Diagrams.Solve.Polynomial:cubForm from diagrams-solve-0.1, J

Percentage Accurate: 79.4% → 90.0%

Time: 15.8s

Alternatives: 20

Speedup: 0.5×

• 37.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	return ((((x * 9.0) * y) - (((z * 4.0) * t) * a)) + b) / (z * c);
}

Fortran

real(8) function code(x, y, z, t, a, b, c)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = ((((x * 9.0d0) * y) - (((z * 4.0d0) * t) * a)) + b) / (z * c)
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c) {
	return ((((x * 9.0) * y) - (((z * 4.0) * t) * a)) + b) / (z * c);
}

Python

def code(x, y, z, t, a, b, c):
	return ((((x * 9.0) * y) - (((z * 4.0) * t) * a)) + b) / (z * c)

Julia

function code(x, y, z, t, a, b, c)
	return Float64(Float64(Float64(Float64(Float64(x * 9.0) * y) - Float64(Float64(Float64(z * 4.0) * t) * a)) + b) / Float64(z * c))
end

MATLAB

function tmp = code(x, y, z, t, a, b, c)
	tmp = ((((x * 9.0) * y) - (((z * 4.0) * t) * a)) + b) / (z * c);
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := N[(N[(N[(N[(N[(x * 9.0), $MachinePrecision] * y), $MachinePrecision] - N[(N[(N[(z * 4.0), $MachinePrecision] * t), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision] / N[(z * c), $MachinePrecision]), $MachinePrecision]

Initial Program: 79.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	return ((((x * 9.0) * y) - (((z * 4.0) * t) * a)) + b) / (z * c);
}

Fortran

real(8) function code(x, y, z, t, a, b, c)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = ((((x * 9.0d0) * y) - (((z * 4.0d0) * t) * a)) + b) / (z * c)
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c) {
	return ((((x * 9.0) * y) - (((z * 4.0) * t) * a)) + b) / (z * c);
}

Python

def code(x, y, z, t, a, b, c):
	return ((((x * 9.0) * y) - (((z * 4.0) * t) * a)) + b) / (z * c)

Julia

function code(x, y, z, t, a, b, c)
	return Float64(Float64(Float64(Float64(Float64(x * 9.0) * y) - Float64(Float64(Float64(z * 4.0) * t) * a)) + b) / Float64(z * c))
end

MATLAB

function tmp = code(x, y, z, t, a, b, c)
	tmp = ((((x * 9.0) * y) - (((z * 4.0) * t) * a)) + b) / (z * c);
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := N[(N[(N[(N[(N[(x * 9.0), $MachinePrecision] * y), $MachinePrecision] - N[(N[(N[(z * 4.0), $MachinePrecision] * t), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision] / N[(z * c), $MachinePrecision]), $MachinePrecision]

Alternative 1: 90.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ c\_s \cdot \begin{array}{l} \mathbf{if}\;c\_m \leq 5.2 \cdot 10^{+44}:\\ \;\;\;\;\frac{b - \left(\left(z \cdot 4\right) \cdot \left(t \cdot a\right) - x \cdot \left(9 \cdot y\right)\right)}{c\_m \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4, z \cdot \left(a \cdot \frac{t}{c\_m}\right), \mathsf{fma}\left(9, x \cdot \frac{y}{c\_m}, \frac{b}{c\_m}\right)\right)}{z}\\ \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (*
  c_s
  (if (<= c_m 5.2e+44)
    (/ (- b (- (* (* z 4.0) (* t a)) (* x (* 9.0 y)))) (* c_m z))
    (/
     (fma -4.0 (* z (* a (/ t c_m))) (fma 9.0 (* x (/ y c_m)) (/ b c_m)))
     z))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double tmp;
	if (c_m <= 5.2e+44) {
		tmp = (b - (((z * 4.0) * (t * a)) - (x * (9.0 * y)))) / (c_m * z);
	} else {
		tmp = fma(-4.0, (z * (a * (t / c_m))), fma(9.0, (x * (y / c_m)), (b / c_m))) / z;
	}
	return c_s * tmp;
}

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
function code(c_s, x, y, z, t, a, b, c_m)
	tmp = 0.0
	if (c_m <= 5.2e+44)
		tmp = Float64(Float64(b - Float64(Float64(Float64(z * 4.0) * Float64(t * a)) - Float64(x * Float64(9.0 * y)))) / Float64(c_m * z));
	else
		tmp = Float64(fma(-4.0, Float64(z * Float64(a * Float64(t / c_m))), fma(9.0, Float64(x * Float64(y / c_m)), Float64(b / c_m))) / z);
	end
	return Float64(c_s * tmp)
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := N[(c$95$s * If[LessEqual[c$95$m, 5.2e+44], N[(N[(b - N[(N[(N[(z * 4.0), $MachinePrecision] * N[(t * a), $MachinePrecision]), $MachinePrecision] - N[(x * N[(9.0 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision], N[(N[(-4.0 * N[(z * N[(a * N[(t / c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(9.0 * N[(x * N[(y / c$95$m), $MachinePrecision]), $MachinePrecision] + N[(b / c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if c < 5.1999999999999998e44

   1. Initial program 83.4%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Step-by-step derivation

      1. +-commutative 83.4%

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)}}{z \cdot c} \]

      2. associate-+r- 83.4%

         \[\leadsto \frac{\color{blue}{\left(b + \left(x \cdot 9\right) \cdot y\right) - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}}{z \cdot c} \]

      3. *-commutative 83.4%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{a \cdot \left(\left(z \cdot 4\right) \cdot t\right)}}{z \cdot c} \]

      4. associate-*r* 85.3%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{\left(a \cdot \left(z \cdot 4\right)\right) \cdot t}}{z \cdot c} \]

      5. *-commutative 85.3%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{\left(\left(z \cdot 4\right) \cdot a\right)} \cdot t}{z \cdot c} \]

      6. associate-+r- 85.3%

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right)}}{z \cdot c} \]

      7. +-commutative 85.3%

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right) + b}}{z \cdot c} \]

      8. associate-*l* 85.3%

         \[\leadsto \frac{\left(\color{blue}{x \cdot \left(9 \cdot y\right)} - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right) + b}{z \cdot c} \]

      9. associate-*l* 86.3%

         \[\leadsto \frac{\left(x \cdot \left(9 \cdot y\right) - \color{blue}{\left(z \cdot 4\right) \cdot \left(a \cdot t\right)}\right) + b}{z \cdot c} \]

      10. *-commutative 86.3%

          \[\leadsto \frac{\left(x \cdot \left(9 \cdot y\right) - \left(z \cdot 4\right) \cdot \color{blue}{\left(t \cdot a\right)}\right) + b}{z \cdot c} \]

   3. Simplified 86.3%

      \[\leadsto \color{blue}{\frac{\left(x \cdot \left(9 \cdot y\right) - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}} \]
   4. Add Preprocessing

   if 5.1999999999999998e44 < c

   1. Initial program 68.7%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Step-by-step derivation

      1. +-commutative 68.7%

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)}}{z \cdot c} \]

      2. associate-+r- 68.7%

         \[\leadsto \frac{\color{blue}{\left(b + \left(x \cdot 9\right) \cdot y\right) - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}}{z \cdot c} \]

      3. *-commutative 68.7%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{a \cdot \left(\left(z \cdot 4\right) \cdot t\right)}}{z \cdot c} \]

      4. associate-*r* 65.6%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{\left(a \cdot \left(z \cdot 4\right)\right) \cdot t}}{z \cdot c} \]

      5. *-commutative 65.6%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{\left(\left(z \cdot 4\right) \cdot a\right)} \cdot t}{z \cdot c} \]

      6. associate-+r- 65.6%

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right)}}{z \cdot c} \]

      7. +-commutative 65.6%

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right) + b}}{z \cdot c} \]

      8. associate-*l* 67.3%

         \[\leadsto \frac{\left(\color{blue}{x \cdot \left(9 \cdot y\right)} - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right) + b}{z \cdot c} \]

      9. associate-*l* 67.4%

         \[\leadsto \frac{\left(x \cdot \left(9 \cdot y\right) - \color{blue}{\left(z \cdot 4\right) \cdot \left(a \cdot t\right)}\right) + b}{z \cdot c} \]

      10. *-commutative 67.4%

          \[\leadsto \frac{\left(x \cdot \left(9 \cdot y\right) - \left(z \cdot 4\right) \cdot \color{blue}{\left(t \cdot a\right)}\right) + b}{z \cdot c} \]

