Quadratic roots, medium range

Percentage Accurate: 31.5% → 99.4%

Time: 3.8s

Alternatives: 5

Speedup: 2.9×

Specification

\[\left(\left(1.1102230246251565 \cdot 10^{-16} < a \land a < 9007199254740992\right) \land \left(1.1102230246251565 \cdot 10^{-16} < b \land b < 9007199254740992\right)\right) \land \left(1.1102230246251565 \cdot 10^{-16} < c \land c < 9007199254740992\right)\]

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: 31.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(c \cdot a, -4, 0\right)}{\left(\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)} + b\right) \cdot \left(a + a\right)} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (fma (* c a) -4.0 0.0)
  (* (+ (sqrt (fma b b (* -4.0 (* a c)))) b) (+ a a))))

C

double code(double a, double b, double c) {
	return fma((c * a), -4.0, 0.0) / ((sqrt(fma(b, b, (-4.0 * (a * c)))) + b) * (a + a));
}

Julia

function code(a, b, c)
	return Float64(fma(Float64(c * a), -4.0, 0.0) / Float64(Float64(sqrt(fma(b, b, Float64(-4.0 * Float64(a * c)))) + b) * Float64(a + a)))
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(c * a), $MachinePrecision] * -4.0 + 0.0), $MachinePrecision] / N[(N[(N[Sqrt[N[(b * b + N[(-4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + b), $MachinePrecision] * N[(a + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.5%

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

   1. lift-/.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. div-add N/A

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

   4. frac-add N/A

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

   5. lower-/.f64 N/A

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

3. Applied rewrites 33.0%

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

5. Applied rewrites 99.4%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(c \cdot a, -4, 0\right)}{\left(\sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)} + b\right) \cdot \left(a \cdot 2\right)}} \]
6. Step-by-step derivation

   1. lift-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(c \cdot a, -4, 0\right)}{\left(\sqrt{\color{blue}{\left(c \cdot a\right) \cdot -4 + b \cdot b}} + b\right) \cdot \left(a \cdot 2\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(c \cdot a, -4, 0\right)}{\left(\sqrt{\color{blue}{\left(c \cdot a\right) \cdot -4} + b \cdot b} + b\right) \cdot \left(a \cdot 2\right)} \]

   3. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(c \cdot a, -4, 0\right)}{\left(\sqrt{\color{blue}{b \cdot b + \left(c \cdot a\right) \cdot -4}} + b\right) \cdot \left(a \cdot 2\right)} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(c \cdot a, -4, 0\right)}{\left(\sqrt{\color{blue}{b \cdot b} + \left(c \cdot a\right) \cdot -4} + b\right) \cdot \left(a \cdot 2\right)} \]

   5. lower-fma.f64 99.4

      \[\leadsto \frac{\mathsf{fma}\left(c \cdot a, -4, 0\right)}{\left(\sqrt{\color{blue}{\mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot -4\right)}} + b\right) \cdot \left(a \cdot 2\right)} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(c \cdot a, -4, 0\right)}{\left(\sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(c \cdot a\right) \cdot -4}\right)} + b\right) \cdot \left(a \cdot 2\right)} \]

   7. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(c \cdot a, -4, 0\right)}{\left(\sqrt{\mathsf{fma}\left(b, b, \color{blue}{-4 \cdot \left(c \cdot a\right)}\right)} + b\right) \cdot \left(a \cdot 2\right)} \]

   8. lower-*.f64 99.4

      \[\leadsto \frac{\mathsf{fma}\left(c \cdot a, -4, 0\right)}{\left(\sqrt{\mathsf{fma}\left(b, b, \color{blue}{-4 \cdot \left(c \cdot a\right)}\right)} + b\right) \cdot \left(a \cdot 2\right)} \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(c \cdot a, -4, 0\right)}{\left(\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \color{blue}{\left(c \cdot a\right)}\right)} + b\right) \cdot \left(a \cdot 2\right)} \]

   10. *-commutative N/A

       \[\leadsto \frac{\mathsf{fma}\left(c \cdot a, -4, 0\right)}{\left(\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \color{blue}{\left(a \cdot c\right)}\right)} + b\right) \cdot \left(a \cdot 2\right)} \]

   11. lower-*.f64 99.4

       \[\leadsto \frac{\mathsf{fma}\left(c \cdot a, -4, 0\right)}{\left(\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \color{blue}{\left(a \cdot c\right)}\right)} + b\right) \cdot \left(a \cdot 2\right)} \]

7. Applied rewrites 99.4%

   \[\leadsto \frac{\mathsf{fma}\left(c \cdot a, -4, 0\right)}{\left(\sqrt{\color{blue}{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)}} + b\right) \cdot \left(a \cdot 2\right)} \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(c \cdot a, -4, 0\right)}{\left(\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)} + b\right) \cdot \color{blue}{\left(a \cdot 2\right)}} \]

   2. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(c \cdot a, -4, 0\right)}{\left(\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)} + b\right) \cdot \color{blue}{\left(2 \cdot a\right)}} \]

   3. count-2-rev N/A

      \[\leadsto \frac{\mathsf{fma}\left(c \cdot a, -4, 0\right)}{\left(\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)} + b\right) \cdot \color{blue}{\left(a + a\right)}} \]

   4. lower-+.f64 99.4

      \[\leadsto \frac{\mathsf{fma}\left(c \cdot a, -4, 0\right)}{\left(\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)} + b\right) \cdot \color{blue}{\left(a + a\right)}} \]

