quad2m (problem 3.2.1, negative)

Percentage Accurate: 52.1% → 85.8%

Time: 4.9s

Alternatives: 8

Speedup: N/A×

Specification

Math

\[\begin{array}{l} \\ \frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))

C

double code(double a, double b_2, double c) {
	return (-b_2 - sqrt(((b_2 * b_2) - (a * c)))) / 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_2, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    code = (-b_2 - sqrt(((b_2 * b_2) - (a * c)))) / a
end function

Java

public static double code(double a, double b_2, double c) {
	return (-b_2 - Math.sqrt(((b_2 * b_2) - (a * c)))) / a;
}

Python

def code(a, b_2, c):
	return (-b_2 - math.sqrt(((b_2 * b_2) - (a * c)))) / a

Julia

function code(a, b_2, c)
	return Float64(Float64(Float64(-b_2) - sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c)))) / a)
end

MATLAB

function tmp = code(a, b_2, c)
	tmp = (-b_2 - sqrt(((b_2 * b_2) - (a * c)))) / a;
end

Wolfram

code[a_, b$95$2_, c_] := N[(N[((-b$95$2) - N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]

Initial Program: 52.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))

C

double code(double a, double b_2, double c) {
	return (-b_2 - sqrt(((b_2 * b_2) - (a * c)))) / 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_2, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    code = (-b_2 - sqrt(((b_2 * b_2) - (a * c)))) / a
end function

Java

public static double code(double a, double b_2, double c) {
	return (-b_2 - Math.sqrt(((b_2 * b_2) - (a * c)))) / a;
}

Python

def code(a, b_2, c):
	return (-b_2 - math.sqrt(((b_2 * b_2) - (a * c)))) / a

Julia

function code(a, b_2, c)
	return Float64(Float64(Float64(-b_2) - sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c)))) / a)
end

MATLAB

function tmp = code(a, b_2, c)
	tmp = (-b_2 - sqrt(((b_2 * b_2) - (a * c)))) / a;
end

Wolfram

code[a_, b$95$2_, c_] := N[(N[((-b$95$2) - N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]

Alternative 1: 85.8% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.26 \cdot 10^{-123}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 7 \cdot 10^{+139}:\\ \;\;\;\;-1 \cdot \frac{b\_2}{a} - \frac{{\left(\mathsf{fma}\left({b\_2}^{1}, {b\_2}^{1}, -1 \cdot \left(c \cdot a\right)\right)\right)}^{0.5}}{a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{b\_2 + b\_2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.26e-123)
   (* -0.5 (/ c b_2))
   (if (<= b_2 7e+139)
     (-
      (* -1.0 (/ b_2 a))
      (/ (pow (fma (pow b_2 1.0) (pow b_2 1.0) (* -1.0 (* c a))) 0.5) a))
     (* -1.0 (/ (+ b_2 b_2) a)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.26e-123) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= 7e+139) {
		tmp = (-1.0 * (b_2 / a)) - (pow(fma(pow(b_2, 1.0), pow(b_2, 1.0), (-1.0 * (c * a))), 0.5) / a);
	} else {
		tmp = -1.0 * ((b_2 + b_2) / a);
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1.26e-123)
		tmp = Float64(-0.5 * Float64(c / b_2));
	elseif (b_2 <= 7e+139)
		tmp = Float64(Float64(-1.0 * Float64(b_2 / a)) - Float64((fma((b_2 ^ 1.0), (b_2 ^ 1.0), Float64(-1.0 * Float64(c * a))) ^ 0.5) / a));
	else
		tmp = Float64(-1.0 * Float64(Float64(b_2 + b_2) / a));
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1.26e-123], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 7e+139], N[(N[(-1.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] - N[(N[Power[N[(N[Power[b$95$2, 1.0], $MachinePrecision] * N[Power[b$95$2, 1.0], $MachinePrecision] + N[(-1.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[(N[(b$95$2 + b$95$2), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -1.26000000000000005e-123

   1. Initial program 15.5%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around -inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \color{blue}{\frac{c}{b\_2}} \]

      2. lower-/.f64 84.9

         \[\leadsto -0.5 \cdot \frac{c}{\color{blue}{b\_2}} \]

   5. Applied rewrites 84.9%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b\_2}} \]

   if -1.26000000000000005e-123 < b_2 < 6.99999999999999957e139

   1. Initial program 83.8%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}} \]

      2. lift-neg.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(b\_2\right)\right)} - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

      3. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(b\_2\right)\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}}{a} \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \color{blue}{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}}{a} \]

      5. lift--.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \sqrt{\color{blue}{b\_2 \cdot b\_2 - a \cdot c}}}{a} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \sqrt{\color{blue}{b\_2 \cdot b\_2} - a \cdot c}}{a} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \sqrt{b\_2 \cdot b\_2 - \color{blue}{a \cdot c}}}{a} \]

      8. div-sub N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(b\_2\right)}{a} - \frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}} \]

      9. mul-1-neg N/A

         \[\leadsto \frac{\color{blue}{-1 \cdot b\_2}}{a} - \frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

      10. associate-*r/ N/A

          \[\leadsto \color{blue}{-1 \cdot \frac{b\_2}{a}} - \frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

      11. lower--.f64 N/A

          \[\leadsto \color{blue}{-1 \cdot \frac{b\_2}{a} - \frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}} \]

      12. lower-*.f64 N/A

          \[\leadsto \color{blue}{-1 \cdot \frac{b\_2}{a}} - \frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

      13. lower-/.f64 N/A

          \[\leadsto -1 \cdot \color{blue}{\frac{b\_2}{a}} - \frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

      14. lower-/.f64 N/A

          \[\leadsto -1 \cdot \frac{b\_2}{a} - \color{blue}{\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}} \]

   4. Applied rewrites 83.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{b\_2}{a} - \frac{{\left(\mathsf{fma}\left({b\_2}^{1}, {b\_2}^{1}, -1 \cdot \left(c \cdot a\right)\right)\right)}^{0.5}}{a}} \]

   if 6.99999999999999957e139 < b_2

   1. Initial program 44.4%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \frac{\left(-b\_2\right) - \color{blue}{b\_2}}{a} \]
   4. Step-by-step derivation

   5. Applied rewrites 98.1%

      \[\leadsto \frac{\left(-b\_2\right) - \color{blue}{b\_2}}{a} \]
3. Recombined 3 regimes into one program.

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.26 \cdot 10^{-123}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 7 \cdot 10^{+139}:\\ \;\;\;\;-1 \cdot \frac{b\_2}{a} - \frac{{\left(\mathsf{fma}\left({b\_2}^{1}, {b\_2}^{1}, -1 \cdot \left(c \cdot a\right)\right)\right)}^{0.5}}{a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{b\_2 + b\_2}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.7% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.26 \cdot 10^{-123}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 7 \cdot 10^{+139}:\\ \;\;\;\;-1 \cdot \frac{b\_2}{a} - \frac{{\left(\mathsf{fma}\left({b\_2}^{1}, {b\_2}^{1}, -1 \cdot \left(c \cdot a\right)\right)\right)}^{0.5}}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{b\_2}, 0.5, \frac{b\_2}{a} \cdot -2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.26e-123)
   (* -0.5 (/ c b_2))
   (if (<= b_2 7e+139)
     (-
      (* -1.0 (/ b_2 a))
      (/ (pow (fma (pow b_2 1.0) (pow b_2 1.0) (* -1.0 (* c a))) 0.5) a))
     (fma (/ c b_2) 0.5 (* (/ b_2 a) -2.0)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.26e-123) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= 7e+139) {
		tmp = (-1.0 * (b_2 / a)) - (pow(fma(pow(b_2, 1.0), pow(b_2, 1.0), (-1.0 * (c * a))), 0.5) / a);
	} else {
		tmp = fma((c / b_2), 0.5, ((b_2 / a) * -2.0));
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1.26e-123)
		tmp = Float64(-0.5 * Float64(c / b_2));
	elseif (b_2 <= 7e+139)
		tmp = Float64(Float64(-1.0 * Float64(b_2 / a)) - Float64((fma((b_2 ^ 1.0), (b_2 ^ 1.0), Float64(-1.0 * Float64(c * a))) ^ 0.5) / a));
	else
		tmp = fma(Float64(c / b_2), 0.5, Float64(Float64(b_2 / a) * -2.0));
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1.26e-123], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 7e+139], N[(N[(-1.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] - N[(N[Power[N[(N[Power[b$95$2, 1.0], $MachinePrecision] * N[Power[b$95$2, 1.0], $MachinePrecision] + N[(-1.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b$95$2), $MachinePrecision] * 0.5 + N[(N[(b$95$2 / a), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -1.26000000000000005e-123

   1. Initial program 15.5%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around -inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \color{blue}{\frac{c}{b\_2}} \]

      2. lower-/.f64 84.9

         \[\leadsto -0.5 \cdot \frac{c}{\color{blue}{b\_2}} \]

   5. Applied rewrites 84.9%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b\_2}} \]

   if -1.26000000000000005e-123 < b_2 < 6.99999999999999957e139

   1. Initial program 83.8%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}} \]

      2. lift-neg.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(b\_2\right)\right)} - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

      3. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(b\_2\right)\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}}{a} \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \color{blue}{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}}{a} \]

      5. lift--.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \sqrt{\color{blue}{b\_2 \cdot b\_2 - a \cdot c}}}{a} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \sqrt{\color{blue}{b\_2 \cdot b\_2} - a \cdot c}}{a} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \sqrt{b\_2 \cdot b\_2 - \color{blue}{a \cdot c}}}{a} \]

      8. div-sub N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(b\_2\right)}{a} - \frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}} \]

      9. mul-1-neg N/A

         \[\leadsto \frac{\color{blue}{-1 \cdot b\_2}}{a} - \frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