   3. Simplified 67.4%

      \[\leadsto \color{blue}{\frac{\left(x \cdot \left(9 \cdot y\right) - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 72.6%

      \[\leadsto \color{blue}{\frac{-4 \cdot \frac{a \cdot \left(t \cdot z\right)}{c} + \left(9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z}} \]
   6. Step-by-step derivation

      1. fma-define 72.6%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-4, \frac{a \cdot \left(t \cdot z\right)}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}}{z} \]

      2. *-commutative 72.6%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{\color{blue}{\left(t \cdot z\right) \cdot a}}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      3. *-commutative 72.6%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{\color{blue}{\left(z \cdot t\right)} \cdot a}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      4. associate-*r* 74.5%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{\color{blue}{z \cdot \left(t \cdot a\right)}}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      5. *-commutative 74.5%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{z \cdot \color{blue}{\left(a \cdot t\right)}}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      6. associate-/l* 81.4%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \color{blue}{z \cdot \frac{a \cdot t}{c}}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      7. associate-/l* 85.7%

         \[\leadsto \frac{\mathsf{fma}\left(-4, z \cdot \color{blue}{\left(a \cdot \frac{t}{c}\right)}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      8. fma-define 85.7%

         \[\leadsto \frac{\mathsf{fma}\left(-4, z \cdot \left(a \cdot \frac{t}{c}\right), \color{blue}{\mathsf{fma}\left(9, \frac{x \cdot y}{c}, \frac{b}{c}\right)}\right)}{z} \]

      9. associate-/l* 92.4%

         \[\leadsto \frac{\mathsf{fma}\left(-4, z \cdot \left(a \cdot \frac{t}{c}\right), \mathsf{fma}\left(9, \color{blue}{x \cdot \frac{y}{c}}, \frac{b}{c}\right)\right)}{z} \]

   7. Simplified 92.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-4, z \cdot \left(a \cdot \frac{t}{c}\right), \mathsf{fma}\left(9, x \cdot \frac{y}{c}, \frac{b}{c}\right)\right)}{z}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq 5.2 \cdot 10^{+44}:\\ \;\;\;\;\frac{b - \left(\left(z \cdot 4\right) \cdot \left(t \cdot a\right) - x \cdot \left(9 \cdot y\right)\right)}{c \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4, z \cdot \left(a \cdot \frac{t}{c}\right), \mathsf{fma}\left(9, x \cdot \frac{y}{c}, \frac{b}{c}\right)\right)}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 86.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ c\_s \cdot \begin{array}{l} \mathbf{if}\;c\_m \leq 5.5 \cdot 10^{+57}:\\ \;\;\;\;\frac{b - \left(\left(z \cdot 4\right) \cdot \left(t \cdot a\right) - x \cdot \left(9 \cdot y\right)\right)}{c\_m \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{\mathsf{fma}\left(a \cdot \frac{t}{c\_m}, -4, \frac{b}{c\_m \cdot z}\right)}{x} - \frac{\frac{y}{c\_m}}{z} \cdot -9\right)\\ \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (*
  c_s
  (if (<= c_m 5.5e+57)
    (/ (- b (- (* (* z 4.0) (* t a)) (* x (* 9.0 y)))) (* c_m z))
    (*
     x
     (-
      (/ (fma (* a (/ t c_m)) -4.0 (/ b (* c_m z))) x)
      (* (/ (/ y c_m) z) -9.0))))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double tmp;
	if (c_m <= 5.5e+57) {
		tmp = (b - (((z * 4.0) * (t * a)) - (x * (9.0 * y)))) / (c_m * z);
	} else {
		tmp = x * ((fma((a * (t / c_m)), -4.0, (b / (c_m * z))) / x) - (((y / c_m) / z) * -9.0));
	}
	return c_s * tmp;
}

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
function code(c_s, x, y, z, t, a, b, c_m)
	tmp = 0.0
	if (c_m <= 5.5e+57)
		tmp = Float64(Float64(b - Float64(Float64(Float64(z * 4.0) * Float64(t * a)) - Float64(x * Float64(9.0 * y)))) / Float64(c_m * z));
	else
		tmp = Float64(x * Float64(Float64(fma(Float64(a * Float64(t / c_m)), -4.0, Float64(b / Float64(c_m * z))) / x) - Float64(Float64(Float64(y / c_m) / z) * -9.0)));
	end
	return Float64(c_s * tmp)
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := N[(c$95$s * If[LessEqual[c$95$m, 5.5e+57], N[(N[(b - N[(N[(N[(z * 4.0), $MachinePrecision] * N[(t * a), $MachinePrecision]), $MachinePrecision] - N[(x * N[(9.0 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(N[(N[(a * N[(t / c$95$m), $MachinePrecision]), $MachinePrecision] * -4.0 + N[(b / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] - N[(N[(N[(y / c$95$m), $MachinePrecision] / z), $MachinePrecision] * -9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if c < 5.5000000000000002e57

   1. Initial program 83.0%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Step-by-step derivation

      1. +-commutative 83.0%

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)}}{z \cdot c} \]

      2. associate-+r- 83.0%

         \[\leadsto \frac{\color{blue}{\left(b + \left(x \cdot 9\right) \cdot y\right) - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}}{z \cdot c} \]

      3. *-commutative 83.0%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{a \cdot \left(\left(z \cdot 4\right) \cdot t\right)}}{z \cdot c} \]

      4. associate-*r* 85.0%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{\left(a \cdot \left(z \cdot 4\right)\right) \cdot t}}{z \cdot c} \]

      5. *-commutative 85.0%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{\left(\left(z \cdot 4\right) \cdot a\right)} \cdot t}{z \cdot c} \]

      6. associate-+r- 85.0%

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right)}}{z \cdot c} \]

      7. +-commutative 85.0%

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right) + b}}{z \cdot c} \]

      8. associate-*l* 85.0%

         \[\leadsto \frac{\left(\color{blue}{x \cdot \left(9 \cdot y\right)} - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right) + b}{z \cdot c} \]

      9. associate-*l* 86.0%

         \[\leadsto \frac{\left(x \cdot \left(9 \cdot y\right) - \color{blue}{\left(z \cdot 4\right) \cdot \left(a \cdot t\right)}\right) + b}{z \cdot c} \]

      10. *-commutative 86.0%

          \[\leadsto \frac{\left(x \cdot \left(9 \cdot y\right) - \left(z \cdot 4\right) \cdot \color{blue}{\left(t \cdot a\right)}\right) + b}{z \cdot c} \]

   3. Simplified 86.0%

      \[\leadsto \color{blue}{\frac{\left(x \cdot \left(9 \cdot y\right) - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}} \]
   4. Add Preprocessing

   if 5.5000000000000002e57 < c

   1. Initial program 69.4%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Step-by-step derivation

      1. +-commutative 69.4%

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)}}{z \cdot c} \]

      2. associate-+r- 69.4%

         \[\leadsto \frac{\color{blue}{\left(b + \left(x \cdot 9\right) \cdot y\right) - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}}{z \cdot c} \]

      3. *-commutative 69.4%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{a \cdot \left(\left(z \cdot 4\right) \cdot t\right)}}{z \cdot c} \]

      4. associate-*r* 66.2%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{\left(a \cdot \left(z \cdot 4\right)\right) \cdot t}}{z \cdot c} \]

      5. *-commutative 66.2%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{\left(\left(z \cdot 4\right) \cdot a\right)} \cdot t}{z \cdot c} \]

      6. associate-+r- 66.2%

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right)}}{z \cdot c} \]

      7. +-commutative 66.2%

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right) + b}}{z \cdot c} \]

      8. associate-*l* 67.9%

         \[\leadsto \frac{\left(\color{blue}{x \cdot \left(9 \cdot y\right)} - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right) + b}{z \cdot c} \]

      9. associate-*l* 68.0%

         \[\leadsto \frac{\left(x \cdot \left(9 \cdot y\right) - \color{blue}{\left(z \cdot 4\right) \cdot \left(a \cdot t\right)}\right) + b}{z \cdot c} \]

      10. *-commutative 68.0%

          \[\leadsto \frac{\left(x \cdot \left(9 \cdot y\right) - \left(z \cdot 4\right) \cdot \color{blue}{\left(t \cdot a\right)}\right) + b}{z \cdot c} \]

