Quadratic roots, narrow range

Percentage Accurate: 55.2% → 91.7%

Time: 5.3s

Alternatives: 8

Speedup: 3.6×

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: 55.2% 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: 91.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -100:\\ \;\;\;\;\frac{\frac{{\left(-b\right)}^{3} + {t\_0}^{3}}{\mathsf{fma}\left(b, b, t\_0 \cdot t\_0 + b \cdot t\_0\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{{a}^{3} \cdot \left(c \cdot c\right)}{{b}^{6}} - \frac{a}{b \cdot b}\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (fma (* -4.0 a) c (* b b)))))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -100.0)
     (/
      (/ (+ (pow (- b) 3.0) (pow t_0 3.0)) (fma b b (+ (* t_0 t_0) (* b t_0))))
      (* 2.0 a))
     (/
      (+
       (fma (/ (* (* a a) (* (* c c) c)) (* (* b b) (* b b))) -2.0 (- c))
       (*
        (* c c)
        (- (* -5.0 (/ (* (pow a 3.0) (* c c)) (pow b 6.0))) (/ a (* b b)))))
      b))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(fma((-4.0 * a), c, (b * b)));
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -100.0) {
		tmp = ((pow(-b, 3.0) + pow(t_0, 3.0)) / fma(b, b, ((t_0 * t_0) + (b * t_0)))) / (2.0 * a);
	} else {
		tmp = (fma((((a * a) * ((c * c) * c)) / ((b * b) * (b * b))), -2.0, -c) + ((c * c) * ((-5.0 * ((pow(a, 3.0) * (c * c)) / pow(b, 6.0))) - (a / (b * b))))) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = sqrt(fma(Float64(-4.0 * a), c, Float64(b * b)))
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a)) <= -100.0)
		tmp = Float64(Float64(Float64((Float64(-b) ^ 3.0) + (t_0 ^ 3.0)) / fma(b, b, Float64(Float64(t_0 * t_0) + Float64(b * t_0)))) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(fma(Float64(Float64(Float64(a * a) * Float64(Float64(c * c) * c)) / Float64(Float64(b * b) * Float64(b * b))), -2.0, Float64(-c)) + Float64(Float64(c * c) * Float64(Float64(-5.0 * Float64(Float64((a ^ 3.0) * Float64(c * c)) / (b ^ 6.0))) - Float64(a / Float64(b * b))))) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(-4.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[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], -100.0], N[(N[(N[(N[Power[(-b), 3.0], $MachinePrecision] + N[Power[t$95$0, 3.0], $MachinePrecision]), $MachinePrecision] / N[(b * b + N[(N[(t$95$0 * t$95$0), $MachinePrecision] + N[(b * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(a * a), $MachinePrecision] * N[(N[(c * c), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -2.0 + (-c)), $MachinePrecision] + N[(N[(c * c), $MachinePrecision] * N[(N[(-5.0 * N[(N[(N[Power[a, 3.0], $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -100

   1. Initial program 94.4%

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

      1. lift-neg.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. flip3-+ N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 94.5%

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

   if -100 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 52.6%

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

   3. Taylor expanded in b around inf

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

   5. Applied rewrites 93.0%

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

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot {c}^{3}}{{b}^{4}}, -2, -c\right) + \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot 20}{a}}{{b}^{6}}, \frac{-1}{4}, -\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}\right)}{b} \]

      2. unpow3 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}}, -2, -c\right) + \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot 20}{a}}{{b}^{6}}, \frac{-1}{4}, -\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}\right)}{b} \]

      3. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left({c}^{2} \cdot c\right)}{{b}^{4}}, -2, -c\right) + \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot 20}{a}}{{b}^{6}}, \frac{-1}{4}, -\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}\right)}{b} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left({c}^{2} \cdot c\right)}{{b}^{4}}, -2, -c\right) + \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot 20}{a}}{{b}^{6}}, \frac{-1}{4}, -\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}\right)}{b} \]

