Quadratic roots, full range

Percentage Accurate: 52.9% → 85.9%

Time: 8.8s

Alternatives: 6

Speedup: 2.5×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 52.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{+154}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 3.85 \cdot 10^{-90}:\\ \;\;\;\;\frac{b - \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot -4\right) \cdot c\right)}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2e+154)
   (/ (- b) a)
   (if (<= b 3.85e-90)
     (/ (- b (sqrt (fma b b (* (* a -4.0) c)))) (* -2.0 a))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e+154) {
		tmp = -b / a;
	} else if (b <= 3.85e-90) {
		tmp = (b - sqrt(fma(b, b, ((a * -4.0) * c)))) / (-2.0 * a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2e+154)
		tmp = Float64(Float64(-b) / a);
	elseif (b <= 3.85e-90)
		tmp = Float64(Float64(b - sqrt(fma(b, b, Float64(Float64(a * -4.0) * c)))) / Float64(-2.0 * a));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2e+154], N[((-b) / a), $MachinePrecision], If[LessEqual[b, 3.85e-90], N[(N[(b - N[Sqrt[N[(b * b + N[(N[(a * -4.0), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(-2.0 * a), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.00000000000000007e154

   1. Initial program 45.5%

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

   3. Taylor expanded in b around -inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 93.8

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

   5. Applied rewrites 93.8%

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

   if -2.00000000000000007e154 < b < 3.84999999999999986e-90

   1. Initial program 84.6%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

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

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

      7. lift-*.f64 N/A

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

      8. sqr-abs-rev N/A

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

      9. sqr-abs-rev N/A

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

      10. fabs-fabs N/A

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

      11. fabs-fabs N/A

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

      12. rem-sqrt-square-rev N/A

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

      13. pow2 N/A

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

      14. sqrt-pow1 N/A

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

      15. metadata-eval N/A

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

      16. unpow1 N/A

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

      17. associate-*l* N/A

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

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

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

      19. *-commutative N/A

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

      20. lift-*.f64 N/A

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

   4. Applied rewrites 84.6%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. lower-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/r* N/A

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

      8. lift-/.f64 N/A

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

      9. frac-2neg N/A

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

      10. lift-/.f64 N/A

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

      11. distribute-frac-neg2 N/A

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

      12. lift-neg.f64 N/A

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

      13. frac-2neg N/A

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

      14. lift-/.f64 N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 N/A

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

      17. lower-/.f64 84.6

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

   6. Applied rewrites 84.6%

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

   8. Applied rewrites 84.6%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

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

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

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

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

      8. lift-*.f64 N/A

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

      9. remove-double-neg N/A

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

      10. *-commutative N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 84.6

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

   10. Applied rewrites 84.6%

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

   if 3.84999999999999986e-90 < b

   1. Initial program 16.6%

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

   3. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 80.0

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

   5. Applied rewrites 80.0%

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

Alternative 2: 80.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -7.4 \cdot 10^{-110}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 3.85 \cdot 10^{-90}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\left(c \cdot a\right) \cdot -4}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -7.4e-110)
   (- (/ c b) (/ b a))
   (if (<= b 3.85e-90)
     (/ (+ (- b) (sqrt (* (* c a) -4.0))) (* 2.0 a))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -7.4e-110) {
		tmp = (c / b) - (b / a);
	} else if (b <= 3.85e-90) {
		tmp = (-b + sqrt(((c * a) * -4.0))) / (2.0 * a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-7.4d-110)) then
        tmp = (c / b) - (b / a)
    else if (b <= 3.85d-90) then
        tmp = (-b + sqrt(((c * a) * (-4.0d0)))) / (2.0d0 * a)
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -7.4e-110) {
		tmp = (c / b) - (b / a);
	} else if (b <= 3.85e-90) {
		tmp = (-b + Math.sqrt(((c * a) * -4.0))) / (2.0 * a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -7.4e-110:
		tmp = (c / b) - (b / a)
	elif b <= 3.85e-90:
		tmp = (-b + math.sqrt(((c * a) * -4.0))) / (2.0 * a)
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -7.4e-110)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 3.85e-90)
		tmp = Float64(Float64(Float64(-b) + sqrt(Float64(Float64(c * a) * -4.0))) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -7.4e-110)
		tmp = (c / b) - (b / a);
	elseif (b <= 3.85e-90)
		tmp = (-b + sqrt(((c * a) * -4.0))) / (2.0 * a);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -7.4e-110], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 3.85e-90], N[(N[((-b) + N[Sqrt[N[(N[(c * a), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -7.40000000000000032e-110

   1. Initial program 72.0%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-*l/ N/A

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

      5. *-lft-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. associate-*r/ N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 87.0

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

   5. Applied rewrites 87.0%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 87.0%

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

   if -7.40000000000000032e-110 < b < 3.84999999999999986e-90

   1. Initial program 76.5%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 75.1

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

   5. Applied rewrites 75.1%

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

   if 3.84999999999999986e-90 < b

   1. Initial program 16.6%

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

   3. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 80.0

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

   5. Applied rewrites 80.0%

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

Alternative 3: 67.4% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 74.2%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-*l/ N/A