9. Applied rewrites 99.4%

   \[\leadsto \frac{\mathsf{fma}\left(c \cdot a, -4, 0\right)}{\left(\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)} + b\right) \cdot \color{blue}{\left(a + a\right)}} \]
10. Add Preprocessing

Alternative 2: 90.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 4.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot -4\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c \cdot a, -4, 0\right)}{a \cdot \mathsf{fma}\left(-4, \frac{a \cdot c}{b}, 4 \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 4.2e-5)
   (/ (- (sqrt (fma b b (* (* c a) -4.0))) b) (* 2.0 a))
   (/ (fma (* c a) -4.0 0.0) (* a (fma -4.0 (/ (* a c) b) (* 4.0 b))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 4.2e-5) {
		tmp = (sqrt(fma(b, b, ((c * a) * -4.0))) - b) / (2.0 * a);
	} else {
		tmp = fma((c * a), -4.0, 0.0) / (a * fma(-4.0, ((a * c) / b), (4.0 * b)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 4.2e-5)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(Float64(c * a) * -4.0))) - b) / Float64(2.0 * a));
	else
		tmp = Float64(fma(Float64(c * a), -4.0, 0.0) / Float64(a * fma(-4.0, Float64(Float64(a * c) / b), Float64(4.0 * b))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 4.2e-5], N[(N[(N[Sqrt[N[(b * b + N[(N[(c * a), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(c * a), $MachinePrecision] * -4.0 + 0.0), $MachinePrecision] / N[(a * N[(-4.0 * N[(N[(a * c), $MachinePrecision] / b), $MachinePrecision] + N[(4.0 * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 4.19999999999999977e-5

   1. Initial program 72.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-negate1-reverse N/A

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

      5. lower--.f64 72.2

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

      6. lift--.f64 N/A

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

      7. sub-negate1 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

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

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

      13. lower-fma.f64 N/A

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

      14. metadata-eval N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 72.2

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

   3. Applied rewrites 72.2%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

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

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. exp-to-pow N/A

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

      7. lift-log.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-exp.f64 N/A

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

      10. metadata-eval N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-exp.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. lift-log.f64 N/A

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

      19. exp-to-pow N/A

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

      20. pow2 N/A

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

      21. lift-*.f64 N/A

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

      22. lift-*.f64 N/A

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

      23. *-commutative N/A

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

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

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

      25. lift-*.f64 N/A

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

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

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

      27. *-commutative N/A

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

      28. lift-*.f64 N/A

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

      29. associate-*l* N/A

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

   5. Applied rewrites 72.3%

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

   if 4.19999999999999977e-5 < b

   1. Initial program 28.8%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. frac-add N/A

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

      5. lower-/.f64 N/A

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

   3. Applied rewrites 30.4%

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

   5. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(c \cdot a, -4, 0\right)}{\left(\sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)} + b\right) \cdot \left(a \cdot 2\right)}} \]
   6. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(c \cdot a, -4, 0\right)}{\left(\sqrt{\color{blue}{\left(c \cdot a\right) \cdot -4 + b \cdot b}} + b\right) \cdot \left(a \cdot 2\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(c \cdot a, -4, 0\right)}{\left(\sqrt{\color{blue}{\left(c \cdot a\right) \cdot -4} + b \cdot b} + b\right) \cdot \left(a \cdot 2\right)} \]

      3. flip-+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(c \cdot a, -4, 0\right)}{\left(\sqrt{\color{blue}{\frac{\left(\left(c \cdot a\right) \cdot -4\right) \cdot \left(\left(c \cdot a\right) \cdot -4\right) - \left(b \cdot b\right) \cdot \left(b \cdot b\right)}{\left(c \cdot a\right) \cdot -4 - b \cdot b}}} + b\right) \cdot \left(a \cdot 2\right)} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(c \cdot a, -4, 0\right)}{\left(\sqrt{\color{blue}{\frac{\left(\left(c \cdot a\right) \cdot -4\right) \cdot \left(\left(c \cdot a\right) \cdot -4\right) - \left(b \cdot b\right) \cdot \left(b \cdot b\right)}{\left(c \cdot a\right) \cdot -4 - b \cdot b}}} + b\right) \cdot \left(a \cdot 2\right)} \]

      5. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(c \cdot a, -4, 0\right)}{\left(\sqrt{\frac{\color{blue}{{\left(\left(c \cdot a\right) \cdot -4\right)}^{2}} - \left(b \cdot b\right) \cdot \left(b \cdot b\right)}{\left(c \cdot a\right) \cdot -4 - b \cdot b}} + b\right) \cdot \left(a \cdot 2\right)} \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(c \cdot a, -4, 0\right)}{\left(\sqrt{\frac{\color{blue}{{\left(\left(c \cdot a\right) \cdot -4\right)}^{2}} - \left(b \cdot b\right) \cdot \left(b \cdot b\right)}{\left(c \cdot a\right) \cdot -4 - b \cdot b}} + b\right) \cdot \left(a \cdot 2\right)} \]

      7. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(c \cdot a, -4, 0\right)}{\left(\sqrt{\frac{\color{blue}{{\left(\left(c \cdot a\right) \cdot -4\right)}^{2} - \left(b \cdot b\right) \cdot \left(b \cdot b\right)}}{\left(c \cdot a\right) \cdot -4 - b \cdot b}} + b\right) \cdot \left(a \cdot 2\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(c \cdot a, -4, 0\right)}{\left(\sqrt{\frac{{\color{blue}{\left(\left(c \cdot a\right) \cdot -4\right)}}^{2} - \left(b \cdot b\right) \cdot \left(b \cdot b\right)}{\left(c \cdot a\right) \cdot -4 - b \cdot b}} + b\right) \cdot \left(a \cdot 2\right)} \]

