Quadratic roots, narrow range

Percentage Accurate: 55.6% → 91.6%

Time: 4.6s

Alternatives: 9

Speedup: 4.3×

Specification

\[\left(\left(1.0536712127723509 \cdot 10^{-8} < a \land a < 94906265.62425156\right) \land \left(1.0536712127723509 \cdot 10^{-8} < b \land b < 94906265.62425156\right)\right) \land \left(1.0536712127723509 \cdot 10^{-8} < c \land c < 94906265.62425156\right)\]

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 55.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 91.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.26:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{a}{{b}^{7}}, -5, -2 \cdot \frac{{b}^{-5}}{c}\right) \cdot {c}^{4}, a, -\frac{c \cdot c}{{b}^{3}}\right), a, \frac{-c}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.26)
   (/ (+ (- b) (sqrt (fma b b (* -4.0 (* c a))))) (* 2.0 a))
   (fma
    (fma
     (* (fma (/ a (pow b 7.0)) -5.0 (* -2.0 (/ (pow b -5.0) c))) (pow c 4.0))
     a
     (- (/ (* c c) (pow b 3.0))))
    a
    (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.26) {
		tmp = (-b + sqrt(fma(b, b, (-4.0 * (c * a))))) / (2.0 * a);
	} else {
		tmp = fma(fma((fma((a / pow(b, 7.0)), -5.0, (-2.0 * (pow(b, -5.0) / c))) * pow(c, 4.0)), a, -((c * c) / pow(b, 3.0))), a, (-c / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.26)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(-4.0 * Float64(c * a))))) / Float64(2.0 * a));
	else
		tmp = fma(fma(Float64(fma(Float64(a / (b ^ 7.0)), -5.0, Float64(-2.0 * Float64((b ^ -5.0) / c))) * (c ^ 4.0)), a, Float64(-Float64(Float64(c * c) / (b ^ 3.0)))), a, Float64(Float64(-c) / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.26], N[(N[((-b) + N[Sqrt[N[(b * b + N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(a / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision] * -5.0 + N[(-2.0 * N[(N[Power[b, -5.0], $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[c, 4.0], $MachinePrecision]), $MachinePrecision] * a + (-N[(N[(c * c), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision])), $MachinePrecision] * a + N[((-c) / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.26000000000000001

   1. Initial program 82.9%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. associate-*r* N/A

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

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

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

      8. pow2 N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 83.0

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

   3. Applied rewrites 83.0%

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

   if 0.26000000000000001 < b

   1. Initial program 51.8%

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

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} + a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{4} \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right)} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{4} \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right) + \color{blue}{-1 \cdot \frac{c}{b}} \]

      2. *-commutative N/A

         \[\leadsto \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{4} \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right) \cdot a + \color{blue}{-1} \cdot \frac{c}{b} \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{4} \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right), \color{blue}{a}, -1 \cdot \frac{c}{b}\right) \]

   4. Applied rewrites 92.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a \cdot \frac{\frac{{c}^{4}}{{b}^{6}} \cdot 20}{b}, -0.25, \frac{-2 \cdot {c}^{3}}{{b}^{5}}\right), a, -\frac{c \cdot c}{{b}^{3}}\right), a, \frac{-c}{b}\right)} \]

   5. Taylor expanded in c around inf

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left({c}^{4} \cdot \left(-5 \cdot \frac{a}{{b}^{7}} - 2 \cdot \frac{1}{{b}^{5} \cdot c}\right), a, -\frac{c \cdot c}{{b}^{3}}\right), a, \frac{-c}{b}\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(-5 \cdot \frac{a}{{b}^{7}} - 2 \cdot \frac{1}{{b}^{5} \cdot c}\right) \cdot {c}^{4}, a, -\frac{c \cdot c}{{b}^{3}}\right), a, \frac{-c}{b}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(-5 \cdot \frac{a}{{b}^{7}} - 2 \cdot \frac{1}{{b}^{5} \cdot c}\right) \cdot {c}^{4}, a, -\frac{c \cdot c}{{b}^{3}}\right), a, \frac{-c}{b}\right) \]