      10. associate-*r/ N/A

          \[\leadsto \color{blue}{-1 \cdot \frac{b\_2}{a}} - \frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

      11. lower--.f64 N/A

          \[\leadsto \color{blue}{-1 \cdot \frac{b\_2}{a} - \frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}} \]

      12. lower-*.f64 N/A

          \[\leadsto \color{blue}{-1 \cdot \frac{b\_2}{a}} - \frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

      13. lower-/.f64 N/A

          \[\leadsto -1 \cdot \color{blue}{\frac{b\_2}{a}} - \frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

      14. lower-/.f64 N/A

          \[\leadsto -1 \cdot \frac{b\_2}{a} - \color{blue}{\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}} \]

   4. Applied rewrites 83.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{b\_2}{a} - \frac{{\left(\mathsf{fma}\left({b\_2}^{1}, {b\_2}^{1}, -1 \cdot \left(c \cdot a\right)\right)\right)}^{0.5}}{a}} \]

   if 6.99999999999999957e139 < b_2

   1. Initial program 44.4%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in c around 0

      \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a} + \frac{1}{2} \cdot \frac{c}{b\_2}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c}{b\_2} + \color{blue}{-2 \cdot \frac{b\_2}{a}} \]

      2. *-commutative N/A

         \[\leadsto \frac{c}{b\_2} \cdot \frac{1}{2} + \color{blue}{-2} \cdot \frac{b\_2}{a} \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{c}{b\_2}, \color{blue}{\frac{1}{2}}, -2 \cdot \frac{b\_2}{a}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{c}{b\_2}, \frac{1}{2}, -2 \cdot \frac{b\_2}{a}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{c}{b\_2}, \frac{1}{2}, \frac{b\_2}{a} \cdot -2\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{c}{b\_2}, \frac{1}{2}, \frac{b\_2}{a} \cdot -2\right) \]

      7. lower-/.f64 98.1

         \[\leadsto \mathsf{fma}\left(\frac{c}{b\_2}, 0.5, \frac{b\_2}{a} \cdot -2\right) \]

   5. Applied rewrites 98.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{c}{b\_2}, 0.5, \frac{b\_2}{a} \cdot -2\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 85.7% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.26 \cdot 10^{-123}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 6.5 \cdot 10^{+139}:\\ \;\;\;\;-1 \cdot \frac{b\_2}{a} - {a}^{-1} \cdot {\left(\mathsf{fma}\left(-1, a \cdot c, b\_2 \cdot b\_2\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{b\_2}, 0.5, \frac{b\_2}{a} \cdot -2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.26e-123)
   (* -0.5 (/ c b_2))
   (if (<= b_2 6.5e+139)
     (-
      (* -1.0 (/ b_2 a))
      (* (pow a -1.0) (pow (fma -1.0 (* a c) (* b_2 b_2)) 0.5)))
     (fma (/ c b_2) 0.5 (* (/ b_2 a) -2.0)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.26e-123) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= 6.5e+139) {
		tmp = (-1.0 * (b_2 / a)) - (pow(a, -1.0) * pow(fma(-1.0, (a * c), (b_2 * b_2)), 0.5));
	} else {
		tmp = fma((c / b_2), 0.5, ((b_2 / a) * -2.0));
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1.26e-123)
		tmp = Float64(-0.5 * Float64(c / b_2));
	elseif (b_2 <= 6.5e+139)
		tmp = Float64(Float64(-1.0 * Float64(b_2 / a)) - Float64((a ^ -1.0) * (fma(-1.0, Float64(a * c), Float64(b_2 * b_2)) ^ 0.5)));
	else
		tmp = fma(Float64(c / b_2), 0.5, Float64(Float64(b_2 / a) * -2.0));
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1.26e-123], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 6.5e+139], N[(N[(-1.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] - N[(N[Power[a, -1.0], $MachinePrecision] * N[Power[N[(-1.0 * N[(a * c), $MachinePrecision] + N[(b$95$2 * b$95$2), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c / b$95$2), $MachinePrecision] * 0.5 + N[(N[(b$95$2 / a), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -1.26000000000000005e-123

   1. Initial program 15.5%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around -inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \color{blue}{\frac{c}{b\_2}} \]

      2. lower-/.f64 84.9

         \[\leadsto -0.5 \cdot \frac{c}{\color{blue}{b\_2}} \]

   5. Applied rewrites 84.9%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b\_2}} \]

   if -1.26000000000000005e-123 < b_2 < 6.5000000000000003e139

   1. Initial program 83.8%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}} \]

      2. lift-neg.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(b\_2\right)\right)} - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

      3. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(b\_2\right)\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}}{a} \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \color{blue}{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}}{a} \]

      5. lift--.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \sqrt{\color{blue}{b\_2 \cdot b\_2 - a \cdot c}}}{a} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \sqrt{\color{blue}{b\_2 \cdot b\_2} - a \cdot c}}{a} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \sqrt{b\_2 \cdot b\_2 - \color{blue}{a \cdot c}}}{a} \]

      8. div-sub N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(b\_2\right)}{a} - \frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}} \]

      9. mul-1-neg N/A

         \[\leadsto \frac{\color{blue}{-1 \cdot b\_2}}{a} - \frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

      10. associate-*r/ N/A

          \[\leadsto \color{blue}{-1 \cdot \frac{b\_2}{a}} - \frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

      11. lower--.f64 N/A

          \[\leadsto \color{blue}{-1 \cdot \frac{b\_2}{a} - \frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}} \]

      12. lower-*.f64 N/A

          \[\leadsto \color{blue}{-1 \cdot \frac{b\_2}{a}} - \frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

      13. lower-/.f64 N/A

          \[\leadsto -1 \cdot \color{blue}{\frac{b\_2}{a}} - \frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

      14. lower-/.f64 N/A

          \[\leadsto -1 \cdot \frac{b\_2}{a} - \color{blue}{\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}} \]

   4. Applied rewrites 83.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{b\_2}{a} - \frac{{\left(\mathsf{fma}\left({b\_2}^{1}, {b\_2}^{1}, -1 \cdot \left(c \cdot a\right)\right)\right)}^{0.5}}{a}} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto -1 \cdot \frac{b\_2}{a} - \frac{\color{blue}{{\left(\mathsf{fma}\left({b\_2}^{1}, {b\_2}^{1}, -1 \cdot \left(c \cdot a\right)\right)\right)}^{\frac{1}{2}}}}{a} \]

      2. lift-pow.f64 N/A

         \[\leadsto -1 \cdot \frac{b\_2}{a} - \frac{{\left(\mathsf{fma}\left(\color{blue}{{b\_2}^{1}}, {b\_2}^{1}, -1 \cdot \left(c \cdot a\right)\right)\right)}^{\frac{1}{2}}}{a} \]

      3. lift-pow.f64 N/A

         \[\leadsto -1 \cdot \frac{b\_2}{a} - \frac{{\left(\mathsf{fma}\left({b\_2}^{1}, \color{blue}{{b\_2}^{1}}, -1 \cdot \left(c \cdot a\right)\right)\right)}^{\frac{1}{2}}}{a} \]

      4. lift-fma.f64 N/A

         \[\leadsto -1 \cdot \frac{b\_2}{a} - \frac{{\color{blue}{\left({b\_2}^{1} \cdot {b\_2}^{1} + -1 \cdot \left(c \cdot a\right)\right)}}^{\frac{1}{2}}}{a} \]

      5. lift-*.f64 N/A

         \[\leadsto -1 \cdot \frac{b\_2}{a} - \frac{{\left({b\_2}^{1} \cdot {b\_2}^{1} + -1 \cdot \color{blue}{\left(c \cdot a\right)}\right)}^{\frac{1}{2}}}{a} \]

      6. lift-*.f64 N/A

         \[\leadsto -1 \cdot \frac{b\_2}{a} - \frac{{\left({b\_2}^{1} \cdot {b\_2}^{1} + \color{blue}{-1 \cdot \left(c \cdot a\right)}\right)}^{\frac{1}{2}}}{a} \]

      7. sqr-pow N/A

         \[\leadsto -1 \cdot \frac{b\_2}{a} - \frac{\color{blue}{{\left({b\_2}^{1} \cdot {b\_2}^{1} + -1 \cdot \left(c \cdot a\right)\right)}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot {\left({b\_2}^{1} \cdot {b\_2}^{1} + -1 \cdot \left(c \cdot a\right)\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}}}{a} \]

      8. lower-*.f64 N/A

         \[\leadsto -1 \cdot \frac{b\_2}{a} - \frac{\color{blue}{{\left({b\_2}^{1} \cdot {b\_2}^{1} + -1 \cdot \left(c \cdot a\right)\right)}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot {\left({b\_2}^{1} \cdot {b\_2}^{1} + -1 \cdot \left(c \cdot a\right)\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}}}{a} \]

   6. Applied rewrites 83.6%

      \[\leadsto -1 \cdot \frac{b\_2}{a} - \frac{\color{blue}{{\left(\mathsf{fma}\left(\left|b\_2\right|, \left|b\_2\right|, -1 \cdot \left(c \cdot a\right)\right)\right)}^{0.25} \cdot {\left(\mathsf{fma}\left(\left|b\_2\right|, \left|b\_2\right|, -1 \cdot \left(c \cdot a\right)\right)\right)}^{0.25}}}{a} \]

   7. Taylor expanded in b_2 around 0

      \[\leadsto -1 \cdot \frac{b\_2}{a} - \color{blue}{\frac{1}{a} \cdot \sqrt{-1 \cdot \left(a \cdot c\right) + {\left(\left|b\_2\right|\right)}^{2}}} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \frac{b\_2}{a} - \frac{1}{a} \cdot \color{blue}{\sqrt{-1 \cdot \left(a \cdot c\right) + {\left(\left|b\_2\right|\right)}^{2}}} \]

