quad2p (problem 3.2.1, positive)

Percentage Accurate: 52.6% → 86.5%

Time: 4.5s

Alternatives: 10

Speedup: 1.7×

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: 52.6% 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: 86.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -5.6 \cdot 10^{+101}:\\ \;\;\;\;\frac{-2 \cdot b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 1.85 \cdot 10^{-121}:\\ \;\;\;\;\frac{\mathsf{hypot}\left({\left(\left|\left(-c\right) \cdot a\right|\right)}^{0.5}, b\_2\right) - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -5.6e+101)
   (/ (* -2.0 b_2) a)
   (if (<= b_2 1.85e-121)
     (/ (- (hypot (pow (fabs (* (- c) a)) 0.5) b_2) b_2) a)
     (* -0.5 (/ c b_2)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5.6e+101) {
		tmp = (-2.0 * b_2) / a;
	} else if (b_2 <= 1.85e-121) {
		tmp = (hypot(pow(fabs((-c * a)), 0.5), b_2) - b_2) / a;
	} else {
		tmp = -0.5 * (c / b_2);
	}
	return tmp;
}

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5.6e+101) {
		tmp = (-2.0 * b_2) / a;
	} else if (b_2 <= 1.85e-121) {
		tmp = (Math.hypot(Math.pow(Math.abs((-c * a)), 0.5), b_2) - b_2) / a;
	} else {
		tmp = -0.5 * (c / b_2);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -5.6e+101:
		tmp = (-2.0 * b_2) / a
	elif b_2 <= 1.85e-121:
		tmp = (math.hypot(math.pow(math.fabs((-c * a)), 0.5), b_2) - b_2) / a
	else:
		tmp = -0.5 * (c / b_2)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -5.6e+101)
		tmp = Float64(Float64(-2.0 * b_2) / a);
	elseif (b_2 <= 1.85e-121)
		tmp = Float64(Float64(hypot((abs(Float64(Float64(-c) * a)) ^ 0.5), b_2) - b_2) / a);
	else
		tmp = Float64(-0.5 * Float64(c / b_2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -5.6e+101)
		tmp = (-2.0 * b_2) / a;
	elseif (b_2 <= 1.85e-121)
		tmp = (hypot((abs((-c * a)) ^ 0.5), b_2) - b_2) / a;
	else
		tmp = -0.5 * (c / b_2);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -5.6e+101], N[(N[(-2.0 * b$95$2), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 1.85e-121], N[(N[(N[Sqrt[N[Power[N[Abs[N[((-c) * a), $MachinePrecision]], $MachinePrecision], 0.5], $MachinePrecision] ^ 2 + b$95$2 ^ 2], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -5.59999999999999962e101

   1. Initial program 53.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-negate1-reverse N/A

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

      5. lower--.f64 53.6

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

      6. lift--.f64 N/A

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

      7. sub-negate1 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

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

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

      12. lower-fma.f64 N/A

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

      13. lower-neg.f64 53.8

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

   3. Applied rewrites 53.8%

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

   5. Applied rewrites 53.0%

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

      1. unpow1 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. sqr-abs-rev N/A

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

      7. mul-fabs N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. pow-prod-up N/A

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

      11. metadata-eval N/A

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

      12. unpow1 N/A

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

      13. lower-fabs.f64 90.4

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 90.4

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

   7. Applied rewrites 90.4%

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

   8. Taylor expanded in b_2 around -inf

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

      1. lower-*.f64 95.7

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

   10. Applied rewrites 95.7%

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

   if -5.59999999999999962e101 < b_2 < 1.8500000000000001e-121

   1. Initial program 83.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-negate1-reverse N/A

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

      5. lower--.f64 83.7

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

      6. lift--.f64 N/A

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

      7. sub-negate1 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

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

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

      12. lower-fma.f64 N/A

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

      13. lower-neg.f64 83.7

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

   3. Applied rewrites 83.7%

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

   5. Applied rewrites 73.5%

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

      1. unpow1 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. sqr-abs-rev N/A

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

      7. mul-fabs N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. pow-prod-up N/A

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

      11. metadata-eval N/A

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

      12. unpow1 N/A

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

      13. lower-fabs.f64 84.5

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 84.5

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

   7. Applied rewrites 84.5%

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

   if 1.8500000000000001e-121 < b_2

   1. Initial program 20.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-negate1-reverse N/A

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

      5. lower--.f64 20.8

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

      6. lift--.f64 N/A

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

      7. sub-negate1 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

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

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

      12. lower-fma.f64 N/A

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

      13. lower-neg.f64 20.8

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

   3. Applied rewrites 20.8%

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

   5. Applied rewrites 14.8%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\left(\mathsf{fma}\left(a, -c, b\_2 \cdot b\_2\right)\right)}^{0.375}, {\left(\mathsf{fma}\left(a, -c, b\_2 \cdot b\_2\right)\right)}^{0.125}, -b\_2\right)}}{a} \]

   7. Applied rewrites 14.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\mathsf{fma}\left(-c, a, b\_2 \cdot b\_2\right)\right)}^{0.125}, \frac{{\left(\mathsf{fma}\left(-c, a, b\_2 \cdot b\_2\right)\right)}^{0.375}}{a}, \frac{-b\_2}{a}\right)} \]

