Quadratic roots, narrow range

Percentage Accurate: 55.4% → 90.8%

Time: 20.1s

Alternatives: 7

Speedup: 29.0×

• could not determine a ground truth

  1. a = 18.183710431050983
  2. b = 6631796.952832928
  3. c = 2.7258471552021383e-7

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

real(8) function code(a, b, c)
    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: 55.4% 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

real(8) function code(a, b, c)
    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: 90.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(\left(-0.25 \cdot \left({\left(a \cdot c\right)}^{4} \cdot \frac{20}{a \cdot {b}^{7}}\right) - \frac{a \cdot {c}^{2}}{{b}^{3}}\right) - \frac{c}{b}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (+
  (* -2.0 (/ (* (pow a 2.0) (pow c 3.0)) (pow b 5.0)))
  (-
   (-
    (* -0.25 (* (pow (* a c) 4.0) (/ 20.0 (* a (pow b 7.0)))))
    (/ (* a (pow c 2.0)) (pow b 3.0)))
   (/ c b))))

C

double code(double a, double b, double c) {
	return (-2.0 * ((pow(a, 2.0) * pow(c, 3.0)) / pow(b, 5.0))) + (((-0.25 * (pow((a * c), 4.0) * (20.0 / (a * pow(b, 7.0))))) - ((a * pow(c, 2.0)) / pow(b, 3.0))) - (c / b));
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = ((-2.0d0) * (((a ** 2.0d0) * (c ** 3.0d0)) / (b ** 5.0d0))) + ((((-0.25d0) * (((a * c) ** 4.0d0) * (20.0d0 / (a * (b ** 7.0d0))))) - ((a * (c ** 2.0d0)) / (b ** 3.0d0))) - (c / b))
end function

Java

public static double code(double a, double b, double c) {
	return (-2.0 * ((Math.pow(a, 2.0) * Math.pow(c, 3.0)) / Math.pow(b, 5.0))) + (((-0.25 * (Math.pow((a * c), 4.0) * (20.0 / (a * Math.pow(b, 7.0))))) - ((a * Math.pow(c, 2.0)) / Math.pow(b, 3.0))) - (c / b));
}

Python

def code(a, b, c):
	return (-2.0 * ((math.pow(a, 2.0) * math.pow(c, 3.0)) / math.pow(b, 5.0))) + (((-0.25 * (math.pow((a * c), 4.0) * (20.0 / (a * math.pow(b, 7.0))))) - ((a * math.pow(c, 2.0)) / math.pow(b, 3.0))) - (c / b))

Julia

function code(a, b, c)
	return Float64(Float64(-2.0 * Float64(Float64((a ^ 2.0) * (c ^ 3.0)) / (b ^ 5.0))) + Float64(Float64(Float64(-0.25 * Float64((Float64(a * c) ^ 4.0) * Float64(20.0 / Float64(a * (b ^ 7.0))))) - Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0))) - Float64(c / b)))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-2.0 * (((a ^ 2.0) * (c ^ 3.0)) / (b ^ 5.0))) + (((-0.25 * (((a * c) ^ 4.0) * (20.0 / (a * (b ^ 7.0))))) - ((a * (c ^ 2.0)) / (b ^ 3.0))) - (c / b));
end

Wolfram

code[a_, b_, c_] := N[(N[(-2.0 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(-0.25 * N[(N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision] * N[(20.0 / N[(a * N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.5%

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

   1. *-commutative 55.5%

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

3. Simplified 55.5%

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

5. Taylor expanded in b around inf 92.6%

   \[\leadsto \color{blue}{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \frac{16 \cdot \left({a}^{4} \cdot {c}^{4}\right) + {\left(-2 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)}^{2}}{a \cdot {b}^{7}}\right)\right)} \]

6. Taylor expanded in c around 0 92.6%

   \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \color{blue}{\frac{{c}^{4} \cdot \left(4 \cdot {a}^{4} + 16 \cdot {a}^{4}\right)}{a \cdot {b}^{7}}}\right)\right) \]
7. Step-by-step derivation

8. Simplified 92.6%

   \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \color{blue}{\left({\left(a \cdot c\right)}^{4} \cdot \frac{20}{a \cdot {b}^{7}}\right)}\right)\right) \]

