quadp (p42, positive)

Percentage Accurate: 52.6% → 85.9%

Time: 4.4s

Alternatives: 6

Speedup: 2.5×

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: 85.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.8 \cdot 10^{+114}:\\ \;\;\;\;-\mathsf{fma}\left(\frac{-c}{b \cdot b}, b, \frac{b}{a}\right)\\ \mathbf{elif}\;b \leq 5.9 \cdot 10^{-47}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - b}{a + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.8e+114)
   (- (fma (/ (- c) (* b b)) b (/ b a)))
   (if (<= b 5.9e-47)
     (/ (- (sqrt (fma -4.0 (* c a) (* b b))) b) (+ a a))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.8e+114) {
		tmp = -fma((-c / (b * b)), b, (b / a));
	} else if (b <= 5.9e-47) {
		tmp = (sqrt(fma(-4.0, (c * a), (b * b))) - b) / (a + a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.8e+114)
		tmp = Float64(-fma(Float64(Float64(-c) / Float64(b * b)), b, Float64(b / a)));
	elseif (b <= 5.9e-47)
		tmp = Float64(Float64(sqrt(fma(-4.0, Float64(c * a), Float64(b * b))) - b) / Float64(a + a));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.8e+114], (-N[(N[((-c) / N[(b * b), $MachinePrecision]), $MachinePrecision] * b + N[(b / a), $MachinePrecision]), $MachinePrecision]), If[LessEqual[b, 5.9e-47], N[(N[(N[Sqrt[N[(-4.0 * N[(c * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.8e114

   1. Initial program 60.1%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-*l/ N/A

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

      5. *-lft-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. associate-*r/ N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 94.5

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

   5. Applied rewrites 94.5%

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

   if -2.8e114 < b < 5.89999999999999973e-47

   1. Initial program 76.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 76.0

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. metadata-eval 76.0

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 76.0

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

   4. Applied rewrites 76.0%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 76.0

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

   6. Applied rewrites 76.0%

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

   if 5.89999999999999973e-47 < b

   1. Initial program 11.3%

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

   3. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 90.6

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

   5. Applied rewrites 90.6%

      \[\leadsto \color{blue}{\frac{-c}{b}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.8 \cdot 10^{+114}:\\ \;\;\;\;-\mathsf{fma}\left(\frac{-c}{b \cdot b}, b, \frac{b}{a}\right)\\ \mathbf{elif}\;b \leq 5.9 \cdot 10^{-47}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - b}{a + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 80.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5.2 \cdot 10^{-35}:\\ \;\;\;\;-\mathsf{fma}\left(\frac{-c}{b \cdot b}, b, \frac{b}{a}\right)\\ \mathbf{elif}\;b \leq 5.9 \cdot 10^{-47}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5.2e-35)
   (- (fma (/ (- c) (* b b)) b (/ b a)))
   (if (<= b 5.9e-47) (/ (- (sqrt (* (* a c) -4.0)) b) (+ a a)) (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5.2e-35) {
		tmp = -fma((-c / (b * b)), b, (b / a));
	} else if (b <= 5.9e-47) {
		tmp = (sqrt(((a * c) * -4.0)) - b) / (a + a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5.2e-35)
		tmp = Float64(-fma(Float64(Float64(-c) / Float64(b * b)), b, Float64(b / a)));
	elseif (b <= 5.9e-47)
		tmp = Float64(Float64(sqrt(Float64(Float64(a * c) * -4.0)) - b) / Float64(a + a));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5.2e-35], (-N[(N[((-c) / N[(b * b), $MachinePrecision]), $MachinePrecision] * b + N[(b / a), $MachinePrecision]), $MachinePrecision]), If[LessEqual[b, 5.9e-47], N[(N[(N[Sqrt[N[(N[(a * c), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -5.20000000000000009e-35

   1. Initial program 73.3%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-*l/ N/A

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

      5. *-lft-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. associate-*r/ N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 94.8