   8. Taylor expanded in b_2 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 83.3

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

   10. Applied rewrites 83.3%

       \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b\_2}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 86.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2 \cdot 10^{+130}:\\ \;\;\;\;\frac{-2 \cdot b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 1.85 \cdot 10^{-121}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-c, a, b\_2 \cdot b\_2\right)} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2e+130)
   (/ (* -2.0 b_2) a)
   (if (<= b_2 1.85e-121)
     (/ (- (sqrt (fma (- c) a (* b_2 b_2))) b_2) a)
     (* -0.5 (/ c b_2)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2e+130) {
		tmp = (-2.0 * b_2) / a;
	} else if (b_2 <= 1.85e-121) {
		tmp = (sqrt(fma(-c, a, (b_2 * b_2))) - b_2) / a;
	} else {
		tmp = -0.5 * (c / b_2);
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2e+130)
		tmp = Float64(Float64(-2.0 * b_2) / a);
	elseif (b_2 <= 1.85e-121)
		tmp = Float64(Float64(sqrt(fma(Float64(-c), a, Float64(b_2 * b_2))) - b_2) / a);
	else
		tmp = Float64(-0.5 * Float64(c / b_2));
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2e+130], N[(N[(-2.0 * b$95$2), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 1.85e-121], N[(N[(N[Sqrt[N[((-c) * a + N[(b$95$2 * b$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -2.0000000000000001e130

   1. Initial program 47.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-negate1-reverse N/A

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

      5. lower--.f64 47.2

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

      6. lift--.f64 N/A

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

      7. sub-negate1 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

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

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

      12. lower-fma.f64 N/A

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

      13. lower-neg.f64 47.4

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

   3. Applied rewrites 47.4%

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

   5. Applied rewrites 52.3%

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

      1. unpow1 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. sqr-abs-rev N/A

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

      7. mul-fabs N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. pow-prod-up N/A

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

      11. metadata-eval N/A

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

      12. unpow1 N/A

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

      13. lower-fabs.f64 90.5

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 90.5

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

   7. Applied rewrites 90.5%

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

   8. Taylor expanded in b_2 around -inf

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

      1. lower-*.f64 97.5

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

   10. Applied rewrites 97.5%

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

   if -2.0000000000000001e130 < b_2 < 1.8500000000000001e-121

   1. Initial program 84.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-negate1-reverse N/A

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

      5. lower--.f64 84.2

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

      6. lift--.f64 N/A

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

      7. sub-negate1 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

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

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

      12. lower-fma.f64 N/A

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

      13. lower-neg.f64 84.2

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

   3. Applied rewrites 84.2%

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

   if 1.8500000000000001e-121 < b_2

   1. Initial program 20.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-negate1-reverse N/A

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

      5. lower--.f64 20.8

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

      6. lift--.f64 N/A

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

      7. sub-negate1 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

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

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

      12. lower-fma.f64 N/A

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

      13. lower-neg.f64 20.8

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

   3. Applied rewrites 20.8%

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

   5. Applied rewrites 14.8%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\left(\mathsf{fma}\left(a, -c, b\_2 \cdot b\_2\right)\right)}^{0.375}, {\left(\mathsf{fma}\left(a, -c, b\_2 \cdot b\_2\right)\right)}^{0.125}, -b\_2\right)}}{a} \]

   7. Applied rewrites 14.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\mathsf{fma}\left(-c, a, b\_2 \cdot b\_2\right)\right)}^{0.125}, \frac{{\left(\mathsf{fma}\left(-c, a, b\_2 \cdot b\_2\right)\right)}^{0.375}}{a}, \frac{-b\_2}{a}\right)} \]

   8. Taylor expanded in b_2 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 83.3

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

   10. Applied rewrites 83.3%

       \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b\_2}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 81.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -4.2 \cdot 10^{-45}:\\ \;\;\;\;\frac{-2 \cdot b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 1.85 \cdot 10^{-121}:\\ \;\;\;\;\frac{\sqrt{\left|-1 \cdot \left(a \cdot c\right)\right|} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -4.2e-45)
   (/ (* -2.0 b_2) a)
   (if (<= b_2 1.85e-121)
     (/ (- (sqrt (fabs (* -1.0 (* a c)))) b_2) a)
     (* -0.5 (/ c b_2)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -4.2e-45) {
		tmp = (-2.0 * b_2) / a;
	} else if (b_2 <= 1.85e-121) {
		tmp = (sqrt(fabs((-1.0 * (a * c)))) - b_2) / a;
	} else {
		tmp = -0.5 * (c / b_2);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b_2, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-4.2d-45)) then
        tmp = ((-2.0d0) * b_2) / a
    else if (b_2 <= 1.85d-121) then
        tmp = (sqrt(abs(((-1.0d0) * (a * c)))) - b_2) / a
    else
        tmp = (-0.5d0) * (c / b_2)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -4.2e-45) {
		tmp = (-2.0 * b_2) / a;
	} else if (b_2 <= 1.85e-121) {
		tmp = (Math.sqrt(Math.abs((-1.0 * (a * c)))) - b_2) / a;
	} else {
		tmp = -0.5 * (c / b_2);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -4.2e-45:
		tmp = (-2.0 * b_2) / a
	elif b_2 <= 1.85e-121:
		tmp = (math.sqrt(math.fabs((-1.0 * (a * c)))) - b_2) / a
	else:
		tmp = -0.5 * (c / b_2)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -4.2e-45)
		tmp = Float64(Float64(-2.0 * b_2) / a);
	elseif (b_2 <= 1.85e-121)
		tmp = Float64(Float64(sqrt(abs(Float64(-1.0 * Float64(a * c)))) - b_2) / a);
	else
		tmp = Float64(-0.5 * Float64(c / b_2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -4.2e-45)
		tmp = (-2.0 * b_2) / a;
	elseif (b_2 <= 1.85e-121)
		tmp = (sqrt(abs((-1.0 * (a * c)))) - b_2) / a;
	else
		tmp = -0.5 * (c / b_2);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -4.2e-45], N[(N[(-2.0 * b$95$2), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 1.85e-121], N[(N[(N[Sqrt[N[Abs[N[(-1.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -4.1999999999999999e-45