      2. inv-pow N/A

         \[\leadsto -1 \cdot \frac{b\_2}{a} - {a}^{-1} \cdot \sqrt{\color{blue}{-1 \cdot \left(a \cdot c\right) + {\left(\left|b\_2\right|\right)}^{2}}} \]

      3. lift-pow.f64 N/A

         \[\leadsto -1 \cdot \frac{b\_2}{a} - {a}^{-1} \cdot \sqrt{\color{blue}{-1 \cdot \left(a \cdot c\right) + {\left(\left|b\_2\right|\right)}^{2}}} \]

      4. pow1/2 N/A

         \[\leadsto -1 \cdot \frac{b\_2}{a} - {a}^{-1} \cdot {\left(-1 \cdot \left(a \cdot c\right) + {\left(\left|b\_2\right|\right)}^{2}\right)}^{\color{blue}{\frac{1}{2}}} \]

      5. pow2 N/A

         \[\leadsto -1 \cdot \frac{b\_2}{a} - {a}^{-1} \cdot {\left(-1 \cdot \left(a \cdot c\right) + \left|b\_2\right| \cdot \left|b\_2\right|\right)}^{\frac{1}{2}} \]

      6. sqr-abs-rev N/A

         \[\leadsto -1 \cdot \frac{b\_2}{a} - {a}^{-1} \cdot {\left(-1 \cdot \left(a \cdot c\right) + b\_2 \cdot b\_2\right)}^{\frac{1}{2}} \]

      7. lower-pow.f64 N/A

         \[\leadsto -1 \cdot \frac{b\_2}{a} - {a}^{-1} \cdot {\left(-1 \cdot \left(a \cdot c\right) + b\_2 \cdot b\_2\right)}^{\color{blue}{\frac{1}{2}}} \]

      8. lift-fma.f64 N/A

         \[\leadsto -1 \cdot \frac{b\_2}{a} - {a}^{-1} \cdot {\left(\mathsf{fma}\left(-1, a \cdot c, b\_2 \cdot b\_2\right)\right)}^{\frac{1}{2}} \]

      9. lift-*.f64 N/A

         \[\leadsto -1 \cdot \frac{b\_2}{a} - {a}^{-1} \cdot {\left(\mathsf{fma}\left(-1, a \cdot c, b\_2 \cdot b\_2\right)\right)}^{\frac{1}{2}} \]

      10. lift-*.f64 83.7

          \[\leadsto -1 \cdot \frac{b\_2}{a} - {a}^{-1} \cdot {\left(\mathsf{fma}\left(-1, a \cdot c, b\_2 \cdot b\_2\right)\right)}^{0.5} \]

   9. Applied rewrites 83.7%

      \[\leadsto -1 \cdot \frac{b\_2}{a} - \color{blue}{{a}^{-1} \cdot {\left(\mathsf{fma}\left(-1, a \cdot c, b\_2 \cdot b\_2\right)\right)}^{0.5}} \]

   if 6.5000000000000003e139 < b_2

   1. Initial program 44.4%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in c around 0

      \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a} + \frac{1}{2} \cdot \frac{c}{b\_2}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c}{b\_2} + \color{blue}{-2 \cdot \frac{b\_2}{a}} \]

      2. *-commutative N/A

         \[\leadsto \frac{c}{b\_2} \cdot \frac{1}{2} + \color{blue}{-2} \cdot \frac{b\_2}{a} \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{c}{b\_2}, \color{blue}{\frac{1}{2}}, -2 \cdot \frac{b\_2}{a}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{c}{b\_2}, \frac{1}{2}, -2 \cdot \frac{b\_2}{a}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{c}{b\_2}, \frac{1}{2}, \frac{b\_2}{a} \cdot -2\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{c}{b\_2}, \frac{1}{2}, \frac{b\_2}{a} \cdot -2\right) \]

      7. lower-/.f64 98.1

         \[\leadsto \mathsf{fma}\left(\frac{c}{b\_2}, 0.5, \frac{b\_2}{a} \cdot -2\right) \]

   5. Applied rewrites 98.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{c}{b\_2}, 0.5, \frac{b\_2}{a} \cdot -2\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 77.7% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.26 \cdot 10^{-123}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 6 \cdot 10^{+136}:\\ \;\;\;\;\left(-1 \cdot b\_2\right) \cdot \mathsf{fma}\left({\left(a \cdot b\_2\right)}^{-1}, {\left(\mathsf{fma}\left(-1, a \cdot c, b\_2 \cdot b\_2\right)\right)}^{0.5}, {a}^{-1}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{b\_2}, 0.5, \frac{b\_2}{a} \cdot -2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.26e-123)
   (* -0.5 (/ c b_2))
   (if (<= b_2 6e+136)
     (*
      (* -1.0 b_2)
      (fma
       (pow (* a b_2) -1.0)
       (pow (fma -1.0 (* a c) (* b_2 b_2)) 0.5)
       (pow a -1.0)))
     (fma (/ c b_2) 0.5 (* (/ b_2 a) -2.0)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.26e-123) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= 6e+136) {
		tmp = (-1.0 * b_2) * fma(pow((a * b_2), -1.0), pow(fma(-1.0, (a * c), (b_2 * b_2)), 0.5), pow(a, -1.0));
	} else {
		tmp = fma((c / b_2), 0.5, ((b_2 / a) * -2.0));
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1.26e-123)
		tmp = Float64(-0.5 * Float64(c / b_2));
	elseif (b_2 <= 6e+136)
		tmp = Float64(Float64(-1.0 * b_2) * fma((Float64(a * b_2) ^ -1.0), (fma(-1.0, Float64(a * c), Float64(b_2 * b_2)) ^ 0.5), (a ^ -1.0)));
	else
		tmp = fma(Float64(c / b_2), 0.5, Float64(Float64(b_2 / a) * -2.0));
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1.26e-123], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 6e+136], N[(N[(-1.0 * b$95$2), $MachinePrecision] * N[(N[Power[N[(a * b$95$2), $MachinePrecision], -1.0], $MachinePrecision] * N[Power[N[(-1.0 * N[(a * c), $MachinePrecision] + N[(b$95$2 * b$95$2), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision] + N[Power[a, -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c / b$95$2), $MachinePrecision] * 0.5 + N[(N[(b$95$2 / a), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -1.26000000000000005e-123

   1. Initial program 15.5%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around -inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \color{blue}{\frac{c}{b\_2}} \]

      2. lower-/.f64 84.9

         \[\leadsto -0.5 \cdot \frac{c}{\color{blue}{b\_2}} \]

   5. Applied rewrites 84.9%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b\_2}} \]

   if -1.26000000000000005e-123 < b_2 < 5.99999999999999958e136

   1. Initial program 83.8%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}} \]

      2. lift-neg.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(b\_2\right)\right)} - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

      3. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(b\_2\right)\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}}{a} \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \color{blue}{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}}{a} \]

      5. lift--.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \sqrt{\color{blue}{b\_2 \cdot b\_2 - a \cdot c}}}{a} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \sqrt{\color{blue}{b\_2 \cdot b\_2} - a \cdot c}}{a} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \sqrt{b\_2 \cdot b\_2 - \color{blue}{a \cdot c}}}{a} \]

      8. div-sub N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(b\_2\right)}{a} - \frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}} \]

      9. mul-1-neg N/A

         \[\leadsto \frac{\color{blue}{-1 \cdot b\_2}}{a} - \frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

      10. associate-*r/ N/A

          \[\leadsto \color{blue}{-1 \cdot \frac{b\_2}{a}} - \frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

      11. lower--.f64 N/A

          \[\leadsto \color{blue}{-1 \cdot \frac{b\_2}{a} - \frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}} \]

      12. lower-*.f64 N/A

          \[\leadsto \color{blue}{-1 \cdot \frac{b\_2}{a}} - \frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

      13. lower-/.f64 N/A

          \[\leadsto -1 \cdot \color{blue}{\frac{b\_2}{a}} - \frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

      14. lower-/.f64 N/A

          \[\leadsto -1 \cdot \frac{b\_2}{a} - \color{blue}{\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}} \]

   4. Applied rewrites 83.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{b\_2}{a} - \frac{{\left(\mathsf{fma}\left({b\_2}^{1}, {b\_2}^{1}, -1 \cdot \left(c \cdot a\right)\right)\right)}^{0.5}}{a}} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto -1 \cdot \frac{b\_2}{a} - \frac{\color{blue}{{\left(\mathsf{fma}\left({b\_2}^{1}, {b\_2}^{1}, -1 \cdot \left(c \cdot a\right)\right)\right)}^{\frac{1}{2}}}}{a} \]

      2. lift-pow.f64 N/A

         \[\leadsto -1 \cdot \frac{b\_2}{a} - \frac{{\left(\mathsf{fma}\left(\color{blue}{{b\_2}^{1}}, {b\_2}^{1}, -1 \cdot \left(c \cdot a\right)\right)\right)}^{\frac{1}{2}}}{a} \]

      3. lift-pow.f64 N/A

         \[\leadsto -1 \cdot \frac{b\_2}{a} - \frac{{\left(\mathsf{fma}\left({b\_2}^{1}, \color{blue}{{b\_2}^{1}}, -1 \cdot \left(c \cdot a\right)\right)\right)}^{\frac{1}{2}}}{a} \]

      4. lift-fma.f64 N/A

         \[\leadsto -1 \cdot \frac{b\_2}{a} - \frac{{\color{blue}{\left({b\_2}^{1} \cdot {b\_2}^{1} + -1 \cdot \left(c \cdot a\right)\right)}}^{\frac{1}{2}}}{a} \]

      5. lift-*.f64 N/A

         \[\leadsto -1 \cdot \frac{b\_2}{a} - \frac{{\left({b\_2}^{1} \cdot {b\_2}^{1} + -1 \cdot \color{blue}{\left(c \cdot a\right)}\right)}^{\frac{1}{2}}}{a} \]

