The quadratic formula (r1)

Percentage Accurate: 51.5% → 83.8%

Time: 7.3s

Alternatives: 8

Speedup: 2.5×

Specification

Math

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

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(Float64(4.0 * 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[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Initial Program: 51.5% accurate, 1.0× speedup

Math

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

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(Float64(4.0 * 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[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Alternative 1: 83.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -8 \cdot 10^{+40}:\\ \;\;\;\;-\mathsf{fma}\left(-c, \frac{b}{b \cdot b}, \frac{b}{a}\right)\\ \mathbf{elif}\;b \leq 1.7 \cdot 10^{-174}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -8e+40)
   (- (fma (- c) (/ b (* b b)) (/ b a)))
   (if (<= b 1.7e-174)
     (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -8e+40) {
		tmp = -fma(-c, (b / (b * b)), (b / a));
	} else if (b <= 1.7e-174) {
		tmp = (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -8e+40)
		tmp = Float64(-fma(Float64(-c), Float64(b / Float64(b * b)), Float64(b / a)));
	elseif (b <= 1.7e-174)
		tmp = Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -8e+40], (-N[((-c) * N[(b / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(b / a), $MachinePrecision]), $MachinePrecision]), If[LessEqual[b, 1.7e-174], N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -8.00000000000000024e40

   1. Initial program 73.0%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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 92.3

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

   5. Applied rewrites 92.3%

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

   7. Applied rewrites 92.3%

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

   if -8.00000000000000024e40 < b < 1.7000000000000001e-174

   1. Initial program 83.8%

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

   if 1.7000000000000001e-174 < b

   1. Initial program 16.0%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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 87.0

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

   5. Applied rewrites 87.0%

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

Alternative 2: 83.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -8 \cdot 10^{+40}:\\ \;\;\;\;-\mathsf{fma}\left(-c, \frac{b}{b \cdot b}, \frac{b}{a}\right)\\ \mathbf{elif}\;b \leq 1.7 \cdot 10^{-174}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -8e+40)
   (- (fma (- c) (/ b (* b b)) (/ b a)))
   (if (<= b 1.7e-174)
     (/ (- (sqrt (fma (* -4.0 a) c (* b b))) b) (* 2.0 a))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -8e+40) {
		tmp = -fma(-c, (b / (b * b)), (b / a));
	} else if (b <= 1.7e-174) {
		tmp = (sqrt(fma((-4.0 * a), c, (b * b))) - b) / (2.0 * a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -8e+40)
		tmp = Float64(-fma(Float64(-c), Float64(b / Float64(b * b)), Float64(b / a)));
	elseif (b <= 1.7e-174)
		tmp = Float64(Float64(sqrt(fma(Float64(-4.0 * a), c, Float64(b * b))) - b) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -8e+40], (-N[((-c) * N[(b / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(b / a), $MachinePrecision]), $MachinePrecision]), If[LessEqual[b, 1.7e-174], N[(N[(N[Sqrt[N[(N[(-4.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -8.00000000000000024e40

   1. Initial program 73.0%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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 92.3

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

   5. Applied rewrites 92.3%

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

   7. Applied rewrites 92.3%

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

   if -8.00000000000000024e40 < b < 1.7000000000000001e-174

   1. Initial program 83.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 83.8

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

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

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

      11. lower-*.f64 N/A

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

      12. metadata-eval 83.8

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

   4. Applied rewrites 83.8%

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

   if 1.7000000000000001e-174 < b

   1. Initial program 16.0%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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 87.0

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

   5. Applied rewrites 87.0%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -8 \cdot 10^{+40}:\\ \;\;\;\;-\mathsf{fma}\left(-c, \frac{b}{b \cdot b}, \frac{b}{a}\right)\\ \mathbf{elif}\;b \leq 1.7 \cdot 10^{-174}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -7.6 \cdot 10^{-60}:\\ \;\;\;\;-\mathsf{fma}\left(-c, \frac{b}{b \cdot b}, \frac{b}{a}\right)\\ \mathbf{elif}\;b \leq 1.7 \cdot 10^{-174}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(c \cdot a\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -7.6e-60)
   (- (fma (- c) (/ b (* b b)) (/ b a)))
   (if (<= b 1.7e-174)
     (/ (+ (- b) (sqrt (* -4.0 (* c a)))) (* 2.0 a))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -7.6e-60) {
		tmp = -fma(-c, (b / (b * b)), (b / a));
	} else if (b <= 1.7e-174) {
		tmp = (-b + sqrt((-4.0 * (c * a)))) / (2.0 * a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -7.6e-60)
		tmp = Float64(-fma(Float64(-c), Float64(b / Float64(b * b)), Float64(b / a)));
	elseif (b <= 1.7e-174)
		tmp = Float64(Float64(Float64(-b) + sqrt(Float64(-4.0 * Float64(c * a)))) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -7.6e-60], (-N[((-c) * N[(b / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(b / a), $MachinePrecision]), $MachinePrecision]), If[LessEqual[b, 1.7e-174], N[(N[((-b) + N[Sqrt[N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -7.59999999999999989e-60