   1. Initial program 68.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-negate1-reverse N/A

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

      5. lower--.f64 68.1

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

      6. lift--.f64 N/A

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

      7. sub-negate1 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

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

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

      12. lower-fma.f64 N/A

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

      13. lower-neg.f64 68.2

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

   3. Applied rewrites 68.2%

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

   5. Applied rewrites 56.9%

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

      1. unpow1 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. sqr-abs-rev N/A

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

      7. mul-fabs N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. pow-prod-up N/A

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

      11. metadata-eval N/A

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

      12. unpow1 N/A

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

      13. lower-fabs.f64 91.3

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 91.3

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

   7. Applied rewrites 91.3%

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

   8. Taylor expanded in b_2 around -inf

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

      1. lower-*.f64 88.6

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

   10. Applied rewrites 88.6%

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

   if -4.1999999999999999e-45 < b_2 < 1.8500000000000001e-121

   1. Initial program 79.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-negate1-reverse N/A

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

      5. lower--.f64 79.3

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

      6. lift--.f64 N/A

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

      7. sub-negate1 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

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

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

      12. lower-fma.f64 N/A

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

      13. lower-neg.f64 79.3

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

   3. Applied rewrites 79.3%

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

   5. Applied rewrites 78.0%

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

      1. unpow1 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. sqr-abs-rev N/A

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

      7. mul-fabs N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. pow-prod-up N/A

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

      11. metadata-eval N/A

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

      12. unpow1 N/A

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

      13. lower-fabs.f64 80.6

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 80.6

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

   7. Applied rewrites 80.6%

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

   8. Taylor expanded in b_2 around 0

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

   10. Applied rewrites 70.8%

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

   if 1.8500000000000001e-121 < b_2

   1. Initial program 20.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-negate1-reverse N/A

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

      5. lower--.f64 20.8

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

      6. lift--.f64 N/A

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

      7. sub-negate1 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

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

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

      12. lower-fma.f64 N/A

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

      13. lower-neg.f64 20.8

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

   3. Applied rewrites 20.8%

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

   5. Applied rewrites 14.8%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\left(\mathsf{fma}\left(a, -c, b\_2 \cdot b\_2\right)\right)}^{0.375}, {\left(\mathsf{fma}\left(a, -c, b\_2 \cdot b\_2\right)\right)}^{0.125}, -b\_2\right)}}{a} \]

   7. Applied rewrites 14.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\mathsf{fma}\left(-c, a, b\_2 \cdot b\_2\right)\right)}^{0.125}, \frac{{\left(\mathsf{fma}\left(-c, a, b\_2 \cdot b\_2\right)\right)}^{0.375}}{a}, \frac{-b\_2}{a}\right)} \]

   8. Taylor expanded in b_2 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 83.3

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

   10. Applied rewrites 83.3%

       \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b\_2}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 81.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.42 \cdot 10^{-46}:\\ \;\;\;\;\frac{-2 \cdot b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 1.85 \cdot 10^{-121}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -1}}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.42e-46)
   (/ (* -2.0 b_2) a)
   (if (<= b_2 1.85e-121) (/ (sqrt (* (* a c) -1.0)) a) (* -0.5 (/ c b_2)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.42e-46) {
		tmp = (-2.0 * b_2) / a;
	} else if (b_2 <= 1.85e-121) {
		tmp = sqrt(((a * c) * -1.0)) / a;
	} else {
		tmp = -0.5 * (c / b_2);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b_2, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-1.42d-46)) then
        tmp = ((-2.0d0) * b_2) / a
    else if (b_2 <= 1.85d-121) then
        tmp = sqrt(((a * c) * (-1.0d0))) / a
    else
        tmp = (-0.5d0) * (c / b_2)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.42e-46) {
		tmp = (-2.0 * b_2) / a;
	} else if (b_2 <= 1.85e-121) {
		tmp = Math.sqrt(((a * c) * -1.0)) / a;
	} else {
		tmp = -0.5 * (c / b_2);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -1.42e-46:
		tmp = (-2.0 * b_2) / a
	elif b_2 <= 1.85e-121:
		tmp = math.sqrt(((a * c) * -1.0)) / a
	else:
		tmp = -0.5 * (c / b_2)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1.42e-46)
		tmp = Float64(Float64(-2.0 * b_2) / a);
	elseif (b_2 <= 1.85e-121)
		tmp = Float64(sqrt(Float64(Float64(a * c) * -1.0)) / a);
	else
		tmp = Float64(-0.5 * Float64(c / b_2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -1.42e-46)
		tmp = (-2.0 * b_2) / a;
	elseif (b_2 <= 1.85e-121)
		tmp = sqrt(((a * c) * -1.0)) / a;
	else
		tmp = -0.5 * (c / b_2);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1.42e-46], N[(N[(-2.0 * b$95$2), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 1.85e-121], N[(N[Sqrt[N[(N[(a * c), $MachinePrecision] * -1.0), $MachinePrecision]], $MachinePrecision] / a), $MachinePrecision], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -1.42e-46

