quadm (p42, negative)

Percentage Accurate: 51.1% → 85.3%

Time: 4.4s

Alternatives: 10

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: 51.1% 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: 85.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -7.2 \cdot 10^{-81}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c}{b \cdot b}, 1\right) \cdot c}{-b}\\ \mathbf{elif}\;b \leq 5.2 \cdot 10^{+89}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{a + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a} + \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -7.2e-81)
   (/ (* (fma a (/ c (* b b)) 1.0) c) (- b))
   (if (<= b 5.2e+89)
     (/ (- (- b) (sqrt (fma (* c a) -4.0 (* b b)))) (+ a a))
     (+ (/ (- b) a) (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -7.2e-81) {
		tmp = (fma(a, (c / (b * b)), 1.0) * c) / -b;
	} else if (b <= 5.2e+89) {
		tmp = (-b - sqrt(fma((c * a), -4.0, (b * b)))) / (a + a);
	} else {
		tmp = (-b / a) + (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -7.2e-81)
		tmp = Float64(Float64(fma(a, Float64(c / Float64(b * b)), 1.0) * c) / Float64(-b));
	elseif (b <= 5.2e+89)
		tmp = Float64(Float64(Float64(-b) - sqrt(fma(Float64(c * a), -4.0, Float64(b * b)))) / Float64(a + a));
	else
		tmp = Float64(Float64(Float64(-b) / a) + Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -7.2e-81], N[(N[(N[(a * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * c), $MachinePrecision] / (-b)), $MachinePrecision], If[LessEqual[b, 5.2e+89], N[(N[((-b) - N[Sqrt[N[(N[(c * a), $MachinePrecision] * -4.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision], N[(N[((-b) / a), $MachinePrecision] + N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -7.1999999999999997e-81

   1. Initial program 16.5%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

         \[\leadsto -\frac{c + \frac{a \cdot {c}^{2}}{{b}^{2}}}{b} \]

      3. lower-/.f64 N/A

         \[\leadsto -\frac{c + \frac{a \cdot {c}^{2}}{{b}^{2}}}{b} \]

      4. +-commutative N/A

         \[\leadsto -\frac{\frac{a \cdot {c}^{2}}{{b}^{2}} + c}{b} \]

      5. associate-/l* N/A

         \[\leadsto -\frac{a \cdot \frac{{c}^{2}}{{b}^{2}} + c}{b} \]

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 69.8

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

   5. Applied rewrites 69.8%

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

   6. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 84.6

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

   8. Applied rewrites 84.6%

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

   if -7.1999999999999997e-81 < b < 5.2000000000000001e89

   1. Initial program 82.0%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

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

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 82.0

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

   4. Applied rewrites 82.0%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 82.0

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

   6. Applied rewrites 82.0%

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

   if 5.2000000000000001e89 < b

   1. Initial program 59.9%

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

   3. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 95.1

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

   5. Applied rewrites 95.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{b}{a}, -1, \frac{c}{b}\right)} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. associate-*r/ N/A

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

      7. mul-1-neg N/A

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

      8. lower-/.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. lift-/.f64 95.1