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

      5. *-lft-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. associate-*r/ N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 66.4

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

   5. Applied rewrites 66.4%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 66.6%

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

   if -4.999999999999985e-310 < b

   1. Initial program 30.0%

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

   3. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 66.6

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

   5. Applied rewrites 66.6%

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

Alternative 4: 67.2% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 74.2%

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

   3. Taylor expanded in b around -inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 65.8

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

   5. Applied rewrites 65.8%

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

   if -4.999999999999985e-310 < b

   1. Initial program 30.0%

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

   3. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 66.6

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

   5. Applied rewrites 66.6%

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

Alternative 5: 43.4% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.15 \cdot 10^{-212}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b 1.15e-212) (/ (- b) a) 0.0))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.15e-212) {
		tmp = -b / a;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 1.15d-212) then
        tmp = -b / a
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.15e-212) {
		tmp = -b / a;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 1.15e-212:
		tmp = -b / a
	else:
		tmp = 0.0
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.15e-212)
		tmp = Float64(Float64(-b) / a);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 1.15e-212)
		tmp = -b / a;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.15e-212], N[((-b) / a), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if b < 1.15e-212

   1. Initial program 75.1%

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

   3. Taylor expanded in b around -inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 60.5

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

   5. Applied rewrites 60.5%

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

   if 1.15e-212 < b

   1. Initial program 24.4%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

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

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

      7. lift-*.f64 N/A

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

      8. sqr-abs-rev N/A

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

      9. sqr-abs-rev N/A

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

      10. fabs-fabs N/A

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

      11. fabs-fabs N/A

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

      12. rem-sqrt-square-rev N/A

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

      13. pow2 N/A

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

      14. sqrt-pow1 N/A

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

      15. metadata-eval N/A

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

      16. unpow1 N/A

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

      17. associate-*l* N/A

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

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

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

      19. *-commutative N/A

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

      20. lift-*.f64 N/A

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

   4. Applied rewrites 24.4%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. lower-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/r* N/A

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

      8. lift-/.f64 N/A

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

      9. frac-2neg N/A

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

      10. lift-/.f64 N/A

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

      11. distribute-frac-neg2 N/A

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

      12. lift-neg.f64 N/A

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

      13. frac-2neg N/A

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

      14. lift-/.f64 N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 N/A

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

      17. lower-/.f64 22.9

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

   6. Applied rewrites 22.9%

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

   7. Taylor expanded in a around 0

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

      1. div-add N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot b}{a} + \frac{\frac{1}{2} \cdot b}{a}} \]

      2. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{b}{a}} + \frac{\frac{1}{2} \cdot b}{a} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{b}{a} + \color{blue}{\frac{1}{2} \cdot \frac{b}{a}} \]

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

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

      5. metadata-eval N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{b}{a} - \color{blue}{\frac{-1}{2}} \cdot \frac{b}{a} \]

      6. +-inverses 23.7

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

   9. Applied rewrites 23.7%

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

Alternative 6: 10.9% accurate, 50.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = 0.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_, c_] := 0.0

Derivation

1. Initial program 52.1%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

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

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

   7. lift-*.f64 N/A

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

   8. sqr-abs-rev N/A

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

   9. sqr-abs-rev N/A

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

   10. fabs-fabs N/A

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

   11. fabs-fabs N/A

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

   12. rem-sqrt-square-rev N/A

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

   13. pow2 N/A

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

   14. sqrt-pow1 N/A

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

   15. metadata-eval N/A

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

   16. unpow1 N/A

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

   17. associate-*l* N/A

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

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

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

   19. *-commutative N/A

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

   20. lift-*.f64 N/A

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

4. Applied rewrites 52.1%

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

   1. lift-/.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. div-add N/A

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

   4. lower-+.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. associate-/r* N/A

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

   8. lift-/.f64 N/A

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

   9. frac-2neg N/A

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

   10. lift-/.f64 N/A

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

   11. distribute-frac-neg2 N/A

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

   12. lift-neg.f64 N/A

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

   13. frac-2neg N/A

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

   14. lift-/.f64 N/A

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

   15. metadata-eval N/A

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

   16. lower-/.f64 N/A

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

   17. lower-/.f64 51.5

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

6. Applied rewrites 51.5%

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

7. Taylor expanded in a around 0

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

   1. div-add N/A

      \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot b}{a} + \frac{\frac{1}{2} \cdot b}{a}} \]

   2. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{b}{a}} + \frac{\frac{1}{2} \cdot b}{a} \]

   3. associate-*r/ N/A

      \[\leadsto \frac{-1}{2} \cdot \frac{b}{a} + \color{blue}{\frac{1}{2} \cdot \frac{b}{a}} \]

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

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

   5. metadata-eval N/A

      \[\leadsto \frac{-1}{2} \cdot \frac{b}{a} - \color{blue}{\frac{-1}{2}} \cdot \frac{b}{a} \]

   6. +-inverses 12.3

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

9. Applied rewrites 12.3%

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

Reproduce

herbie shell --seed 2024354 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))