   3. Simplified 68.0%

      \[\leadsto \color{blue}{\frac{\left(x \cdot \left(9 \cdot y\right) - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around -inf 75.3%

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(-9 \cdot \frac{y}{c \cdot z} + -1 \cdot \frac{\frac{b}{c \cdot z} - 4 \cdot \frac{a \cdot t}{c}}{x}\right)\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg 75.3%

         \[\leadsto \color{blue}{-x \cdot \left(-9 \cdot \frac{y}{c \cdot z} + -1 \cdot \frac{\frac{b}{c \cdot z} - 4 \cdot \frac{a \cdot t}{c}}{x}\right)} \]

      2. *-commutative 75.3%

         \[\leadsto -\color{blue}{\left(-9 \cdot \frac{y}{c \cdot z} + -1 \cdot \frac{\frac{b}{c \cdot z} - 4 \cdot \frac{a \cdot t}{c}}{x}\right) \cdot x} \]

      3. distribute-rgt-neg-in 75.3%

         \[\leadsto \color{blue}{\left(-9 \cdot \frac{y}{c \cdot z} + -1 \cdot \frac{\frac{b}{c \cdot z} - 4 \cdot \frac{a \cdot t}{c}}{x}\right) \cdot \left(-x\right)} \]

   7. Simplified 84.3%

      \[\leadsto \color{blue}{\left(\frac{\frac{y}{c}}{z} \cdot -9 - \frac{\mathsf{fma}\left(a \cdot \frac{t}{c}, -4, \frac{b}{z \cdot c}\right)}{x}\right) \cdot \left(-x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq 5.5 \cdot 10^{+57}:\\ \;\;\;\;\frac{b - \left(\left(z \cdot 4\right) \cdot \left(t \cdot a\right) - x \cdot \left(9 \cdot y\right)\right)}{c \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{\mathsf{fma}\left(a \cdot \frac{t}{c}, -4, \frac{b}{c \cdot z}\right)}{x} - \frac{\frac{y}{c}}{z} \cdot -9\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ c\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1.32 \cdot 10^{+135} \lor \neg \left(z \leq 1.5 \cdot 10^{+20}\right):\\ \;\;\;\;y \cdot \left(\frac{\frac{b}{c\_m \cdot z} - 4 \cdot \frac{t \cdot a}{c\_m}}{y} - -9 \cdot \frac{x}{c\_m \cdot z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{b + \mathsf{fma}\left(x, 9 \cdot y, t \cdot \left(a \cdot \left(z \cdot -4\right)\right)\right)}{c\_m \cdot z}\\ \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (*
  c_s
  (if (or (<= z -1.32e+135) (not (<= z 1.5e+20)))
    (*
     y
     (-
      (/ (- (/ b (* c_m z)) (* 4.0 (/ (* t a) c_m))) y)
      (* -9.0 (/ x (* c_m z)))))
    (/ (+ b (fma x (* 9.0 y) (* t (* a (* z -4.0))))) (* c_m z)))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double tmp;
	if ((z <= -1.32e+135) || !(z <= 1.5e+20)) {
		tmp = y * ((((b / (c_m * z)) - (4.0 * ((t * a) / c_m))) / y) - (-9.0 * (x / (c_m * z))));
	} else {
		tmp = (b + fma(x, (9.0 * y), (t * (a * (z * -4.0))))) / (c_m * z);
	}
	return c_s * tmp;
}

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
function code(c_s, x, y, z, t, a, b, c_m)
	tmp = 0.0
	if ((z <= -1.32e+135) || !(z <= 1.5e+20))
		tmp = Float64(y * Float64(Float64(Float64(Float64(b / Float64(c_m * z)) - Float64(4.0 * Float64(Float64(t * a) / c_m))) / y) - Float64(-9.0 * Float64(x / Float64(c_m * z)))));
	else
		tmp = Float64(Float64(b + fma(x, Float64(9.0 * y), Float64(t * Float64(a * Float64(z * -4.0))))) / Float64(c_m * z));
	end
	return Float64(c_s * tmp)
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := N[(c$95$s * If[Or[LessEqual[z, -1.32e+135], N[Not[LessEqual[z, 1.5e+20]], $MachinePrecision]], N[(y * N[(N[(N[(N[(b / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision] - N[(4.0 * N[(N[(t * a), $MachinePrecision] / c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] - N[(-9.0 * N[(x / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b + N[(x * N[(9.0 * y), $MachinePrecision] + N[(t * N[(a * N[(z * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -1.32e135 or 1.5e20 < z

   1. Initial program 50.7%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Step-by-step derivation

      1. +-commutative 50.7%

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)}}{z \cdot c} \]

      2. associate-+r- 50.7%

         \[\leadsto \frac{\color{blue}{\left(b + \left(x \cdot 9\right) \cdot y\right) - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}}{z \cdot c} \]

      3. *-commutative 50.7%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{a \cdot \left(\left(z \cdot 4\right) \cdot t\right)}}{z \cdot c} \]

      4. associate-*r* 54.4%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{\left(a \cdot \left(z \cdot 4\right)\right) \cdot t}}{z \cdot c} \]

      5. *-commutative 54.4%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{\left(\left(z \cdot 4\right) \cdot a\right)} \cdot t}{z \cdot c} \]

      6. associate-+r- 54.4%

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right)}}{z \cdot c} \]

      7. +-commutative 54.4%

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right) + b}}{z \cdot c} \]

      8. associate-*l* 54.3%

         \[\leadsto \frac{\left(\color{blue}{x \cdot \left(9 \cdot y\right)} - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right) + b}{z \cdot c} \]

      9. associate-*l* 58.2%

         \[\leadsto \frac{\left(x \cdot \left(9 \cdot y\right) - \color{blue}{\left(z \cdot 4\right) \cdot \left(a \cdot t\right)}\right) + b}{z \cdot c} \]

      10. *-commutative 58.2%

          \[\leadsto \frac{\left(x \cdot \left(9 \cdot y\right) - \left(z \cdot 4\right) \cdot \color{blue}{\left(t \cdot a\right)}\right) + b}{z \cdot c} \]

   3. Simplified 58.2%

      \[\leadsto \color{blue}{\frac{\left(x \cdot \left(9 \cdot y\right) - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around -inf 73.5%

      \[\leadsto \color{blue}{-1 \cdot \left(y \cdot \left(-9 \cdot \frac{x}{c \cdot z} + -1 \cdot \frac{\frac{b}{c \cdot z} - 4 \cdot \frac{a \cdot t}{c}}{y}\right)\right)} \]

   if -1.32e135 < z < 1.5e20

   1. Initial program 93.7%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]

   3. Simplified 93.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, 9 \cdot y, t \cdot \left(a \cdot \left(z \cdot -4\right)\right)\right) + b}{z \cdot c}} \]
   4. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.32 \cdot 10^{+135} \lor \neg \left(z \leq 1.5 \cdot 10^{+20}\right):\\ \;\;\;\;y \cdot \left(\frac{\frac{b}{c \cdot z} - 4 \cdot \frac{t \cdot a}{c}}{y} - -9 \cdot \frac{x}{c \cdot z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{b + \mathsf{fma}\left(x, 9 \cdot y, t \cdot \left(a \cdot \left(z \cdot -4\right)\right)\right)}{c \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ \begin{array}{l} t_1 := \frac{b + \left(y \cdot \left(x \cdot 9\right) - a \cdot \left(\left(z \cdot 4\right) \cdot t\right)\right)}{c\_m \cdot z}\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+274}:\\ \;\;\;\;\frac{b - \left(\left(z \cdot 4\right) \cdot \left(t \cdot a\right) - x \cdot \left(9 \cdot y\right)\right)}{c\_m \cdot z}\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{\frac{b + \left(-4 \cdot \left(a \cdot \left(z \cdot t\right)\right) + 9 \cdot \left(x \cdot y\right)\right)}{c\_m}}{z}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{\frac{b}{c\_m \cdot t} - a \cdot \left(4 \cdot \frac{z}{c\_m}\right)}{z}\\ \end{array} \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (let* ((t_1 (/ (+ b (- (* y (* x 9.0)) (* a (* (* z 4.0) t)))) (* c_m z))))
   (*
    c_s
    (if (<= t_1 -5e+274)
      (/ (- b (- (* (* z 4.0) (* t a)) (* x (* 9.0 y)))) (* c_m z))
      (if (<= t_1 0.0)
        (/ (/ (+ b (+ (* -4.0 (* a (* z t))) (* 9.0 (* x y)))) c_m) z)
        (if (<= t_1 INFINITY)
          t_1
          (* t (/ (- (/ b (* c_m t)) (* a (* 4.0 (/ z c_m)))) z))))))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double t_1 = (b + ((y * (x * 9.0)) - (a * ((z * 4.0) * t)))) / (c_m * z);
	double tmp;
	if (t_1 <= -5e+274) {
		tmp = (b - (((z * 4.0) * (t * a)) - (x * (9.0 * y)))) / (c_m * z);
	} else if (t_1 <= 0.0) {
		tmp = ((b + ((-4.0 * (a * (z * t))) + (9.0 * (x * y)))) / c_m) / z;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = t * (((b / (c_m * t)) - (a * (4.0 * (z / c_m)))) / z);
	}
	return c_s * tmp;
}