      6. lift-*.f64 N/A

         \[\leadsto -1 \cdot \frac{b\_2}{a} - \frac{{\left({b\_2}^{1} \cdot {b\_2}^{1} + \color{blue}{-1 \cdot \left(c \cdot a\right)}\right)}^{\frac{1}{2}}}{a} \]

      7. sqr-pow N/A

         \[\leadsto -1 \cdot \frac{b\_2}{a} - \frac{\color{blue}{{\left({b\_2}^{1} \cdot {b\_2}^{1} + -1 \cdot \left(c \cdot a\right)\right)}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot {\left({b\_2}^{1} \cdot {b\_2}^{1} + -1 \cdot \left(c \cdot a\right)\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}}}{a} \]

      8. lower-*.f64 N/A

         \[\leadsto -1 \cdot \frac{b\_2}{a} - \frac{\color{blue}{{\left({b\_2}^{1} \cdot {b\_2}^{1} + -1 \cdot \left(c \cdot a\right)\right)}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot {\left({b\_2}^{1} \cdot {b\_2}^{1} + -1 \cdot \left(c \cdot a\right)\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}}}{a} \]

   6. Applied rewrites 83.6%

      \[\leadsto -1 \cdot \frac{b\_2}{a} - \frac{\color{blue}{{\left(\mathsf{fma}\left(\left|b\_2\right|, \left|b\_2\right|, -1 \cdot \left(c \cdot a\right)\right)\right)}^{0.25} \cdot {\left(\mathsf{fma}\left(\left|b\_2\right|, \left|b\_2\right|, -1 \cdot \left(c \cdot a\right)\right)\right)}^{0.25}}}{a} \]

   7. Taylor expanded in b_2 around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(b\_2 \cdot \left(\frac{1}{a \cdot b\_2} \cdot \sqrt{-1 \cdot \left(a \cdot c\right) + {\left(\left|b\_2\right|\right)}^{2}} + \frac{1}{a}\right)\right)} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(b\_2 \cdot \left(\frac{1}{a \cdot b\_2} \cdot \sqrt{-1 \cdot \left(a \cdot c\right) + {\left(\left|b\_2\right|\right)}^{2}} + \frac{1}{a}\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(b\_2 \cdot \color{blue}{\left(\frac{1}{a \cdot b\_2} \cdot \sqrt{-1 \cdot \left(a \cdot c\right) + {\left(\left|b\_2\right|\right)}^{2}} + \frac{1}{a}\right)}\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto -1 \cdot \left(b\_2 \cdot \mathsf{fma}\left(\frac{1}{a \cdot b\_2}, \color{blue}{\sqrt{-1 \cdot \left(a \cdot c\right) + {\left(\left|b\_2\right|\right)}^{2}}}, \frac{1}{a}\right)\right) \]

   9. Applied rewrites 63.8%

      \[\leadsto \color{blue}{-1 \cdot \left(b\_2 \cdot \mathsf{fma}\left({\left(a \cdot b\_2\right)}^{-1}, {\left(\mathsf{fma}\left(-1, a \cdot c, b\_2 \cdot b\_2\right)\right)}^{0.5}, {a}^{-1}\right)\right)} \]

   if 5.99999999999999958e136 < b_2

   1. Initial program 44.4%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in c around 0

      \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a} + \frac{1}{2} \cdot \frac{c}{b\_2}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c}{b\_2} + \color{blue}{-2 \cdot \frac{b\_2}{a}} \]

      2. *-commutative N/A

         \[\leadsto \frac{c}{b\_2} \cdot \frac{1}{2} + \color{blue}{-2} \cdot \frac{b\_2}{a} \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{c}{b\_2}, \color{blue}{\frac{1}{2}}, -2 \cdot \frac{b\_2}{a}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{c}{b\_2}, \frac{1}{2}, -2 \cdot \frac{b\_2}{a}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{c}{b\_2}, \frac{1}{2}, \frac{b\_2}{a} \cdot -2\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{c}{b\_2}, \frac{1}{2}, \frac{b\_2}{a} \cdot -2\right) \]

      7. lower-/.f64 98.1

         \[\leadsto \mathsf{fma}\left(\frac{c}{b\_2}, 0.5, \frac{b\_2}{a} \cdot -2\right) \]

   5. Applied rewrites 98.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{c}{b\_2}, 0.5, \frac{b\_2}{a} \cdot -2\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.26 \cdot 10^{-123}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 6 \cdot 10^{+136}:\\ \;\;\;\;\left(-1 \cdot b\_2\right) \cdot \mathsf{fma}\left({\left(a \cdot b\_2\right)}^{-1}, {\left(\mathsf{fma}\left(-1, a \cdot c, b\_2 \cdot b\_2\right)\right)}^{0.5}, {a}^{-1}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{b\_2}, 0.5, \frac{b\_2}{a} \cdot -2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.7% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.26 \cdot 10^{-123}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 6 \cdot 10^{+136}:\\ \;\;\;\;\left(-1 \cdot b\_2\right) \cdot \mathsf{fma}\left({\left(a \cdot b\_2\right)}^{-1}, {\left(\mathsf{fma}\left(-1, a \cdot c, b\_2 \cdot b\_2\right)\right)}^{0.5}, {a}^{-1}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{b\_2 \cdot b\_2}, 0.5, -2 \cdot {a}^{-1}\right) \cdot b\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.26e-123)
   (* -0.5 (/ c b_2))
   (if (<= b_2 6e+136)
     (*
      (* -1.0 b_2)
      (fma
       (pow (* a b_2) -1.0)
       (pow (fma -1.0 (* a c) (* b_2 b_2)) 0.5)
       (pow a -1.0)))
     (* (fma (/ c (* b_2 b_2)) 0.5 (* -2.0 (pow a -1.0))) b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.26e-123) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= 6e+136) {
		tmp = (-1.0 * b_2) * fma(pow((a * b_2), -1.0), pow(fma(-1.0, (a * c), (b_2 * b_2)), 0.5), pow(a, -1.0));
	} else {
		tmp = fma((c / (b_2 * b_2)), 0.5, (-2.0 * pow(a, -1.0))) * b_2;
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1.26e-123)
		tmp = Float64(-0.5 * Float64(c / b_2));
	elseif (b_2 <= 6e+136)
		tmp = Float64(Float64(-1.0 * b_2) * fma((Float64(a * b_2) ^ -1.0), (fma(-1.0, Float64(a * c), Float64(b_2 * b_2)) ^ 0.5), (a ^ -1.0)));
	else
		tmp = Float64(fma(Float64(c / Float64(b_2 * b_2)), 0.5, Float64(-2.0 * (a ^ -1.0))) * b_2);
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1.26e-123], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 6e+136], N[(N[(-1.0 * b$95$2), $MachinePrecision] * N[(N[Power[N[(a * b$95$2), $MachinePrecision], -1.0], $MachinePrecision] * N[Power[N[(-1.0 * N[(a * c), $MachinePrecision] + N[(b$95$2 * b$95$2), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision] + N[Power[a, -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(c / N[(b$95$2 * b$95$2), $MachinePrecision]), $MachinePrecision] * 0.5 + N[(-2.0 * N[Power[a, -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * b$95$2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -1.26000000000000005e-123

   1. Initial program 15.5%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around -inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \color{blue}{\frac{c}{b\_2}} \]

      2. lower-/.f64 84.9

         \[\leadsto -0.5 \cdot \frac{c}{\color{blue}{b\_2}} \]

   5. Applied rewrites 84.9%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b\_2}} \]

   if -1.26000000000000005e-123 < b_2 < 5.99999999999999958e136

   1. Initial program 83.8%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}} \]

      2. lift-neg.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(b\_2\right)\right)} - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

      3. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(b\_2\right)\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}}{a} \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \color{blue}{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}}{a} \]

      5. lift--.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \sqrt{\color{blue}{b\_2 \cdot b\_2 - a \cdot c}}}{a} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \sqrt{\color{blue}{b\_2 \cdot b\_2} - a \cdot c}}{a} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \sqrt{b\_2 \cdot b\_2 - \color{blue}{a \cdot c}}}{a} \]

      8. div-sub N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(b\_2\right)}{a} - \frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}} \]

      9. mul-1-neg N/A

         \[\leadsto \frac{\color{blue}{-1 \cdot b\_2}}{a} - \frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

      10. associate-*r/ N/A

          \[\leadsto \color{blue}{-1 \cdot \frac{b\_2}{a}} - \frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

      11. lower--.f64 N/A

          \[\leadsto \color{blue}{-1 \cdot \frac{b\_2}{a} - \frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}} \]

      12. lower-*.f64 N/A

          \[\leadsto \color{blue}{-1 \cdot \frac{b\_2}{a}} - \frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

      13. lower-/.f64 N/A

          \[\leadsto -1 \cdot \color{blue}{\frac{b\_2}{a}} - \frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

      14. lower-/.f64 N/A

          \[\leadsto -1 \cdot \frac{b\_2}{a} - \color{blue}{\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}} \]

   4. Applied rewrites 83.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{b\_2}{a} - \frac{{\left(\mathsf{fma}\left({b\_2}^{1}, {b\_2}^{1}, -1 \cdot \left(c \cdot a\right)\right)\right)}^{0.5}}{a}} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto -1 \cdot \frac{b\_2}{a} - \frac{\color{blue}{{\left(\mathsf{fma}\left({b\_2}^{1}, {b\_2}^{1}, -1 \cdot \left(c \cdot a\right)\right)\right)}^{\frac{1}{2}}}}{a} \]

      2. lift-pow.f64 N/A

         \[\leadsto -1 \cdot \frac{b\_2}{a} - \frac{{\left(\mathsf{fma}\left(\color{blue}{{b\_2}^{1}}, {b\_2}^{1}, -1 \cdot \left(c \cdot a\right)\right)\right)}^{\frac{1}{2}}}{a} \]