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

   5. Applied rewrites 94.8%

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

   if -5.20000000000000009e-35 < b < 5.89999999999999973e-47

   1. Initial program 65.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 65.6

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. metadata-eval 65.6

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 65.6

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

   4. Applied rewrites 65.6%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 65.6

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

   6. Applied rewrites 65.6%

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

   7. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 60.1

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

   9. Applied rewrites 60.1%

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

   if 5.89999999999999973e-47 < b

   1. Initial program 11.3%

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

   3. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 90.6

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

   5. Applied rewrites 90.6%

      \[\leadsto \color{blue}{\frac{-c}{b}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.2 \cdot 10^{-35}:\\ \;\;\;\;-\mathsf{fma}\left(\frac{-c}{b \cdot b}, b, \frac{b}{a}\right)\\ \mathbf{elif}\;b \leq 5.9 \cdot 10^{-47}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 67.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.5 \cdot 10^{-300}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.5e-300) (/ (- b) a) (/ (- c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.5e-300) {
		tmp = -b / a;
	} else {
		tmp = -c / b;
	}
	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 <= 1.5d-300) then
        tmp = -b / a
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.5e-300) {
		tmp = -b / a;
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 1.5e-300:
		tmp = -b / a
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.5e-300)
		tmp = Float64(Float64(-b) / a);
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 1.5e-300)
		tmp = -b / a;
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.5e-300], N[((-b) / a), $MachinePrecision], N[((-c) / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.50000000000000012e-300

   1. Initial program 70.6%

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

   3. Taylor expanded in b around -inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 76.0

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

   5. Applied rewrites 76.0%

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

   if 1.50000000000000012e-300 < b

   1. Initial program 29.7%

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

   3. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 66.5

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

   5. Applied rewrites 66.5%

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

Alternative 4: 43.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.3 \cdot 10^{-290}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.3e-290) (/ (- b) a) (/ 0.0 a)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.3e-290) {
		tmp = -b / a;
	} else {
		tmp = 0.0 / 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) :: tmp
    if (b <= (-1.3d-290)) then
        tmp = -b / a
    else
        tmp = 0.0d0 / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.3e-290) {
		tmp = -b / a;
	} else {
		tmp = 0.0 / a;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.3e-290:
		tmp = -b / a
	else:
		tmp = 0.0 / a
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.3e-290)
		tmp = Float64(Float64(-b) / a);
	else
		tmp = Float64(0.0 / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.3e-290)
		tmp = -b / a;
	else
		tmp = 0.0 / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.3e-290], N[((-b) / a), $MachinePrecision], N[(0.0 / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.3e-290

   1. Initial program 71.8%

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

   3. Taylor expanded in b around -inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 78.7

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

   5. Applied rewrites 78.7%

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

   if -1.3e-290 < b

   1. Initial program 30.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 30.2

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. metadata-eval 30.2

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 30.2

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

   4. Applied rewrites 30.2%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. frac-add N/A

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

      5. lower-/.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 11.9

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

   6. Applied rewrites 11.9%

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

   7. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

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

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

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

      3. metadata-eval N/A

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

      4. +-inverses N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \color{blue}{0}}{a} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{0}}{a} \]

      6. +-inverses N/A

         \[\leadsto \frac{\color{blue}{-2 \cdot b - -2 \cdot b}}{a} \]

      7. metadata-eval N/A

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

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

         \[\leadsto \frac{\color{blue}{-2 \cdot b + 2 \cdot b}}{a} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-2 \cdot b + 2 \cdot b}{a}} \]

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

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

      11. metadata-eval N/A

          \[\leadsto \frac{-2 \cdot b - \color{blue}{-2} \cdot b}{a} \]

      12. +-inverses 17.3

          \[\leadsto \frac{\color{blue}{0}}{a} \]

   9. Applied rewrites 17.3%

      \[\leadsto \color{blue}{\frac{0}{a}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 10.9% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \frac{0}{a} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (/ 0.0 a))

C

double code(double a, double b, double c) {
	return 0.0 / 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 = 0.0d0 / a
end function