Java

c\_m = Math.abs(c);
c\_s = Math.copySign(1.0, c);
public static double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double t_1 = (b + ((y * (x * 9.0)) - (a * ((z * 4.0) * t)))) / (c_m * z);
	double tmp;
	if (t_1 <= -5e+274) {
		tmp = (b - (((z * 4.0) * (t * a)) - (x * (9.0 * y)))) / (c_m * z);
	} else if (t_1 <= 0.0) {
		tmp = ((b + ((-4.0 * (a * (z * t))) + (9.0 * (x * y)))) / c_m) / z;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else {
		tmp = t * (((b / (c_m * t)) - (a * (4.0 * (z / c_m)))) / z);
	}
	return c_s * tmp;
}

Python

c\_m = math.fabs(c)
c\_s = math.copysign(1.0, c)
def code(c_s, x, y, z, t, a, b, c_m):
	t_1 = (b + ((y * (x * 9.0)) - (a * ((z * 4.0) * t)))) / (c_m * z)
	tmp = 0
	if t_1 <= -5e+274:
		tmp = (b - (((z * 4.0) * (t * a)) - (x * (9.0 * y)))) / (c_m * z)
	elif t_1 <= 0.0:
		tmp = ((b + ((-4.0 * (a * (z * t))) + (9.0 * (x * y)))) / c_m) / z
	elif t_1 <= math.inf:
		tmp = t_1
	else:
		tmp = t * (((b / (c_m * t)) - (a * (4.0 * (z / c_m)))) / z)
	return c_s * tmp

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
function code(c_s, x, y, z, t, a, b, c_m)
	t_1 = Float64(Float64(b + Float64(Float64(y * Float64(x * 9.0)) - Float64(a * Float64(Float64(z * 4.0) * t)))) / Float64(c_m * z))
	tmp = 0.0
	if (t_1 <= -5e+274)
		tmp = Float64(Float64(b - Float64(Float64(Float64(z * 4.0) * Float64(t * a)) - Float64(x * Float64(9.0 * y)))) / Float64(c_m * z));
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(Float64(b + Float64(Float64(-4.0 * Float64(a * Float64(z * t))) + Float64(9.0 * Float64(x * y)))) / c_m) / z);
	elseif (t_1 <= Inf)
		tmp = t_1;
	else
		tmp = Float64(t * Float64(Float64(Float64(b / Float64(c_m * t)) - Float64(a * Float64(4.0 * Float64(z / c_m)))) / z));
	end
	return Float64(c_s * tmp)
end

MATLAB

c\_m = abs(c);
c\_s = sign(c) * abs(1.0);
function tmp_2 = code(c_s, x, y, z, t, a, b, c_m)
	t_1 = (b + ((y * (x * 9.0)) - (a * ((z * 4.0) * t)))) / (c_m * z);
	tmp = 0.0;
	if (t_1 <= -5e+274)
		tmp = (b - (((z * 4.0) * (t * a)) - (x * (9.0 * y)))) / (c_m * z);
	elseif (t_1 <= 0.0)
		tmp = ((b + ((-4.0 * (a * (z * t))) + (9.0 * (x * y)))) / c_m) / z;
	elseif (t_1 <= Inf)
		tmp = t_1;
	else
		tmp = t * (((b / (c_m * t)) - (a * (4.0 * (z / c_m)))) / z);
	end
	tmp_2 = c_s * tmp;
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := Block[{t$95$1 = N[(N[(b + N[(N[(y * N[(x * 9.0), $MachinePrecision]), $MachinePrecision] - N[(a * N[(N[(z * 4.0), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision]}, N[(c$95$s * If[LessEqual[t$95$1, -5e+274], N[(N[(b - N[(N[(N[(z * 4.0), $MachinePrecision] * N[(t * a), $MachinePrecision]), $MachinePrecision] - N[(x * N[(9.0 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(N[(N[(b + N[(N[(-4.0 * N[(a * N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(9.0 * N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c$95$m), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$1, Infinity], t$95$1, N[(t * N[(N[(N[(b / N[(c$95$m * t), $MachinePrecision]), $MachinePrecision] - N[(a * N[(4.0 * N[(z / c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x #s(literal 9 binary64)) y) (*.f64 (*.f64 (*.f64 z #s(literal 4 binary64)) t) a)) b) (*.f64 z c)) < -4.9999999999999998e274

   1. Initial program 85.4%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Step-by-step derivation

      1. +-commutative 85.4%

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)}}{z \cdot c} \]

      2. associate-+r- 85.4%

         \[\leadsto \frac{\color{blue}{\left(b + \left(x \cdot 9\right) \cdot y\right) - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}}{z \cdot c} \]

      3. *-commutative 85.4%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{a \cdot \left(\left(z \cdot 4\right) \cdot t\right)}}{z \cdot c} \]

      4. associate-*r* 90.3%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{\left(a \cdot \left(z \cdot 4\right)\right) \cdot t}}{z \cdot c} \]

      5. *-commutative 90.3%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{\left(\left(z \cdot 4\right) \cdot a\right)} \cdot t}{z \cdot c} \]

      6. associate-+r- 90.3%

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right)}}{z \cdot c} \]

      7. +-commutative 90.3%

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right) + b}}{z \cdot c} \]

      8. associate-*l* 90.3%

         \[\leadsto \frac{\left(\color{blue}{x \cdot \left(9 \cdot y\right)} - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right) + b}{z \cdot c} \]

      9. associate-*l* 90.3%

         \[\leadsto \frac{\left(x \cdot \left(9 \cdot y\right) - \color{blue}{\left(z \cdot 4\right) \cdot \left(a \cdot t\right)}\right) + b}{z \cdot c} \]

      10. *-commutative 90.3%

          \[\leadsto \frac{\left(x \cdot \left(9 \cdot y\right) - \left(z \cdot 4\right) \cdot \color{blue}{\left(t \cdot a\right)}\right) + b}{z \cdot c} \]

   3. Simplified 90.3%

      \[\leadsto \color{blue}{\frac{\left(x \cdot \left(9 \cdot y\right) - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}} \]
   4. Add Preprocessing

   if -4.9999999999999998e274 < (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x #s(literal 9 binary64)) y) (*.f64 (*.f64 (*.f64 z #s(literal 4 binary64)) t) a)) b) (*.f64 z c)) < -0.0

   1. Initial program 87.9%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Step-by-step derivation

      1. +-commutative 87.9%

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)}}{z \cdot c} \]

      2. associate-+r- 87.9%

         \[\leadsto \frac{\color{blue}{\left(b + \left(x \cdot 9\right) \cdot y\right) - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}}{z \cdot c} \]

      3. *-commutative 87.9%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{a \cdot \left(\left(z \cdot 4\right) \cdot t\right)}}{z \cdot c} \]

      4. associate-*r* 86.0%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{\left(a \cdot \left(z \cdot 4\right)\right) \cdot t}}{z \cdot c} \]

      5. *-commutative 86.0%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{\left(\left(z \cdot 4\right) \cdot a\right)} \cdot t}{z \cdot c} \]

      6. associate-+r- 86.0%

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right)}}{z \cdot c} \]

      7. +-commutative 86.0%

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right) + b}}{z \cdot c} \]

      8. associate-*l* 86.0%

         \[\leadsto \frac{\left(\color{blue}{x \cdot \left(9 \cdot y\right)} - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right) + b}{z \cdot c} \]