   1. Initial program 68.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-negate1-reverse N/A

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

      5. lower--.f64 68.3

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

      6. lift--.f64 N/A

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

      7. sub-negate1 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

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

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

      12. lower-fma.f64 N/A

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

      13. lower-neg.f64 68.3

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

   3. Applied rewrites 68.3%

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

   5. Applied rewrites 56.9%

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

      1. unpow1 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. sqr-abs-rev N/A

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

      7. mul-fabs N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. pow-prod-up N/A

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

      11. metadata-eval N/A

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

      12. unpow1 N/A

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

      13. lower-fabs.f64 91.3

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 91.3

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

   7. Applied rewrites 91.3%

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

   8. Taylor expanded in b_2 around -inf

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

      1. lower-*.f64 88.5

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

   10. Applied rewrites 88.5%

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

   if -1.42e-46 < b_2 < 1.8500000000000001e-121

   1. Initial program 79.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-negate1-reverse N/A

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

      5. lower--.f64 79.2

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

      6. lift--.f64 N/A

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

      7. sub-negate1 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

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

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

      12. lower-fma.f64 N/A

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

      13. lower-neg.f64 79.2

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

   3. Applied rewrites 79.2%

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

   5. Applied rewrites 2.0%

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

   6. Taylor expanded in a around -inf

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

      1. sqrt-unprod N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 68.8

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

   8. Applied rewrites 68.8%

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

   if 1.8500000000000001e-121 < b_2

   1. Initial program 20.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-negate1-reverse N/A

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

      5. lower--.f64 20.8

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

      6. lift--.f64 N/A

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

      7. sub-negate1 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

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

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

      12. lower-fma.f64 N/A

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

      13. lower-neg.f64 20.8

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

   3. Applied rewrites 20.8%

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

   5. Applied rewrites 14.8%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\left(\mathsf{fma}\left(a, -c, b\_2 \cdot b\_2\right)\right)}^{0.375}, {\left(\mathsf{fma}\left(a, -c, b\_2 \cdot b\_2\right)\right)}^{0.125}, -b\_2\right)}}{a} \]

   7. Applied rewrites 14.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\mathsf{fma}\left(-c, a, b\_2 \cdot b\_2\right)\right)}^{0.125}, \frac{{\left(\mathsf{fma}\left(-c, a, b\_2 \cdot b\_2\right)\right)}^{0.375}}{a}, \frac{-b\_2}{a}\right)} \]

   8. Taylor expanded in b_2 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 83.3

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

   10. Applied rewrites 83.3%

       \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b\_2}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 72.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.1 \cdot 10^{-181}:\\ \;\;\;\;\frac{-2 \cdot b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 3.4 \cdot 10^{-121}:\\ \;\;\;\;-1 \cdot \sqrt{\frac{c}{a} \cdot -1}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.1e-181)
   (/ (* -2.0 b_2) a)
   (if (<= b_2 3.4e-121) (* -1.0 (sqrt (* (/ c a) -1.0))) (* -0.5 (/ c b_2)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.1e-181) {
		tmp = (-2.0 * b_2) / a;
	} else if (b_2 <= 3.4e-121) {
		tmp = -1.0 * sqrt(((c / a) * -1.0));
	} else {
		tmp = -0.5 * (c / b_2);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b_2, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-2.1d-181)) then
        tmp = ((-2.0d0) * b_2) / a
    else if (b_2 <= 3.4d-121) then
        tmp = (-1.0d0) * sqrt(((c / a) * (-1.0d0)))
    else
        tmp = (-0.5d0) * (c / b_2)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.1e-181) {
		tmp = (-2.0 * b_2) / a;
	} else if (b_2 <= 3.4e-121) {
		tmp = -1.0 * Math.sqrt(((c / a) * -1.0));
	} else {
		tmp = -0.5 * (c / b_2);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2.1e-181:
		tmp = (-2.0 * b_2) / a
	elif b_2 <= 3.4e-121:
		tmp = -1.0 * math.sqrt(((c / a) * -1.0))
	else:
		tmp = -0.5 * (c / b_2)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2.1e-181)
		tmp = Float64(Float64(-2.0 * b_2) / a);
	elseif (b_2 <= 3.4e-121)
		tmp = Float64(-1.0 * sqrt(Float64(Float64(c / a) * -1.0)));
	else
		tmp = Float64(-0.5 * Float64(c / b_2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -2.1e-181)
		tmp = (-2.0 * b_2) / a;
	elseif (b_2 <= 3.4e-121)
		tmp = -1.0 * sqrt(((c / a) * -1.0));
	else
		tmp = -0.5 * (c / b_2);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2.1e-181], N[(N[(-2.0 * b$95$2), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 3.4e-121], N[(-1.0 * N[Sqrt[N[(N[(c / a), $MachinePrecision] * -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -2.10000000000000003e-181