      3. lift-pow.f64 N/A

         \[\leadsto -1 \cdot \frac{b\_2}{a} - \frac{{\left(\mathsf{fma}\left({b\_2}^{1}, \color{blue}{{b\_2}^{1}}, -1 \cdot \left(c \cdot a\right)\right)\right)}^{\frac{1}{2}}}{a} \]

      4. lift-fma.f64 N/A

         \[\leadsto -1 \cdot \frac{b\_2}{a} - \frac{{\color{blue}{\left({b\_2}^{1} \cdot {b\_2}^{1} + -1 \cdot \left(c \cdot a\right)\right)}}^{\frac{1}{2}}}{a} \]

      5. lift-*.f64 N/A

         \[\leadsto -1 \cdot \frac{b\_2}{a} - \frac{{\left({b\_2}^{1} \cdot {b\_2}^{1} + -1 \cdot \color{blue}{\left(c \cdot a\right)}\right)}^{\frac{1}{2}}}{a} \]

      6. lift-*.f64 N/A

         \[\leadsto -1 \cdot \frac{b\_2}{a} - \frac{{\left({b\_2}^{1} \cdot {b\_2}^{1} + \color{blue}{-1 \cdot \left(c \cdot a\right)}\right)}^{\frac{1}{2}}}{a} \]

      7. sqr-pow N/A

         \[\leadsto -1 \cdot \frac{b\_2}{a} - \frac{\color{blue}{{\left({b\_2}^{1} \cdot {b\_2}^{1} + -1 \cdot \left(c \cdot a\right)\right)}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot {\left({b\_2}^{1} \cdot {b\_2}^{1} + -1 \cdot \left(c \cdot a\right)\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}}}{a} \]

      8. lower-*.f64 N/A

         \[\leadsto -1 \cdot \frac{b\_2}{a} - \frac{\color{blue}{{\left({b\_2}^{1} \cdot {b\_2}^{1} + -1 \cdot \left(c \cdot a\right)\right)}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot {\left({b\_2}^{1} \cdot {b\_2}^{1} + -1 \cdot \left(c \cdot a\right)\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}}}{a} \]

   6. Applied rewrites 83.6%

      \[\leadsto -1 \cdot \frac{b\_2}{a} - \frac{\color{blue}{{\left(\mathsf{fma}\left(\left|b\_2\right|, \left|b\_2\right|, -1 \cdot \left(c \cdot a\right)\right)\right)}^{0.25} \cdot {\left(\mathsf{fma}\left(\left|b\_2\right|, \left|b\_2\right|, -1 \cdot \left(c \cdot a\right)\right)\right)}^{0.25}}}{a} \]

   7. Taylor expanded in b_2 around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(b\_2 \cdot \left(\frac{1}{a \cdot b\_2} \cdot \sqrt{-1 \cdot \left(a \cdot c\right) + {\left(\left|b\_2\right|\right)}^{2}} + \frac{1}{a}\right)\right)} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(b\_2 \cdot \left(\frac{1}{a \cdot b\_2} \cdot \sqrt{-1 \cdot \left(a \cdot c\right) + {\left(\left|b\_2\right|\right)}^{2}} + \frac{1}{a}\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(b\_2 \cdot \color{blue}{\left(\frac{1}{a \cdot b\_2} \cdot \sqrt{-1 \cdot \left(a \cdot c\right) + {\left(\left|b\_2\right|\right)}^{2}} + \frac{1}{a}\right)}\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto -1 \cdot \left(b\_2 \cdot \mathsf{fma}\left(\frac{1}{a \cdot b\_2}, \color{blue}{\sqrt{-1 \cdot \left(a \cdot c\right) + {\left(\left|b\_2\right|\right)}^{2}}}, \frac{1}{a}\right)\right) \]

   9. Applied rewrites 63.8%

      \[\leadsto \color{blue}{-1 \cdot \left(b\_2 \cdot \mathsf{fma}\left({\left(a \cdot b\_2\right)}^{-1}, {\left(\mathsf{fma}\left(-1, a \cdot c, b\_2 \cdot b\_2\right)\right)}^{0.5}, {a}^{-1}\right)\right)} \]

   if 5.99999999999999958e136 < b_2

   1. Initial program 44.4%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around inf

      \[\leadsto \color{blue}{b\_2 \cdot \left(\frac{1}{2} \cdot \frac{c}{{b\_2}^{2}} - 2 \cdot \frac{1}{a}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \frac{c}{{b\_2}^{2}} - 2 \cdot \frac{1}{a}\right) \cdot \color{blue}{b\_2} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \frac{c}{{b\_2}^{2}} - 2 \cdot \frac{1}{a}\right) \cdot \color{blue}{b\_2} \]

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

         \[\leadsto \left(\frac{1}{2} \cdot \frac{c}{{b\_2}^{2}} + \left(\mathsf{neg}\left(2\right)\right) \cdot \frac{1}{a}\right) \cdot b\_2 \]

      4. *-commutative N/A

         \[\leadsto \left(\frac{c}{{b\_2}^{2}} \cdot \frac{1}{2} + \left(\mathsf{neg}\left(2\right)\right) \cdot \frac{1}{a}\right) \cdot b\_2 \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{c}{{b\_2}^{2}}, \frac{1}{2}, \left(\mathsf{neg}\left(2\right)\right) \cdot \frac{1}{a}\right) \cdot b\_2 \]

      6. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{c}{{b\_2}^{2}}, \frac{1}{2}, \left(\mathsf{neg}\left(2\right)\right) \cdot \frac{1}{a}\right) \cdot b\_2 \]

      7. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{c}{b\_2 \cdot b\_2}, \frac{1}{2}, \left(\mathsf{neg}\left(2\right)\right) \cdot \frac{1}{a}\right) \cdot b\_2 \]

      8. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{c}{b\_2 \cdot b\_2}, \frac{1}{2}, \left(\mathsf{neg}\left(2\right)\right) \cdot \frac{1}{a}\right) \cdot b\_2 \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{c}{b\_2 \cdot b\_2}, \frac{1}{2}, -2 \cdot \frac{1}{a}\right) \cdot b\_2 \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{c}{b\_2 \cdot b\_2}, \frac{1}{2}, -2 \cdot \frac{1}{a}\right) \cdot b\_2 \]

      11. inv-pow N/A

          \[\leadsto \mathsf{fma}\left(\frac{c}{b\_2 \cdot b\_2}, \frac{1}{2}, -2 \cdot {a}^{-1}\right) \cdot b\_2 \]

      12. lower-pow.f64 97.8

          \[\leadsto \mathsf{fma}\left(\frac{c}{b\_2 \cdot b\_2}, 0.5, -2 \cdot {a}^{-1}\right) \cdot b\_2 \]

   5. Applied rewrites 97.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{c}{b\_2 \cdot b\_2}, 0.5, -2 \cdot {a}^{-1}\right) \cdot b\_2} \]
3. Recombined 3 regimes into one program.

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.26 \cdot 10^{-123}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 6 \cdot 10^{+136}:\\ \;\;\;\;\left(-1 \cdot b\_2\right) \cdot \mathsf{fma}\left({\left(a \cdot b\_2\right)}^{-1}, {\left(\mathsf{fma}\left(-1, a \cdot c, b\_2 \cdot b\_2\right)\right)}^{0.5}, {a}^{-1}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{b\_2 \cdot b\_2}, 0.5, -2 \cdot {a}^{-1}\right) \cdot b\_2\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 68.0% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.26 \cdot 10^{-123}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;\left(-1 \cdot b\_2\right) \cdot \mathsf{fma}\left({\left(a \cdot b\_2\right)}^{-1}, {\left(\mathsf{fma}\left(-1, a \cdot c, b\_2 \cdot b\_2\right)\right)}^{0.5}, {a}^{-1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.26e-123)
   (* -0.5 (/ c b_2))
   (*
    (* -1.0 b_2)
    (fma
     (pow (* a b_2) -1.0)
     (pow (fma -1.0 (* a c) (* b_2 b_2)) 0.5)
     (pow a -1.0)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.26e-123) {
		tmp = -0.5 * (c / b_2);
	} else {
		tmp = (-1.0 * b_2) * fma(pow((a * b_2), -1.0), pow(fma(-1.0, (a * c), (b_2 * b_2)), 0.5), pow(a, -1.0));
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1.26e-123)
		tmp = Float64(-0.5 * Float64(c / b_2));
	else
		tmp = Float64(Float64(-1.0 * b_2) * fma((Float64(a * b_2) ^ -1.0), (fma(-1.0, Float64(a * c), Float64(b_2 * b_2)) ^ 0.5), (a ^ -1.0)));
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1.26e-123], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 * b$95$2), $MachinePrecision] * N[(N[Power[N[(a * b$95$2), $MachinePrecision], -1.0], $MachinePrecision] * N[Power[N[(-1.0 * N[(a * c), $MachinePrecision] + N[(b$95$2 * b$95$2), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision] + N[Power[a, -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -1.26000000000000005e-123

   1. Initial program 15.5%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around -inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \color{blue}{\frac{c}{b\_2}} \]

      2. lower-/.f64 84.9

         \[\leadsto -0.5 \cdot \frac{c}{\color{blue}{b\_2}} \]

   5. Applied rewrites 84.9%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b\_2}} \]

   if -1.26000000000000005e-123 < b_2

   1. Initial program 72.2%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}} \]

      2. lift-neg.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(b\_2\right)\right)} - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

      3. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(b\_2\right)\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}}{a} \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \color{blue}{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}}{a} \]

      5. lift--.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \sqrt{\color{blue}{b\_2 \cdot b\_2 - a \cdot c}}}{a} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \sqrt{\color{blue}{b\_2 \cdot b\_2} - a \cdot c}}{a} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \sqrt{b\_2 \cdot b\_2 - \color{blue}{a \cdot c}}}{a} \]