      9. associate-*l* 88.0%

         \[\leadsto \frac{\left(x \cdot \left(9 \cdot y\right) - \color{blue}{\left(z \cdot 4\right) \cdot \left(a \cdot t\right)}\right) + b}{z \cdot c} \]

      10. *-commutative 88.0%

          \[\leadsto \frac{\left(x \cdot \left(9 \cdot y\right) - \left(z \cdot 4\right) \cdot \color{blue}{\left(t \cdot a\right)}\right) + b}{z \cdot c} \]

   3. Simplified 88.0%

      \[\leadsto \color{blue}{\frac{\left(x \cdot \left(9 \cdot y\right) - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 95.5%

      \[\leadsto \color{blue}{\frac{-4 \cdot \frac{a \cdot \left(t \cdot z\right)}{c} + \left(9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z}} \]
   6. Step-by-step derivation

      1. fma-define 95.5%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-4, \frac{a \cdot \left(t \cdot z\right)}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}}{z} \]

      2. *-commutative 95.5%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{\color{blue}{\left(t \cdot z\right) \cdot a}}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      3. *-commutative 95.5%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{\color{blue}{\left(z \cdot t\right)} \cdot a}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      4. associate-*r* 95.6%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{\color{blue}{z \cdot \left(t \cdot a\right)}}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      5. *-commutative 95.6%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{z \cdot \color{blue}{\left(a \cdot t\right)}}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      6. associate-/l* 95.5%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \color{blue}{z \cdot \frac{a \cdot t}{c}}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      7. associate-/l* 94.8%

         \[\leadsto \frac{\mathsf{fma}\left(-4, z \cdot \color{blue}{\left(a \cdot \frac{t}{c}\right)}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      8. fma-define 94.8%

         \[\leadsto \frac{\mathsf{fma}\left(-4, z \cdot \left(a \cdot \frac{t}{c}\right), \color{blue}{\mathsf{fma}\left(9, \frac{x \cdot y}{c}, \frac{b}{c}\right)}\right)}{z} \]

      9. associate-/l* 94.8%

         \[\leadsto \frac{\mathsf{fma}\left(-4, z \cdot \left(a \cdot \frac{t}{c}\right), \mathsf{fma}\left(9, \color{blue}{x \cdot \frac{y}{c}}, \frac{b}{c}\right)\right)}{z} \]

   7. Simplified 94.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-4, z \cdot \left(a \cdot \frac{t}{c}\right), \mathsf{fma}\left(9, x \cdot \frac{y}{c}, \frac{b}{c}\right)\right)}{z}} \]

   8. Taylor expanded in c around 0 95.6%

      \[\leadsto \frac{\color{blue}{\frac{b + \left(-4 \cdot \left(a \cdot \left(t \cdot z\right)\right) + 9 \cdot \left(x \cdot y\right)\right)}{c}}}{z} \]

   if -0.0 < (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x #s(literal 9 binary64)) y) (*.f64 (*.f64 (*.f64 z #s(literal 4 binary64)) t) a)) b) (*.f64 z c)) < +inf.0

   1. Initial program 87.1%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing

   if +inf.0 < (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x #s(literal 9 binary64)) y) (*.f64 (*.f64 (*.f64 z #s(literal 4 binary64)) t) a)) b) (*.f64 z c))

   1. Initial program 0.0%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Step-by-step derivation

      1. +-commutative 0.0%

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)}}{z \cdot c} \]

      2. associate-+r- 0.0%

         \[\leadsto \frac{\color{blue}{\left(b + \left(x \cdot 9\right) \cdot y\right) - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}}{z \cdot c} \]

      3. *-commutative 0.0%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{a \cdot \left(\left(z \cdot 4\right) \cdot t\right)}}{z \cdot c} \]

      4. associate-*r* 6.2%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{\left(a \cdot \left(z \cdot 4\right)\right) \cdot t}}{z \cdot c} \]

      5. *-commutative 6.2%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{\left(\left(z \cdot 4\right) \cdot a\right)} \cdot t}{z \cdot c} \]

      6. associate-+r- 6.2%

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right)}}{z \cdot c} \]

      7. +-commutative 6.2%

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right) + b}}{z \cdot c} \]

      8. associate-*l* 6.2%

         \[\leadsto \frac{\left(\color{blue}{x \cdot \left(9 \cdot y\right)} - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right) + b}{z \cdot c} \]

      9. associate-*l* 6.2%

         \[\leadsto \frac{\left(x \cdot \left(9 \cdot y\right) - \color{blue}{\left(z \cdot 4\right) \cdot \left(a \cdot t\right)}\right) + b}{z \cdot c} \]

      10. *-commutative 6.2%

          \[\leadsto \frac{\left(x \cdot \left(9 \cdot y\right) - \left(z \cdot 4\right) \cdot \color{blue}{\left(t \cdot a\right)}\right) + b}{z \cdot c} \]

   3. Simplified 6.2%

      \[\leadsto \color{blue}{\frac{\left(x \cdot \left(9 \cdot y\right) - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 0.0%

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

   6. Taylor expanded in t around -inf 65.8%

      \[\leadsto \color{blue}{-1 \cdot \left(t \cdot \left(-1 \cdot \frac{b}{c \cdot \left(t \cdot z\right)} + 4 \cdot \frac{a}{c}\right)\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg 65.8%

         \[\leadsto \color{blue}{-t \cdot \left(-1 \cdot \frac{b}{c \cdot \left(t \cdot z\right)} + 4 \cdot \frac{a}{c}\right)} \]

      2. distribute-rgt-neg-in 65.8%

         \[\leadsto \color{blue}{t \cdot \left(-\left(-1 \cdot \frac{b}{c \cdot \left(t \cdot z\right)} + 4 \cdot \frac{a}{c}\right)\right)} \]

      3. +-commutative 65.8%

         \[\leadsto t \cdot \left(-\color{blue}{\left(4 \cdot \frac{a}{c} + -1 \cdot \frac{b}{c \cdot \left(t \cdot z\right)}\right)}\right) \]

      4. mul-1-neg 65.8%

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

      5. unsub-neg 65.8%

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

      6. associate-/r* 65.8%

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

   8. Simplified 65.8%

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

   9. Taylor expanded in z around 0 27.5%

      \[\leadsto t \cdot \left(-\color{blue}{\frac{4 \cdot \frac{a \cdot z}{c} - \frac{b}{c \cdot t}}{z}}\right) \]
   10. Step-by-step derivation

       1. *-commutative 27.5%

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

       2. associate-/l* 65.8%

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

       3. associate-*l* 65.8%

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

       4. *-commutative 65.8%

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

   11. Simplified 65.8%

       \[\leadsto t \cdot \left(-\color{blue}{\frac{a \cdot \left(\frac{z}{c} \cdot 4\right) - \frac{b}{t \cdot c}}{z}}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{b + \left(y \cdot \left(x \cdot 9\right) - a \cdot \left(\left(z \cdot 4\right) \cdot t\right)\right)}{c \cdot z} \leq -5 \cdot 10^{+274}:\\ \;\;\;\;\frac{b - \left(\left(z \cdot 4\right) \cdot \left(t \cdot a\right) - x \cdot \left(9 \cdot y\right)\right)}{c \cdot z}\\ \mathbf{elif}\;\frac{b + \left(y \cdot \left(x \cdot 9\right) - a \cdot \left(\left(z \cdot 4\right) \cdot t\right)\right)}{c \cdot z} \leq 0:\\ \;\;\;\;\frac{\frac{b + \left(-4 \cdot \left(a \cdot \left(z \cdot t\right)\right) + 9 \cdot \left(x \cdot y\right)\right)}{c}}{z}\\ \mathbf{elif}\;\frac{b + \left(y \cdot \left(x \cdot 9\right) - a \cdot \left(\left(z \cdot 4\right) \cdot t\right)\right)}{c \cdot z} \leq \infty:\\ \;\;\;\;\frac{b + \left(y \cdot \left(x \cdot 9\right) - a \cdot \left(\left(z \cdot 4\right) \cdot t\right)\right)}{c \cdot z}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{\frac{b}{c \cdot t} - a \cdot \left(4 \cdot \frac{z}{c}\right)}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ c\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{b + \left(y \cdot \left(x \cdot 9\right) - a \cdot \left(\left(z \cdot 4\right) \cdot t\right)\right)}{c\_m \cdot z} \leq \infty:\\ \;\;\;\;\frac{b - \left(\left(z \cdot 4\right) \cdot \left(t \cdot a\right) - x \cdot \left(9 \cdot y\right)\right)}{c\_m \cdot z}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{\frac{b}{c\_m \cdot t} - a \cdot \left(4 \cdot \frac{z}{c\_m}\right)}{z}\\ \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (*
  c_s
  (if (<=
       (/ (+ b (- (* y (* x 9.0)) (* a (* (* z 4.0) t)))) (* c_m z))
       INFINITY)
    (/ (- b (- (* (* z 4.0) (* t a)) (* x (* 9.0 y)))) (* c_m z))
    (* t (/ (- (/ b (* c_m t)) (* a (* 4.0 (/ z c_m)))) z)))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double tmp;
	if (((b + ((y * (x * 9.0)) - (a * ((z * 4.0) * t)))) / (c_m * z)) <= ((double) INFINITY)) {
		tmp = (b - (((z * 4.0) * (t * a)) - (x * (9.0 * y)))) / (c_m * z);
	} else {
		tmp = t * (((b / (c_m * t)) - (a * (4.0 * (z / c_m)))) / z);
	}
	return c_s * tmp;
}