   1. Initial program 72.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-negate1-reverse N/A

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

      5. lower--.f64 72.7

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

      6. lift--.f64 N/A

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

      7. sub-negate1 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

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

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

      12. lower-fma.f64 N/A

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

      13. lower-neg.f64 72.8

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

   3. Applied rewrites 72.8%

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

   5. Applied rewrites 62.4%

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

      1. unpow1 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. sqr-abs-rev N/A

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

      7. mul-fabs N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. pow-prod-up N/A

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

      11. metadata-eval N/A

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

      12. unpow1 N/A

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

      13. lower-fabs.f64 91.2

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 91.2

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

   7. Applied rewrites 91.2%

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

   8. Taylor expanded in b_2 around -inf

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

      1. lower-*.f64 77.1

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

   10. Applied rewrites 77.1%

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

   if -2.10000000000000003e-181 < b_2 < 3.40000000000000001e-121

   1. Initial program 74.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-negate1-reverse N/A

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

      5. lower--.f64 74.1

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

      6. lift--.f64 N/A

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

      7. sub-negate1 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

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

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

      12. lower-fma.f64 N/A

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

      13. lower-neg.f64 74.1

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

   3. Applied rewrites 74.1%

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

   5. Applied rewrites 2.1%

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

   6. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. sqrt-unprod N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 35.7

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

   8. Applied rewrites 35.7%

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

   if 3.40000000000000001e-121 < b_2

   1. Initial program 20.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-negate1-reverse N/A

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

      5. lower--.f64 20.8

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

      6. lift--.f64 N/A

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

      7. sub-negate1 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

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

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

      12. lower-fma.f64 N/A

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

      13. lower-neg.f64 20.8

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

   3. Applied rewrites 20.8%

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

   5. Applied rewrites 14.8%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\left(\mathsf{fma}\left(a, -c, b\_2 \cdot b\_2\right)\right)}^{0.375}, {\left(\mathsf{fma}\left(a, -c, b\_2 \cdot b\_2\right)\right)}^{0.125}, -b\_2\right)}}{a} \]

   7. Applied rewrites 14.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\mathsf{fma}\left(-c, a, b\_2 \cdot b\_2\right)\right)}^{0.125}, \frac{{\left(\mathsf{fma}\left(-c, a, b\_2 \cdot b\_2\right)\right)}^{0.375}}{a}, \frac{-b\_2}{a}\right)} \]

   8. Taylor expanded in b_2 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 83.3

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

   10. Applied rewrites 83.3%

       \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b\_2}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 71.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -9 \cdot 10^{-212}:\\ \;\;\;\;\frac{-2 \cdot b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 2.9 \cdot 10^{-123}:\\ \;\;\;\;\sqrt{\frac{c}{a} \cdot -1}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -9e-212)
   (/ (* -2.0 b_2) a)
   (if (<= b_2 2.9e-123) (sqrt (* (/ c a) -1.0)) (* -0.5 (/ c b_2)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -9e-212) {
		tmp = (-2.0 * b_2) / a;
	} else if (b_2 <= 2.9e-123) {
		tmp = sqrt(((c / a) * -1.0));
	} else {
		tmp = -0.5 * (c / b_2);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b_2, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-9d-212)) then
        tmp = ((-2.0d0) * b_2) / a
    else if (b_2 <= 2.9d-123) then
        tmp = sqrt(((c / a) * (-1.0d0)))
    else
        tmp = (-0.5d0) * (c / b_2)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -9e-212) {
		tmp = (-2.0 * b_2) / a;
	} else if (b_2 <= 2.9e-123) {
		tmp = Math.sqrt(((c / a) * -1.0));
	} else {
		tmp = -0.5 * (c / b_2);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -9e-212:
		tmp = (-2.0 * b_2) / a
	elif b_2 <= 2.9e-123:
		tmp = math.sqrt(((c / a) * -1.0))
	else:
		tmp = -0.5 * (c / b_2)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -9e-212)
		tmp = Float64(Float64(-2.0 * b_2) / a);
	elseif (b_2 <= 2.9e-123)
		tmp = sqrt(Float64(Float64(c / a) * -1.0));
	else
		tmp = Float64(-0.5 * Float64(c / b_2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -9e-212)
		tmp = (-2.0 * b_2) / a;
	elseif (b_2 <= 2.9e-123)
		tmp = sqrt(((c / a) * -1.0));
	else
		tmp = -0.5 * (c / b_2);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -9e-212], N[(N[(-2.0 * b$95$2), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 2.9e-123], N[Sqrt[N[(N[(c / a), $MachinePrecision] * -1.0), $MachinePrecision]], $MachinePrecision], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -8.9999999999999997e-212