      5. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}}, -2, -c\right) + \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot 20}{a}}{{b}^{6}}, \frac{-1}{4}, -\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}\right)}{b} \]

      6. lift-*.f64 93.0

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

   7. Applied rewrites 93.0%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}}, -2, -c\right) + \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot 20}{a}}{{b}^{6}}, -0.25, -\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}\right)}{b} \]
   8. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}}, -2, -c\right) + \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot 20}{a}}{{b}^{6}}, \frac{-1}{4}, -\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}\right)}{b} \]

      2. sqr-pow N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{\left(\frac{4}{2}\right)} \cdot {b}^{\left(\frac{4}{2}\right)}}, -2, -c\right) + \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot 20}{a}}{{b}^{6}}, \frac{-1}{4}, -\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}\right)}{b} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{2} \cdot {b}^{\left(\frac{4}{2}\right)}}, -2, -c\right) + \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot 20}{a}}{{b}^{6}}, \frac{-1}{4}, -\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}\right)}{b} \]

      4. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{2} \cdot {b}^{2}}, -2, -c\right) + \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot 20}{a}}{{b}^{6}}, \frac{-1}{4}, -\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}\right)}{b} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{2} \cdot {b}^{2}}, -2, -c\right) + \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot 20}{a}}{{b}^{6}}, \frac{-1}{4}, -\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}\right)}{b} \]

      6. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot {b}^{2}}, -2, -c\right) + \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot 20}{a}}{{b}^{6}}, \frac{-1}{4}, -\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}\right)}{b} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot {b}^{2}}, -2, -c\right) + \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot 20}{a}}{{b}^{6}}, \frac{-1}{4}, -\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}\right)}{b} \]

      8. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot 20}{a}}{{b}^{6}}, \frac{-1}{4}, -\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}\right)}{b} \]

      9. lift-*.f64 93.0

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

   9. Applied rewrites 93.0%

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

   10. Taylor expanded in c around 0

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

       1. lower-*.f64 N/A

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

       2. pow2 N/A

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

       3. lift-*.f64 N/A

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

       4. lower--.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-/.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. lift-pow.f64 N/A

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

       9. pow2 N/A

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

       10. lift-*.f64 N/A

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

       11. lift-pow.f64 N/A

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

       12. lower-/.f64 N/A

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

       13. pow2 N/A

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

       14. lift-*.f64 93.0

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

   12. Applied rewrites 93.0%

       \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{{a}^{3} \cdot \left(c \cdot c\right)}{{b}^{6}} - \frac{a}{b \cdot b}\right)}{b} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -100:\\ \;\;\;\;\frac{\frac{{\left(-b\right)}^{3} + {\left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}\right)}^{3}}{\mathsf{fma}\left(b, b, \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} \cdot \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} + b \cdot \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{{a}^{3} \cdot \left(c \cdot c\right)}{{b}^{6}} - \frac{a}{b \cdot b}\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 91.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -100:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{{a}^{3} \cdot \left(c \cdot c\right)}{{b}^{6}} - \frac{a}{b \cdot b}\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -100.0)
   (/ (+ (- b) (sqrt (fma b b (* -4.0 (* c a))))) (* 2.0 a))
   (/
    (+
     (fma (/ (* (* a a) (* (* c c) c)) (* (* b b) (* b b))) -2.0 (- c))
     (*
      (* c c)
      (- (* -5.0 (/ (* (pow a 3.0) (* c c)) (pow b 6.0))) (/ a (* b b)))))
    b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -100.0) {
		tmp = (-b + sqrt(fma(b, b, (-4.0 * (c * a))))) / (2.0 * a);
	} else {
		tmp = (fma((((a * a) * ((c * c) * c)) / ((b * b) * (b * b))), -2.0, -c) + ((c * c) * ((-5.0 * ((pow(a, 3.0) * (c * c)) / pow(b, 6.0))) - (a / (b * b))))) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a)) <= -100.0)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(-4.0 * Float64(c * a))))) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(fma(Float64(Float64(Float64(a * a) * Float64(Float64(c * c) * c)) / Float64(Float64(b * b) * Float64(b * b))), -2.0, Float64(-c)) + Float64(Float64(c * c) * Float64(Float64(-5.0 * Float64(Float64((a ^ 3.0) * Float64(c * c)) / (b ^ 6.0))) - Float64(a / Float64(b * b))))) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[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], -100.0], 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[(a * a), $MachinePrecision] * N[(N[(c * c), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -2.0 + (-c)), $MachinePrecision] + N[(N[(c * c), $MachinePrecision] * N[(N[(-5.0 * N[(N[(N[Power[a, 3.0], $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -100