      9. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(c \cdot a, -4, 0\right)}{\left(\sqrt{\frac{{\color{blue}{\left(-4 \cdot \left(c \cdot a\right)\right)}}^{2} - \left(b \cdot b\right) \cdot \left(b \cdot b\right)}{\left(c \cdot a\right) \cdot -4 - b \cdot b}} + b\right) \cdot \left(a \cdot 2\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(c \cdot a, -4, 0\right)}{\left(\sqrt{\frac{{\color{blue}{\left(-4 \cdot \left(c \cdot a\right)\right)}}^{2} - \left(b \cdot b\right) \cdot \left(b \cdot b\right)}{\left(c \cdot a\right) \cdot -4 - b \cdot b}} + b\right) \cdot \left(a \cdot 2\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(c \cdot a, -4, 0\right)}{\left(\sqrt{\frac{{\left(-4 \cdot \color{blue}{\left(c \cdot a\right)}\right)}^{2} - \left(b \cdot b\right) \cdot \left(b \cdot b\right)}{\left(c \cdot a\right) \cdot -4 - b \cdot b}} + b\right) \cdot \left(a \cdot 2\right)} \]

      12. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(c \cdot a, -4, 0\right)}{\left(\sqrt{\frac{{\left(-4 \cdot \color{blue}{\left(a \cdot c\right)}\right)}^{2} - \left(b \cdot b\right) \cdot \left(b \cdot b\right)}{\left(c \cdot a\right) \cdot -4 - b \cdot b}} + b\right) \cdot \left(a \cdot 2\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(c \cdot a, -4, 0\right)}{\left(\sqrt{\frac{{\left(-4 \cdot \color{blue}{\left(a \cdot c\right)}\right)}^{2} - \left(b \cdot b\right) \cdot \left(b \cdot b\right)}{\left(c \cdot a\right) \cdot -4 - b \cdot b}} + b\right) \cdot \left(a \cdot 2\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(c \cdot a, -4, 0\right)}{\left(\sqrt{\frac{{\left(-4 \cdot \left(a \cdot c\right)\right)}^{2} - \color{blue}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}}{\left(c \cdot a\right) \cdot -4 - b \cdot b}} + b\right) \cdot \left(a \cdot 2\right)} \]

      15. lower--.f64 99.4

          \[\leadsto \frac{\mathsf{fma}\left(c \cdot a, -4, 0\right)}{\left(\sqrt{\frac{{\left(-4 \cdot \left(a \cdot c\right)\right)}^{2} - \left(b \cdot b\right) \cdot \left(b \cdot b\right)}{\color{blue}{\left(c \cdot a\right) \cdot -4 - b \cdot b}}} + b\right) \cdot \left(a \cdot 2\right)} \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(c \cdot a, -4, 0\right)}{\left(\sqrt{\frac{{\left(-4 \cdot \left(a \cdot c\right)\right)}^{2} - \left(b \cdot b\right) \cdot \left(b \cdot b\right)}{\color{blue}{\left(c \cdot a\right) \cdot -4} - b \cdot b}} + b\right) \cdot \left(a \cdot 2\right)} \]

      17. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(c \cdot a, -4, 0\right)}{\left(\sqrt{\frac{{\left(-4 \cdot \left(a \cdot c\right)\right)}^{2} - \left(b \cdot b\right) \cdot \left(b \cdot b\right)}{\color{blue}{-4 \cdot \left(c \cdot a\right)} - b \cdot b}} + b\right) \cdot \left(a \cdot 2\right)} \]

      18. lower-*.f64 99.4

          \[\leadsto \frac{\mathsf{fma}\left(c \cdot a, -4, 0\right)}{\left(\sqrt{\frac{{\left(-4 \cdot \left(a \cdot c\right)\right)}^{2} - \left(b \cdot b\right) \cdot \left(b \cdot b\right)}{\color{blue}{-4 \cdot \left(c \cdot a\right)} - b \cdot b}} + b\right) \cdot \left(a \cdot 2\right)} \]

      19. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(c \cdot a, -4, 0\right)}{\left(\sqrt{\frac{{\left(-4 \cdot \left(a \cdot c\right)\right)}^{2} - \left(b \cdot b\right) \cdot \left(b \cdot b\right)}{-4 \cdot \color{blue}{\left(c \cdot a\right)} - b \cdot b}} + b\right) \cdot \left(a \cdot 2\right)} \]

      20. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(c \cdot a, -4, 0\right)}{\left(\sqrt{\frac{{\left(-4 \cdot \left(a \cdot c\right)\right)}^{2} - \left(b \cdot b\right) \cdot \left(b \cdot b\right)}{-4 \cdot \color{blue}{\left(a \cdot c\right)} - b \cdot b}} + b\right) \cdot \left(a \cdot 2\right)} \]

      21. lower-*.f64 99.4

          \[\leadsto \frac{\mathsf{fma}\left(c \cdot a, -4, 0\right)}{\left(\sqrt{\frac{{\left(-4 \cdot \left(a \cdot c\right)\right)}^{2} - \left(b \cdot b\right) \cdot \left(b \cdot b\right)}{-4 \cdot \color{blue}{\left(a \cdot c\right)} - b \cdot b}} + b\right) \cdot \left(a \cdot 2\right)} \]