   1. Initial program 72.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-negate1-reverse N/A

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

      5. lower--.f64 72.8

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

      6. lift--.f64 N/A

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

      7. sub-negate1 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

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

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

      12. lower-fma.f64 N/A

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

      13. lower-neg.f64 72.9

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

   3. Applied rewrites 72.9%

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

   5. Applied rewrites 63.1%

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

      1. unpow1 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. sqr-abs-rev N/A

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

      7. mul-fabs N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. pow-prod-up N/A

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

      11. metadata-eval N/A

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

      12. unpow1 N/A

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

      13. lower-fabs.f64 90.9

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 90.9

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

   7. Applied rewrites 90.9%

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

   8. Taylor expanded in b_2 around -inf

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

      1. lower-*.f64 75.1

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

   10. Applied rewrites 75.1%

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

   if -8.9999999999999997e-212 < b_2 < 2.90000000000000004e-123

   1. Initial program 74.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-negate1-reverse N/A

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

      5. lower--.f64 74.1

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

      6. lift--.f64 N/A

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

      7. sub-negate1 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

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

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

      12. lower-fma.f64 N/A

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

      13. lower-neg.f64 74.1

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

   3. Applied rewrites 74.1%

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

   5. Applied rewrites 2.2%

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

   6. Taylor expanded in a around -inf

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

      1. sqrt-unprod N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 35.2

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

   8. Applied rewrites 35.2%

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

   if 2.90000000000000004e-123 < b_2

   1. Initial program 21.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-negate1-reverse N/A

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

      5. lower--.f64 21.0

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

      6. lift--.f64 N/A

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

      7. sub-negate1 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

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

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

      12. lower-fma.f64 N/A

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

      13. lower-neg.f64 21.0

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

   3. Applied rewrites 21.0%

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

   5. Applied rewrites 15.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\left(\mathsf{fma}\left(a, -c, b\_2 \cdot b\_2\right)\right)}^{0.375}, {\left(\mathsf{fma}\left(a, -c, b\_2 \cdot b\_2\right)\right)}^{0.125}, -b\_2\right)}}{a} \]

   7. Applied rewrites 14.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\mathsf{fma}\left(-c, a, b\_2 \cdot b\_2\right)\right)}^{0.125}, \frac{{\left(\mathsf{fma}\left(-c, a, b\_2 \cdot b\_2\right)\right)}^{0.375}}{a}, \frac{-b\_2}{a}\right)} \]

   8. Taylor expanded in b_2 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 83.1

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

   10. Applied rewrites 83.1%

       \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b\_2}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 68.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq 8 \cdot 10^{-297}:\\ \;\;\;\;\frac{-2 \cdot b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 8e-297) (/ (* -2.0 b_2) a) (* -0.5 (/ c b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 8e-297) {
		tmp = (-2.0 * b_2) / a;
	} else {
		tmp = -0.5 * (c / b_2);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b_2, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= 8d-297) then
        tmp = ((-2.0d0) * b_2) / a
    else
        tmp = (-0.5d0) * (c / b_2)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 8e-297) {
		tmp = (-2.0 * b_2) / a;
	} else {
		tmp = -0.5 * (c / b_2);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= 8e-297:
		tmp = (-2.0 * b_2) / a
	else:
		tmp = -0.5 * (c / b_2)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= 8e-297)
		tmp = Float64(Float64(-2.0 * b_2) / a);
	else
		tmp = Float64(-0.5 * Float64(c / b_2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= 8e-297)
		tmp = (-2.0 * b_2) / a;
	else
		tmp = -0.5 * (c / b_2);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, 8e-297], N[(N[(-2.0 * b$95$2), $MachinePrecision] / a), $MachinePrecision], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < 8.00000000000000032e-297

   1. Initial program 73.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-negate1-reverse N/A

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

      5. lower--.f64 73.1

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

      6. lift--.f64 N/A

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

      7. sub-negate1 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

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

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

      12. lower-fma.f64 N/A

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

      13. lower-neg.f64 73.1

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

   3. Applied rewrites 73.1%

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

   5. Applied rewrites 64.8%

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

      1. unpow1 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. sqr-abs-rev N/A

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

      7. mul-fabs N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. pow-prod-up N/A

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

      11. metadata-eval N/A

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

      12. unpow1 N/A

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

      13. lower-fabs.f64 89.4

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 89.4

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

   7. Applied rewrites 89.4%

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

   8. Taylor expanded in b_2 around -inf

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

      1. lower-*.f64 67.2

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

   10. Applied rewrites 67.2%

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

   if 8.00000000000000032e-297 < b_2

   1. Initial program 32.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-negate1-reverse N/A

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

      5. lower--.f64 32.1

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

      6. lift--.f64 N/A

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

      7. sub-negate1 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

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

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

      12. lower-fma.f64 N/A

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

      13. lower-neg.f64 32.1

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

   3. Applied rewrites 32.1%

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

   5. Applied rewrites 27.4%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\left(\mathsf{fma}\left(a, -c, b\_2 \cdot b\_2\right)\right)}^{0.375}, {\left(\mathsf{fma}\left(a, -c, b\_2 \cdot b\_2\right)\right)}^{0.125}, -b\_2\right)}}{a} \]