9. Final simplification 92.6%

   \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(\left(-0.25 \cdot \left({\left(a \cdot c\right)}^{4} \cdot \frac{20}{a \cdot {b}^{7}}\right) - \frac{a \cdot {c}^{2}}{{b}^{3}}\right) - \frac{c}{b}\right) \]
10. Add Preprocessing

Alternative 2: 84.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -0.8:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b} - a \cdot \left(\frac{c}{b} \cdot \frac{c}{{b}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0)) -0.8)
   (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* a 2.0))
   (- (/ c (- b)) (* a (* (/ c b) (/ c (pow b 2.0)))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)) <= -0.8) {
		tmp = (sqrt(fma(b, b, (c * (a * -4.0)))) - b) / (a * 2.0);
	} else {
		tmp = (c / -b) - (a * ((c / b) * (c / pow(b, 2.0))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0)) <= -0.8)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(c / Float64(-b)) - Float64(a * Float64(Float64(c / b) * Float64(c / (b ^ 2.0)))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -0.8], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(c / (-b)), $MachinePrecision] - N[(a * N[(N[(c / b), $MachinePrecision] * N[(c / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) < -0.80000000000000004

   1. Initial program 82.4%

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

      1. *-commutative 82.4%

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

      2. +-commutative 82.4%

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

      3. sqr-neg 82.4%

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

      4. unsub-neg 82.4%

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

      5. sqr-neg 82.4%

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

      6. fma-neg 82.6%

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

      7. distribute-lft-neg-in 82.6%

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

      8. *-commutative 82.6%

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

      9. *-commutative 82.6%

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

      10. distribute-rgt-neg-in 82.6%

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

      11. metadata-eval 82.6%

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

   3. Simplified 82.6%

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

   if -0.80000000000000004 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))

   1. Initial program 49.4%

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

      1. *-commutative 49.4%

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

   3. Simplified 49.4%

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

   5. Taylor expanded in b around inf 88.4%

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

      1. distribute-lft-out 88.4%

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

      2. associate-/l* 88.4%

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

   7. Simplified 88.4%

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

      1. unpow2 88.4%

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

      2. unpow3 88.4%

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

      3. times-frac 88.4%

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

      4. pow2 88.4%

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

   9. Applied egg-rr 88.4%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -0.8:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b} - a \cdot \left(\frac{c}{b} \cdot \frac{c}{{b}^{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.6:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} - \left(\frac{c}{b} + \frac{c}{b \cdot \frac{b}{c}} \cdot \frac{a}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 2.6)
   (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0))
   (-
    (* -2.0 (/ (* (pow a 2.0) (pow c 3.0)) (pow b 5.0)))
    (+ (/ c b) (* (/ c (* b (/ b c))) (/ a b))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.6) {
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	} else {
		tmp = (-2.0 * ((pow(a, 2.0) * pow(c, 3.0)) / pow(b, 5.0))) - ((c / b) + ((c / (b * (b / c))) * (a / b)));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 2.6d0) then
        tmp = (sqrt(((b * b) - (c * (a * 4.0d0)))) - b) / (a * 2.0d0)
    else
        tmp = ((-2.0d0) * (((a ** 2.0d0) * (c ** 3.0d0)) / (b ** 5.0d0))) - ((c / b) + ((c / (b * (b / c))) * (a / b)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.6) {
		tmp = (Math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	} else {
		tmp = (-2.0 * ((Math.pow(a, 2.0) * Math.pow(c, 3.0)) / Math.pow(b, 5.0))) - ((c / b) + ((c / (b * (b / c))) * (a / b)));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 2.6:
		tmp = (math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)
	else:
		tmp = (-2.0 * ((math.pow(a, 2.0) * math.pow(c, 3.0)) / math.pow(b, 5.0))) - ((c / b) + ((c / (b * (b / c))) * (a / b)))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 2.6)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(-2.0 * Float64(Float64((a ^ 2.0) * (c ^ 3.0)) / (b ^ 5.0))) - Float64(Float64(c / b) + Float64(Float64(c / Float64(b * Float64(b / c))) * Float64(a / b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 2.6)
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	else
		tmp = (-2.0 * (((a ^ 2.0) * (c ^ 3.0)) / (b ^ 5.0))) - ((c / b) + ((c / (b * (b / c))) * (a / b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 2.6], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(c / b), $MachinePrecision] + N[(N[(c / N[(b * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.60000000000000009