      8. div-sub N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(b\_2\right)}{a} - \frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}} \]

      9. mul-1-neg N/A

         \[\leadsto \frac{\color{blue}{-1 \cdot b\_2}}{a} - \frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

      10. associate-*r/ N/A

          \[\leadsto \color{blue}{-1 \cdot \frac{b\_2}{a}} - \frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

      11. lower--.f64 N/A

          \[\leadsto \color{blue}{-1 \cdot \frac{b\_2}{a} - \frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}} \]

      12. lower-*.f64 N/A

          \[\leadsto \color{blue}{-1 \cdot \frac{b\_2}{a}} - \frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

      13. lower-/.f64 N/A

          \[\leadsto -1 \cdot \color{blue}{\frac{b\_2}{a}} - \frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

      14. lower-/.f64 N/A

          \[\leadsto -1 \cdot \frac{b\_2}{a} - \color{blue}{\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}} \]

   4. Applied rewrites 72.3%

      \[\leadsto \color{blue}{-1 \cdot \frac{b\_2}{a} - \frac{{\left(\mathsf{fma}\left({b\_2}^{1}, {b\_2}^{1}, -1 \cdot \left(c \cdot a\right)\right)\right)}^{0.5}}{a}} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto -1 \cdot \frac{b\_2}{a} - \frac{\color{blue}{{\left(\mathsf{fma}\left({b\_2}^{1}, {b\_2}^{1}, -1 \cdot \left(c \cdot a\right)\right)\right)}^{\frac{1}{2}}}}{a} \]

      2. lift-pow.f64 N/A

         \[\leadsto -1 \cdot \frac{b\_2}{a} - \frac{{\left(\mathsf{fma}\left(\color{blue}{{b\_2}^{1}}, {b\_2}^{1}, -1 \cdot \left(c \cdot a\right)\right)\right)}^{\frac{1}{2}}}{a} \]

      3. lift-pow.f64 N/A

         \[\leadsto -1 \cdot \frac{b\_2}{a} - \frac{{\left(\mathsf{fma}\left({b\_2}^{1}, \color{blue}{{b\_2}^{1}}, -1 \cdot \left(c \cdot a\right)\right)\right)}^{\frac{1}{2}}}{a} \]

      4. lift-fma.f64 N/A

         \[\leadsto -1 \cdot \frac{b\_2}{a} - \frac{{\color{blue}{\left({b\_2}^{1} \cdot {b\_2}^{1} + -1 \cdot \left(c \cdot a\right)\right)}}^{\frac{1}{2}}}{a} \]

      5. lift-*.f64 N/A

         \[\leadsto -1 \cdot \frac{b\_2}{a} - \frac{{\left({b\_2}^{1} \cdot {b\_2}^{1} + -1 \cdot \color{blue}{\left(c \cdot a\right)}\right)}^{\frac{1}{2}}}{a} \]

      6. lift-*.f64 N/A

         \[\leadsto -1 \cdot \frac{b\_2}{a} - \frac{{\left({b\_2}^{1} \cdot {b\_2}^{1} + \color{blue}{-1 \cdot \left(c \cdot a\right)}\right)}^{\frac{1}{2}}}{a} \]

      7. sqr-pow N/A

         \[\leadsto -1 \cdot \frac{b\_2}{a} - \frac{\color{blue}{{\left({b\_2}^{1} \cdot {b\_2}^{1} + -1 \cdot \left(c \cdot a\right)\right)}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot {\left({b\_2}^{1} \cdot {b\_2}^{1} + -1 \cdot \left(c \cdot a\right)\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}}}{a} \]

      8. lower-*.f64 N/A

         \[\leadsto -1 \cdot \frac{b\_2}{a} - \frac{\color{blue}{{\left({b\_2}^{1} \cdot {b\_2}^{1} + -1 \cdot \left(c \cdot a\right)\right)}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot {\left({b\_2}^{1} \cdot {b\_2}^{1} + -1 \cdot \left(c \cdot a\right)\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}}}{a} \]

   6. Applied rewrites 72.1%

      \[\leadsto -1 \cdot \frac{b\_2}{a} - \frac{\color{blue}{{\left(\mathsf{fma}\left(\left|b\_2\right|, \left|b\_2\right|, -1 \cdot \left(c \cdot a\right)\right)\right)}^{0.25} \cdot {\left(\mathsf{fma}\left(\left|b\_2\right|, \left|b\_2\right|, -1 \cdot \left(c \cdot a\right)\right)\right)}^{0.25}}}{a} \]

   7. Taylor expanded in b_2 around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(b\_2 \cdot \left(\frac{1}{a \cdot b\_2} \cdot \sqrt{-1 \cdot \left(a \cdot c\right) + {\left(\left|b\_2\right|\right)}^{2}} + \frac{1}{a}\right)\right)} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(b\_2 \cdot \left(\frac{1}{a \cdot b\_2} \cdot \sqrt{-1 \cdot \left(a \cdot c\right) + {\left(\left|b\_2\right|\right)}^{2}} + \frac{1}{a}\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(b\_2 \cdot \color{blue}{\left(\frac{1}{a \cdot b\_2} \cdot \sqrt{-1 \cdot \left(a \cdot c\right) + {\left(\left|b\_2\right|\right)}^{2}} + \frac{1}{a}\right)}\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto -1 \cdot \left(b\_2 \cdot \mathsf{fma}\left(\frac{1}{a \cdot b\_2}, \color{blue}{\sqrt{-1 \cdot \left(a \cdot c\right) + {\left(\left|b\_2\right|\right)}^{2}}}, \frac{1}{a}\right)\right) \]

   9. Applied rewrites 57.8%

      \[\leadsto \color{blue}{-1 \cdot \left(b\_2 \cdot \mathsf{fma}\left({\left(a \cdot b\_2\right)}^{-1}, {\left(\mathsf{fma}\left(-1, a \cdot c, b\_2 \cdot b\_2\right)\right)}^{0.5}, {a}^{-1}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.26 \cdot 10^{-123}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;\left(-1 \cdot b\_2\right) \cdot \mathsf{fma}\left({\left(a \cdot b\_2\right)}^{-1}, {\left(\mathsf{fma}\left(-1, a \cdot c, b\_2 \cdot b\_2\right)\right)}^{0.5}, {a}^{-1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 41.0% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \left(-1 \cdot b\_2\right) \cdot \mathsf{fma}\left({\left(a \cdot b\_2\right)}^{-1}, {\left(\mathsf{fma}\left(-1, a \cdot c, b\_2 \cdot b\_2\right)\right)}^{0.5}, {a}^{-1}\right) \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (*
  (* -1.0 b_2)
  (fma
   (pow (* a b_2) -1.0)
   (pow (fma -1.0 (* a c) (* b_2 b_2)) 0.5)
   (pow a -1.0))))

C

double code(double a, double b_2, double c) {
	return (-1.0 * b_2) * fma(pow((a * b_2), -1.0), pow(fma(-1.0, (a * c), (b_2 * b_2)), 0.5), pow(a, -1.0));
}

Julia

function code(a, b_2, c)
	return Float64(Float64(-1.0 * b_2) * fma((Float64(a * b_2) ^ -1.0), (fma(-1.0, Float64(a * c), Float64(b_2 * b_2)) ^ 0.5), (a ^ -1.0)))
end

Wolfram

code[a_, b$95$2_, c_] := N[(N[(-1.0 * b$95$2), $MachinePrecision] * N[(N[Power[N[(a * b$95$2), $MachinePrecision], -1.0], $MachinePrecision] * N[Power[N[(-1.0 * N[(a * c), $MachinePrecision] + N[(b$95$2 * b$95$2), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision] + N[Power[a, -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 49.4%

   \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}} \]

   2. lift-neg.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(b\_2\right)\right)} - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

   3. lift--.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(b\_2\right)\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}}{a} \]

   4. lift-sqrt.f64 N/A

      \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \color{blue}{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}}{a} \]

   5. lift--.f64 N/A

      \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \sqrt{\color{blue}{b\_2 \cdot b\_2 - a \cdot c}}}{a} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \sqrt{\color{blue}{b\_2 \cdot b\_2} - a \cdot c}}{a} \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \sqrt{b\_2 \cdot b\_2 - \color{blue}{a \cdot c}}}{a} \]

   8. div-sub N/A

      \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(b\_2\right)}{a} - \frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}} \]

   9. mul-1-neg N/A

      \[\leadsto \frac{\color{blue}{-1 \cdot b\_2}}{a} - \frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

   10. associate-*r/ N/A

       \[\leadsto \color{blue}{-1 \cdot \frac{b\_2}{a}} - \frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

   11. lower--.f64 N/A

       \[\leadsto \color{blue}{-1 \cdot \frac{b\_2}{a} - \frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}} \]

   12. lower-*.f64 N/A

       \[\leadsto \color{blue}{-1 \cdot \frac{b\_2}{a}} - \frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

   13. lower-/.f64 N/A

       \[\leadsto -1 \cdot \color{blue}{\frac{b\_2}{a}} - \frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

   14. lower-/.f64 N/A

       \[\leadsto -1 \cdot \frac{b\_2}{a} - \color{blue}{\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}} \]

4. Applied rewrites 48.7%

   \[\leadsto \color{blue}{-1 \cdot \frac{b\_2}{a} - \frac{{\left(\mathsf{fma}\left({b\_2}^{1}, {b\_2}^{1}, -1 \cdot \left(c \cdot a\right)\right)\right)}^{0.5}}{a}} \]
5. Step-by-step derivation