Java

public static double code(double a, double b, double c) {
	return 0.0 / a;
}

Python

def code(a, b, c):
	return 0.0 / a

Julia

function code(a, b, c)
	return Float64(0.0 / a)
end

MATLAB

function tmp = code(a, b, c)
	tmp = 0.0 / a;
end

Wolfram

code[a_, b_, c_] := N[(0.0 / a), $MachinePrecision]

Derivation

1. Initial program 51.6%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f64 51.6

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

   4. lift--.f64 N/A

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

   5. lift-*.f64 N/A

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

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

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

   7. +-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. metadata-eval 51.6

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 51.6

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

4. Applied rewrites 51.6%

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

   1. lift-/.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. div-add N/A

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

   4. frac-add N/A

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

   5. lower-/.f64 N/A

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

   6. lower-fma.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. associate-*r* N/A

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

   11. lower-fma.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-*.f64 34.2

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

6. Applied rewrites 34.2%

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

7. Taylor expanded in a around 0

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

   1. associate-*r/ N/A

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

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

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

   3. metadata-eval N/A

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

   4. +-inverses N/A

      \[\leadsto \frac{\frac{1}{4} \cdot \color{blue}{0}}{a} \]

   5. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{0}}{a} \]

   6. +-inverses N/A

      \[\leadsto \frac{\color{blue}{-2 \cdot b - -2 \cdot b}}{a} \]

   7. metadata-eval N/A

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

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

      \[\leadsto \frac{\color{blue}{-2 \cdot b + 2 \cdot b}}{a} \]

   9. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{-2 \cdot b + 2 \cdot b}{a}} \]

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

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

   11. metadata-eval N/A

       \[\leadsto \frac{-2 \cdot b - \color{blue}{-2} \cdot b}{a} \]

   12. +-inverses 9.7

       \[\leadsto \frac{\color{blue}{0}}{a} \]

9. Applied rewrites 9.7%

   \[\leadsto \color{blue}{\frac{0}{a}} \]
10. Add Preprocessing

Alternative 6: 10.7% accurate, 4.2× 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 51.6%

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

3. Taylor expanded in b around -inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. distribute-rgt-in N/A

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

   4. associate-*l/ N/A

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

   5. *-lft-identity N/A

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

   6. lower-fma.f64 N/A

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

   7. associate-*r/ N/A

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

   8. mul-1-neg N/A

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

   9. lower-/.f64 N/A

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

   10. lower-neg.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-/.f64 40.5

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

5. Applied rewrites 40.5%

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

6. Taylor expanded in a around inf

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

8. Applied rewrites 9.3%

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

Developer Target 1: 99.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|\frac{b}{2}\right|\\ t_1 := \sqrt{\left|a\right|} \cdot \sqrt{\left|c\right|}\\ t_2 := \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(a, c\right) = a:\\ \;\;\;\;\sqrt{t\_0 - t\_1} \cdot \sqrt{t\_0 + t\_1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\frac{b}{2}, t\_1\right)\\ \end{array}\\ \mathbf{if}\;b < 0:\\ \;\;\;\;\frac{t\_2 - \frac{b}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{\frac{b}{2} + t\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fabs (/ b 2.0)))
        (t_1 (* (sqrt (fabs a)) (sqrt (fabs c))))
        (t_2
         (if (== (copysign a c) a)
           (* (sqrt (- t_0 t_1)) (sqrt (+ t_0 t_1)))
           (hypot (/ b 2.0) t_1))))
   (if (< b 0.0) (/ (- t_2 (/ b 2.0)) a) (/ (- c) (+ (/ b 2.0) t_2)))))

C

double code(double a, double b, double c) {
	double t_0 = fabs((b / 2.0));
	double t_1 = sqrt(fabs(a)) * sqrt(fabs(c));
	double tmp;
	if (copysign(a, c) == a) {
		tmp = sqrt((t_0 - t_1)) * sqrt((t_0 + t_1));
	} else {
		tmp = hypot((b / 2.0), t_1);
	}
	double t_2 = tmp;
	double tmp_1;
	if (b < 0.0) {
		tmp_1 = (t_2 - (b / 2.0)) / a;
	} else {
		tmp_1 = -c / ((b / 2.0) + t_2);
	}
	return tmp_1;
}