   7. Applied rewrites 27.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\mathsf{fma}\left(-c, a, b\_2 \cdot b\_2\right)\right)}^{0.125}, \frac{{\left(\mathsf{fma}\left(-c, a, b\_2 \cdot b\_2\right)\right)}^{0.375}}{a}, \frac{-b\_2}{a}\right)} \]

   8. Taylor expanded in b_2 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 68.7

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

   10. Applied rewrites 68.7%

       \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b\_2}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 35.5% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ -0.5 \cdot \frac{c}{b\_2} \end{array} \]

FPCore

(FPCore (a b_2 c) :precision binary64 (* -0.5 (/ c b_2)))

C

double code(double a, double b_2, double c) {
	return -0.5 * (c / b_2);
}

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 = (-0.5d0) * (c / b_2)
end function

Java

public static double code(double a, double b_2, double c) {
	return -0.5 * (c / b_2);
}

Python

def code(a, b_2, c):
	return -0.5 * (c / b_2)

Julia

function code(a, b_2, c)
	return Float64(-0.5 * Float64(c / b_2))
end

MATLAB

function tmp = code(a, b_2, c)
	tmp = -0.5 * (c / b_2);
end

Wolfram

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

Derivation

1. Initial program 52.6%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-neg.f64 N/A

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

   4. sub-negate1-reverse N/A

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

   5. lower--.f64 52.6

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

   6. lift--.f64 N/A

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

   7. sub-negate1 N/A

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

   8. +-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

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

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

   12. lower-fma.f64 N/A

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

   13. lower-neg.f64 52.7

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

3. Applied rewrites 52.7%

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

5. Applied rewrites 50.2%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\left(\mathsf{fma}\left(a, -c, b\_2 \cdot b\_2\right)\right)}^{0.375}, {\left(\mathsf{fma}\left(a, -c, b\_2 \cdot b\_2\right)\right)}^{0.125}, -b\_2\right)}}{a} \]

7. Applied rewrites 50.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\mathsf{fma}\left(-c, a, b\_2 \cdot b\_2\right)\right)}^{0.125}, \frac{{\left(\mathsf{fma}\left(-c, a, b\_2 \cdot b\_2\right)\right)}^{0.375}}{a}, \frac{-b\_2}{a}\right)} \]

8. Taylor expanded in b_2 around inf

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 35.5

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

10. Applied rewrites 35.5%

    \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b\_2}} \]
11. Add Preprocessing

Alternative 9: 11.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \frac{b\_2 - b\_2}{a} \end{array} \]

FPCore

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

C

double code(double a, double b_2, double c) {
	return (b_2 - b_2) / 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 - b_2) / a
end function

Java

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

Python

def code(a, b_2, c):
	return (b_2 - b_2) / a

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.6%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-neg.f64 N/A

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

   4. sub-negate1-reverse N/A

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

   5. lower--.f64 52.6

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

   6. lift--.f64 N/A

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

   7. sub-negate1 N/A

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

   8. +-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

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

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

   12. lower-fma.f64 N/A

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

   13. lower-neg.f64 52.7

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

3. Applied rewrites 52.7%

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

5. Applied rewrites 4.0%

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

6. Taylor expanded in a around 0

   \[\leadsto \frac{b\_2 - \color{blue}{b\_2}}{a} \]
7. Step-by-step derivation

8. Applied rewrites 11.4%

   \[\leadsto \frac{b\_2 - \color{blue}{b\_2}}{a} \]
9. Add Preprocessing

Alternative 10: 2.6% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \frac{b\_2 + b\_2}{a} \end{array} \]

FPCore

(FPCore (a b_2 c) :precision binary64 (/ (+ b_2 b_2) a))

C

double code(double a, double b_2, double c) {
	return (b_2 + b_2) / 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 + b_2) / a
end function

Java

public static double code(double a, double b_2, double c) {
	return (b_2 + b_2) / a;
}

Python

def code(a, b_2, c):
	return (b_2 + b_2) / a

Julia

function code(a, b_2, c)
	return Float64(Float64(b_2 + b_2) / a)
end

MATLAB

function tmp = code(a, b_2, c)
	tmp = (b_2 + b_2) / a;
end

Wolfram

code[a_, b$95$2_, c_] := N[(N[(b$95$2 + b$95$2), $MachinePrecision] / a), $MachinePrecision]

Derivation

1. Initial program 52.6%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-neg.f64 N/A

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

   4. sub-negate1-reverse N/A

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

   5. lower--.f64 52.6

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

   6. lift--.f64 N/A

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

   7. sub-negate1 N/A

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

   8. +-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

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

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

   12. lower-fma.f64 N/A

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

   13. lower-neg.f64 52.7

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

3. Applied rewrites 52.7%

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

5. Applied rewrites 32.9%

   \[\leadsto \frac{\color{blue}{\sqrt{\mathsf{fma}\left(a, -c, b\_2 \cdot b\_2\right)} + b\_2}}{a} \]
6. Step-by-step derivation