   7. Applied rewrites 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(c \cdot a, -4, 0\right)}{\left(\sqrt{\color{blue}{\frac{{\left(-4 \cdot \left(a \cdot c\right)\right)}^{2} - \left(b \cdot b\right) \cdot \left(b \cdot b\right)}{-4 \cdot \left(a \cdot c\right) - b \cdot b}}} + b\right) \cdot \left(a \cdot 2\right)} \]

   8. Taylor expanded in a around 0

      \[\leadsto \frac{\mathsf{fma}\left(c \cdot a, -4, 0\right)}{\color{blue}{a \cdot \left(-4 \cdot \frac{a \cdot c}{b} + 4 \cdot b\right)}} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(c \cdot a, -4, 0\right)}{a \cdot \color{blue}{\left(-4 \cdot \frac{a \cdot c}{b} + 4 \cdot b\right)}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(c \cdot a, -4, 0\right)}{a \cdot \mathsf{fma}\left(-4, \color{blue}{\frac{a \cdot c}{b}}, 4 \cdot b\right)} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(c \cdot a, -4, 0\right)}{a \cdot \mathsf{fma}\left(-4, \frac{a \cdot c}{\color{blue}{b}}, 4 \cdot b\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(c \cdot a, -4, 0\right)}{a \cdot \mathsf{fma}\left(-4, \frac{a \cdot c}{b}, 4 \cdot b\right)} \]

      5. lower-*.f64 91.9

         \[\leadsto \frac{\mathsf{fma}\left(c \cdot a, -4, 0\right)}{a \cdot \mathsf{fma}\left(-4, \frac{a \cdot c}{b}, 4 \cdot b\right)} \]

   10. Applied rewrites 91.9%

       \[\leadsto \frac{\mathsf{fma}\left(c \cdot a, -4, 0\right)}{\color{blue}{a \cdot \mathsf{fma}\left(-4, \frac{a \cdot c}{b}, 4 \cdot b\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 82.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 8 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot -4\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 8e-5)
   (/ (- (sqrt (fma b b (* (* c a) -4.0))) b) (* 2.0 a))
   (* -1.0 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 8e-5) {
		tmp = (sqrt(fma(b, b, ((c * a) * -4.0))) - b) / (2.0 * a);
	} else {
		tmp = -1.0 * (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 8e-5)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(Float64(c * a) * -4.0))) - b) / Float64(2.0 * a));
	else
		tmp = Float64(-1.0 * Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 8e-5], N[(N[(N[Sqrt[N[(b * b + N[(N[(c * a), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 8.00000000000000065e-5

   1. Initial program 71.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-negate1-reverse N/A

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

      5. lower--.f64 71.6

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

      6. lift--.f64 N/A

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

      7. sub-negate1 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

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

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

      13. lower-fma.f64 N/A

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

      14. metadata-eval N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 71.6

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

   3. Applied rewrites 71.6%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

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

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. exp-to-pow N/A

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

      7. lift-log.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-exp.f64 N/A

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

      10. metadata-eval N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-exp.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. lift-log.f64 N/A

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

      19. exp-to-pow N/A

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

      20. pow2 N/A

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

      21. lift-*.f64 N/A

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

      22. lift-*.f64 N/A

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

      23. *-commutative N/A

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

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

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

      25. lift-*.f64 N/A

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

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

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

      27. *-commutative N/A

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

      28. lift-*.f64 N/A

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

      29. associate-*l* N/A

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

   5. Applied rewrites 71.7%

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

   if 8.00000000000000065e-5 < b

   1. Initial program 28.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. flip-+ N/A

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

      4. lower-/.f64 N/A

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

   3. Applied rewrites 29.4%

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

      1. lift--.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate--l+ N/A

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

      4. flip3-+ N/A

         \[\leadsto \frac{\frac{\color{blue}{\frac{{\left(-4 \cdot \left(c \cdot a\right)\right)}^{3} + {\left(b \cdot b - b \cdot b\right)}^{3}}{\left(-4 \cdot \left(c \cdot a\right)\right) \cdot \left(-4 \cdot \left(c \cdot a\right)\right) + \left(\left(b \cdot b - b \cdot b\right) \cdot \left(b \cdot b - b \cdot b\right) - \left(-4 \cdot \left(c \cdot a\right)\right) \cdot \left(b \cdot b - b \cdot b\right)\right)}}}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - \left(-b\right)}}{2 \cdot a} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\frac{{\left(-4 \cdot \left(c \cdot a\right)\right)}^{3} + {\left(b \cdot b - b \cdot b\right)}^{3}}{\left(-4 \cdot \left(c \cdot a\right)\right) \cdot \left(-4 \cdot \left(c \cdot a\right)\right) + \left(\left(b \cdot b - b \cdot b\right) \cdot \left(b \cdot b - b \cdot b\right) - \left(-4 \cdot \left(c \cdot a\right)\right) \cdot \left(b \cdot b - b \cdot b\right)\right)}}}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - \left(-b\right)}}{2 \cdot a} \]