   1. Initial program 94.4%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Add Preprocessing
   3. 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 94.4

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

   4. Applied rewrites 94.4%

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

   if -100 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 52.6%

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

   3. Taylor expanded in b around inf

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

   5. Applied rewrites 93.0%

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

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot {c}^{3}}{{b}^{4}}, -2, -c\right) + \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot 20}{a}}{{b}^{6}}, \frac{-1}{4}, -\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}\right)}{b} \]

      2. unpow3 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}}, -2, -c\right) + \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot 20}{a}}{{b}^{6}}, \frac{-1}{4}, -\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}\right)}{b} \]

      3. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left({c}^{2} \cdot c\right)}{{b}^{4}}, -2, -c\right) + \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot 20}{a}}{{b}^{6}}, \frac{-1}{4}, -\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}\right)}{b} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left({c}^{2} \cdot c\right)}{{b}^{4}}, -2, -c\right) + \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot 20}{a}}{{b}^{6}}, \frac{-1}{4}, -\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}\right)}{b} \]

      5. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}}, -2, -c\right) + \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot 20}{a}}{{b}^{6}}, \frac{-1}{4}, -\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}\right)}{b} \]

      6. lift-*.f64 93.0

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

   7. Applied rewrites 93.0%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}}, -2, -c\right) + \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot 20}{a}}{{b}^{6}}, -0.25, -\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}\right)}{b} \]
   8. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}}, -2, -c\right) + \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot 20}{a}}{{b}^{6}}, \frac{-1}{4}, -\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}\right)}{b} \]

      2. sqr-pow N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{\left(\frac{4}{2}\right)} \cdot {b}^{\left(\frac{4}{2}\right)}}, -2, -c\right) + \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot 20}{a}}{{b}^{6}}, \frac{-1}{4}, -\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}\right)}{b} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{2} \cdot {b}^{\left(\frac{4}{2}\right)}}, -2, -c\right) + \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot 20}{a}}{{b}^{6}}, \frac{-1}{4}, -\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}\right)}{b} \]

      4. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{2} \cdot {b}^{2}}, -2, -c\right) + \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot 20}{a}}{{b}^{6}}, \frac{-1}{4}, -\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}\right)}{b} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{2} \cdot {b}^{2}}, -2, -c\right) + \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot 20}{a}}{{b}^{6}}, \frac{-1}{4}, -\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}\right)}{b} \]

      6. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot {b}^{2}}, -2, -c\right) + \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot 20}{a}}{{b}^{6}}, \frac{-1}{4}, -\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}\right)}{b} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot {b}^{2}}, -2, -c\right) + \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot 20}{a}}{{b}^{6}}, \frac{-1}{4}, -\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}\right)}{b} \]

      8. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot 20}{a}}{{b}^{6}}, \frac{-1}{4}, -\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}\right)}{b} \]

      9. lift-*.f64 93.0

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

   9. Applied rewrites 93.0%

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

   10. Taylor expanded in c around 0

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

       1. lower-*.f64 N/A

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

       2. pow2 N/A

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

       3. lift-*.f64 N/A

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

       4. lower--.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-/.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. lift-pow.f64 N/A