Java

c\_m = Math.abs(c);
c\_s = Math.copySign(1.0, c);
public static double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double tmp;
	if (((b + ((y * (x * 9.0)) - (a * ((z * 4.0) * t)))) / (c_m * z)) <= Double.POSITIVE_INFINITY) {
		tmp = (b - (((z * 4.0) * (t * a)) - (x * (9.0 * y)))) / (c_m * z);
	} else {
		tmp = t * (((b / (c_m * t)) - (a * (4.0 * (z / c_m)))) / z);
	}
	return c_s * tmp;
}

Python

c\_m = math.fabs(c)
c\_s = math.copysign(1.0, c)
def code(c_s, x, y, z, t, a, b, c_m):
	tmp = 0
	if ((b + ((y * (x * 9.0)) - (a * ((z * 4.0) * t)))) / (c_m * z)) <= math.inf:
		tmp = (b - (((z * 4.0) * (t * a)) - (x * (9.0 * y)))) / (c_m * z)
	else:
		tmp = t * (((b / (c_m * t)) - (a * (4.0 * (z / c_m)))) / z)
	return c_s * tmp

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
function code(c_s, x, y, z, t, a, b, c_m)
	tmp = 0.0
	if (Float64(Float64(b + Float64(Float64(y * Float64(x * 9.0)) - Float64(a * Float64(Float64(z * 4.0) * t)))) / Float64(c_m * z)) <= Inf)
		tmp = Float64(Float64(b - Float64(Float64(Float64(z * 4.0) * Float64(t * a)) - Float64(x * Float64(9.0 * y)))) / Float64(c_m * z));
	else
		tmp = Float64(t * Float64(Float64(Float64(b / Float64(c_m * t)) - Float64(a * Float64(4.0 * Float64(z / c_m)))) / z));
	end
	return Float64(c_s * tmp)
end

MATLAB

c\_m = abs(c);
c\_s = sign(c) * abs(1.0);
function tmp_2 = code(c_s, x, y, z, t, a, b, c_m)
	tmp = 0.0;
	if (((b + ((y * (x * 9.0)) - (a * ((z * 4.0) * t)))) / (c_m * z)) <= Inf)
		tmp = (b - (((z * 4.0) * (t * a)) - (x * (9.0 * y)))) / (c_m * z);
	else
		tmp = t * (((b / (c_m * t)) - (a * (4.0 * (z / c_m)))) / z);
	end
	tmp_2 = c_s * tmp;
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := N[(c$95$s * If[LessEqual[N[(N[(b + N[(N[(y * N[(x * 9.0), $MachinePrecision]), $MachinePrecision] - N[(a * N[(N[(z * 4.0), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(b - N[(N[(N[(z * 4.0), $MachinePrecision] * N[(t * a), $MachinePrecision]), $MachinePrecision] - N[(x * N[(9.0 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision], N[(t * N[(N[(N[(b / N[(c$95$m * t), $MachinePrecision]), $MachinePrecision] - N[(a * N[(4.0 * N[(z / c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x #s(literal 9 binary64)) y) (*.f64 (*.f64 (*.f64 z #s(literal 4 binary64)) t) a)) b) (*.f64 z c)) < +inf.0

   1. Initial program 86.9%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Step-by-step derivation

      1. +-commutative 86.9%

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)}}{z \cdot c} \]

      2. associate-+r- 86.9%

         \[\leadsto \frac{\color{blue}{\left(b + \left(x \cdot 9\right) \cdot y\right) - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}}{z \cdot c} \]

      3. *-commutative 86.9%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{a \cdot \left(\left(z \cdot 4\right) \cdot t\right)}}{z \cdot c} \]

      4. associate-*r* 87.2%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{\left(a \cdot \left(z \cdot 4\right)\right) \cdot t}}{z \cdot c} \]

      5. *-commutative 87.2%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{\left(\left(z \cdot 4\right) \cdot a\right)} \cdot t}{z \cdot c} \]

      6. associate-+r- 87.2%

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right)}}{z \cdot c} \]

      7. +-commutative 87.2%

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right) + b}}{z \cdot c} \]

      8. associate-*l* 87.7%

         \[\leadsto \frac{\left(\color{blue}{x \cdot \left(9 \cdot y\right)} - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right) + b}{z \cdot c} \]

      9. associate-*l* 88.6%

         \[\leadsto \frac{\left(x \cdot \left(9 \cdot y\right) - \color{blue}{\left(z \cdot 4\right) \cdot \left(a \cdot t\right)}\right) + b}{z \cdot c} \]

      10. *-commutative 88.6%

          \[\leadsto \frac{\left(x \cdot \left(9 \cdot y\right) - \left(z \cdot 4\right) \cdot \color{blue}{\left(t \cdot a\right)}\right) + b}{z \cdot c} \]

   3. Simplified 88.6%

      \[\leadsto \color{blue}{\frac{\left(x \cdot \left(9 \cdot y\right) - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}} \]
   4. Add Preprocessing

   if +inf.0 < (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x #s(literal 9 binary64)) y) (*.f64 (*.f64 (*.f64 z #s(literal 4 binary64)) t) a)) b) (*.f64 z c))

   1. Initial program 0.0%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Step-by-step derivation

      1. +-commutative 0.0%

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)}}{z \cdot c} \]

      2. associate-+r- 0.0%

         \[\leadsto \frac{\color{blue}{\left(b + \left(x \cdot 9\right) \cdot y\right) - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}}{z \cdot c} \]

      3. *-commutative 0.0%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{a \cdot \left(\left(z \cdot 4\right) \cdot t\right)}}{z \cdot c} \]

      4. associate-*r* 6.2%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{\left(a \cdot \left(z \cdot 4\right)\right) \cdot t}}{z \cdot c} \]

      5. *-commutative 6.2%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{\left(\left(z \cdot 4\right) \cdot a\right)} \cdot t}{z \cdot c} \]

      6. associate-+r- 6.2%

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right)}}{z \cdot c} \]

      7. +-commutative 6.2%

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right) + b}}{z \cdot c} \]

      8. associate-*l* 6.2%

         \[\leadsto \frac{\left(\color{blue}{x \cdot \left(9 \cdot y\right)} - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right) + b}{z \cdot c} \]

      9. associate-*l* 6.2%

         \[\leadsto \frac{\left(x \cdot \left(9 \cdot y\right) - \color{blue}{\left(z \cdot 4\right) \cdot \left(a \cdot t\right)}\right) + b}{z \cdot c} \]

      10. *-commutative 6.2%

          \[\leadsto \frac{\left(x \cdot \left(9 \cdot y\right) - \left(z \cdot 4\right) \cdot \color{blue}{\left(t \cdot a\right)}\right) + b}{z \cdot c} \]

   3. Simplified 6.2%

      \[\leadsto \color{blue}{\frac{\left(x \cdot \left(9 \cdot y\right) - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 0.0%

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

   6. Taylor expanded in t around -inf 65.8%

      \[\leadsto \color{blue}{-1 \cdot \left(t \cdot \left(-1 \cdot \frac{b}{c \cdot \left(t \cdot z\right)} + 4 \cdot \frac{a}{c}\right)\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg 65.8%

         \[\leadsto \color{blue}{-t \cdot \left(-1 \cdot \frac{b}{c \cdot \left(t \cdot z\right)} + 4 \cdot \frac{a}{c}\right)} \]