   1. lift-pow.f64 N/A

      \[\leadsto -1 \cdot \frac{b\_2}{a} - \frac{\color{blue}{{\left(\mathsf{fma}\left({b\_2}^{1}, {b\_2}^{1}, -1 \cdot \left(c \cdot a\right)\right)\right)}^{\frac{1}{2}}}}{a} \]

   2. lift-pow.f64 N/A

      \[\leadsto -1 \cdot \frac{b\_2}{a} - \frac{{\left(\mathsf{fma}\left(\color{blue}{{b\_2}^{1}}, {b\_2}^{1}, -1 \cdot \left(c \cdot a\right)\right)\right)}^{\frac{1}{2}}}{a} \]

   3. lift-pow.f64 N/A

      \[\leadsto -1 \cdot \frac{b\_2}{a} - \frac{{\left(\mathsf{fma}\left({b\_2}^{1}, \color{blue}{{b\_2}^{1}}, -1 \cdot \left(c \cdot a\right)\right)\right)}^{\frac{1}{2}}}{a} \]

   4. lift-fma.f64 N/A

      \[\leadsto -1 \cdot \frac{b\_2}{a} - \frac{{\color{blue}{\left({b\_2}^{1} \cdot {b\_2}^{1} + -1 \cdot \left(c \cdot a\right)\right)}}^{\frac{1}{2}}}{a} \]

   5. lift-*.f64 N/A

      \[\leadsto -1 \cdot \frac{b\_2}{a} - \frac{{\left({b\_2}^{1} \cdot {b\_2}^{1} + -1 \cdot \color{blue}{\left(c \cdot a\right)}\right)}^{\frac{1}{2}}}{a} \]

   6. lift-*.f64 N/A

      \[\leadsto -1 \cdot \frac{b\_2}{a} - \frac{{\left({b\_2}^{1} \cdot {b\_2}^{1} + \color{blue}{-1 \cdot \left(c \cdot a\right)}\right)}^{\frac{1}{2}}}{a} \]

   7. sqr-pow N/A

      \[\leadsto -1 \cdot \frac{b\_2}{a} - \frac{\color{blue}{{\left({b\_2}^{1} \cdot {b\_2}^{1} + -1 \cdot \left(c \cdot a\right)\right)}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot {\left({b\_2}^{1} \cdot {b\_2}^{1} + -1 \cdot \left(c \cdot a\right)\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}}}{a} \]

   8. lower-*.f64 N/A

      \[\leadsto -1 \cdot \frac{b\_2}{a} - \frac{\color{blue}{{\left({b\_2}^{1} \cdot {b\_2}^{1} + -1 \cdot \left(c \cdot a\right)\right)}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot {\left({b\_2}^{1} \cdot {b\_2}^{1} + -1 \cdot \left(c \cdot a\right)\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}}}{a} \]

6. Applied rewrites 48.0%

   \[\leadsto -1 \cdot \frac{b\_2}{a} - \frac{\color{blue}{{\left(\mathsf{fma}\left(\left|b\_2\right|, \left|b\_2\right|, -1 \cdot \left(c \cdot a\right)\right)\right)}^{0.25} \cdot {\left(\mathsf{fma}\left(\left|b\_2\right|, \left|b\_2\right|, -1 \cdot \left(c \cdot a\right)\right)\right)}^{0.25}}}{a} \]

7. Taylor expanded in b_2 around -inf

   \[\leadsto \color{blue}{-1 \cdot \left(b\_2 \cdot \left(\frac{1}{a \cdot b\_2} \cdot \sqrt{-1 \cdot \left(a \cdot c\right) + {\left(\left|b\_2\right|\right)}^{2}} + \frac{1}{a}\right)\right)} \]
8. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto -1 \cdot \color{blue}{\left(b\_2 \cdot \left(\frac{1}{a \cdot b\_2} \cdot \sqrt{-1 \cdot \left(a \cdot c\right) + {\left(\left|b\_2\right|\right)}^{2}} + \frac{1}{a}\right)\right)} \]

   2. lower-*.f64 N/A

      \[\leadsto -1 \cdot \left(b\_2 \cdot \color{blue}{\left(\frac{1}{a \cdot b\_2} \cdot \sqrt{-1 \cdot \left(a \cdot c\right) + {\left(\left|b\_2\right|\right)}^{2}} + \frac{1}{a}\right)}\right) \]

   3. lower-fma.f64 N/A

      \[\leadsto -1 \cdot \left(b\_2 \cdot \mathsf{fma}\left(\frac{1}{a \cdot b\_2}, \color{blue}{\sqrt{-1 \cdot \left(a \cdot c\right) + {\left(\left|b\_2\right|\right)}^{2}}}, \frac{1}{a}\right)\right) \]

9. Applied rewrites 39.2%

   \[\leadsto \color{blue}{-1 \cdot \left(b\_2 \cdot \mathsf{fma}\left({\left(a \cdot b\_2\right)}^{-1}, {\left(\mathsf{fma}\left(-1, a \cdot c, b\_2 \cdot b\_2\right)\right)}^{0.5}, {a}^{-1}\right)\right)} \]

10. Final simplification 39.2%

    \[\leadsto \left(-1 \cdot b\_2\right) \cdot \mathsf{fma}\left({\left(a \cdot b\_2\right)}^{-1}, {\left(\mathsf{fma}\left(-1, a \cdot c, b\_2 \cdot b\_2\right)\right)}^{0.5}, {a}^{-1}\right) \]
11. Add Preprocessing

Alternative 8: 5.7% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \left(-1 \cdot b\_2\right) \cdot \mathsf{fma}\left({\left(a \cdot b\_2\right)}^{-1}, b\_2 \cdot \left(-1 - \mathsf{fma}\left(-0.5, \frac{a \cdot c}{b\_2 \cdot b\_2}, \mathsf{fma}\left(-0.125, \frac{{\left(a \cdot c\right)}^{2}}{{b\_2}^{4}}, -0.0625 \cdot \frac{{\left(a \cdot c\right)}^{3}}{{b\_2}^{6}}\right)\right)\right), {a}^{-1}\right) \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (*
  (* -1.0 b_2)
  (fma
   (pow (* a b_2) -1.0)
   (*
    b_2
    (-
     -1.0
     (fma
      -0.5
      (/ (* a c) (* b_2 b_2))
      (fma
       -0.125
       (/ (pow (* a c) 2.0) (pow b_2 4.0))
       (* -0.0625 (/ (pow (* a c) 3.0) (pow b_2 6.0)))))))
   (pow a -1.0))))

C

double code(double a, double b_2, double c) {
	return (-1.0 * b_2) * fma(pow((a * b_2), -1.0), (b_2 * (-1.0 - fma(-0.5, ((a * c) / (b_2 * b_2)), fma(-0.125, (pow((a * c), 2.0) / pow(b_2, 4.0)), (-0.0625 * (pow((a * c), 3.0) / pow(b_2, 6.0))))))), pow(a, -1.0));
}

Julia

function code(a, b_2, c)
	return Float64(Float64(-1.0 * b_2) * fma((Float64(a * b_2) ^ -1.0), Float64(b_2 * Float64(-1.0 - fma(-0.5, Float64(Float64(a * c) / Float64(b_2 * b_2)), fma(-0.125, Float64((Float64(a * c) ^ 2.0) / (b_2 ^ 4.0)), Float64(-0.0625 * Float64((Float64(a * c) ^ 3.0) / (b_2 ^ 6.0))))))), (a ^ -1.0)))
end

Wolfram

code[a_, b$95$2_, c_] := N[(N[(-1.0 * b$95$2), $MachinePrecision] * N[(N[Power[N[(a * b$95$2), $MachinePrecision], -1.0], $MachinePrecision] * N[(b$95$2 * N[(-1.0 - N[(-0.5 * N[(N[(a * c), $MachinePrecision] / N[(b$95$2 * b$95$2), $MachinePrecision]), $MachinePrecision] + N[(-0.125 * N[(N[Power[N[(a * c), $MachinePrecision], 2.0], $MachinePrecision] / N[Power[b$95$2, 4.0], $MachinePrecision]), $MachinePrecision] + N[(-0.0625 * N[(N[Power[N[(a * c), $MachinePrecision], 3.0], $MachinePrecision] / N[Power[b$95$2, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[a, -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 49.4%

   \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}} \]

   2. lift-neg.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(b\_2\right)\right)} - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

   3. lift--.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(b\_2\right)\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}}{a} \]

   4. lift-sqrt.f64 N/A

      \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \color{blue}{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}}{a} \]

   5. lift--.f64 N/A

      \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \sqrt{\color{blue}{b\_2 \cdot b\_2 - a \cdot c}}}{a} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \sqrt{\color{blue}{b\_2 \cdot b\_2} - a \cdot c}}{a} \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \sqrt{b\_2 \cdot b\_2 - \color{blue}{a \cdot c}}}{a} \]

   8. div-sub N/A

      \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(b\_2\right)}{a} - \frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}} \]

   9. mul-1-neg N/A

      \[\leadsto \frac{\color{blue}{-1 \cdot b\_2}}{a} - \frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

   10. associate-*r/ N/A

       \[\leadsto \color{blue}{-1 \cdot \frac{b\_2}{a}} - \frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

   11. lower--.f64 N/A

       \[\leadsto \color{blue}{-1 \cdot \frac{b\_2}{a} - \frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}} \]

   12. lower-*.f64 N/A

       \[\leadsto \color{blue}{-1 \cdot \frac{b\_2}{a}} - \frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

   13. lower-/.f64 N/A

       \[\leadsto -1 \cdot \color{blue}{\frac{b\_2}{a}} - \frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

   14. lower-/.f64 N/A

       \[\leadsto -1 \cdot \frac{b\_2}{a} - \color{blue}{\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}} \]

4. Applied rewrites 48.7%

   \[\leadsto \color{blue}{-1 \cdot \frac{b\_2}{a} - \frac{{\left(\mathsf{fma}\left({b\_2}^{1}, {b\_2}^{1}, -1 \cdot \left(c \cdot a\right)\right)\right)}^{0.5}}{a}} \]
5. Step-by-step derivation

