quad2m (problem 3.2.1, negative)

Percentage Accurate: 53.3% → 85.9%

Time: 4.8s

Alternatives: 7

Speedup: N/A×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.9% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -80000000000:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq -4.3 \cdot 10^{-74}:\\ \;\;\;\;\frac{\frac{a \cdot c}{\mathsf{fma}\left(-1, b\_2, {\left(\mathsf{fma}\left({b\_2}^{1}, {b\_2}^{1}, -1 \cdot \left(c \cdot a\right)\right)\right)}^{0.5}\right)}}{a}\\ \mathbf{elif}\;b\_2 \leq 1.36 \cdot 10^{+78}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\_2\right) - b\_2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -80000000000.0)
   (* -0.5 (/ c b_2))
   (if (<= b_2 -4.3e-74)
     (/
      (/
       (* a c)
       (fma
        -1.0
        b_2
        (pow (fma (pow b_2 1.0) (pow b_2 1.0) (* -1.0 (* c a))) 0.5)))
      a)
     (if (<= b_2 1.36e+78)
       (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a)
       (/ (- (- b_2) b_2) a)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -80000000000.0) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= -4.3e-74) {
		tmp = ((a * c) / fma(-1.0, b_2, pow(fma(pow(b_2, 1.0), pow(b_2, 1.0), (-1.0 * (c * a))), 0.5))) / a;
	} else if (b_2 <= 1.36e+78) {
		tmp = (-b_2 - sqrt(((b_2 * b_2) - (a * c)))) / a;
	} else {
		tmp = (-b_2 - b_2) / a;
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -80000000000.0)
		tmp = Float64(-0.5 * Float64(c / b_2));
	elseif (b_2 <= -4.3e-74)
		tmp = Float64(Float64(Float64(a * c) / fma(-1.0, b_2, (fma((b_2 ^ 1.0), (b_2 ^ 1.0), Float64(-1.0 * Float64(c * a))) ^ 0.5))) / a);
	elseif (b_2 <= 1.36e+78)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c)))) / a);
	else
		tmp = Float64(Float64(Float64(-b_2) - b_2) / a);
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -80000000000.0], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, -4.3e-74], N[(N[(N[(a * c), $MachinePrecision] / N[(-1.0 * b$95$2 + N[Power[N[(N[Power[b$95$2, 1.0], $MachinePrecision] * N[Power[b$95$2, 1.0], $MachinePrecision] + N[(-1.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 1.36e+78], 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], N[(N[((-b$95$2) - b$95$2), $MachinePrecision] / a), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b_2 < -8e10