   7. Applied rewrites 92.8%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{a}{{b}^{7}}, -5, -2 \cdot \frac{{b}^{-5}}{c}\right) \cdot {c}^{4}, a, -\frac{c \cdot c}{{b}^{3}}\right), a, \frac{-c}{b}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 91.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.26:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(-5, {a}^{3} \cdot c, -2 \cdot {\left(a \cdot b\right)}^{2}\right)}{{b}^{7}}, c, -\frac{a}{{b}^{3}}\right) \cdot c - {b}^{-1}\right) \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.26)
   (/ (+ (- b) (sqrt (fma b b (* -4.0 (* c a))))) (* 2.0 a))
   (*
    (-
     (*
      (fma
       (/ (fma -5.0 (* (pow a 3.0) c) (* -2.0 (pow (* a b) 2.0))) (pow b 7.0))
       c
       (- (/ a (pow b 3.0))))
      c)
     (pow b -1.0))
    c)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.26) {
		tmp = (-b + sqrt(fma(b, b, (-4.0 * (c * a))))) / (2.0 * a);
	} else {
		tmp = ((fma((fma(-5.0, (pow(a, 3.0) * c), (-2.0 * pow((a * b), 2.0))) / pow(b, 7.0)), c, -(a / pow(b, 3.0))) * c) - pow(b, -1.0)) * c;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.26)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(-4.0 * Float64(c * a))))) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(Float64(fma(Float64(fma(-5.0, Float64((a ^ 3.0) * c), Float64(-2.0 * (Float64(a * b) ^ 2.0))) / (b ^ 7.0)), c, Float64(-Float64(a / (b ^ 3.0)))) * c) - (b ^ -1.0)) * c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.26], N[(N[((-b) + N[Sqrt[N[(b * b + N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(-5.0 * N[(N[Power[a, 3.0], $MachinePrecision] * c), $MachinePrecision] + N[(-2.0 * N[Power[N[(a * b), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision] * c + (-N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision])), $MachinePrecision] * c), $MachinePrecision] - N[Power[b, -1.0], $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.26000000000000001

   1. Initial program 82.9%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. associate-*r* N/A

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

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

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

      8. pow2 N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 83.0

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

   3. Applied rewrites 83.0%

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

   if 0.26000000000000001 < b

   1. Initial program 51.8%

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

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} + a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{4} \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right)} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{4} \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right) + \color{blue}{-1 \cdot \frac{c}{b}} \]

      2. *-commutative N/A

         \[\leadsto \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{4} \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right) \cdot a + \color{blue}{-1} \cdot \frac{c}{b} \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{4} \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right), \color{blue}{a}, -1 \cdot \frac{c}{b}\right) \]

   4. Applied rewrites 92.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a \cdot \frac{\frac{{c}^{4}}{{b}^{6}} \cdot 20}{b}, -0.25, \frac{-2 \cdot {c}^{3}}{{b}^{5}}\right), a, -\frac{c \cdot c}{{b}^{3}}\right), a, \frac{-c}{b}\right)} \]

   5. Taylor expanded in c around 0

      \[\leadsto c \cdot \color{blue}{\left(c \cdot \left(-1 \cdot \frac{a}{{b}^{3}} + c \cdot \left(-5 \cdot \frac{{a}^{3} \cdot c}{{b}^{7}} + -2 \cdot \frac{{a}^{2}}{{b}^{5}}\right)\right) - \frac{1}{b}\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(c \cdot \left(-1 \cdot \frac{a}{{b}^{3}} + c \cdot \left(-5 \cdot \frac{{a}^{3} \cdot c}{{b}^{7}} + -2 \cdot \frac{{a}^{2}}{{b}^{5}}\right)\right) - \frac{1}{b}\right) \cdot c \]

      2. lower-*.f64 N/A

         \[\leadsto \left(c \cdot \left(-1 \cdot \frac{a}{{b}^{3}} + c \cdot \left(-5 \cdot \frac{{a}^{3} \cdot c}{{b}^{7}} + -2 \cdot \frac{{a}^{2}}{{b}^{5}}\right)\right) - \frac{1}{b}\right) \cdot c \]

   7. Applied rewrites 92.6%

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left({a}^{3} \cdot \frac{c}{{b}^{7}}, -5, \frac{-2 \cdot \left(a \cdot a\right)}{{b}^{5}}\right), c, -\frac{a}{{b}^{3}}\right) \cdot c - {b}^{-1}\right) \cdot \color{blue}{c} \]