      2. distribute-rgt-neg-in 65.8%

         \[\leadsto \color{blue}{t \cdot \left(-\left(-1 \cdot \frac{b}{c \cdot \left(t \cdot z\right)} + 4 \cdot \frac{a}{c}\right)\right)} \]

      3. +-commutative 65.8%

         \[\leadsto t \cdot \left(-\color{blue}{\left(4 \cdot \frac{a}{c} + -1 \cdot \frac{b}{c \cdot \left(t \cdot z\right)}\right)}\right) \]

      4. mul-1-neg 65.8%

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

      5. unsub-neg 65.8%

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

      6. associate-/r* 65.8%

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

   8. Simplified 65.8%

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

   9. Taylor expanded in z around 0 27.5%

      \[\leadsto t \cdot \left(-\color{blue}{\frac{4 \cdot \frac{a \cdot z}{c} - \frac{b}{c \cdot t}}{z}}\right) \]
   10. Step-by-step derivation

       1. *-commutative 27.5%

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

       2. associate-/l* 65.8%

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

       3. associate-*l* 65.8%

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

       4. *-commutative 65.8%

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

   11. Simplified 65.8%

       \[\leadsto t \cdot \left(-\color{blue}{\frac{a \cdot \left(\frac{z}{c} \cdot 4\right) - \frac{b}{t \cdot c}}{z}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{b + \left(y \cdot \left(x \cdot 9\right) - a \cdot \left(\left(z \cdot 4\right) \cdot t\right)\right)}{c \cdot z} \leq \infty:\\ \;\;\;\;\frac{b - \left(\left(z \cdot 4\right) \cdot \left(t \cdot a\right) - x \cdot \left(9 \cdot y\right)\right)}{c \cdot z}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{\frac{b}{c \cdot t} - a \cdot \left(4 \cdot \frac{z}{c}\right)}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 70.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ c\_s \cdot \begin{array}{l} \mathbf{if}\;a \leq -4.15 \cdot 10^{-94} \lor \neg \left(a \leq 4.5 \cdot 10^{+63}\right):\\ \;\;\;\;a \cdot \left(-4 \cdot \frac{t}{c\_m} + \frac{b}{a \cdot \left(c\_m \cdot z\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot \left(9 \cdot y\right) + b}{c\_m}}{z}\\ \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (*
  c_s
  (if (or (<= a -4.15e-94) (not (<= a 4.5e+63)))
    (* a (+ (* -4.0 (/ t c_m)) (/ b (* a (* c_m z)))))
    (/ (/ (+ (* x (* 9.0 y)) b) c_m) z))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double tmp;
	if ((a <= -4.15e-94) || !(a <= 4.5e+63)) {
		tmp = a * ((-4.0 * (t / c_m)) + (b / (a * (c_m * z))));
	} else {
		tmp = (((x * (9.0 * y)) + b) / c_m) / z;
	}
	return c_s * tmp;
}

Fortran

c\_m = abs(c)
c\_s = copysign(1.0d0, c)
real(8) function code(c_s, x, y, z, t, a, b, c_m)
    real(8), intent (in) :: c_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c_m
    real(8) :: tmp
    if ((a <= (-4.15d-94)) .or. (.not. (a <= 4.5d+63))) then
        tmp = a * (((-4.0d0) * (t / c_m)) + (b / (a * (c_m * z))))
    else
        tmp = (((x * (9.0d0 * y)) + b) / c_m) / z
    end if
    code = c_s * tmp
end function

Java

c\_m = Math.abs(c);
c\_s = Math.copySign(1.0, c);
public static double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double tmp;
	if ((a <= -4.15e-94) || !(a <= 4.5e+63)) {
		tmp = a * ((-4.0 * (t / c_m)) + (b / (a * (c_m * z))));
	} else {
		tmp = (((x * (9.0 * y)) + b) / c_m) / z;
	}
	return c_s * tmp;
}

Python

c\_m = math.fabs(c)
c\_s = math.copysign(1.0, c)
def code(c_s, x, y, z, t, a, b, c_m):
	tmp = 0
	if (a <= -4.15e-94) or not (a <= 4.5e+63):
		tmp = a * ((-4.0 * (t / c_m)) + (b / (a * (c_m * z))))
	else:
		tmp = (((x * (9.0 * y)) + b) / c_m) / z
	return c_s * tmp

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
function code(c_s, x, y, z, t, a, b, c_m)
	tmp = 0.0
	if ((a <= -4.15e-94) || !(a <= 4.5e+63))
		tmp = Float64(a * Float64(Float64(-4.0 * Float64(t / c_m)) + Float64(b / Float64(a * Float64(c_m * z)))));
	else
		tmp = Float64(Float64(Float64(Float64(x * Float64(9.0 * y)) + b) / c_m) / z);
	end
	return Float64(c_s * tmp)
end

MATLAB

c\_m = abs(c);
c\_s = sign(c) * abs(1.0);
function tmp_2 = code(c_s, x, y, z, t, a, b, c_m)
	tmp = 0.0;
	if ((a <= -4.15e-94) || ~((a <= 4.5e+63)))
		tmp = a * ((-4.0 * (t / c_m)) + (b / (a * (c_m * z))));
	else
		tmp = (((x * (9.0 * y)) + b) / c_m) / z;
	end
	tmp_2 = c_s * tmp;
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := N[(c$95$s * If[Or[LessEqual[a, -4.15e-94], N[Not[LessEqual[a, 4.5e+63]], $MachinePrecision]], N[(a * N[(N[(-4.0 * N[(t / c$95$m), $MachinePrecision]), $MachinePrecision] + N[(b / N[(a * N[(c$95$m * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(x * N[(9.0 * y), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision] / c$95$m), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if a < -4.1499999999999998e-94 or 4.50000000000000017e63 < a

   1. Initial program 78.1%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Step-by-step derivation

      1. +-commutative 78.1%

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)}}{z \cdot c} \]

      2. associate-+r- 78.1%

         \[\leadsto \frac{\color{blue}{\left(b + \left(x \cdot 9\right) \cdot y\right) - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}}{z \cdot c} \]

      3. *-commutative 78.1%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{a \cdot \left(\left(z \cdot 4\right) \cdot t\right)}}{z \cdot c} \]

      4. associate-*r* 73.4%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{\left(a \cdot \left(z \cdot 4\right)\right) \cdot t}}{z \cdot c} \]

      5. *-commutative 73.4%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{\left(\left(z \cdot 4\right) \cdot a\right)} \cdot t}{z \cdot c} \]

      6. associate-+r- 73.4%

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right)}}{z \cdot c} \]

      7. +-commutative 73.4%

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right) + b}}{z \cdot c} \]

      8. associate-*l* 74.2%

         \[\leadsto \frac{\left(\color{blue}{x \cdot \left(9 \cdot y\right)} - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right) + b}{z \cdot c} \]

      9. associate-*l* 76.7%

         \[\leadsto \frac{\left(x \cdot \left(9 \cdot y\right) - \color{blue}{\left(z \cdot 4\right) \cdot \left(a \cdot t\right)}\right) + b}{z \cdot c} \]

      10. *-commutative 76.7%

          \[\leadsto \frac{\left(x \cdot \left(9 \cdot y\right) - \left(z \cdot 4\right) \cdot \color{blue}{\left(t \cdot a\right)}\right) + b}{z \cdot c} \]

   3. Simplified 76.7%

      \[\leadsto \color{blue}{\frac{\left(x \cdot \left(9 \cdot y\right) - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 66.1%

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

   6. Taylor expanded in a around inf 74.4%

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

   if -4.1499999999999998e-94 < a < 4.50000000000000017e63

   1. Initial program 81.9%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Step-by-step derivation

      1. +-commutative 81.9%

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)}}{z \cdot c} \]

      2. associate-+r- 81.9%

         \[\leadsto \frac{\color{blue}{\left(b + \left(x \cdot 9\right) \cdot y\right) - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}}{z \cdot c} \]

      3. *-commutative 81.9%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{a \cdot \left(\left(z \cdot 4\right) \cdot t\right)}}{z \cdot c} \]

      4. associate-*r* 87.8%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{\left(a \cdot \left(z \cdot 4\right)\right) \cdot t}}{z \cdot c} \]

      5. *-commutative 87.8%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{\left(\left(z \cdot 4\right) \cdot a\right)} \cdot t}{z \cdot c} \]