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

   7. Applied rewrites 95.1%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -7.2 \cdot 10^{-81}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c}{b \cdot b}, 1\right) \cdot c}{-b}\\ \mathbf{elif}\;b \leq 5.2 \cdot 10^{+89}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{a + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a} + \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 73.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.45 \cdot 10^{-178}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{elif}\;b \leq -1.2 \cdot 10^{-275}:\\ \;\;\;\;\sqrt{\frac{-c}{a}}\\ \mathbf{elif}\;b \leq 1.65 \cdot 10^{-105}:\\ \;\;\;\;\frac{\sqrt{-c}}{-\sqrt{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a} + \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.45e-178)
   (/ c (- b))
   (if (<= b -1.2e-275)
     (sqrt (/ (- c) a))
     (if (<= b 1.65e-105)
       (/ (sqrt (- c)) (- (sqrt a)))
       (+ (/ (- b) a) (/ c b))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.45e-178) {
		tmp = c / -b;
	} else if (b <= -1.2e-275) {
		tmp = sqrt((-c / a));
	} else if (b <= 1.65e-105) {
		tmp = sqrt(-c) / -sqrt(a);
	} else {
		tmp = (-b / a) + (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 <= (-2.45d-178)) then
        tmp = c / -b
    else if (b <= (-1.2d-275)) then
        tmp = sqrt((-c / a))
    else if (b <= 1.65d-105) then
        tmp = sqrt(-c) / -sqrt(a)
    else
        tmp = (-b / a) + (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.45e-178) {
		tmp = c / -b;
	} else if (b <= -1.2e-275) {
		tmp = Math.sqrt((-c / a));
	} else if (b <= 1.65e-105) {
		tmp = Math.sqrt(-c) / -Math.sqrt(a);
	} else {
		tmp = (-b / a) + (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.45e-178:
		tmp = c / -b
	elif b <= -1.2e-275:
		tmp = math.sqrt((-c / a))
	elif b <= 1.65e-105:
		tmp = math.sqrt(-c) / -math.sqrt(a)
	else:
		tmp = (-b / a) + (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.45e-178)
		tmp = Float64(c / Float64(-b));
	elseif (b <= -1.2e-275)
		tmp = sqrt(Float64(Float64(-c) / a));
	elseif (b <= 1.65e-105)
		tmp = Float64(sqrt(Float64(-c)) / Float64(-sqrt(a)));
	else
		tmp = Float64(Float64(Float64(-b) / a) + Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.45e-178)
		tmp = c / -b;
	elseif (b <= -1.2e-275)
		tmp = sqrt((-c / a));
	elseif (b <= 1.65e-105)
		tmp = sqrt(-c) / -sqrt(a);
	else
		tmp = (-b / a) + (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.45e-178], N[(c / (-b)), $MachinePrecision], If[LessEqual[b, -1.2e-275], N[Sqrt[N[((-c) / a), $MachinePrecision]], $MachinePrecision], If[LessEqual[b, 1.65e-105], N[(N[Sqrt[(-c)], $MachinePrecision] / (-N[Sqrt[a], $MachinePrecision])), $MachinePrecision], N[(N[((-b) / a), $MachinePrecision] + N[(c / b), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b < -2.4500000000000001e-178

   1. Initial program 22.3%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 78.1

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

   5. Applied rewrites 78.1%

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

   if -2.4500000000000001e-178 < b < -1.19999999999999995e-275

   1. Initial program 84.9%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

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

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 84.9

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

   4. Applied rewrites 84.9%

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

   5. Taylor expanded in a around -inf

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

      1. div-sub N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. sqrt-unprod N/A

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

      6. *-commutative N/A

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

      7. lower-sqrt.f64 N/A

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

      8. associate-*r/ N/A

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

      9. mul-1-neg N/A

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

      10. lower-/.f64 N/A

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

      11. lower-neg.f64 56.1

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

   7. Applied rewrites 56.1%

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

   if -1.19999999999999995e-275 < b < 1.6499999999999999e-105

   1. Initial program 82.6%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sqrt-unprod N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 35.2

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

   5. Applied rewrites 35.2%

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

   6. Taylor expanded in a around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. sqrt-unprod N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lift-/.f64 35.2

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

   8. Applied rewrites 35.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. *-commutative N/A

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

      5. mul-1-neg N/A

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

      6. distribute-frac-neg N/A

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

      7. sqrt-div N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. lower-sqrt.f64 46.0

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

   10. Applied rewrites 46.0%

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

   if 1.6499999999999999e-105 < b

   1. Initial program 71.9%

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

   3. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 79.6

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

   5. Applied rewrites 79.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{b}{a}, -1, \frac{c}{b}\right)} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. associate-*r/ N/A