      6. unpow-prod-down N/A

         \[\leadsto \frac{\frac{\frac{\color{blue}{{-4}^{3} \cdot {\left(c \cdot a\right)}^{3}} + {\left(b \cdot b - b \cdot b\right)}^{3}}{\left(-4 \cdot \left(c \cdot a\right)\right) \cdot \left(-4 \cdot \left(c \cdot a\right)\right) + \left(\left(b \cdot b - b \cdot b\right) \cdot \left(b \cdot b - b \cdot b\right) - \left(-4 \cdot \left(c \cdot a\right)\right) \cdot \left(b \cdot b - b \cdot b\right)\right)}}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - \left(-b\right)}}{2 \cdot a} \]

      7. +-inverses N/A

         \[\leadsto \frac{\frac{\frac{{-4}^{3} \cdot {\left(c \cdot a\right)}^{3} + {\color{blue}{0}}^{3}}{\left(-4 \cdot \left(c \cdot a\right)\right) \cdot \left(-4 \cdot \left(c \cdot a\right)\right) + \left(\left(b \cdot b - b \cdot b\right) \cdot \left(b \cdot b - b \cdot b\right) - \left(-4 \cdot \left(c \cdot a\right)\right) \cdot \left(b \cdot b - b \cdot b\right)\right)}}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - \left(-b\right)}}{2 \cdot a} \]

      8. metadata-eval N/A

         \[\leadsto \frac{\frac{\frac{{-4}^{3} \cdot {\left(c \cdot a\right)}^{3} + \color{blue}{0}}{\left(-4 \cdot \left(c \cdot a\right)\right) \cdot \left(-4 \cdot \left(c \cdot a\right)\right) + \left(\left(b \cdot b - b \cdot b\right) \cdot \left(b \cdot b - b \cdot b\right) - \left(-4 \cdot \left(c \cdot a\right)\right) \cdot \left(b \cdot b - b \cdot b\right)\right)}}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - \left(-b\right)}}{2 \cdot a} \]

      9. +-inverses N/A

         \[\leadsto \frac{\frac{\frac{{-4}^{3} \cdot {\left(c \cdot a\right)}^{3} + \color{blue}{\left(b \cdot b - b \cdot b\right)}}{\left(-4 \cdot \left(c \cdot a\right)\right) \cdot \left(-4 \cdot \left(c \cdot a\right)\right) + \left(\left(b \cdot b - b \cdot b\right) \cdot \left(b \cdot b - b \cdot b\right) - \left(-4 \cdot \left(c \cdot a\right)\right) \cdot \left(b \cdot b - b \cdot b\right)\right)}}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - \left(-b\right)}}{2 \cdot a} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\frac{\frac{\color{blue}{\mathsf{fma}\left({-4}^{3}, {\left(c \cdot a\right)}^{3}, b \cdot b - b \cdot b\right)}}{\left(-4 \cdot \left(c \cdot a\right)\right) \cdot \left(-4 \cdot \left(c \cdot a\right)\right) + \left(\left(b \cdot b - b \cdot b\right) \cdot \left(b \cdot b - b \cdot b\right) - \left(-4 \cdot \left(c \cdot a\right)\right) \cdot \left(b \cdot b - b \cdot b\right)\right)}}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - \left(-b\right)}}{2 \cdot a} \]

      11. metadata-eval N/A

          \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(\color{blue}{-64}, {\left(c \cdot a\right)}^{3}, b \cdot b - b \cdot b\right)}{\left(-4 \cdot \left(c \cdot a\right)\right) \cdot \left(-4 \cdot \left(c \cdot a\right)\right) + \left(\left(b \cdot b - b \cdot b\right) \cdot \left(b \cdot b - b \cdot b\right) - \left(-4 \cdot \left(c \cdot a\right)\right) \cdot \left(b \cdot b - b \cdot b\right)\right)}}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - \left(-b\right)}}{2 \cdot a} \]

      12. lower-pow.f64 N/A

          \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(-64, \color{blue}{{\left(c \cdot a\right)}^{3}}, b \cdot b - b \cdot b\right)}{\left(-4 \cdot \left(c \cdot a\right)\right) \cdot \left(-4 \cdot \left(c \cdot a\right)\right) + \left(\left(b \cdot b - b \cdot b\right) \cdot \left(b \cdot b - b \cdot b\right) - \left(-4 \cdot \left(c \cdot a\right)\right) \cdot \left(b \cdot b - b \cdot b\right)\right)}}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - \left(-b\right)}}{2 \cdot a} \]

      13. +-inverses N/A

          \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(-64, {\left(c \cdot a\right)}^{3}, \color{blue}{0}\right)}{\left(-4 \cdot \left(c \cdot a\right)\right) \cdot \left(-4 \cdot \left(c \cdot a\right)\right) + \left(\left(b \cdot b - b \cdot b\right) \cdot \left(b \cdot b - b \cdot b\right) - \left(-4 \cdot \left(c \cdot a\right)\right) \cdot \left(b \cdot b - b \cdot b\right)\right)}}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - \left(-b\right)}}{2 \cdot a} \]

   5. Applied rewrites 99.2%

      \[\leadsto \frac{\frac{\color{blue}{\frac{\mathsf{fma}\left(-64, {\left(c \cdot a\right)}^{3}, 0\right)}{{\left(\left(c \cdot a\right) \cdot -4\right)}^{2} + \left(0 - \left(\left(c \cdot a\right) \cdot -4\right) \cdot 0\right)}}}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - \left(-b\right)}}{2 \cdot a} \]