   1. lift-sqrt.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. unpow1 N/A

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

   5. metadata-eval N/A

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

   6. pow-prod-up N/A

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

   7. lift-pow.f64 N/A

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

   8. lift-pow.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-hypot.f64 26.4

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

   11. lift-pow.f64 N/A

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

   12. unpow1/2 N/A

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

   13. lower-sqrt.f64 26.4

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

   14. lift-*.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 26.4

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

7. Applied rewrites 26.4%

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

8. Taylor expanded in a around 0

   \[\leadsto \frac{\color{blue}{b\_2} + b\_2}{a} \]
9. Step-by-step derivation

10. Applied rewrites 2.6%

    \[\leadsto \frac{\color{blue}{b\_2} + b\_2}{a} \]
11. Add Preprocessing

Developer Target 1: 99.7% accurate, 0.2× speedup

Math

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

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (let* ((t_0 (* (sqrt (fabs a)) (sqrt (fabs c))))
        (t_1
         (if (== (copysign a c) a)
           (* (sqrt (- (fabs b_2) t_0)) (sqrt (+ (fabs b_2) t_0)))
           (hypot b_2 t_0))))
   (if (< b_2 0.0) (/ (- t_1 b_2) a) (/ (- c) (+ b_2 t_1)))))

C

double code(double a, double b_2, double c) {
	double t_0 = sqrt(fabs(a)) * sqrt(fabs(c));
	double tmp;
	if (copysign(a, c) == a) {
		tmp = sqrt((fabs(b_2) - t_0)) * sqrt((fabs(b_2) + t_0));
	} else {
		tmp = hypot(b_2, t_0);
	}
	double t_1 = tmp;
	double tmp_1;
	if (b_2 < 0.0) {
		tmp_1 = (t_1 - b_2) / a;
	} else {
		tmp_1 = -c / (b_2 + t_1);
	}
	return tmp_1;
}

Java

public static double code(double a, double b_2, double c) {
	double t_0 = Math.sqrt(Math.abs(a)) * Math.sqrt(Math.abs(c));
	double tmp;
	if (Math.copySign(a, c) == a) {
		tmp = Math.sqrt((Math.abs(b_2) - t_0)) * Math.sqrt((Math.abs(b_2) + t_0));
	} else {
		tmp = Math.hypot(b_2, t_0);
	}
	double t_1 = tmp;
	double tmp_1;
	if (b_2 < 0.0) {
		tmp_1 = (t_1 - b_2) / a;
	} else {
		tmp_1 = -c / (b_2 + t_1);
	}
	return tmp_1;
}

Python

def code(a, b_2, c):
	t_0 = math.sqrt(math.fabs(a)) * math.sqrt(math.fabs(c))
	tmp = 0
	if math.copysign(a, c) == a:
		tmp = math.sqrt((math.fabs(b_2) - t_0)) * math.sqrt((math.fabs(b_2) + t_0))
	else:
		tmp = math.hypot(b_2, t_0)
	t_1 = tmp
	tmp_1 = 0
	if b_2 < 0.0:
		tmp_1 = (t_1 - b_2) / a
	else:
		tmp_1 = -c / (b_2 + t_1)
	return tmp_1

Julia

function code(a, b_2, c)
	t_0 = Float64(sqrt(abs(a)) * sqrt(abs(c)))
	tmp = 0.0
	if (copysign(a, c) == a)
		tmp = Float64(sqrt(Float64(abs(b_2) - t_0)) * sqrt(Float64(abs(b_2) + t_0)));
	else
		tmp = hypot(b_2, t_0);
	end
	t_1 = tmp
	tmp_1 = 0.0
	if (b_2 < 0.0)
		tmp_1 = Float64(Float64(t_1 - b_2) / a);
	else
		tmp_1 = Float64(Float64(-c) / Float64(b_2 + t_1));
	end
	return tmp_1
end

MATLAB

function tmp_3 = code(a, b_2, c)
	t_0 = sqrt(abs(a)) * sqrt(abs(c));
	tmp = 0.0;
	if ((sign(c) * abs(a)) == a)
		tmp = sqrt((abs(b_2) - t_0)) * sqrt((abs(b_2) + t_0));
	else
		tmp = hypot(b_2, t_0);
	end
	t_1 = tmp;
	tmp_2 = 0.0;
	if (b_2 < 0.0)
		tmp_2 = (t_1 - b_2) / a;
	else
		tmp_2 = -c / (b_2 + t_1);
	end
	tmp_3 = tmp_2;
end

Wolfram

code[a_, b$95$2_, c_] := Block[{t$95$0 = N[(N[Sqrt[N[Abs[a], $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[Abs[c], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = If[Equal[N[With[{TMP1 = Abs[a], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], a], N[(N[Sqrt[N[(N[Abs[b$95$2], $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(N[Abs[b$95$2], $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[b$95$2 ^ 2 + t$95$0 ^ 2], $MachinePrecision]]}, If[Less[b$95$2, 0.0], N[(N[(t$95$1 - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[((-c) / N[(b$95$2 + t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Reproduce

herbie shell --seed 2025107 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  :herbie-expected 10

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

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