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

      7. mul-1-neg N/A

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

      8. lower-/.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. lift-/.f64 79.6

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

   7. Applied rewrites 79.6%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.45 \cdot 10^{-178}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{elif}\;b \leq -1.2 \cdot 10^{-275}:\\ \;\;\;\;\sqrt{\frac{-c}{a}}\\ \mathbf{elif}\;b \leq 1.65 \cdot 10^{-105}:\\ \;\;\;\;\frac{\sqrt{-c}}{-\sqrt{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a} + \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -6.2 \cdot 10^{-84}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c}{b \cdot b}, 1\right) \cdot c}{-b}\\ \mathbf{elif}\;b \leq 7.5 \cdot 10^{-49}:\\ \;\;\;\;\frac{b + \sqrt{\left(a \cdot c\right) \cdot -4}}{-\left(a + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a} + \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -6.2e-84)
   (/ (* (fma a (/ c (* b b)) 1.0) c) (- b))
   (if (<= b 7.5e-49)
     (/ (+ b (sqrt (* (* a c) -4.0))) (- (+ a a)))
     (+ (/ (- b) a) (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -6.2e-84) {
		tmp = (fma(a, (c / (b * b)), 1.0) * c) / -b;
	} else if (b <= 7.5e-49) {
		tmp = (b + sqrt(((a * c) * -4.0))) / -(a + a);
	} else {
		tmp = (-b / a) + (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -6.2e-84)
		tmp = Float64(Float64(fma(a, Float64(c / Float64(b * b)), 1.0) * c) / Float64(-b));
	elseif (b <= 7.5e-49)
		tmp = Float64(Float64(b + sqrt(Float64(Float64(a * c) * -4.0))) / Float64(-Float64(a + a)));
	else
		tmp = Float64(Float64(Float64(-b) / a) + Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -6.2e-84], N[(N[(N[(a * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * c), $MachinePrecision] / (-b)), $MachinePrecision], If[LessEqual[b, 7.5e-49], N[(N[(b + N[Sqrt[N[(N[(a * c), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / (-N[(a + a), $MachinePrecision])), $MachinePrecision], N[(N[((-b) / a), $MachinePrecision] + N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -6.20000000000000003e-84

   1. Initial program 16.5%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

         \[\leadsto -\frac{c + \frac{a \cdot {c}^{2}}{{b}^{2}}}{b} \]

      3. lower-/.f64 N/A

         \[\leadsto -\frac{c + \frac{a \cdot {c}^{2}}{{b}^{2}}}{b} \]

      4. +-commutative N/A

         \[\leadsto -\frac{\frac{a \cdot {c}^{2}}{{b}^{2}} + c}{b} \]

      5. associate-/l* N/A

         \[\leadsto -\frac{a \cdot \frac{{c}^{2}}{{b}^{2}} + c}{b} \]

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 69.8

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

   5. Applied rewrites 69.8%

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

   6. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 84.6

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

   8. Applied rewrites 84.6%

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

   if -6.20000000000000003e-84 < b < 7.4999999999999998e-49

   1. Initial program 79.3%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

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

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 79.3

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

   4. Applied rewrites 79.3%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 79.3

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

   6. Applied rewrites 79.3%

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

   7. Taylor expanded in a around inf

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

      1. sqrt-unprod N/A

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

      2. *-commutative N/A

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

      3. lower-sqrt.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 72.3

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

   9. Applied rewrites 72.3%

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

   if 7.4999999999999998e-49 < b

   1. Initial program 70.3%

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

   3. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 84.6

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

   5. Applied rewrites 84.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{b}{a}, -1, \frac{c}{b}\right)} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. associate-*r/ N/A

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

      7. mul-1-neg N/A

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

      8. lower-/.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. lift-/.f64 84.6