   6. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 83.4

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

   8. Applied rewrites 83.4%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 82.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 8 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - b}{a + a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 8e-5)
   (/ (- (sqrt (fma -4.0 (* c a) (* b b))) b) (+ a a))
   (* -1.0 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 8e-5) {
		tmp = (sqrt(fma(-4.0, (c * a), (b * b))) - b) / (a + a);
	} else {
		tmp = -1.0 * (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 8e-5)
		tmp = Float64(Float64(sqrt(fma(-4.0, Float64(c * a), Float64(b * b))) - b) / Float64(a + a));
	else
		tmp = Float64(-1.0 * Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 8e-5], N[(N[(N[Sqrt[N[(-4.0 * N[(c * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 8.00000000000000065e-5

   1. Initial program 71.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-negate1-reverse N/A

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

      5. lower--.f64 71.6

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

      6. lift--.f64 N/A

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

      7. sub-negate1 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

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

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

      13. lower-fma.f64 N/A

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

      14. metadata-eval N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 71.6

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

   3. Applied rewrites 71.6%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 71.6

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

   5. Applied rewrites 71.6%

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

   if 8.00000000000000065e-5 < b

   1. Initial program 28.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. flip-+ N/A

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

      4. lower-/.f64 N/A

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

   3. Applied rewrites 29.4%

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

      1. lift--.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate--l+ N/A

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

      4. flip3-+ N/A

         \[\leadsto \frac{\frac{\color{blue}{\frac{{\left(-4 \cdot \left(c \cdot a\right)\right)}^{3} + {\left(b \cdot b - b \cdot b\right)}^{3}}{\left(-4 \cdot \left(c \cdot a\right)\right) \cdot \left(-4 \cdot \left(c \cdot a\right)\right) + \left(\left(b \cdot b - b \cdot b\right) \cdot \left(b \cdot b - b \cdot b\right) - \left(-4 \cdot \left(c \cdot a\right)\right) \cdot \left(b \cdot b - b \cdot b\right)\right)}}}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - \left(-b\right)}}{2 \cdot a} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\frac{{\left(-4 \cdot \left(c \cdot a\right)\right)}^{3} + {\left(b \cdot b - b \cdot b\right)}^{3}}{\left(-4 \cdot \left(c \cdot a\right)\right) \cdot \left(-4 \cdot \left(c \cdot a\right)\right) + \left(\left(b \cdot b - b \cdot b\right) \cdot \left(b \cdot b - b \cdot b\right) - \left(-4 \cdot \left(c \cdot a\right)\right) \cdot \left(b \cdot b - b \cdot b\right)\right)}}}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - \left(-b\right)}}{2 \cdot a} \]

      6. unpow-prod-down N/A

         \[\leadsto \frac{\frac{\frac{\color{blue}{{-4}^{3} \cdot {\left(c \cdot a\right)}^{3}} + {\left(b \cdot b - b \cdot b\right)}^{3}}{\left(-4 \cdot \left(c \cdot a\right)\right) \cdot \left(-4 \cdot \left(c \cdot a\right)\right) + \left(\left(b \cdot b - b \cdot b\right) \cdot \left(b \cdot b - b \cdot b\right) - \left(-4 \cdot \left(c \cdot a\right)\right) \cdot \left(b \cdot b - b \cdot b\right)\right)}}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - \left(-b\right)}}{2 \cdot a} \]

      7. +-inverses N/A

         \[\leadsto \frac{\frac{\frac{{-4}^{3} \cdot {\left(c \cdot a\right)}^{3} + {\color{blue}{0}}^{3}}{\left(-4 \cdot \left(c \cdot a\right)\right) \cdot \left(-4 \cdot \left(c \cdot a\right)\right) + \left(\left(b \cdot b - b \cdot b\right) \cdot \left(b \cdot b - b \cdot b\right) - \left(-4 \cdot \left(c \cdot a\right)\right) \cdot \left(b \cdot b - b \cdot b\right)\right)}}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - \left(-b\right)}}{2 \cdot a} \]

      8. metadata-eval N/A

         \[\leadsto \frac{\frac{\frac{{-4}^{3} \cdot {\left(c \cdot a\right)}^{3} + \color{blue}{0}}{\left(-4 \cdot \left(c \cdot a\right)\right) \cdot \left(-4 \cdot \left(c \cdot a\right)\right) + \left(\left(b \cdot b - b \cdot b\right) \cdot \left(b \cdot b - b \cdot b\right) - \left(-4 \cdot \left(c \cdot a\right)\right) \cdot \left(b \cdot b - b \cdot b\right)\right)}}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - \left(-b\right)}}{2 \cdot a} \]

      9. +-inverses N/A

         \[\leadsto \frac{\frac{\frac{{-4}^{3} \cdot {\left(c \cdot a\right)}^{3} + \color{blue}{\left(b \cdot b - b \cdot b\right)}}{\left(-4 \cdot \left(c \cdot a\right)\right) \cdot \left(-4 \cdot \left(c \cdot a\right)\right) + \left(\left(b \cdot b - b \cdot b\right) \cdot \left(b \cdot b - b \cdot b\right) - \left(-4 \cdot \left(c \cdot a\right)\right) \cdot \left(b \cdot b - b \cdot b\right)\right)}}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - \left(-b\right)}}{2 \cdot a} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\frac{\frac{\color{blue}{\mathsf{fma}\left({-4}^{3}, {\left(c \cdot a\right)}^{3}, b \cdot b - b \cdot b\right)}}{\left(-4 \cdot \left(c \cdot a\right)\right) \cdot \left(-4 \cdot \left(c \cdot a\right)\right) + \left(\left(b \cdot b - b \cdot b\right) \cdot \left(b \cdot b - b \cdot b\right) - \left(-4 \cdot \left(c \cdot a\right)\right) \cdot \left(b \cdot b - b \cdot b\right)\right)}}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - \left(-b\right)}}{2 \cdot a} \]