   8. Taylor expanded in b around 0

      \[\leadsto \left(\mathsf{fma}\left(\frac{-5 \cdot \left({a}^{3} \cdot c\right) + -2 \cdot \left({a}^{2} \cdot {b}^{2}\right)}{{b}^{7}}, c, -\frac{a}{{b}^{3}}\right) \cdot c - {b}^{-1}\right) \cdot c \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-5 \cdot \left({a}^{3} \cdot c\right) + -2 \cdot \left({a}^{2} \cdot {b}^{2}\right)}{{b}^{7}}, c, -\frac{a}{{b}^{3}}\right) \cdot c - {b}^{-1}\right) \cdot c \]

      2. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(-5, {a}^{3} \cdot c, -2 \cdot \left({a}^{2} \cdot {b}^{2}\right)\right)}{{b}^{7}}, c, -\frac{a}{{b}^{3}}\right) \cdot c - {b}^{-1}\right) \cdot c \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(-5, {a}^{3} \cdot c, -2 \cdot \left({a}^{2} \cdot {b}^{2}\right)\right)}{{b}^{7}}, c, -\frac{a}{{b}^{3}}\right) \cdot c - {b}^{-1}\right) \cdot c \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(-5, {a}^{3} \cdot c, -2 \cdot \left({a}^{2} \cdot {b}^{2}\right)\right)}{{b}^{7}}, c, -\frac{a}{{b}^{3}}\right) \cdot c - {b}^{-1}\right) \cdot c \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(-5, {a}^{3} \cdot c, -2 \cdot \left({a}^{2} \cdot {b}^{2}\right)\right)}{{b}^{7}}, c, -\frac{a}{{b}^{3}}\right) \cdot c - {b}^{-1}\right) \cdot c \]

      6. pow-prod-down N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(-5, {a}^{3} \cdot c, -2 \cdot {\left(a \cdot b\right)}^{2}\right)}{{b}^{7}}, c, -\frac{a}{{b}^{3}}\right) \cdot c - {b}^{-1}\right) \cdot c \]

      7. lower-pow.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(-5, {a}^{3} \cdot c, -2 \cdot {\left(a \cdot b\right)}^{2}\right)}{{b}^{7}}, c, -\frac{a}{{b}^{3}}\right) \cdot c - {b}^{-1}\right) \cdot c \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(-5, {a}^{3} \cdot c, -2 \cdot {\left(a \cdot b\right)}^{2}\right)}{{b}^{7}}, c, -\frac{a}{{b}^{3}}\right) \cdot c - {b}^{-1}\right) \cdot c \]

      9. lift-pow.f64 92.6

         \[\leadsto \left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(-5, {a}^{3} \cdot c, -2 \cdot {\left(a \cdot b\right)}^{2}\right)}{{b}^{7}}, c, -\frac{a}{{b}^{3}}\right) \cdot c - {b}^{-1}\right) \cdot c \]

   10. Applied rewrites 92.6%

       \[\leadsto \left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(-5, {a}^{3} \cdot c, -2 \cdot {\left(a \cdot b\right)}^{2}\right)}{{b}^{7}}, c, -\frac{a}{{b}^{3}}\right) \cdot c - {b}^{-1}\right) \cdot c \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 89.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.6:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\frac{\left(c \cdot a\right) \cdot -2}{{b}^{5}} - {b}^{-3}\right) \cdot \left(c \cdot c\right), a, \frac{-c}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.6)
   (/ (+ (- b) (sqrt (fma b b (* -4.0 (* c a))))) (* 2.0 a))
   (fma
    (* (- (/ (* (* c a) -2.0) (pow b 5.0)) (pow b -3.0)) (* c c))
    a
    (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.6) {
		tmp = (-b + sqrt(fma(b, b, (-4.0 * (c * a))))) / (2.0 * a);
	} else {
		tmp = fma((((((c * a) * -2.0) / pow(b, 5.0)) - pow(b, -3.0)) * (c * c)), a, (-c / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.6)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(-4.0 * Float64(c * a))))) / Float64(2.0 * a));
	else
		tmp = fma(Float64(Float64(Float64(Float64(Float64(c * a) * -2.0) / (b ^ 5.0)) - (b ^ -3.0)) * Float64(c * c)), a, Float64(Float64(-c) / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.6], N[(N[((-b) + N[Sqrt[N[(b * b + N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(c * a), $MachinePrecision] * -2.0), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] - N[Power[b, -3.0], $MachinePrecision]), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision] * a + N[((-c) / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.6000000000000001