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

   7. Applied rewrites 84.6%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6.2 \cdot 10^{-84}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c}{b \cdot b}, 1\right) \cdot c}{-b}\\ \mathbf{elif}\;b \leq 7.5 \cdot 10^{-49}:\\ \;\;\;\;\frac{b + \sqrt{\left(a \cdot c\right) \cdot -4}}{-\left(a + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a} + \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -6.2 \cdot 10^{-84}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{elif}\;b \leq 7.5 \cdot 10^{-49}:\\ \;\;\;\;\frac{b + \sqrt{\left(a \cdot c\right) \cdot -4}}{-\left(a + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a} + \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -6.2e-84)
   (/ c (- b))
   (if (<= b 7.5e-49)
     (/ (+ b (sqrt (* (* a c) -4.0))) (- (+ a a)))
     (+ (/ (- b) a) (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -6.2e-84) {
		tmp = c / -b;
	} else if (b <= 7.5e-49) {
		tmp = (b + sqrt(((a * c) * -4.0))) / -(a + a);
	} else {
		tmp = (-b / a) + (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 <= (-6.2d-84)) then
        tmp = c / -b
    else if (b <= 7.5d-49) then
        tmp = (b + sqrt(((a * c) * (-4.0d0)))) / -(a + a)
    else
        tmp = (-b / a) + (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -6.2e-84) {
		tmp = c / -b;
	} else if (b <= 7.5e-49) {
		tmp = (b + Math.sqrt(((a * c) * -4.0))) / -(a + a);
	} else {
		tmp = (-b / a) + (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -6.2e-84:
		tmp = c / -b
	elif b <= 7.5e-49:
		tmp = (b + math.sqrt(((a * c) * -4.0))) / -(a + a)
	else:
		tmp = (-b / a) + (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -6.2e-84)
		tmp = Float64(c / Float64(-b));
	elseif (b <= 7.5e-49)
		tmp = Float64(Float64(b + sqrt(Float64(Float64(a * c) * -4.0))) / Float64(-Float64(a + a)));
	else
		tmp = Float64(Float64(Float64(-b) / a) + Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -6.2e-84)
		tmp = c / -b;
	elseif (b <= 7.5e-49)
		tmp = (b + sqrt(((a * c) * -4.0))) / -(a + a);
	else
		tmp = (-b / a) + (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -6.2e-84], N[(c / (-b)), $MachinePrecision], If[LessEqual[b, 7.5e-49], N[(N[(b + N[Sqrt[N[(N[(a * c), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / (-N[(a + a), $MachinePrecision])), $MachinePrecision], N[(N[((-b) / a), $MachinePrecision] + N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -6.20000000000000003e-84

   1. Initial program 16.5%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 84.4

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

   5. Applied rewrites 84.4%

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

   if -6.20000000000000003e-84 < b < 7.4999999999999998e-49

   1. Initial program 79.3%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

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

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 79.3

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

   4. Applied rewrites 79.3%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 79.3

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

   6. Applied rewrites 79.3%

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

   7. Taylor expanded in a around inf

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

      1. sqrt-unprod N/A

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

      2. *-commutative N/A

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

      3. lower-sqrt.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 72.3

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

   9. Applied rewrites 72.3%

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

   if 7.4999999999999998e-49 < b

   1. Initial program 70.3%

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

   3. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 84.6

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

   5. Applied rewrites 84.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{b}{a}, -1, \frac{c}{b}\right)} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. associate-*r/ N/A

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

      7. mul-1-neg N/A

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

      8. lower-/.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. lift-/.f64 84.6