   1. Initial program 77.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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 89.8

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

   5. Applied rewrites 89.8%

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

   7. Applied rewrites 89.8%

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

   if -7.59999999999999989e-60 < b < 1.7000000000000001e-174

   1. Initial program 79.4%

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

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 69.3

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

   5. Applied rewrites 69.3%

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

   if 1.7000000000000001e-174 < b

   1. Initial program 16.0%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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 87.0

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

   5. Applied rewrites 87.0%

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

Alternative 4: 79.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -7.6 \cdot 10^{-60}:\\ \;\;\;\;-\mathsf{fma}\left(-c, \frac{b}{b \cdot b}, \frac{b}{a}\right)\\ \mathbf{elif}\;b \leq 1.7 \cdot 10^{-174}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -7.6e-60)
   (- (fma (- c) (/ b (* b b)) (/ b a)))
   (if (<= b 1.7e-174)
     (/ (- (sqrt (* c (* a -4.0))) b) (* 2.0 a))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -7.6e-60) {
		tmp = -fma(-c, (b / (b * b)), (b / a));
	} else if (b <= 1.7e-174) {
		tmp = (sqrt((c * (a * -4.0))) - b) / (2.0 * a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -7.6e-60)
		tmp = Float64(-fma(Float64(-c), Float64(b / Float64(b * b)), Float64(b / a)));
	elseif (b <= 1.7e-174)
		tmp = Float64(Float64(sqrt(Float64(c * Float64(a * -4.0))) - b) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -7.6e-60], (-N[((-c) * N[(b / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(b / a), $MachinePrecision]), $MachinePrecision]), If[LessEqual[b, 1.7e-174], N[(N[(N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -7.59999999999999989e-60

   1. Initial program 77.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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 89.8

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

   5. Applied rewrites 89.8%

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

   7. Applied rewrites 89.8%

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

   if -7.59999999999999989e-60 < b < 1.7000000000000001e-174

   1. Initial program 79.4%

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

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 69.3

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

   5. Applied rewrites 69.3%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. flip-+ N/A

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

   7. Applied rewrites 61.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

   9. Applied rewrites 69.3%

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

   if 1.7000000000000001e-174 < b

   1. Initial program 16.0%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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 87.0

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

   5. Applied rewrites 87.0%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -7.6 \cdot 10^{-60}:\\ \;\;\;\;-\mathsf{fma}\left(-c, \frac{b}{b \cdot b}, \frac{b}{a}\right)\\ \mathbf{elif}\;b \leq 1.7 \cdot 10^{-174}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 69.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1e-310) (- (/ c b) (/ b a)) (/ (- c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e-310) {
		tmp = (c / b) - (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 <= (-1d-310)) then
        tmp = (c / b) - (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 <= -1e-310) {
		tmp = (c / b) - (b / a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1e-310:
		tmp = (c / b) - (b / a)
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1e-310)
		tmp = Float64(Float64(c / b) - 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 <= -1e-310)
		tmp = (c / b) - (b / a);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1e-310], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -9.999999999999969e-311

   1. Initial program 79.3%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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 69.0

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

   5. Applied rewrites 69.0%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 70.6%

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

   if -9.999999999999969e-311 < b

   1. Initial program 22.1%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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 77.9

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

   5. Applied rewrites 77.9%

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

Alternative 6: 68.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1e-310) (/ (- b) a) (/ (- c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e-310) {
		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 <= (-1d-310)) 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 <= -1e-310) {
		tmp = -b / a;
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1e-310:
		tmp = -b / a
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1e-310)
		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 <= -1e-310)
		tmp = -b / a;
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1e-310], N[((-b) / a), $MachinePrecision], N[((-c) / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -9.999999999999969e-311

   1. Initial program 79.3%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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 70.6

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

   5. Applied rewrites 70.6%

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

   if -9.999999999999969e-311 < b

   1. Initial program 22.1%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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 77.9

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

   5. Applied rewrites 77.9%

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

Alternative 7: 44.0% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-308}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b -5e-308) (/ (- b) a) 0.0))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-308) {
		tmp = -b / a;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-5d-308)) then
        tmp = -b / a
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-308) {
		tmp = -b / a;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e-308:
		tmp = -b / a
	else:
		tmp = 0.0
	return tmp

Julia

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

MATLAB

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

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-308], N[((-b) / a), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if b < -4.99999999999999955e-308

   1. Initial program 79.1%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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 71.1

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

   5. Applied rewrites 71.1%

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

   if -4.99999999999999955e-308 < b

   1. Initial program 22.7%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. pow-to-exp N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-log.f64 17.2

         \[\leadsto \frac{\left(-b\right) + e^{\color{blue}{\log \left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)} \cdot 0.5}}{2 \cdot a} \]

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

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

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-*.f64 N/A

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

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

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

      14. lower-*.f64 N/A

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

      15. metadata-eval 17.2

          \[\leadsto \frac{\left(-b\right) + e^{\log \left(\mathsf{fma}\left(\color{blue}{-4} \cdot a, c, b \cdot b\right)\right) \cdot 0.5}}{2 \cdot a} \]