   1. Initial program 83.8%

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

   if 2.60000000000000009 < b

   1. Initial program 50.4%

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

      1. *-commutative 50.4%

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

   3. Simplified 50.4%

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

   5. Taylor expanded in b around inf 93.5%

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

      1. *-commutative 93.5%

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

      2. unpow3 93.5%

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

      3. times-frac 93.5%

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

      4. unpow2 93.5%

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

      5. frac-times 93.5%

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

      6. pow2 93.5%

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

   7. Applied egg-rr 93.5%

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

      1. unpow2 93.5%

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

      2. clear-num 93.5%

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

      3. frac-times 93.5%

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

      4. *-un-lft-identity 93.5%

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

   9. Applied egg-rr 93.5%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.6:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} - \left(\frac{c}{b} + \frac{c}{b \cdot \frac{b}{c}} \cdot \frac{a}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{if}\;t\_0 \leq -0.8:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b} - a \cdot \left(\frac{c}{b} \cdot \frac{c}{{b}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0))))
   (if (<= t_0 -0.8) t_0 (- (/ c (- b)) (* a (* (/ c b) (/ c (pow b 2.0))))))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	double tmp;
	if (t_0 <= -0.8) {
		tmp = t_0;
	} else {
		tmp = (c / -b) - (a * ((c / b) * (c / pow(b, 2.0))));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (sqrt(((b * b) - (c * (a * 4.0d0)))) - b) / (a * 2.0d0)
    if (t_0 <= (-0.8d0)) then
        tmp = t_0
    else
        tmp = (c / -b) - (a * ((c / b) * (c / (b ** 2.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (Math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	double tmp;
	if (t_0 <= -0.8) {
		tmp = t_0;
	} else {
		tmp = (c / -b) - (a * ((c / b) * (c / Math.pow(b, 2.0))));
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)
	tmp = 0
	if t_0 <= -0.8:
		tmp = t_0
	else:
		tmp = (c / -b) - (a * ((c / b) * (c / math.pow(b, 2.0))))
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0))
	tmp = 0.0
	if (t_0 <= -0.8)
		tmp = t_0;
	else
		tmp = Float64(Float64(c / Float64(-b)) - Float64(a * Float64(Float64(c / b) * Float64(c / (b ^ 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	tmp = 0.0;
	if (t_0 <= -0.8)
		tmp = t_0;
	else
		tmp = (c / -b) - (a * ((c / b) * (c / (b ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.8], t$95$0, N[(N[(c / (-b)), $MachinePrecision] - N[(a * N[(N[(c / b), $MachinePrecision] * N[(c / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) < -0.80000000000000004

   1. Initial program 82.4%

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

   if -0.80000000000000004 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))

   1. Initial program 49.4%

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

      1. *-commutative 49.4%

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

   3. Simplified 49.4%

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

   5. Taylor expanded in b around inf 88.4%

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

      1. distribute-lft-out 88.4%

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

      2. associate-/l* 88.4%

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

   7. Simplified 88.4%

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

      1. unpow2 88.4%

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

      2. unpow3 88.4%

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

      3. times-frac 88.4%

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

      4. pow2 88.4%

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

   9. Applied egg-rr 88.4%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -0.8:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b} - a \cdot \left(\frac{c}{b} \cdot \frac{c}{{b}^{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.5% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (- (/ c (- b)) (* a (* (/ c b) (/ c (pow b 2.0))))))

C

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

Fortran

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

Java

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

Python

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

Julia

function code(a, b, c)
	return Float64(Float64(c / Float64(-b)) - Float64(a * Float64(Float64(c / b) * Float64(c / (b ^ 2.0)))))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.5%

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

   1. *-commutative 55.5%

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

3. Simplified 55.5%

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

5. Taylor expanded in b around inf 83.0%

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

   1. distribute-lft-out 83.0%

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

   2. associate-/l* 83.0%

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

7. Simplified 83.0%

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

   1. unpow2 83.0%

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

   2. unpow3 83.0%

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

   3. times-frac 83.0%

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

   4. pow2 83.0%

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

9. Applied egg-rr 83.0%

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

10. Final simplification 83.0%

    \[\leadsto \frac{c}{-b} - a \cdot \left(\frac{c}{b} \cdot \frac{c}{{b}^{2}}\right) \]
11. Add Preprocessing