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

   7. Applied rewrites 84.6%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6.2 \cdot 10^{-84}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{elif}\;b \leq 7.5 \cdot 10^{-49}:\\ \;\;\;\;\frac{b + \sqrt{\left(a \cdot c\right) \cdot -4}}{-\left(a + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a} + \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -6.2 \cdot 10^{-84}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{elif}\;b \leq 3.5 \cdot 10^{-99}:\\ \;\;\;\;\frac{-\sqrt{\left(a \cdot c\right) \cdot -4}}{a + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a} + \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -6.2e-84)
   (/ c (- b))
   (if (<= b 3.5e-99)
     (/ (- (sqrt (* (* a c) -4.0))) (+ a a))
     (+ (/ (- b) a) (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -6.2e-84) {
		tmp = c / -b;
	} else if (b <= 3.5e-99) {
		tmp = -sqrt(((a * c) * -4.0)) / (a + a);
	} else {
		tmp = (-b / a) + (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 <= (-6.2d-84)) then
        tmp = c / -b
    else if (b <= 3.5d-99) then
        tmp = -sqrt(((a * c) * (-4.0d0))) / (a + a)
    else
        tmp = (-b / a) + (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -6.2e-84) {
		tmp = c / -b;
	} else if (b <= 3.5e-99) {
		tmp = -Math.sqrt(((a * c) * -4.0)) / (a + a);
	} else {
		tmp = (-b / a) + (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -6.2e-84:
		tmp = c / -b
	elif b <= 3.5e-99:
		tmp = -math.sqrt(((a * c) * -4.0)) / (a + a)
	else:
		tmp = (-b / a) + (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -6.2e-84)
		tmp = Float64(c / Float64(-b));
	elseif (b <= 3.5e-99)
		tmp = Float64(Float64(-sqrt(Float64(Float64(a * c) * -4.0))) / Float64(a + a));
	else
		tmp = Float64(Float64(Float64(-b) / a) + Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -6.2e-84)
		tmp = c / -b;
	elseif (b <= 3.5e-99)
		tmp = -sqrt(((a * c) * -4.0)) / (a + a);
	else
		tmp = (-b / a) + (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -6.2e-84], N[(c / (-b)), $MachinePrecision], If[LessEqual[b, 3.5e-99], N[((-N[Sqrt[N[(N[(a * c), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision]) / N[(a + a), $MachinePrecision]), $MachinePrecision], N[(N[((-b) / a), $MachinePrecision] + N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -6.20000000000000003e-84

   1. Initial program 16.5%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 84.4

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

   5. Applied rewrites 84.4%

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

   if -6.20000000000000003e-84 < b < 3.4999999999999999e-99

   1. Initial program 78.6%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

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

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 78.6

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

   4. Applied rewrites 78.6%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 78.6

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

   6. Applied rewrites 78.6%

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

   7. Taylor expanded in a around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. sqrt-unprod N/A

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

      4. *-commutative N/A

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

      5. lower-sqrt.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 75.8

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

   9. Applied rewrites 75.8%

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

   if 3.4999999999999999e-99 < b

   1. Initial program 71.9%

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

   3. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 79.6

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

   5. Applied rewrites 79.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{b}{a}, -1, \frac{c}{b}\right)} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. associate-*r/ N/A

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

      7. mul-1-neg N/A

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

      8. lower-/.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. lift-/.f64 79.6