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.abs((b / 2.0));
	double t_1 = Math.sqrt(Math.abs(a)) * Math.sqrt(Math.abs(c));
	double tmp;
	if (Math.copySign(a, c) == a) {
		tmp = Math.sqrt((t_0 - t_1)) * Math.sqrt((t_0 + t_1));
	} else {
		tmp = Math.hypot((b / 2.0), t_1);
	}
	double t_2 = tmp;
	double tmp_1;
	if (b < 0.0) {
		tmp_1 = (t_2 - (b / 2.0)) / a;
	} else {
		tmp_1 = -c / ((b / 2.0) + t_2);
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = math.fabs((b / 2.0))
	t_1 = math.sqrt(math.fabs(a)) * math.sqrt(math.fabs(c))
	tmp = 0
	if math.copysign(a, c) == a:
		tmp = math.sqrt((t_0 - t_1)) * math.sqrt((t_0 + t_1))
	else:
		tmp = math.hypot((b / 2.0), t_1)
	t_2 = tmp
	tmp_1 = 0
	if b < 0.0:
		tmp_1 = (t_2 - (b / 2.0)) / a
	else:
		tmp_1 = -c / ((b / 2.0) + t_2)
	return tmp_1

Julia

function code(a, b, c)
	t_0 = abs(Float64(b / 2.0))
	t_1 = Float64(sqrt(abs(a)) * sqrt(abs(c)))
	tmp = 0.0
	if (copysign(a, c) == a)
		tmp = Float64(sqrt(Float64(t_0 - t_1)) * sqrt(Float64(t_0 + t_1)));
	else
		tmp = hypot(Float64(b / 2.0), t_1);
	end
	t_2 = tmp
	tmp_1 = 0.0
	if (b < 0.0)
		tmp_1 = Float64(Float64(t_2 - Float64(b / 2.0)) / a);
	else
		tmp_1 = Float64(Float64(-c) / Float64(Float64(b / 2.0) + t_2));
	end
	return tmp_1
end

MATLAB

function tmp_3 = code(a, b, c)
	t_0 = abs((b / 2.0));
	t_1 = sqrt(abs(a)) * sqrt(abs(c));
	tmp = 0.0;
	if ((sign(c) * abs(a)) == a)
		tmp = sqrt((t_0 - t_1)) * sqrt((t_0 + t_1));
	else
		tmp = hypot((b / 2.0), t_1);
	end
	t_2 = tmp;
	tmp_2 = 0.0;
	if (b < 0.0)
		tmp_2 = (t_2 - (b / 2.0)) / a;
	else
		tmp_2 = -c / ((b / 2.0) + t_2);
	end
	tmp_3 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Abs[N[(b / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[Abs[a], $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[Abs[c], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = If[Equal[N[With[{TMP1 = Abs[a], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], a], N[(N[Sqrt[N[(t$95$0 - t$95$1), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(t$95$0 + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(b / 2.0), $MachinePrecision] ^ 2 + t$95$1 ^ 2], $MachinePrecision]]}, If[Less[b, 0.0], N[(N[(t$95$2 - N[(b / 2.0), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[((-c) / N[(N[(b / 2.0), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]]]]]

Reproduce

herbie shell --seed 2025017 
(FPCore (a b c)
  :name "quadp (p42, positive)"
  :precision binary64
  :herbie-expected 10

  :alt
  (! :herbie-platform default (let ((sqtD (let ((x (* (sqrt (fabs a)) (sqrt (fabs c))))) (if (== (copysign a c) a) (* (sqrt (- (fabs (/ b 2)) x)) (sqrt (+ (fabs (/ b 2)) x))) (hypot (/ b 2) x))))) (if (< b 0) (/ (- sqtD (/ b 2)) a) (/ (- c) (+ (/ b 2) sqtD)))))

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