Alternative 6: 64.2% accurate, 29.0× 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

real(8) function code(a, b, c)
    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(c / Float64(-b))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.5%

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

   1. *-commutative 55.5%

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

3. Simplified 55.5%

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

5. Taylor expanded in b around inf 64.3%

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

   1. mul-1-neg 64.3%

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

   2. distribute-neg-frac 64.3%

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

7. Simplified 64.3%

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

8. Final simplification 64.3%

   \[\leadsto \frac{c}{-b} \]
9. Add Preprocessing

Alternative 7: 3.2% accurate, 116.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 0.0)

C

double code(double a, double b, double c) {
	return 0.0;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = 0.0d0
end function

Java

public static double code(double a, double b, double c) {
	return 0.0;
}

Python

def code(a, b, c):
	return 0.0

Julia

function code(a, b, c)
	return 0.0
end

MATLAB

function tmp = code(a, b, c)
	tmp = 0.0;
end

Wolfram

code[a_, b_, c_] := 0.0

Derivation

1. Initial program 55.5%

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

   1. *-commutative 55.5%

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

3. Simplified 55.5%

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

5. Taylor expanded in b around inf 64.2%

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

   1. *-commutative 64.2%

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

   2. associate-/l* 64.2%

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

   3. associate-*l* 64.2%

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

7. Simplified 64.2%

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

   1. expm1-log1p-u 57.1%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{a \cdot \left(\frac{c}{b} \cdot -2\right)}{a \cdot 2}\right)\right)} \]

   2. expm1-undefine 44.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{a \cdot \left(\frac{c}{b} \cdot -2\right)}{a \cdot 2}\right)} - 1} \]

   3. times-frac 44.8%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{a}{a} \cdot \frac{\frac{c}{b} \cdot -2}{2}}\right)} - 1 \]

   4. pow1 44.8%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{{a}^{1}}}{a} \cdot \frac{\frac{c}{b} \cdot -2}{2}\right)} - 1 \]

   5. pow1 44.8%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{{a}^{1}}{\color{blue}{{a}^{1}}} \cdot \frac{\frac{c}{b} \cdot -2}{2}\right)} - 1 \]

   6. pow-div 44.8%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{{a}^{\left(1 - 1\right)}} \cdot \frac{\frac{c}{b} \cdot -2}{2}\right)} - 1 \]

   7. metadata-eval 44.8%

      \[\leadsto e^{\mathsf{log1p}\left({a}^{\color{blue}{0}} \cdot \frac{\frac{c}{b} \cdot -2}{2}\right)} - 1 \]

   8. metadata-eval 44.8%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{1} \cdot \frac{\frac{c}{b} \cdot -2}{2}\right)} - 1 \]

9. Applied egg-rr 44.8%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(1 \cdot \frac{\frac{c}{b} \cdot -2}{2}\right)} - 1} \]
10. Step-by-step derivation

    1. sub-neg 44.8%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(1 \cdot \frac{\frac{c}{b} \cdot -2}{2}\right)} + \left(-1\right)} \]

    2. metadata-eval 44.8%

       \[\leadsto e^{\mathsf{log1p}\left(1 \cdot \frac{\frac{c}{b} \cdot -2}{2}\right)} + \color{blue}{-1} \]

    3. +-commutative 44.8%

       \[\leadsto \color{blue}{-1 + e^{\mathsf{log1p}\left(1 \cdot \frac{\frac{c}{b} \cdot -2}{2}\right)}} \]

    4. log1p-undefine 44.8%

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

    5. rem-exp-log 52.0%

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

    6. *-lft-identity 52.0%

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

    7. associate-/l* 52.0%

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

    8. metadata-eval 52.0%

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

    9. *-commutative 52.0%

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

    10. mul-1-neg 52.0%

        \[\leadsto -1 + \left(1 + \color{blue}{\left(-\frac{c}{b}\right)}\right) \]

    11. unsub-neg 52.0%

        \[\leadsto -1 + \color{blue}{\left(1 - \frac{c}{b}\right)} \]

11. Simplified 52.0%

    \[\leadsto \color{blue}{-1 + \left(1 - \frac{c}{b}\right)} \]

12. Taylor expanded in c around 0 3.2%

    \[\leadsto -1 + \color{blue}{1} \]

13. Final simplification 3.2%

    \[\leadsto 0 \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024046 
(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)))