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

   7. Applied rewrites 79.6%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6.2 \cdot 10^{-84}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{elif}\;b \leq 3.5 \cdot 10^{-99}:\\ \;\;\;\;\frac{-\sqrt{\left(a \cdot c\right) \cdot -4}}{a + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a} + \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 72.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.45 \cdot 10^{-178}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{elif}\;b \leq 5.8 \cdot 10^{-162}:\\ \;\;\;\;\sqrt{\frac{-c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a} + \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.45e-178)
   (/ c (- b))
   (if (<= b 5.8e-162) (sqrt (/ (- c) a)) (+ (/ (- b) a) (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.45e-178) {
		tmp = c / -b;
	} else if (b <= 5.8e-162) {
		tmp = sqrt((-c / a));
	} else {
		tmp = (-b / a) + (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 <= (-2.45d-178)) then
        tmp = c / -b
    else if (b <= 5.8d-162) then
        tmp = sqrt((-c / a))
    else
        tmp = (-b / a) + (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.45e-178) {
		tmp = c / -b;
	} else if (b <= 5.8e-162) {
		tmp = Math.sqrt((-c / a));
	} else {
		tmp = (-b / a) + (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.45e-178:
		tmp = c / -b
	elif b <= 5.8e-162:
		tmp = math.sqrt((-c / a))
	else:
		tmp = (-b / a) + (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.45e-178)
		tmp = Float64(c / Float64(-b));
	elseif (b <= 5.8e-162)
		tmp = sqrt(Float64(Float64(-c) / a));
	else
		tmp = Float64(Float64(Float64(-b) / a) + Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.45e-178)
		tmp = c / -b;
	elseif (b <= 5.8e-162)
		tmp = sqrt((-c / a));
	else
		tmp = (-b / a) + (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.45e-178], N[(c / (-b)), $MachinePrecision], If[LessEqual[b, 5.8e-162], N[Sqrt[N[((-c) / a), $MachinePrecision]], $MachinePrecision], N[(N[((-b) / a), $MachinePrecision] + N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.4500000000000001e-178

   1. Initial program 22.3%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 78.1

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

   5. Applied rewrites 78.1%

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

   if -2.4500000000000001e-178 < b < 5.8000000000000002e-162

   1. Initial program 82.4%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

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

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 82.4

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

   4. Applied rewrites 82.4%

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

   5. Taylor expanded in a around -inf

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

      1. div-sub N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. sqrt-unprod N/A

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

      6. *-commutative N/A

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

      7. lower-sqrt.f64 N/A

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

      8. associate-*r/ N/A

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

      9. mul-1-neg N/A

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

      10. lower-/.f64 N/A

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

      11. lower-neg.f64 43.8

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

   7. Applied rewrites 43.8%

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

   if 5.8000000000000002e-162 < b

   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 c around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 75.0

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

   5. Applied rewrites 75.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{b}{a}, -1, \frac{c}{b}\right)} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. associate-*r/ N/A

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

      7. mul-1-neg N/A

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

      8. lower-/.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. lift-/.f64 75.0

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

   7. Applied rewrites 75.0%

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

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.45 \cdot 10^{-178}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{elif}\;b \leq 5.8 \cdot 10^{-162}:\\ \;\;\;\;\sqrt{\frac{-c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a} + \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 72.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.45 \cdot 10^{-178}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{elif}\;b \leq 5.8 \cdot 10^{-162}:\\ \;\;\;\;\sqrt{\frac{-c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.45e-178)
   (/ c (- b))
   (if (<= b 5.8e-162) (sqrt (/ (- c) a)) (/ (- b) a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.45e-178) {
		tmp = c / -b;
	} else if (b <= 5.8e-162) {
		tmp = sqrt((-c / a));
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-2.45d-178)) then
        tmp = c / -b
    else if (b <= 5.8d-162) then
        tmp = sqrt((-c / a))
    else
        tmp = -b / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.45e-178) {
		tmp = c / -b;
	} else if (b <= 5.8e-162) {
		tmp = Math.sqrt((-c / a));
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.45e-178:
		tmp = c / -b
	elif b <= 5.8e-162:
		tmp = math.sqrt((-c / a))
	else:
		tmp = -b / a
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.45e-178)
		tmp = Float64(c / Float64(-b));
	elseif (b <= 5.8e-162)
		tmp = sqrt(Float64(Float64(-c) / a));
	else
		tmp = Float64(Float64(-b) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.45e-178)
		tmp = c / -b;
	elseif (b <= 5.8e-162)
		tmp = sqrt((-c / a));
	else
		tmp = -b / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.45e-178], N[(c / (-b)), $MachinePrecision], If[LessEqual[b, 5.8e-162], N[Sqrt[N[((-c) / a), $MachinePrecision]], $MachinePrecision], N[((-b) / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.4500000000000001e-178