   1. Initial program 14.5%

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

   2. Taylor expanded in b_2 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 90.7

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

   4. Applied rewrites 90.7%

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

   if -8e10 < b_2 < -4.29999999999999972e-74

   1. Initial program 40.3%

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

      1. lift-neg.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. flip-- N/A

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

      8. lower-/.f64 N/A

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

   3. Applied rewrites 39.9%

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

   4. Taylor expanded in a around 0

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

      1. lower-*.f64 75.4

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

   6. Applied rewrites 75.4%

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

   if -4.29999999999999972e-74 < b_2 < 1.35999999999999999e78

   1. Initial program 79.3%

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

   if 1.35999999999999999e78 < b_2

   1. Initial program 57.4%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 94.5%

      \[\leadsto \frac{\left(-b\_2\right) - \color{blue}{b\_2}}{a} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 2: 84.7% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -9 \cdot 10^{-73}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 1.36 \cdot 10^{+78}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\_2\right) - b\_2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -9e-73)
   (* -0.5 (/ c b_2))
   (if (<= b_2 1.36e+78)
     (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a)
     (/ (- (- b_2) b_2) a))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -9e-73) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= 1.36e+78) {
		tmp = (-b_2 - sqrt(((b_2 * b_2) - (a * c)))) / a;
	} else {
		tmp = (-b_2 - b_2) / a;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b_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-73)) then
        tmp = (-0.5d0) * (c / b_2)
    else if (b_2 <= 1.36d+78) then
        tmp = (-b_2 - sqrt(((b_2 * b_2) - (a * c)))) / a
    else
        tmp = (-b_2 - b_2) / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -9e-73) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= 1.36e+78) {
		tmp = (-b_2 - Math.sqrt(((b_2 * b_2) - (a * c)))) / a;
	} else {
		tmp = (-b_2 - b_2) / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -9e-73:
		tmp = -0.5 * (c / b_2)
	elif b_2 <= 1.36e+78:
		tmp = (-b_2 - math.sqrt(((b_2 * b_2) - (a * c)))) / a
	else:
		tmp = (-b_2 - b_2) / a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -9e-73)
		tmp = Float64(-0.5 * Float64(c / b_2));
	elseif (b_2 <= 1.36e+78)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c)))) / a);
	else
		tmp = Float64(Float64(Float64(-b_2) - b_2) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -9e-73)
		tmp = -0.5 * (c / b_2);
	elseif (b_2 <= 1.36e+78)
		tmp = (-b_2 - sqrt(((b_2 * b_2) - (a * c)))) / a;
	else
		tmp = (-b_2 - b_2) / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -9e-73], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 1.36e+78], 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], N[(N[((-b$95$2) - b$95$2), $MachinePrecision] / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -9e-73

   1. Initial program 19.1%

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

   2. Taylor expanded in b_2 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 84.5

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

   4. Applied rewrites 84.5%

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

   if -9e-73 < b_2 < 1.35999999999999999e78

   1. Initial program 79.2%

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

   if 1.35999999999999999e78 < b_2

   1. Initial program 57.4%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 94.5%

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

Alternative 3: 79.5% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -8.5 \cdot 10^{-73}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 2.35 \cdot 10^{-101}:\\ \;\;\;\;-1 \cdot \left({a}^{-1} \cdot {\left(\mathsf{fma}\left(-1, a \cdot c, b\_2 \cdot b\_2\right)\right)}^{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{b\_2}, 0.5, \frac{b\_2}{a} \cdot -2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -8.5e-73)
   (* -0.5 (/ c b_2))
   (if (<= b_2 2.35e-101)
     (* -1.0 (* (pow a -1.0) (pow (fma -1.0 (* a c) (* b_2 b_2)) 0.5)))
     (fma (/ c b_2) 0.5 (* (/ b_2 a) -2.0)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -8.5e-73) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= 2.35e-101) {
		tmp = -1.0 * (pow(a, -1.0) * pow(fma(-1.0, (a * c), (b_2 * b_2)), 0.5));
	} else {
		tmp = fma((c / b_2), 0.5, ((b_2 / a) * -2.0));
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -8.5e-73)
		tmp = Float64(-0.5 * Float64(c / b_2));
	elseif (b_2 <= 2.35e-101)
		tmp = Float64(-1.0 * Float64((a ^ -1.0) * (fma(-1.0, Float64(a * c), Float64(b_2 * b_2)) ^ 0.5)));
	else
		tmp = fma(Float64(c / b_2), 0.5, Float64(Float64(b_2 / a) * -2.0));
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -8.5e-73], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 2.35e-101], N[(-1.0 * N[(N[Power[a, -1.0], $MachinePrecision] * N[Power[N[(-1.0 * N[(a * c), $MachinePrecision] + N[(b$95$2 * b$95$2), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c / b$95$2), $MachinePrecision] * 0.5 + N[(N[(b$95$2 / a), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -8.4999999999999996e-73

   1. Initial program 19.1%

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

   2. Taylor expanded in b_2 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 84.5

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

   4. Applied rewrites 84.5%

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

   if -8.4999999999999996e-73 < b_2 < 2.35e-101

   1. Initial program 72.5%

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

      1. lift-neg.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. flip-- N/A

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

      8. lower-/.f64 N/A

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

   3. Applied rewrites 67.2%

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

      1. lift-pow.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. sqr-pow N/A

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

      8. lower-*.f64 N/A

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

   5. Applied rewrites 67.0%

      \[\leadsto \frac{\frac{b\_2 \cdot b\_2 - {\left(\mathsf{fma}\left({b\_2}^{1}, {b\_2}^{1}, -1 \cdot \left(c \cdot a\right)\right)\right)}^{0.5} \cdot \color{blue}{\left({\left(\mathsf{fma}\left(\left|b\_2\right|, \left|b\_2\right|, -1 \cdot \left(c \cdot a\right)\right)\right)}^{0.25} \cdot {\left(\mathsf{fma}\left(\left|b\_2\right|, \left|b\_2\right|, -1 \cdot \left(c \cdot a\right)\right)\right)}^{0.25}\right)}}{\mathsf{fma}\left(-1, b\_2, {\left(\mathsf{fma}\left({b\_2}^{1}, {b\_2}^{1}, -1 \cdot \left(c \cdot a\right)\right)\right)}^{0.5}\right)}}{a} \]