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

       9. pow2 N/A

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

       10. lift-*.f64 N/A

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

       11. lift-pow.f64 N/A

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

       12. lower-/.f64 N/A

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

       13. pow2 N/A

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

       14. lift-*.f64 93.0

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

   12. Applied rewrites 93.0%

       \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \left(c \cdot c\right) \cdot \left(-5 \cdot \frac{{a}^{3} \cdot \left(c \cdot c\right)}{{b}^{6}} - \frac{a}{b \cdot b}\right)}{b} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 89.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.2:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\frac{\left(c \cdot a\right) \cdot -2}{{b}^{5}} - {b}^{-3}\right) \cdot \left(c \cdot c\right), a, \frac{-c}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.2)
   (/ (+ (- 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 <= 1.2) {
		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 <= 1.2)
		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, 1.2], 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 < 1.19999999999999996

   1. Initial program 82.1%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Add Preprocessing
   3. 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 82.3

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

   4. Applied rewrites 82.3%

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

   if 1.19999999999999996 < b

   1. Initial program 47.2%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 92.3%

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

   6. Taylor expanded in c around 0

      \[\leadsto \mathsf{fma}\left({c}^{2} \cdot \left(-2 \cdot \frac{a \cdot c}{{b}^{5}} - \frac{1}{{b}^{3}}\right), a, \frac{-c}{b}\right) \]
   7. 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 92.3

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

   8. Applied rewrites 92.3%

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

Alternative 4: 89.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.2:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(-2 \cdot a, c, \left(-b\right) \cdot b\right)}{{b}^{5}} \cdot \left(c \cdot c\right), a, \frac{-c}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.2)
   (/ (+ (- b) (sqrt (fma b b (* -4.0 (* c a))))) (* 2.0 a))
   (fma
    (* (/ (fma (* -2.0 a) c (* (- b) b)) (pow b 5.0)) (* c c))
    a
    (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.2) {
		tmp = (-b + sqrt(fma(b, b, (-4.0 * (c * a))))) / (2.0 * a);
	} else {
		tmp = fma(((fma((-2.0 * a), c, (-b * b)) / pow(b, 5.0)) * (c * c)), a, (-c / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.2)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(-4.0 * Float64(c * a))))) / Float64(2.0 * a));
	else
		tmp = fma(Float64(Float64(fma(Float64(-2.0 * a), c, Float64(Float64(-b) * b)) / (b ^ 5.0)) * Float64(c * c)), a, Float64(Float64(-c) / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.2], 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[(-2.0 * a), $MachinePrecision] * c + N[((-b) * b), $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision] * a + N[((-c) / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.19999999999999996

   1. Initial program 82.1%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Add Preprocessing
   3. 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 82.3

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

   4. Applied rewrites 82.3%

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

   if 1.19999999999999996 < b

   1. Initial program 47.2%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 92.3%

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

   6. Taylor expanded in c around 0

      \[\leadsto \mathsf{fma}\left({c}^{2} \cdot \left(-2 \cdot \frac{a \cdot c}{{b}^{5}} - \frac{1}{{b}^{3}}\right), a, \frac{-c}{b}\right) \]
   7. 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 92.3