   1. Initial program 22.3%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 78.1

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

   5. Applied rewrites 78.1%

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

   if -2.4500000000000001e-178 < b < 5.8000000000000002e-162

   1. Initial program 82.4%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

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

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 82.4

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

   4. Applied rewrites 82.4%

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

   5. Taylor expanded in a around -inf

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

      1. div-sub N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. sqrt-unprod N/A

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

      6. *-commutative N/A

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

      7. lower-sqrt.f64 N/A

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

      8. associate-*r/ N/A

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

      9. mul-1-neg N/A

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

      10. lower-/.f64 N/A

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

      11. lower-neg.f64 43.8

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

   7. Applied rewrites 43.8%

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

   if 5.8000000000000002e-162 < b

   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 a around 0

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lift-neg.f64 74.9

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

   5. Applied rewrites 74.9%

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

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.45 \cdot 10^{-178}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{elif}\;b \leq 5.8 \cdot 10^{-162}:\\ \;\;\;\;\sqrt{\frac{-c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 68.2% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e-310) (/ c (- b)) (/ (- b) a)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = c / -b;
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-5d-310)) then
        tmp = c / -b
    else
        tmp = -b / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = c / -b;
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e-310:
		tmp = c / -b
	else:
		tmp = -b / a
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e-310)
		tmp = Float64(c / Float64(-b));
	else
		tmp = Float64(Float64(-b) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5e-310)
		tmp = c / -b;
	else
		tmp = -b / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-310], N[(c / (-b)), $MachinePrecision], N[((-b) / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 35.0%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 63.0

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

   5. Applied rewrites 63.0%

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

   if -4.999999999999985e-310 < b

   1. Initial program 74.6%

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

   3. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lift-neg.f64 62.5

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

   5. Applied rewrites 62.5%

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

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 35.8% accurate, 3.6× 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 / Float64(-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 55.0%

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

3. Taylor expanded in b around -inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. lower-/.f64 32.4

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

5. Applied rewrites 32.4%

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

6. Final simplification 32.4%

   \[\leadsto \frac{c}{-b} \]
7. Add Preprocessing

Alternative 10: 10.8% 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 55.0%

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

3. Taylor expanded in c around 0

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

   1. *-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-/.f64 32.7

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

5. Applied rewrites 32.7%

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

6. Taylor expanded in a around inf

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

   1. lift-/.f64 11.4

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

8. Applied rewrites 11.4%

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

Developer Target 1: 99.7% accurate, 0.2× speedup

Math

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

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) (/ c (- t_2 (/ b 2.0))) (/ (+ (/ b 2.0) t_2) (- a)))))

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 = c / (t_2 - (b / 2.0));
	} else {
		tmp_1 = ((b / 2.0) + t_2) / -a;
	}
	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 = c / (t_2 - (b / 2.0));
	} else {
		tmp_1 = ((b / 2.0) + t_2) / -a;
	}
	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 = c / (t_2 - (b / 2.0))
	else:
		tmp_1 = ((b / 2.0) + t_2) / -a
	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(c / Float64(t_2 - Float64(b / 2.0)));
	else
		tmp_1 = Float64(Float64(Float64(b / 2.0) + t_2) / Float64(-a));
	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 = c / (t_2 - (b / 2.0));
	else
		tmp_2 = ((b / 2.0) + t_2) / -a;
	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[(c / N[(t$95$2 - N[(b / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b / 2.0), $MachinePrecision] + t$95$2), $MachinePrecision] / (-a)), $MachinePrecision]]]]]

Reproduce

herbie shell --seed 2025051 
(FPCore (a b c)
  :name "quadm (p42, negative)"
  :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) (/ c (- sqtD (/ b 2))) (/ (+ (/ b 2) sqtD) (- a)))))

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