   1. Initial program 81.3%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. associate-*r* N/A

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

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

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

      8. pow2 N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 81.5

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

   3. Applied rewrites 81.5%

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

   if 1.6000000000000001 < b

   1. Initial program 50.3%

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

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} + a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{4} \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right)} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{4} \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right) + \color{blue}{-1 \cdot \frac{c}{b}} \]

      2. *-commutative N/A

         \[\leadsto \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{4} \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right) \cdot a + \color{blue}{-1} \cdot \frac{c}{b} \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{4} \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right), \color{blue}{a}, -1 \cdot \frac{c}{b}\right) \]

   4. Applied rewrites 93.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a \cdot \frac{\frac{{c}^{4}}{{b}^{6}} \cdot 20}{b}, -0.25, \frac{-2 \cdot {c}^{3}}{{b}^{5}}\right), a, -\frac{c \cdot c}{{b}^{3}}\right), a, \frac{-c}{b}\right)} \]

   5. Taylor expanded in c around 0

      \[\leadsto \mathsf{fma}\left({c}^{2} \cdot \left(-2 \cdot \frac{a \cdot c}{{b}^{5}} - \frac{1}{{b}^{3}}\right), a, \frac{-c}{b}\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(-2 \cdot \frac{a \cdot c}{{b}^{5}} - \frac{1}{{b}^{3}}\right) \cdot {c}^{2}, a, \frac{-c}{b}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(-2 \cdot \frac{a \cdot c}{{b}^{5}} - \frac{1}{{b}^{3}}\right) \cdot {c}^{2}, a, \frac{-c}{b}\right) \]

      3. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(-2 \cdot \frac{a \cdot c}{{b}^{5}} - \frac{1}{{b}^{3}}\right) \cdot {c}^{2}, a, \frac{-c}{b}\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{-2 \cdot \left(a \cdot c\right)}{{b}^{5}} - \frac{1}{{b}^{3}}\right) \cdot {c}^{2}, a, \frac{-c}{b}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{-2 \cdot \left(a \cdot c\right)}{{b}^{5}} - \frac{1}{{b}^{3}}\right) \cdot {c}^{2}, a, \frac{-c}{b}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{\left(a \cdot c\right) \cdot -2}{{b}^{5}} - \frac{1}{{b}^{3}}\right) \cdot {c}^{2}, a, \frac{-c}{b}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{\left(a \cdot c\right) \cdot -2}{{b}^{5}} - \frac{1}{{b}^{3}}\right) \cdot {c}^{2}, a, \frac{-c}{b}\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{\left(c \cdot a\right) \cdot -2}{{b}^{5}} - \frac{1}{{b}^{3}}\right) \cdot {c}^{2}, a, \frac{-c}{b}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{\left(c \cdot a\right) \cdot -2}{{b}^{5}} - \frac{1}{{b}^{3}}\right) \cdot {c}^{2}, a, \frac{-c}{b}\right) \]

      10. lift-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{\left(c \cdot a\right) \cdot -2}{{b}^{5}} - \frac{1}{{b}^{3}}\right) \cdot {c}^{2}, a, \frac{-c}{b}\right) \]

      11. pow-flip N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{\left(c \cdot a\right) \cdot -2}{{b}^{5}} - {b}^{\left(\mathsf{neg}\left(3\right)\right)}\right) \cdot {c}^{2}, a, \frac{-c}{b}\right) \]

      12. lower-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{\left(c \cdot a\right) \cdot -2}{{b}^{5}} - {b}^{\left(\mathsf{neg}\left(3\right)\right)}\right) \cdot {c}^{2}, a, \frac{-c}{b}\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{\left(c \cdot a\right) \cdot -2}{{b}^{5}} - {b}^{-3}\right) \cdot {c}^{2}, a, \frac{-c}{b}\right) \]

      14. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{\left(c \cdot a\right) \cdot -2}{{b}^{5}} - {b}^{-3}\right) \cdot \left(c \cdot c\right), a, \frac{-c}{b}\right) \]