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

   8. Applied rewrites 92.3%

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

   9. Taylor expanded in b around 0

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

       1. lower-/.f64 N/A

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

       2. associate-*r* N/A

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

       3. mul-1-neg N/A

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

       4. lower-fma.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-neg.f64 N/A

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

       7. pow2 N/A

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

       8. lift-*.f64 N/A

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

       9. lift-pow.f64 92.3

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

   11. Applied rewrites 92.3%

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

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.2:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(-2 \cdot a, c, \left(-b\right) \cdot b\right)}{{b}^{5}} \cdot \left(c \cdot c\right), a, \frac{-c}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.2:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \frac{a \cdot \left(c \cdot c\right)}{\left(-b\right) \cdot b}}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.2)
   (/ (+ (- b) (sqrt (fma b b (* -4.0 (* c a))))) (* 2.0 a))
   (/
    (+
     (fma (/ (* (* a a) (* (* c c) c)) (* (* b b) (* b b))) -2.0 (- c))
     (/ (* a (* c c)) (* (- b) b)))
    b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.2) {
		tmp = (-b + sqrt(fma(b, b, (-4.0 * (c * a))))) / (2.0 * a);
	} else {
		tmp = (fma((((a * a) * ((c * c) * c)) / ((b * b) * (b * b))), -2.0, -c) + ((a * (c * c)) / (-b * b))) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.2)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(-4.0 * Float64(c * a))))) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(fma(Float64(Float64(Float64(a * a) * Float64(Float64(c * c) * c)) / Float64(Float64(b * b) * Float64(b * b))), -2.0, Float64(-c)) + Float64(Float64(a * Float64(c * c)) / Float64(Float64(-b) * b))) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.2], 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[(a * a), $MachinePrecision] * N[(N[(c * c), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -2.0 + (-c)), $MachinePrecision] + N[(N[(a * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[((-b) * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.19999999999999996

   1. Initial program 82.1%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Add Preprocessing
   3. 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 82.3

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

   4. Applied rewrites 82.3%

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

   if 1.19999999999999996 < b

   1. Initial program 47.2%

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

   3. Taylor expanded in b around inf

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

   5. Applied rewrites 94.7%

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

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot {c}^{3}}{{b}^{4}}, -2, -c\right) + \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot 20}{a}}{{b}^{6}}, \frac{-1}{4}, -\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}\right)}{b} \]

      2. unpow3 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}}, -2, -c\right) + \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot 20}{a}}{{b}^{6}}, \frac{-1}{4}, -\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}\right)}{b} \]

      3. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left({c}^{2} \cdot c\right)}{{b}^{4}}, -2, -c\right) + \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot 20}{a}}{{b}^{6}}, \frac{-1}{4}, -\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}\right)}{b} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left({c}^{2} \cdot c\right)}{{b}^{4}}, -2, -c\right) + \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot 20}{a}}{{b}^{6}}, \frac{-1}{4}, -\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}\right)}{b} \]

      5. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}}, -2, -c\right) + \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot 20}{a}}{{b}^{6}}, \frac{-1}{4}, -\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}\right)}{b} \]

      6. lift-*.f64 94.7

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

   7. Applied rewrites 94.7%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}}, -2, -c\right) + \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot 20}{a}}{{b}^{6}}, -0.25, -\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}\right)}{b} \]
   8. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}}, -2, -c\right) + \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot 20}{a}}{{b}^{6}}, \frac{-1}{4}, -\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}\right)}{b} \]

      2. sqr-pow N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{\left(\frac{4}{2}\right)} \cdot {b}^{\left(\frac{4}{2}\right)}}, -2, -c\right) + \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot 20}{a}}{{b}^{6}}, \frac{-1}{4}, -\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}\right)}{b} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{2} \cdot {b}^{\left(\frac{4}{2}\right)}}, -2, -c\right) + \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot 20}{a}}{{b}^{6}}, \frac{-1}{4}, -\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}\right)}{b} \]

      4. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{2} \cdot {b}^{2}}, -2, -c\right) + \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot 20}{a}}{{b}^{6}}, \frac{-1}{4}, -\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}\right)}{b} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{2} \cdot {b}^{2}}, -2, -c\right) + \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot 20}{a}}{{b}^{6}}, \frac{-1}{4}, -\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}\right)}{b} \]

      6. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot {b}^{2}}, -2, -c\right) + \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot 20}{a}}{{b}^{6}}, \frac{-1}{4}, -\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}\right)}{b} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot {b}^{2}}, -2, -c\right) + \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot 20}{a}}{{b}^{6}}, \frac{-1}{4}, -\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}\right)}{b} \]

      8. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot 20}{a}}{{b}^{6}}, \frac{-1}{4}, -\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}\right)}{b} \]

