqrt(-(c / a));
	else
		tmp = (c / b) + (-b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.3e-158], (-N[(c / b), $MachinePrecision]), If[LessEqual[b, 1.85e-196], (-N[Sqrt[(-N[(c / a), $MachinePrecision])], $MachinePrecision]), N[(N[(c / b), $MachinePrecision] + N[((-b) / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.2999999999999999e-158

   1. Initial program 24.2%

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

   2. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 79.1

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

   4. Applied rewrites 79.1%

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

   if -2.2999999999999999e-158 < b < 1.85000000000000005e-196

   1. Initial program 74.0%

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

   2. Taylor expanded in a around -inf

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. sqrt-unprod N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 36.7

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

   4. Applied rewrites 36.7%

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

   5. Taylor expanded in c around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. sqrt-prod N/A

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

      4. lower-sqrt.f64 N/A

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

      5. *-commutative N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

         \[\leadsto -\sqrt{-\frac{c}{a}} \]

      8. lift-/.f64 35.6

         \[\leadsto -\sqrt{-\frac{c}{a}} \]

   7. Applied rewrites 35.6%

      \[\leadsto -\sqrt{-\frac{c}{a}} \]

   if 1.85000000000000005e-196 < b

   1. Initial program 71.7%

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

   2. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. mul-1-neg N/A

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

      6. lift-neg.f64 N/A

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

      7. lower-/.f64 75.3

         \[\leadsto \frac{c}{b} + \frac{-b}{\color{blue}{a}} \]

   4. Applied rewrites 75.3%

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

Alternative 8: 70.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.3 \cdot 10^{-158}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \leq 1.85 \cdot 10^{-196}:\\ \;\;\;\;-\sqrt{-\frac{c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.3e-158)
   (- (/ c b))
   (if (<= b 1.85e-196) (- (sqrt (- (/ c a)))) (/ (- b) a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.3e-158) {
		tmp = -(c / b);
	} else if (b <= 1.85e-196) {
		tmp = -sqrt(-(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 <= (-2.3d-158)) then
        tmp = -(c / b)
    else if (b <= 1.85d-196) then
        tmp = -sqrt(-(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 <= -2.3e-158) {
		tmp = -(c / b);
	} else if (b <= 1.85e-196) {
		tmp = -Math.sqrt(-(c / a));
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.3e-158:
		tmp = -(c / b)
	elif b <= 1.85e-196:
		tmp = -math.sqrt(-(c / a))
	else:
		tmp = -b / a
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.3e-158)
		tmp = Float64(-Float64(c / b));
	elseif (b <= 1.85e-196)
		tmp = Float64(-sqrt(Float64(-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 <= -2.3e-158)
		tmp = -(c / b);
	elseif (b <= 1.85e-196)
		tmp = -sqrt(-(c / a));
	else
		tmp = -b / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.3e-158], (-N[(c / b), $MachinePrecision]), If[LessEqual[b, 1.85e-196], (-N[Sqrt[(-N[(c / a), $MachinePrecision])], $MachinePrecision]), N[((-b) / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.2999999999999999e-158

   1. Initial program 24.2%

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

   2. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 79.1

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

   4. Applied rewrites 79.1%

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

   if -2.2999999999999999e-158 < b < 1.85000000000000005e-196

   1. Initial program 74.0%

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

   2. Taylor expanded in a around -inf

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. sqrt-unprod N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 36.7

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

   4. Applied rewrites 36.7%

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

   5. Taylor expanded in c around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. sqrt-prod N/A

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

      4. lower-sqrt.f64 N/A

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

      5. *-commutative N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

         \[\leadsto -\sqrt{-\frac{c}{a}} \]

      8. lift-/.f64 35.6

         \[\leadsto -\sqrt{-\frac{c}{a}} \]