      15. lift-*.f64 90.9

          \[\leadsto \mathsf{fma}\left(\left(\frac{\left(c \cdot a\right) \cdot -2}{{b}^{5}} - {b}^{-3}\right) \cdot \left(c \cdot c\right), a, \frac{-c}{b}\right) \]

   7. Applied rewrites 90.9%

      \[\leadsto \mathsf{fma}\left(\left(\frac{\left(c \cdot a\right) \cdot -2}{{b}^{5}} - {b}^{-3}\right) \cdot \left(c \cdot c\right), a, \frac{-c}{b}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 89.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.6:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a \cdot \left(\frac{-2 \cdot \left(a \cdot c\right)}{{b}^{5}} - {b}^{-3}\right)\right) \cdot c - {b}^{-1}\right) \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.6)
   (/ (+ (- b) (sqrt (fma b b (* -4.0 (* c a))))) (* 2.0 a))
   (*
    (-
     (* (* a (- (/ (* -2.0 (* a c)) (pow b 5.0)) (pow b -3.0))) c)
     (pow b -1.0))
    c)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.6) {
		tmp = (-b + sqrt(fma(b, b, (-4.0 * (c * a))))) / (2.0 * a);
	} else {
		tmp = (((a * (((-2.0 * (a * c)) / pow(b, 5.0)) - pow(b, -3.0))) * c) - pow(b, -1.0)) * c;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.6)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(-4.0 * Float64(c * a))))) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(Float64(Float64(a * Float64(Float64(Float64(-2.0 * Float64(a * c)) / (b ^ 5.0)) - (b ^ -3.0))) * c) - (b ^ -1.0)) * c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.6], N[(N[((-b) + N[Sqrt[N[(b * b + N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(a * N[(N[(N[(-2.0 * N[(a * c), $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] - N[Power[b, -3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] - N[Power[b, -1.0], $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.6000000000000001

   1. Initial program 81.3%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. associate-*r* N/A

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

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

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

      8. pow2 N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 81.5

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

   3. Applied rewrites 81.5%

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

   if 1.6000000000000001 < b

   1. Initial program 50.3%

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

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} + a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{4} \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right)} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{4} \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right) + \color{blue}{-1 \cdot \frac{c}{b}} \]

      2. *-commutative N/A

         \[\leadsto \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{4} \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right) \cdot a + \color{blue}{-1} \cdot \frac{c}{b} \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{4} \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right), \color{blue}{a}, -1 \cdot \frac{c}{b}\right) \]

   4. Applied rewrites 93.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a \cdot \frac{\frac{{c}^{4}}{{b}^{6}} \cdot 20}{b}, -0.25, \frac{-2 \cdot {c}^{3}}{{b}^{5}}\right), a, -\frac{c \cdot c}{{b}^{3}}\right), a, \frac{-c}{b}\right)} \]

   5. Taylor expanded in c around 0

      \[\leadsto c \cdot \color{blue}{\left(c \cdot \left(-1 \cdot \frac{a}{{b}^{3}} + c \cdot \left(-5 \cdot \frac{{a}^{3} \cdot c}{{b}^{7}} + -2 \cdot \frac{{a}^{2}}{{b}^{5}}\right)\right) - \frac{1}{b}\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(c \cdot \left(-1 \cdot \frac{a}{{b}^{3}} + c \cdot \left(-5 \cdot \frac{{a}^{3} \cdot c}{{b}^{7}} + -2 \cdot \frac{{a}^{2}}{{b}^{5}}\right)\right) - \frac{1}{b}\right) \cdot c \]

      2. lower-*.f64 N/A

         \[\leadsto \left(c \cdot \left(-1 \cdot \frac{a}{{b}^{3}} + c \cdot \left(-5 \cdot \frac{{a}^{3} \cdot c}{{b}^{7}} + -2 \cdot \frac{{a}^{2}}{{b}^{5}}\right)\right) - \frac{1}{b}\right) \cdot c \]

   7. Applied rewrites 93.3%

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left({a}^{3} \cdot \frac{c}{{b}^{7}}, -5, \frac{-2 \cdot \left(a \cdot a\right)}{{b}^{5}}\right), c, -\frac{a}{{b}^{3}}\right) \cdot c - {b}^{-1}\right) \cdot \color{blue}{c} \]