      9. lift-*.f64 94.7

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

   9. Applied rewrites 94.7%

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

   10. Taylor expanded in a around 0

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

       1. associate-*r/ N/A

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

       2. lower-/.f64 N/A

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

       3. associate-*r* N/A

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

       4. mul-1-neg N/A

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

       5. lower-*.f64 N/A

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

       6. lower-neg.f64 N/A

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

       7. pow2 N/A

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

       8. lift-*.f64 N/A

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

       9. pow2 N/A

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

       10. lift-*.f64 92.3

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

   12. Applied rewrites 92.3%

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

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.2:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \frac{a \cdot \left(c \cdot c\right)}{\left(-b\right) \cdot b}}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 70:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\mathsf{fma}\left(c \cdot c, \frac{a}{b \cdot b}, c\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 70.0)
   (/ (+ (- b) (sqrt (fma b b (* -4.0 (* c a))))) (* 2.0 a))
   (/ (- (fma (* c c) (/ a (* b b)) c)) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 70.0) {
		tmp = (-b + sqrt(fma(b, b, (-4.0 * (c * a))))) / (2.0 * a);
	} else {
		tmp = -fma((c * c), (a / (b * b)), c) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 70.0)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(-4.0 * Float64(c * a))))) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(-fma(Float64(c * c), Float64(a / Float64(b * b)), c)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 70.0], 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[(a / N[(b * b), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision]) / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 70

   1. Initial program 79.3%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Add Preprocessing
   3. 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 79.5

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

   4. Applied rewrites 79.5%

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

   if 70 < b

   1. Initial program 43.3%

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

   3. Taylor expanded in b around inf

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

   5. Applied rewrites 96.1%

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

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot {c}^{3}}{{b}^{4}}, -2, -c\right) + \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot 20}{a}}{{b}^{6}}, \frac{-1}{4}, -\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}\right)}{b} \]

      2. unpow3 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}}, -2, -c\right) + \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot 20}{a}}{{b}^{6}}, \frac{-1}{4}, -\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}\right)}{b} \]

      3. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left({c}^{2} \cdot c\right)}{{b}^{4}}, -2, -c\right) + \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot 20}{a}}{{b}^{6}}, \frac{-1}{4}, -\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}\right)}{b} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left({c}^{2} \cdot c\right)}{{b}^{4}}, -2, -c\right) + \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot 20}{a}}{{b}^{6}}, \frac{-1}{4}, -\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}\right)}{b} \]

      5. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}}, -2, -c\right) + \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot 20}{a}}{{b}^{6}}, \frac{-1}{4}, -\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}\right)}{b} \]

      6. lift-*.f64 96.1

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

   7. Applied rewrites 96.1%

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

   8. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   10. Applied rewrites 89.5%

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

Alternative 7: 81.4% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.9%

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

3. Taylor expanded in b around inf

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

5. Applied rewrites 90.8%

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

   1. lift-pow.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot {c}^{3}}{{b}^{4}}, -2, -c\right) + \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot 20}{a}}{{b}^{6}}, \frac{-1}{4}, -\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}\right)}{b} \]

   2. unpow3 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}}, -2, -c\right) + \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot 20}{a}}{{b}^{6}}, \frac{-1}{4}, -\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}\right)}{b} \]

   3. pow2 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left({c}^{2} \cdot c\right)}{{b}^{4}}, -2, -c\right) + \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot 20}{a}}{{b}^{6}}, \frac{-1}{4}, -\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}\right)}{b} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left({c}^{2} \cdot c\right)}{{b}^{4}}, -2, -c\right) + \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot 20}{a}}{{b}^{6}}, \frac{-1}{4}, -\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}\right)}{b} \]

   5. pow2 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}}, -2, -c\right) + \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot 20}{a}}{{b}^{6}}, \frac{-1}{4}, -\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}\right)}{b} \]

   6. lift-*.f64 90.8

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

7. Applied rewrites 90.8%

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

8. Taylor expanded in b around inf

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

   1. lower-/.f64 N/A

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

10. Applied rewrites 81.0%

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

Alternative 8: 64.5% accurate, 3.6× 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 54.9%

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

3. Taylor expanded in a around 0

   \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
4. 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 64.5

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

5. Applied rewrites 64.5%

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

Reproduce

herbie shell --seed 2025082 
(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)))