      11. metadata-eval N/A

          \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(\color{blue}{-64}, {\left(c \cdot a\right)}^{3}, b \cdot b - b \cdot b\right)}{\left(-4 \cdot \left(c \cdot a\right)\right) \cdot \left(-4 \cdot \left(c \cdot a\right)\right) + \left(\left(b \cdot b - b \cdot b\right) \cdot \left(b \cdot b - b \cdot b\right) - \left(-4 \cdot \left(c \cdot a\right)\right) \cdot \left(b \cdot b - b \cdot b\right)\right)}}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - \left(-b\right)}}{2 \cdot a} \]

      12. lower-pow.f64 N/A

          \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(-64, \color{blue}{{\left(c \cdot a\right)}^{3}}, b \cdot b - b \cdot b\right)}{\left(-4 \cdot \left(c \cdot a\right)\right) \cdot \left(-4 \cdot \left(c \cdot a\right)\right) + \left(\left(b \cdot b - b \cdot b\right) \cdot \left(b \cdot b - b \cdot b\right) - \left(-4 \cdot \left(c \cdot a\right)\right) \cdot \left(b \cdot b - b \cdot b\right)\right)}}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - \left(-b\right)}}{2 \cdot a} \]

      13. +-inverses N/A

          \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(-64, {\left(c \cdot a\right)}^{3}, \color{blue}{0}\right)}{\left(-4 \cdot \left(c \cdot a\right)\right) \cdot \left(-4 \cdot \left(c \cdot a\right)\right) + \left(\left(b \cdot b - b \cdot b\right) \cdot \left(b \cdot b - b \cdot b\right) - \left(-4 \cdot \left(c \cdot a\right)\right) \cdot \left(b \cdot b - b \cdot b\right)\right)}}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - \left(-b\right)}}{2 \cdot a} \]

   5. Applied rewrites 99.2%

      \[\leadsto \frac{\frac{\color{blue}{\frac{\mathsf{fma}\left(-64, {\left(c \cdot a\right)}^{3}, 0\right)}{{\left(\left(c \cdot a\right) \cdot -4\right)}^{2} + \left(0 - \left(\left(c \cdot a\right) \cdot -4\right) \cdot 0\right)}}}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - \left(-b\right)}}{2 \cdot a} \]

   6. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 83.4

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

   8. Applied rewrites 83.4%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 81.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ -1 \cdot \frac{c}{b} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (* -1.0 (/ c b)))

C

double code(double a, double b, double c) {
	return -1.0 * (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 = (-1.0d0) * (c / b)
end function

Java

public static double code(double a, double b, double c) {
	return -1.0 * (c / b);
}

Python

def code(a, b, c):
	return -1.0 * (c / b)

Julia

function code(a, b, c)
	return Float64(-1.0 * Float64(c / b))
end

MATLAB

function tmp = code(a, b, c)
	tmp = -1.0 * (c / b);
end

Wolfram

code[a_, b_, c_] := N[(-1.0 * N[(c / b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.5%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. flip-+ N/A

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

   4. lower-/.f64 N/A

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

3. Applied rewrites 32.4%

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

   1. lift--.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. associate--l+ N/A

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

   4. flip3-+ N/A

      \[\leadsto \frac{\frac{\color{blue}{\frac{{\left(-4 \cdot \left(c \cdot a\right)\right)}^{3} + {\left(b \cdot b - b \cdot b\right)}^{3}}{\left(-4 \cdot \left(c \cdot a\right)\right) \cdot \left(-4 \cdot \left(c \cdot a\right)\right) + \left(\left(b \cdot b - b \cdot b\right) \cdot \left(b \cdot b - b \cdot b\right) - \left(-4 \cdot \left(c \cdot a\right)\right) \cdot \left(b \cdot b - b \cdot b\right)\right)}}}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - \left(-b\right)}}{2 \cdot a} \]

   5. lower-/.f64 N/A

      \[\leadsto \frac{\frac{\color{blue}{\frac{{\left(-4 \cdot \left(c \cdot a\right)\right)}^{3} + {\left(b \cdot b - b \cdot b\right)}^{3}}{\left(-4 \cdot \left(c \cdot a\right)\right) \cdot \left(-4 \cdot \left(c \cdot a\right)\right) + \left(\left(b \cdot b - b \cdot b\right) \cdot \left(b \cdot b - b \cdot b\right) - \left(-4 \cdot \left(c \cdot a\right)\right) \cdot \left(b \cdot b - b \cdot b\right)\right)}}}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - \left(-b\right)}}{2 \cdot a} \]

   6. unpow-prod-down N/A

      \[\leadsto \frac{\frac{\frac{\color{blue}{{-4}^{3} \cdot {\left(c \cdot a\right)}^{3}} + {\left(b \cdot b - b \cdot b\right)}^{3}}{\left(-4 \cdot \left(c \cdot a\right)\right) \cdot \left(-4 \cdot \left(c \cdot a\right)\right) + \left(\left(b \cdot b - b \cdot b\right) \cdot \left(b \cdot b - b \cdot b\right) - \left(-4 \cdot \left(c \cdot a\right)\right) \cdot \left(b \cdot b - b \cdot b\right)\right)}}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - \left(-b\right)}}{2 \cdot a} \]