   8. Taylor expanded in a around 0

      \[\leadsto \left(\left(a \cdot \left(-2 \cdot \frac{a \cdot c}{{b}^{5}} - \frac{1}{{b}^{3}}\right)\right) \cdot c - {b}^{-1}\right) \cdot c \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\left(a \cdot \left(-2 \cdot \frac{a \cdot c}{{b}^{5}} - \frac{1}{{b}^{3}}\right)\right) \cdot c - {b}^{-1}\right) \cdot c \]

      2. lower--.f64 N/A

         \[\leadsto \left(\left(a \cdot \left(-2 \cdot \frac{a \cdot c}{{b}^{5}} - \frac{1}{{b}^{3}}\right)\right) \cdot c - {b}^{-1}\right) \cdot c \]

      3. associate-*r/ N/A

         \[\leadsto \left(\left(a \cdot \left(\frac{-2 \cdot \left(a \cdot c\right)}{{b}^{5}} - \frac{1}{{b}^{3}}\right)\right) \cdot c - {b}^{-1}\right) \cdot c \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\left(a \cdot \left(\frac{-2 \cdot \left(a \cdot c\right)}{{b}^{5}} - \frac{1}{{b}^{3}}\right)\right) \cdot c - {b}^{-1}\right) \cdot c \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\left(a \cdot \left(\frac{-2 \cdot \left(a \cdot c\right)}{{b}^{5}} - \frac{1}{{b}^{3}}\right)\right) \cdot c - {b}^{-1}\right) \cdot c \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\left(a \cdot \left(\frac{-2 \cdot \left(a \cdot c\right)}{{b}^{5}} - \frac{1}{{b}^{3}}\right)\right) \cdot c - {b}^{-1}\right) \cdot c \]

      7. lift-pow.f64 N/A

         \[\leadsto \left(\left(a \cdot \left(\frac{-2 \cdot \left(a \cdot c\right)}{{b}^{5}} - \frac{1}{{b}^{3}}\right)\right) \cdot c - {b}^{-1}\right) \cdot c \]

      8. pow-flip N/A

         \[\leadsto \left(\left(a \cdot \left(\frac{-2 \cdot \left(a \cdot c\right)}{{b}^{5}} - {b}^{\left(\mathsf{neg}\left(3\right)\right)}\right)\right) \cdot c - {b}^{-1}\right) \cdot c \]

      9. metadata-eval N/A

         \[\leadsto \left(\left(a \cdot \left(\frac{-2 \cdot \left(a \cdot c\right)}{{b}^{5}} - {b}^{-3}\right)\right) \cdot c - {b}^{-1}\right) \cdot c \]

      10. lift-pow.f64 90.8

          \[\leadsto \left(\left(a \cdot \left(\frac{-2 \cdot \left(a \cdot c\right)}{{b}^{5}} - {b}^{-3}\right)\right) \cdot c - {b}^{-1}\right) \cdot c \]

   10. Applied rewrites 90.8%

       \[\leadsto \left(\left(a \cdot \left(\frac{-2 \cdot \left(a \cdot c\right)}{{b}^{5}} - {b}^{-3}\right)\right) \cdot c - {b}^{-1}\right) \cdot c \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 84.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.6:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-c \cdot c}{{b}^{3}}, a, \frac{-c}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.6)
   (/ (+ (- b) (sqrt (fma b b (* -4.0 (* c a))))) (* 2.0 a))
   (fma (/ (- (* c c)) (pow b 3.0)) a (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.6) {
		tmp = (-b + sqrt(fma(b, b, (-4.0 * (c * a))))) / (2.0 * a);
	} else {
		tmp = fma((-(c * c) / pow(b, 3.0)), a, (-c / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.6)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(-4.0 * Float64(c * a))))) / Float64(2.0 * a));
	else
		tmp = fma(Float64(Float64(-Float64(c * c)) / (b ^ 3.0)), a, Float64(Float64(-c) / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.6], N[(N[((-b) + N[Sqrt[N[(b * b + N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[((-N[(c * c), $MachinePrecision]) / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] * a + N[((-c) / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.6000000000000001