   7. Applied rewrites 35.6%

      \[\leadsto -\sqrt{-\frac{c}{a}} \]

   if 1.85000000000000005e-196 < b

   1. Initial program 71.7%

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

   2. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lift-neg.f64 N/A

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

      4. lower-/.f64 74.8

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

   4. Applied rewrites 74.8%

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

Alternative 9: 70.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.25 \cdot 10^{-91}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \leq 1.2 \cdot 10^{-223}:\\ \;\;\;\;\sqrt{-\frac{c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.25e-91)
   (- (/ c b))
   (if (<= b 1.2e-223) (sqrt (- (/ c a))) (/ (- b) a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.25e-91) {
		tmp = -(c / b);
	} else if (b <= 1.2e-223) {
		tmp = sqrt(-(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 <= (-1.25d-91)) then
        tmp = -(c / b)
    else if (b <= 1.2d-223) then
        tmp = sqrt(-(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 <= -1.25e-91) {
		tmp = -(c / b);
	} else if (b <= 1.2e-223) {
		tmp = Math.sqrt(-(c / a));
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.25e-91:
		tmp = -(c / b)
	elif b <= 1.2e-223:
		tmp = math.sqrt(-(c / a))
	else:
		tmp = -b / a
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.25e-91)
		tmp = Float64(-Float64(c / b));
	elseif (b <= 1.2e-223)
		tmp = sqrt(Float64(-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 <= -1.25e-91)
		tmp = -(c / b);
	elseif (b <= 1.2e-223)
		tmp = sqrt(-(c / a));
	else
		tmp = -b / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.25e-91], (-N[(c / b), $MachinePrecision]), If[LessEqual[b, 1.2e-223], N[Sqrt[(-N[(c / a), $MachinePrecision])], $MachinePrecision], N[((-b) / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.24999999999999999e-91

   1. Initial program 19.5%

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

   2. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 84.4

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

   4. Applied rewrites 84.4%

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

   if -1.24999999999999999e-91 < b < 1.19999999999999993e-223

   1. Initial program 71.7%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 71.8

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

      14. lift-*.f64 N/A

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

      15. count-2-rev N/A

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

      16. lower-+.f64 71.8

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

   3. Applied rewrites 71.8%

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

   4. Taylor expanded in a around -inf

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

   6. Applied rewrites 34.3%

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

   if 1.19999999999999993e-223 < b

   1. Initial program 72.0%

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

   2. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lift-neg.f64 N/A

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

      4. lower-/.f64 72.3

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

   4. Applied rewrites 72.3%

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

Alternative 10: 67.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.3 \cdot 10^{-296}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.3e-296) (- (/ c b)) (/ (- b) a)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.3e-296) {
		tmp = -(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 <= (-1.3d-296)) then
        tmp = -(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 <= -1.3e-296) {
		tmp = -(c / b);
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.3e-296:
		tmp = -(c / b)
	else:
		tmp = -b / a
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.3e-296)
		tmp = Float64(-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 <= -1.3e-296)
		tmp = -(c / b);
	else
		tmp = -b / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.3e-296], (-N[(c / b), $MachinePrecision]), N[((-b) / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.3e-296

   1. Initial program 31.8%

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

   2. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 68.8

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

   4. Applied rewrites 68.8%

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

   if -1.3e-296 < b

   1. Initial program 72.2%

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

   2. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lift-neg.f64 N/A

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

      4. lower-/.f64 65.6

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

   4. Applied rewrites 65.6%

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

Alternative 11: 34.6% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ -\frac{c}{b} \end{array} \]

FPCore

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

C

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

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 = -(c / b)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.6%

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

2. Taylor expanded in b around -inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. lower-/.f64 34.6

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

4. Applied rewrites 34.6%

   \[\leadsto \color{blue}{-\frac{c}{b}} \]
5. Add Preprocessing

Alternative 12: 10.8% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \frac{c}{b} \end{array} \]

FPCore

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

C

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

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 = c / b
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.6%

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

2. Taylor expanded in c around 0

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

   1. +-commutative N/A

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

   2. lower-+.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. associate-*r/ N/A

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

   5. mul-1-neg N/A

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

   6. lift-neg.f64 N/A

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

   7. lower-/.f64 35.0

      \[\leadsto \frac{c}{b} + \frac{-b}{\color{blue}{a}} \]

4. Applied rewrites 35.0%

   \[\leadsto \color{blue}{\frac{c}{b} + \frac{-b}{a}} \]

5. Taylor expanded in a around inf

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

   1. lift-/.f64 10.8

      \[\leadsto \frac{c}{b} \]

7. Applied rewrites 10.8%

   \[\leadsto \frac{c}{\color{blue}{b}} \]
8. Add Preprocessing

Developer Target 1: 70.5% accurate, 0.7× speedup

Math

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

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 2025117 
(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)))