   7. +-inverses N/A

      \[\leadsto \frac{\frac{\frac{{-4}^{3} \cdot {\left(c \cdot a\right)}^{3} + {\color{blue}{0}}^{3}}{\left(-4 \cdot \left(c \cdot a\right)\right) \cdot \left(-4 \cdot \left(c \cdot a\right)\right) + \left(\left(b \cdot b - b \cdot b\right) \cdot \left(b \cdot b - b \cdot b\right) - \left(-4 \cdot \left(c \cdot a\right)\right) \cdot \left(b \cdot b - b \cdot b\right)\right)}}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - \left(-b\right)}}{2 \cdot a} \]

   8. metadata-eval N/A

      \[\leadsto \frac{\frac{\frac{{-4}^{3} \cdot {\left(c \cdot a\right)}^{3} + \color{blue}{0}}{\left(-4 \cdot \left(c \cdot a\right)\right) \cdot \left(-4 \cdot \left(c \cdot a\right)\right) + \left(\left(b \cdot b - b \cdot b\right) \cdot \left(b \cdot b - b \cdot b\right) - \left(-4 \cdot \left(c \cdot a\right)\right) \cdot \left(b \cdot b - b \cdot b\right)\right)}}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - \left(-b\right)}}{2 \cdot a} \]

   9. +-inverses N/A

      \[\leadsto \frac{\frac{\frac{{-4}^{3} \cdot {\left(c \cdot a\right)}^{3} + \color{blue}{\left(b \cdot b - b \cdot b\right)}}{\left(-4 \cdot \left(c \cdot a\right)\right) \cdot \left(-4 \cdot \left(c \cdot a\right)\right) + \left(\left(b \cdot b - b \cdot b\right) \cdot \left(b \cdot b - b \cdot b\right) - \left(-4 \cdot \left(c \cdot a\right)\right) \cdot \left(b \cdot b - b \cdot b\right)\right)}}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - \left(-b\right)}}{2 \cdot a} \]

   10. lower-fma.f64 N/A

       \[\leadsto \frac{\frac{\frac{\color{blue}{\mathsf{fma}\left({-4}^{3}, {\left(c \cdot a\right)}^{3}, b \cdot b - b \cdot b\right)}}{\left(-4 \cdot \left(c \cdot a\right)\right) \cdot \left(-4 \cdot \left(c \cdot a\right)\right) + \left(\left(b \cdot b - b \cdot b\right) \cdot \left(b \cdot b - b \cdot b\right) - \left(-4 \cdot \left(c \cdot a\right)\right) \cdot \left(b \cdot b - b \cdot b\right)\right)}}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - \left(-b\right)}}{2 \cdot a} \]

   11. metadata-eval N/A

       \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(\color{blue}{-64}, {\left(c \cdot a\right)}^{3}, b \cdot b - b \cdot b\right)}{\left(-4 \cdot \left(c \cdot a\right)\right) \cdot \left(-4 \cdot \left(c \cdot a\right)\right) + \left(\left(b \cdot b - b \cdot b\right) \cdot \left(b \cdot b - b \cdot b\right) - \left(-4 \cdot \left(c \cdot a\right)\right) \cdot \left(b \cdot b - b \cdot b\right)\right)}}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - \left(-b\right)}}{2 \cdot a} \]

   12. lower-pow.f64 N/A

       \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(-64, \color{blue}{{\left(c \cdot a\right)}^{3}}, b \cdot b - b \cdot b\right)}{\left(-4 \cdot \left(c \cdot a\right)\right) \cdot \left(-4 \cdot \left(c \cdot a\right)\right) + \left(\left(b \cdot b - b \cdot b\right) \cdot \left(b \cdot b - b \cdot b\right) - \left(-4 \cdot \left(c \cdot a\right)\right) \cdot \left(b \cdot b - b \cdot b\right)\right)}}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - \left(-b\right)}}{2 \cdot a} \]

   13. +-inverses N/A

       \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(-64, {\left(c \cdot a\right)}^{3}, \color{blue}{0}\right)}{\left(-4 \cdot \left(c \cdot a\right)\right) \cdot \left(-4 \cdot \left(c \cdot a\right)\right) + \left(\left(b \cdot b - b \cdot b\right) \cdot \left(b \cdot b - b \cdot b\right) - \left(-4 \cdot \left(c \cdot a\right)\right) \cdot \left(b \cdot b - b \cdot b\right)\right)}}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - \left(-b\right)}}{2 \cdot a} \]

5. Applied rewrites 99.2%

   \[\leadsto \frac{\frac{\color{blue}{\frac{\mathsf{fma}\left(-64, {\left(c \cdot a\right)}^{3}, 0\right)}{{\left(\left(c \cdot a\right) \cdot -4\right)}^{2} + \left(0 - \left(\left(c \cdot a\right) \cdot -4\right) \cdot 0\right)}}}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - \left(-b\right)}}{2 \cdot a} \]

6. Taylor expanded in a around 0

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 81.1

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

8. Applied rewrites 81.1%

   \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025111 
(FPCore (a b c)
  :name "Quadratic roots, medium range"
  :precision binary64
  :pre (and (and (and (< 1.1102230246251565e-16 a) (< a 9007199254740992.0)) (and (< 1.1102230246251565e-16 b) (< b 9007199254740992.0))) (and (< 1.1102230246251565e-16 c) (< c 9007199254740992.0)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))