   6. Taylor expanded in b_2 around 0

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

   8. Applied rewrites 67.5%

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

   if 2.35e-101 < b_2

   1. Initial program 70.4%

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

   2. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 83.6

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

   4. Applied rewrites 83.6%

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

Alternative 4: 66.7% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{b\_2}, 0.5, \frac{b\_2}{a} \cdot -2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -5e-310)
   (* -0.5 (/ c b_2))
   (fma (/ c b_2) 0.5 (* (/ b_2 a) -2.0))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5e-310) {
		tmp = -0.5 * (c / b_2);
	} else {
		tmp = fma((c / b_2), 0.5, ((b_2 / a) * -2.0));
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -5e-310)
		tmp = Float64(-0.5 * Float64(c / b_2));
	else
		tmp = fma(Float64(c / b_2), 0.5, Float64(Float64(b_2 / a) * -2.0));
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -5e-310], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision], N[(N[(c / b$95$2), $MachinePrecision] * 0.5 + N[(N[(b$95$2 / a), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -4.999999999999985e-310

   1. Initial program 33.0%

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

   2. Taylor expanded in b_2 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 66.0

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

   4. Applied rewrites 66.0%

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

   if -4.999999999999985e-310 < b_2

   1. Initial program 72.8%

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

   2. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 67.4

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

   4. Applied rewrites 67.4%

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

Alternative 5: 65.8% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{e^{\log b\_2 \cdot 2}}, 0.5, -2 \cdot {a}^{-1}\right) \cdot b\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -5e-310)
   (* -0.5 (/ c b_2))
   (* (fma (/ c (exp (* (log b_2) 2.0))) 0.5 (* -2.0 (pow a -1.0))) b_2)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5e-310) {
		tmp = -0.5 * (c / b_2);
	} else {
		tmp = fma((c / exp((log(b_2) * 2.0))), 0.5, (-2.0 * pow(a, -1.0))) * b_2;
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -5e-310)
		tmp = Float64(-0.5 * Float64(c / b_2));
	else
		tmp = Float64(fma(Float64(c / exp(Float64(log(b_2) * 2.0))), 0.5, Float64(-2.0 * (a ^ -1.0))) * b_2);
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -5e-310], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision], N[(N[(N[(c / N[Exp[N[(N[Log[b$95$2], $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5 + N[(-2.0 * N[Power[a, -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * b$95$2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -4.999999999999985e-310

   1. Initial program 33.0%

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

   2. Taylor expanded in b_2 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 66.0

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

   4. Applied rewrites 66.0%

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

   if -4.999999999999985e-310 < b_2

   1. Initial program 72.8%

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

   2. Taylor expanded in b_2 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. metadata-eval N/A

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

      10. lower-*.f64 N/A

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

      11. inv-pow N/A

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

      12. lower-pow.f64 65.6

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

   4. Applied rewrites 65.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{c}{b\_2 \cdot b\_2}, 0.5, -2 \cdot {a}^{-1}\right) \cdot b\_2} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. pow2 N/A

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

      3. pow-to-exp N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-log.f64 65.6

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

   6. Applied rewrites 65.6%

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

Alternative 6: 44.9% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {a}^{-1} \cdot 2\\ t_1 := \frac{\frac{c}{b\_2}}{b\_2} \cdot 0.5\\ \mathbf{if}\;b\_2 \leq 7.2 \cdot 10^{-125}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{{t\_1}^{3} - {t\_0}^{3}}{\mathsf{fma}\left(t\_1, t\_1, \mathsf{fma}\left(t\_0, t\_0, t\_1 \cdot t\_0\right)\right)} \cdot b\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (let* ((t_0 (* (pow a -1.0) 2.0)) (t_1 (* (/ (/ c b_2) b_2) 0.5)))
   (if (<= b_2 7.2e-125)
     (* -0.5 (/ c b_2))
     (*
      (/
       (- (pow t_1 3.0) (pow t_0 3.0))
       (fma t_1 t_1 (fma t_0 t_0 (* t_1 t_0))))
      b_2))))