   1. Initial program 81.3%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. associate-*r* N/A

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

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

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

      8. pow2 N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 81.5

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

   3. Applied rewrites 81.5%

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

   if 1.6000000000000001 < b

   1. Initial program 50.3%

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

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} + a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{4} \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right)} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{4} \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right) + \color{blue}{-1 \cdot \frac{c}{b}} \]

      2. *-commutative N/A

         \[\leadsto \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{4} \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right) \cdot a + \color{blue}{-1} \cdot \frac{c}{b} \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{4} \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right), \color{blue}{a}, -1 \cdot \frac{c}{b}\right) \]

   4. Applied rewrites 93.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a \cdot \frac{\frac{{c}^{4}}{{b}^{6}} \cdot 20}{b}, -0.25, \frac{-2 \cdot {c}^{3}}{{b}^{5}}\right), a, -\frac{c \cdot c}{{b}^{3}}\right), a, \frac{-c}{b}\right)} \]

   5. Taylor expanded in a around 0

      \[\leadsto \mathsf{fma}\left(-1 \cdot \frac{{c}^{2}}{{b}^{3}}, a, \frac{-c}{b}\right) \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\frac{\mathsf{neg}\left({c}^{2}\right)}{{b}^{3}}, a, \frac{-c}{b}\right) \]

      4. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-{c}^{2}}{{b}^{3}}, a, \frac{-c}{b}\right) \]

      5. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-c \cdot c}{{b}^{3}}, a, \frac{-c}{b}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-c \cdot c}{{b}^{3}}, a, \frac{-c}{b}\right) \]

      7. lift-pow.f64 85.5

         \[\leadsto \mathsf{fma}\left(\frac{-c \cdot c}{{b}^{3}}, a, \frac{-c}{b}\right) \]

   7. Applied rewrites 85.5%

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

Alternative 6: 81.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{-c \cdot c}{{b}^{3}}, a, \frac{-c}{b}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (fma (/ (- (* c c)) (pow b 3.0)) a (/ (- c) b)))

C

double code(double a, double b, double c) {
	return fma((-(c * c) / pow(b, 3.0)), a, (-c / b));
}

Julia

function code(a, b, c)
	return fma(Float64(Float64(-Float64(c * c)) / (b ^ 3.0)), a, Float64(Float64(-c) / b))
end

Wolfram

code[a_, b_, c_] := N[(N[((-N[(c * c), $MachinePrecision]) / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] * a + N[((-c) / b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.6%

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

2. Taylor expanded in a around 0

   \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} + a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{4} \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right)} \]
3. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{4} \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right) + \color{blue}{-1 \cdot \frac{c}{b}} \]

   2. *-commutative N/A

      \[\leadsto \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{4} \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right) \cdot a + \color{blue}{-1} \cdot \frac{c}{b} \]

   3. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{4} \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right), \color{blue}{a}, -1 \cdot \frac{c}{b}\right) \]

4. Applied rewrites 90.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a \cdot \frac{\frac{{c}^{4}}{{b}^{6}} \cdot 20}{b}, -0.25, \frac{-2 \cdot {c}^{3}}{{b}^{5}}\right), a, -\frac{c \cdot c}{{b}^{3}}\right), a, \frac{-c}{b}\right)} \]

5. Taylor expanded in a around 0

   \[\leadsto \mathsf{fma}\left(-1 \cdot \frac{{c}^{2}}{{b}^{3}}, a, \frac{-c}{b}\right) \]
6. Step-by-step derivation

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. mul-1-neg N/A

      \[\leadsto \mathsf{fma}\left(\frac{\mathsf{neg}\left({c}^{2}\right)}{{b}^{3}}, a, \frac{-c}{b}\right) \]

   4. lower-neg.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{-{c}^{2}}{{b}^{3}}, a, \frac{-c}{b}\right) \]

   5. pow2 N/A

      \[\leadsto \mathsf{fma}\left(\frac{-c \cdot c}{{b}^{3}}, a, \frac{-c}{b}\right) \]

   6. lift-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{-c \cdot c}{{b}^{3}}, a, \frac{-c}{b}\right) \]

   7. lift-pow.f64 81.2

      \[\leadsto \mathsf{fma}\left(\frac{-c \cdot c}{{b}^{3}}, a, \frac{-c}{b}\right) \]