   4. Applied rewrites 17.2%

      \[\leadsto \frac{\left(-b\right) + \color{blue}{e^{\log \left(\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\right) \cdot 0.5}}}{2 \cdot a} \]
   5. Step-by-step derivation

      1. lift-exp.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. exp-prod N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-exp.f64 17.1

         \[\leadsto \frac{\left(-b\right) + {\color{blue}{\left(e^{0.5}\right)}}^{\log \left(\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\right)}}{2 \cdot a} \]

      7. lift-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 17.1

         \[\leadsto \frac{\left(-b\right) + {\left(e^{0.5}\right)}^{\log \color{blue}{\left(\mathsf{fma}\left(c, -4 \cdot a, b \cdot b\right)\right)}}}{2 \cdot a} \]

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 17.1

          \[\leadsto \frac{\left(-b\right) + {\left(e^{0.5}\right)}^{\log \left(\mathsf{fma}\left(c, \color{blue}{a \cdot -4}, b \cdot b\right)\right)}}{2 \cdot a} \]

   6. Applied rewrites 17.1%

      \[\leadsto \frac{\left(-b\right) + \color{blue}{{\left(e^{0.5}\right)}^{\log \left(\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)\right)}}}{2 \cdot a} \]

   7. Taylor expanded in b around inf

      \[\leadsto \color{blue}{0} \]
   8. Step-by-step derivation

   9. Applied rewrites 24.7%

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

Alternative 8: 11.4% accurate, 50.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(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
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_, c_] := 0.0

Derivation

1. Initial program 50.9%

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

   1. lift-sqrt.f64 N/A

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

   2. pow1/2 N/A

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

   3. pow-to-exp N/A

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

   4. lower-exp.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-log.f64 46.6

      \[\leadsto \frac{\left(-b\right) + e^{\color{blue}{\log \left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)} \cdot 0.5}}{2 \cdot a} \]

   7. lift--.f64 N/A

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

   8. lift-*.f64 N/A

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

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

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

   10. +-commutative N/A

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

   11. lower-fma.f64 N/A

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

   12. lift-*.f64 N/A

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

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

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

   14. lower-*.f64 N/A

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

   15. metadata-eval 46.6

       \[\leadsto \frac{\left(-b\right) + e^{\log \left(\mathsf{fma}\left(\color{blue}{-4} \cdot a, c, b \cdot b\right)\right) \cdot 0.5}}{2 \cdot a} \]

4. Applied rewrites 46.6%

   \[\leadsto \frac{\left(-b\right) + \color{blue}{e^{\log \left(\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\right) \cdot 0.5}}}{2 \cdot a} \]
5. Step-by-step derivation

   1. lift-exp.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. exp-prod N/A

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

   5. lower-pow.f64 N/A

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

   6. lower-exp.f64 46.3

      \[\leadsto \frac{\left(-b\right) + {\color{blue}{\left(e^{0.5}\right)}}^{\log \left(\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\right)}}{2 \cdot a} \]

   7. lift-fma.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 46.3

      \[\leadsto \frac{\left(-b\right) + {\left(e^{0.5}\right)}^{\log \color{blue}{\left(\mathsf{fma}\left(c, -4 \cdot a, b \cdot b\right)\right)}}}{2 \cdot a} \]

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 46.3

       \[\leadsto \frac{\left(-b\right) + {\left(e^{0.5}\right)}^{\log \left(\mathsf{fma}\left(c, \color{blue}{a \cdot -4}, b \cdot b\right)\right)}}{2 \cdot a} \]

6. Applied rewrites 46.3%

   \[\leadsto \frac{\left(-b\right) + \color{blue}{{\left(e^{0.5}\right)}^{\log \left(\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)\right)}}}{2 \cdot a} \]

7. Taylor expanded in b around inf

   \[\leadsto \color{blue}{0} \]
8. Step-by-step derivation

9. Applied rewrites 13.6%

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

Developer Target 1: 70.2% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (- (* b b) (* (* 4.0 a) c)))))
   (if (< b 0.0)
     (/ (+ (- b) t_0) (* 2.0 a))
     (/ c (* 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 = (-b + t_0) / (2.0 * a);
	} else {
		tmp = c / (a * ((-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 = (-b + t_0) / (2.0d0 * a)
    else
        tmp = c / (a * ((-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 = (-b + t_0) / (2.0 * a);
	} else {
		tmp = c / (a * ((-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 = (-b + t_0) / (2.0 * a)
	else:
		tmp = c / (a * ((-b - t_0) / (2.0 * a)))
	return tmp

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))
	tmp = 0.0
	if (b < 0.0)
		tmp = Float64(Float64(Float64(-b) + t_0) / Float64(2.0 * a));
	else
		tmp = Float64(c / Float64(a * 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 = (-b + t_0) / (2.0 * a);
	else
		tmp = c / (a * ((-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[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[Less[b, 0.0], N[(N[((-b) + t$95$0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(c / N[(a * N[(N[((-b) - t$95$0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Reproduce

herbie shell --seed 2025007 
(FPCore (a b c)
  :name "The quadratic formula (r1)"
  :precision binary64

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

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