C

double code(double a, double b_2, double c) {
	double t_0 = pow(a, -1.0) * 2.0;
	double t_1 = ((c / b_2) / b_2) * 0.5;
	double tmp;
	if (b_2 <= 7.2e-125) {
		tmp = -0.5 * (c / b_2);
	} else {
		tmp = ((pow(t_1, 3.0) - pow(t_0, 3.0)) / fma(t_1, t_1, fma(t_0, t_0, (t_1 * t_0)))) * b_2;
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	t_0 = Float64((a ^ -1.0) * 2.0)
	t_1 = Float64(Float64(Float64(c / b_2) / b_2) * 0.5)
	tmp = 0.0
	if (b_2 <= 7.2e-125)
		tmp = Float64(-0.5 * Float64(c / b_2));
	else
		tmp = Float64(Float64(Float64((t_1 ^ 3.0) - (t_0 ^ 3.0)) / fma(t_1, t_1, fma(t_0, t_0, Float64(t_1 * t_0)))) * b_2);
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := Block[{t$95$0 = N[(N[Power[a, -1.0], $MachinePrecision] * 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(c / b$95$2), $MachinePrecision] / b$95$2), $MachinePrecision] * 0.5), $MachinePrecision]}, If[LessEqual[b$95$2, 7.2e-125], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Power[t$95$1, 3.0], $MachinePrecision] - N[Power[t$95$0, 3.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$1 * t$95$1 + N[(t$95$0 * t$95$0 + N[(t$95$1 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * b$95$2), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if b_2 < 7.2000000000000004e-125

   1. Initial program 41.6%

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

   2. Taylor expanded in b_2 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 53.9

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

   4. Applied rewrites 53.9%

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

   if 7.2000000000000004e-125 < b_2

   1. Initial program 71.0%

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

   2. Taylor expanded in b_2 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. metadata-eval N/A

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

      10. lower-*.f64 N/A

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

      11. inv-pow N/A

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

      12. lower-pow.f64 81.5

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

   4. Applied rewrites 81.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{c}{b\_2 \cdot b\_2}, 0.5, -2 \cdot {a}^{-1}\right) \cdot b\_2} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. *-commutative N/A

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

      7. pow2 N/A

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

      8. inv-pow N/A

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

      9. metadata-eval N/A

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

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

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

      11. flip3-- N/A

          \[\leadsto \frac{{\left(\frac{1}{2} \cdot \frac{c}{{b\_2}^{2}}\right)}^{3} - {\left(2 \cdot \frac{1}{a}\right)}^{3}}{\left(\frac{1}{2} \cdot \frac{c}{{b\_2}^{2}}\right) \cdot \left(\frac{1}{2} \cdot \frac{c}{{b\_2}^{2}}\right) + \left(\left(2 \cdot \frac{1}{a}\right) \cdot \left(2 \cdot \frac{1}{a}\right) + \left(\frac{1}{2} \cdot \frac{c}{{b\_2}^{2}}\right) \cdot \left(2 \cdot \frac{1}{a}\right)\right)} \cdot b\_2 \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{{\left(\frac{1}{2} \cdot \frac{c}{{b\_2}^{2}}\right)}^{3} - {\left(2 \cdot \frac{1}{a}\right)}^{3}}{\left(\frac{1}{2} \cdot \frac{c}{{b\_2}^{2}}\right) \cdot \left(\frac{1}{2} \cdot \frac{c}{{b\_2}^{2}}\right) + \left(\left(2 \cdot \frac{1}{a}\right) \cdot \left(2 \cdot \frac{1}{a}\right) + \left(\frac{1}{2} \cdot \frac{c}{{b\_2}^{2}}\right) \cdot \left(2 \cdot \frac{1}{a}\right)\right)} \cdot b\_2 \]

   6. Applied rewrites 31.3%

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

Alternative 7: 33.4% accurate, N/A× 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 53.3%

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

2. Taylor expanded in b_2 around -inf

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 33.4

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

4. Applied rewrites 33.4%

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

Reproduce

herbie shell --seed 2025093 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  :herbie-expected 10
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))