7. Applied rewrites 81.2%

   \[\leadsto \mathsf{fma}\left(\frac{-c \cdot c}{{b}^{3}}, a, \frac{-c}{b}\right) \]
8. Add Preprocessing

Alternative 7: 81.2% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 55.6%

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

2. Taylor expanded in a around 0

   \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} + a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{4} \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right)} \]
3. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{4} \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right) + \color{blue}{-1 \cdot \frac{c}{b}} \]

   2. *-commutative N/A

      \[\leadsto \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{4} \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right) \cdot a + \color{blue}{-1} \cdot \frac{c}{b} \]

   3. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{4} \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right), \color{blue}{a}, -1 \cdot \frac{c}{b}\right) \]

4. Applied rewrites 90.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a \cdot \frac{\frac{{c}^{4}}{{b}^{6}} \cdot 20}{b}, -0.25, \frac{-2 \cdot {c}^{3}}{{b}^{5}}\right), a, -\frac{c \cdot c}{{b}^{3}}\right), a, \frac{-c}{b}\right)} \]

5. Taylor expanded in b around inf

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

7. Applied rewrites 81.2%

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

Alternative 8: 81.1% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 55.6%

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

2. Taylor expanded in a around 0

   \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} + a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{4} \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right)} \]
3. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{4} \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right) + \color{blue}{-1 \cdot \frac{c}{b}} \]

   2. *-commutative N/A

      \[\leadsto \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{4} \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right) \cdot a + \color{blue}{-1} \cdot \frac{c}{b} \]

   3. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{4} \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right), \color{blue}{a}, -1 \cdot \frac{c}{b}\right) \]

4. Applied rewrites 90.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a \cdot \frac{\frac{{c}^{4}}{{b}^{6}} \cdot 20}{b}, -0.25, \frac{-2 \cdot {c}^{3}}{{b}^{5}}\right), a, -\frac{c \cdot c}{{b}^{3}}\right), a, \frac{-c}{b}\right)} \]

5. Taylor expanded in c around 0

   \[\leadsto c \cdot \color{blue}{\left(c \cdot \left(-1 \cdot \frac{a}{{b}^{3}} + c \cdot \left(-5 \cdot \frac{{a}^{3} \cdot c}{{b}^{7}} + -2 \cdot \frac{{a}^{2}}{{b}^{5}}\right)\right) - \frac{1}{b}\right)} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(c \cdot \left(-1 \cdot \frac{a}{{b}^{3}} + c \cdot \left(-5 \cdot \frac{{a}^{3} \cdot c}{{b}^{7}} + -2 \cdot \frac{{a}^{2}}{{b}^{5}}\right)\right) - \frac{1}{b}\right) \cdot c \]

   2. lower-*.f64 N/A

      \[\leadsto \left(c \cdot \left(-1 \cdot \frac{a}{{b}^{3}} + c \cdot \left(-5 \cdot \frac{{a}^{3} \cdot c}{{b}^{7}} + -2 \cdot \frac{{a}^{2}}{{b}^{5}}\right)\right) - \frac{1}{b}\right) \cdot c \]

7. Applied rewrites 90.6%

   \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left({a}^{3} \cdot \frac{c}{{b}^{7}}, -5, \frac{-2 \cdot \left(a \cdot a\right)}{{b}^{5}}\right), c, -\frac{a}{{b}^{3}}\right) \cdot c - {b}^{-1}\right) \cdot \color{blue}{c} \]

8. Taylor expanded in b around -inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. +-commutative N/A

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

   5. associate-/l* N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. pow2 N/A

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

   9. lower-*.f64 81.1

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

10. Applied rewrites 81.1%

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

Alternative 9: 64.2% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.6%

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

2. Taylor expanded in a around 0

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

   1. associate-*r/ N/A

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

   2. mul-1-neg N/A

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

   3. lower-/.f64 N/A

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

   4. lower-neg.f64 64.2

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

4. Applied rewrites 64.2%

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

Reproduce

herbie shell --seed 2025101 
(FPCore (a b c)
  :name "Quadratic roots, narrow range"
  :precision binary64
  :pre (and (and (and (< 1.0536712127723509e-8 a) (< a 94906265.62425156)) (and (< 1.0536712127723509e-8 b) (< b 94906265.62425156))) (and (< 1.0536712127723509e-8 c) (< c 94906265.62425156)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))