   1. lift-pow.f64 N/A

      \[\leadsto -1 \cdot \frac{b\_2}{a} - \frac{\color{blue}{{\left(\mathsf{fma}\left({b\_2}^{1}, {b\_2}^{1}, -1 \cdot \left(c \cdot a\right)\right)\right)}^{\frac{1}{2}}}}{a} \]

   2. lift-pow.f64 N/A

      \[\leadsto -1 \cdot \frac{b\_2}{a} - \frac{{\left(\mathsf{fma}\left(\color{blue}{{b\_2}^{1}}, {b\_2}^{1}, -1 \cdot \left(c \cdot a\right)\right)\right)}^{\frac{1}{2}}}{a} \]

   3. lift-pow.f64 N/A

      \[\leadsto -1 \cdot \frac{b\_2}{a} - \frac{{\left(\mathsf{fma}\left({b\_2}^{1}, \color{blue}{{b\_2}^{1}}, -1 \cdot \left(c \cdot a\right)\right)\right)}^{\frac{1}{2}}}{a} \]

   4. lift-fma.f64 N/A

      \[\leadsto -1 \cdot \frac{b\_2}{a} - \frac{{\color{blue}{\left({b\_2}^{1} \cdot {b\_2}^{1} + -1 \cdot \left(c \cdot a\right)\right)}}^{\frac{1}{2}}}{a} \]

   5. lift-*.f64 N/A

      \[\leadsto -1 \cdot \frac{b\_2}{a} - \frac{{\left({b\_2}^{1} \cdot {b\_2}^{1} + -1 \cdot \color{blue}{\left(c \cdot a\right)}\right)}^{\frac{1}{2}}}{a} \]

   6. lift-*.f64 N/A

      \[\leadsto -1 \cdot \frac{b\_2}{a} - \frac{{\left({b\_2}^{1} \cdot {b\_2}^{1} + \color{blue}{-1 \cdot \left(c \cdot a\right)}\right)}^{\frac{1}{2}}}{a} \]

   7. sqr-pow N/A

      \[\leadsto -1 \cdot \frac{b\_2}{a} - \frac{\color{blue}{{\left({b\_2}^{1} \cdot {b\_2}^{1} + -1 \cdot \left(c \cdot a\right)\right)}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot {\left({b\_2}^{1} \cdot {b\_2}^{1} + -1 \cdot \left(c \cdot a\right)\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}}}{a} \]

   8. lower-*.f64 N/A

      \[\leadsto -1 \cdot \frac{b\_2}{a} - \frac{\color{blue}{{\left({b\_2}^{1} \cdot {b\_2}^{1} + -1 \cdot \left(c \cdot a\right)\right)}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot {\left({b\_2}^{1} \cdot {b\_2}^{1} + -1 \cdot \left(c \cdot a\right)\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}}}{a} \]

6. Applied rewrites 48.0%

   \[\leadsto -1 \cdot \frac{b\_2}{a} - \frac{\color{blue}{{\left(\mathsf{fma}\left(\left|b\_2\right|, \left|b\_2\right|, -1 \cdot \left(c \cdot a\right)\right)\right)}^{0.25} \cdot {\left(\mathsf{fma}\left(\left|b\_2\right|, \left|b\_2\right|, -1 \cdot \left(c \cdot a\right)\right)\right)}^{0.25}}}{a} \]

7. Taylor expanded in b_2 around -inf

   \[\leadsto \color{blue}{-1 \cdot \left(b\_2 \cdot \left(\frac{1}{a \cdot b\_2} \cdot \sqrt{-1 \cdot \left(a \cdot c\right) + {\left(\left|b\_2\right|\right)}^{2}} + \frac{1}{a}\right)\right)} \]
8. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto -1 \cdot \color{blue}{\left(b\_2 \cdot \left(\frac{1}{a \cdot b\_2} \cdot \sqrt{-1 \cdot \left(a \cdot c\right) + {\left(\left|b\_2\right|\right)}^{2}} + \frac{1}{a}\right)\right)} \]

   2. lower-*.f64 N/A

      \[\leadsto -1 \cdot \left(b\_2 \cdot \color{blue}{\left(\frac{1}{a \cdot b\_2} \cdot \sqrt{-1 \cdot \left(a \cdot c\right) + {\left(\left|b\_2\right|\right)}^{2}} + \frac{1}{a}\right)}\right) \]

   3. lower-fma.f64 N/A

      \[\leadsto -1 \cdot \left(b\_2 \cdot \mathsf{fma}\left(\frac{1}{a \cdot b\_2}, \color{blue}{\sqrt{-1 \cdot \left(a \cdot c\right) + {\left(\left|b\_2\right|\right)}^{2}}}, \frac{1}{a}\right)\right) \]

9. Applied rewrites 39.2%

   \[\leadsto \color{blue}{-1 \cdot \left(b\_2 \cdot \mathsf{fma}\left({\left(a \cdot b\_2\right)}^{-1}, {\left(\mathsf{fma}\left(-1, a \cdot c, b\_2 \cdot b\_2\right)\right)}^{0.5}, {a}^{-1}\right)\right)} \]

10. Taylor expanded in b_2 around -inf

    \[\leadsto -1 \cdot \left(b\_2 \cdot \mathsf{fma}\left({\left(a \cdot b\_2\right)}^{-1}, -1 \cdot \color{blue}{\left(b\_2 \cdot \left(1 + \left(\frac{-1}{2} \cdot \frac{a \cdot c}{{b\_2}^{2}} + \left(\frac{-1}{8} \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b\_2}^{4}} + \frac{-1}{16} \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b\_2}^{6}}\right)\right)\right)\right)}, {a}^{-1}\right)\right) \]
11. Step-by-step derivation

    1. lower-*.f64 N/A

       \[\leadsto -1 \cdot \left(b\_2 \cdot \mathsf{fma}\left({\left(a \cdot b\_2\right)}^{-1}, -1 \cdot \left(b\_2 \cdot \color{blue}{\left(1 + \left(\frac{-1}{2} \cdot \frac{a \cdot c}{{b\_2}^{2}} + \left(\frac{-1}{8} \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b\_2}^{4}} + \frac{-1}{16} \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b\_2}^{6}}\right)\right)\right)}\right), {a}^{-1}\right)\right) \]

    2. lower-*.f64 N/A

       \[\leadsto -1 \cdot \left(b\_2 \cdot \mathsf{fma}\left({\left(a \cdot b\_2\right)}^{-1}, -1 \cdot \left(b\_2 \cdot \left(1 + \color{blue}{\left(\frac{-1}{2} \cdot \frac{a \cdot c}{{b\_2}^{2}} + \left(\frac{-1}{8} \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b\_2}^{4}} + \frac{-1}{16} \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b\_2}^{6}}\right)\right)}\right)\right), {a}^{-1}\right)\right) \]

    3. lower-+.f64 N/A

       \[\leadsto -1 \cdot \left(b\_2 \cdot \mathsf{fma}\left({\left(a \cdot b\_2\right)}^{-1}, -1 \cdot \left(b\_2 \cdot \left(1 + \left(\frac{-1}{2} \cdot \frac{a \cdot c}{{b\_2}^{2}} + \color{blue}{\left(\frac{-1}{8} \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b\_2}^{4}} + \frac{-1}{16} \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b\_2}^{6}}\right)}\right)\right)\right), {a}^{-1}\right)\right) \]

    4. lower-fma.f64 N/A

       \[\leadsto -1 \cdot \left(b\_2 \cdot \mathsf{fma}\left({\left(a \cdot b\_2\right)}^{-1}, -1 \cdot \left(b\_2 \cdot \left(1 + \mathsf{fma}\left(\frac{-1}{2}, \frac{a \cdot c}{\color{blue}{{b\_2}^{2}}}, \frac{-1}{8} \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b\_2}^{4}} + \frac{-1}{16} \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b\_2}^{6}}\right)\right)\right), {a}^{-1}\right)\right) \]

12. Applied rewrites 7.0%

    \[\leadsto -1 \cdot \left(b\_2 \cdot \mathsf{fma}\left({\left(a \cdot b\_2\right)}^{-1}, -1 \cdot \color{blue}{\left(b\_2 \cdot \left(1 + \mathsf{fma}\left(-0.5, \frac{a \cdot c}{b\_2 \cdot b\_2}, \mathsf{fma}\left(-0.125, \frac{{\left(a \cdot c\right)}^{2}}{{b\_2}^{4}}, -0.0625 \cdot \frac{{\left(a \cdot c\right)}^{3}}{{b\_2}^{6}}\right)\right)\right)\right)}, {a}^{-1}\right)\right) \]

13. Final simplification 7.0%

    \[\leadsto \left(-1 \cdot b\_2\right) \cdot \mathsf{fma}\left({\left(a \cdot b\_2\right)}^{-1}, b\_2 \cdot \left(-1 - \mathsf{fma}\left(-0.5, \frac{a \cdot c}{b\_2 \cdot b\_2}, \mathsf{fma}\left(-0.125, \frac{{\left(a \cdot c\right)}^{2}}{{b\_2}^{4}}, -0.0625 \cdot \frac{{\left(a \cdot c\right)}^{3}}{{b\_2}^{6}}\right)\right)\right), {a}^{-1}\right) \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2025064 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  :herbie-expected 10

  :alt
  (! :herbie-platform c (let ((sqtD (let ((x (* (sqrt (fabs a)) (sqrt (fabs c))))) (if (== (copysign a c) a) (* (sqrt (- (fabs b_2) x)) (sqrt (+ (fabs b_2) x))) (hypot b_2 x))))) (if (< b_2 0) (/ c (- sqtD b_2)) (/ (+ b_2 sqtD) (- a)))))

  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))