      6. associate-+r- 87.8%

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right)}}{z \cdot c} \]

      7. +-commutative 87.8%

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right) + b}}{z \cdot c} \]

      8. associate-*l* 87.8%

         \[\leadsto \frac{\left(\color{blue}{x \cdot \left(9 \cdot y\right)} - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right) + b}{z \cdot c} \]

      9. associate-*l* 87.1%

         \[\leadsto \frac{\left(x \cdot \left(9 \cdot y\right) - \color{blue}{\left(z \cdot 4\right) \cdot \left(a \cdot t\right)}\right) + b}{z \cdot c} \]

      10. *-commutative 87.1%

          \[\leadsto \frac{\left(x \cdot \left(9 \cdot y\right) - \left(z \cdot 4\right) \cdot \color{blue}{\left(t \cdot a\right)}\right) + b}{z \cdot c} \]

   3. Simplified 87.1%

      \[\leadsto \color{blue}{\frac{\left(x \cdot \left(9 \cdot y\right) - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 80.1%

      \[\leadsto \color{blue}{\frac{-4 \cdot \frac{a \cdot \left(t \cdot z\right)}{c} + \left(9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z}} \]
   6. Step-by-step derivation

      1. fma-define 80.1%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-4, \frac{a \cdot \left(t \cdot z\right)}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}}{z} \]

      2. *-commutative 80.1%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{\color{blue}{\left(t \cdot z\right) \cdot a}}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      3. *-commutative 80.1%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{\color{blue}{\left(z \cdot t\right)} \cdot a}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      4. associate-*r* 84.7%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{\color{blue}{z \cdot \left(t \cdot a\right)}}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      5. *-commutative 84.7%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{z \cdot \color{blue}{\left(a \cdot t\right)}}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      6. associate-/l* 83.9%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \color{blue}{z \cdot \frac{a \cdot t}{c}}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      7. associate-/l* 81.6%

         \[\leadsto \frac{\mathsf{fma}\left(-4, z \cdot \color{blue}{\left(a \cdot \frac{t}{c}\right)}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      8. fma-define 81.6%

         \[\leadsto \frac{\mathsf{fma}\left(-4, z \cdot \left(a \cdot \frac{t}{c}\right), \color{blue}{\mathsf{fma}\left(9, \frac{x \cdot y}{c}, \frac{b}{c}\right)}\right)}{z} \]

      9. associate-/l* 83.7%

         \[\leadsto \frac{\mathsf{fma}\left(-4, z \cdot \left(a \cdot \frac{t}{c}\right), \mathsf{fma}\left(9, \color{blue}{x \cdot \frac{y}{c}}, \frac{b}{c}\right)\right)}{z} \]

   7. Simplified 83.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-4, z \cdot \left(a \cdot \frac{t}{c}\right), \mathsf{fma}\left(9, x \cdot \frac{y}{c}, \frac{b}{c}\right)\right)}{z}} \]

   8. Taylor expanded in c around 0 83.9%

      \[\leadsto \frac{\color{blue}{\frac{b + \left(-4 \cdot \left(a \cdot \left(t \cdot z\right)\right) + 9 \cdot \left(x \cdot y\right)\right)}{c}}}{z} \]

   9. Taylor expanded in a around 0 76.2%

      \[\leadsto \frac{\frac{b + \color{blue}{9 \cdot \left(x \cdot y\right)}}{c}}{z} \]
   10. Step-by-step derivation

       1. associate-*r* 76.2%

          \[\leadsto \frac{\frac{b + \color{blue}{\left(9 \cdot x\right) \cdot y}}{c}}{z} \]

       2. *-commutative 76.2%

          \[\leadsto \frac{\frac{b + \color{blue}{\left(x \cdot 9\right)} \cdot y}{c}}{z} \]

       3. associate-*r* 76.2%

          \[\leadsto \frac{\frac{b + \color{blue}{x \cdot \left(9 \cdot y\right)}}{c}}{z} \]

   11. Simplified 76.2%

       \[\leadsto \frac{\frac{b + \color{blue}{x \cdot \left(9 \cdot y\right)}}{c}}{z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -4.15 \cdot 10^{-94} \lor \neg \left(a \leq 4.5 \cdot 10^{+63}\right):\\ \;\;\;\;a \cdot \left(-4 \cdot \frac{t}{c} + \frac{b}{a \cdot \left(c \cdot z\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot \left(9 \cdot y\right) + b}{c}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ c\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -3.1 \cdot 10^{-85}:\\ \;\;\;\;t \cdot \left(-4 \cdot \frac{a}{c\_m} + \frac{b}{c\_m \cdot \left(z \cdot t\right)}\right)\\ \mathbf{elif}\;t \leq 10^{-141}:\\ \;\;\;\;\frac{b + 9 \cdot \left(x \cdot y\right)}{c\_m \cdot z}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(-4 \cdot \frac{t}{c\_m} + \frac{b}{a \cdot \left(c\_m \cdot z\right)}\right)\\ \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (*
  c_s
  (if (<= t -3.1e-85)
    (* t (+ (* -4.0 (/ a c_m)) (/ b (* c_m (* z t)))))
    (if (<= t 1e-141)
      (/ (+ b (* 9.0 (* x y))) (* c_m z))
      (* a (+ (* -4.0 (/ t c_m)) (/ b (* a (* c_m z)))))))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double tmp;
	if (t <= -3.1e-85) {
		tmp = t * ((-4.0 * (a / c_m)) + (b / (c_m * (z * t))));
	} else if (t <= 1e-141) {
		tmp = (b + (9.0 * (x * y))) / (c_m * z);
	} else {
		tmp = a * ((-4.0 * (t / c_m)) + (b / (a * (c_m * z))));
	}
	return c_s * tmp;
}

Fortran

c\_m = abs(c)
c\_s = copysign(1.0d0, c)
real(8) function code(c_s, x, y, z, t, a, b, c_m)
    real(8), intent (in) :: c_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c_m
    real(8) :: tmp
    if (t <= (-3.1d-85)) then
        tmp = t * (((-4.0d0) * (a / c_m)) + (b / (c_m * (z * t))))
    else if (t <= 1d-141) then
        tmp = (b + (9.0d0 * (x * y))) / (c_m * z)
    else
        tmp = a * (((-4.0d0) * (t / c_m)) + (b / (a * (c_m * z))))
    end if
    code = c_s * tmp
end function

Java

c\_m = Math.abs(c);
c\_s = Math.copySign(1.0, c);
public static double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double tmp;
	if (t <= -3.1e-85) {
		tmp = t * ((-4.0 * (a / c_m)) + (b / (c_m * (z * t))));
	} else if (t <= 1e-141) {
		tmp = (b + (9.0 * (x * y))) / (c_m * z);
	} else {
		tmp = a * ((-4.0 * (t / c_m)) + (b / (a * (c_m * z))));
	}
	return c_s * tmp;
}

Python

c\_m = math.fabs(c)
c\_s = math.copysign(1.0, c)
def code(c_s, x, y, z, t, a, b, c_m):
	tmp = 0
	if t <= -3.1e-85:
		tmp = t * ((-4.0 * (a / c_m)) + (b / (c_m * (z * t))))
	elif t <= 1e-141:
		tmp = (b + (9.0 * (x * y))) / (c_m * z)
	else:
		tmp = a * ((-4.0 * (t / c_m)) + (b / (a * (c_m * z))))
	return c_s * tmp

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
function code(c_s, x, y, z, t, a, b, c_m)
	tmp = 0.0
	if (t <= -3.1e-85)
		tmp = Float64(t * Float64(Float64(-4.0 * Float64(a / c_m)) + Float64(b / Float64(c_m * Float64(z * t)))));
	elseif (t <= 1e-141)
		tmp = Float64(Float64(b + Float64(9.0 * Float64(x * y))) / Float64(c_m * z));
	else
		tmp = Float64(a * Float64(Float64(-4.0 * Float64(t / c_m)) + Float64(b / Float64(a * Float64(c_m * z)))));
	end
	return Float64(c_s * tmp)
end

MATLAB

c\_m = abs(c);
c\_s = sign(c) * abs(1.0);
function tmp_2 = code(c_s, x, y, z, t, a, b, c_m)
	tmp = 0.0;
	if (t <= -3.1e-85)
		tmp = t * ((-4.0 * (a / c_m)) + (b / (c_m * (z * t))));
	elseif (t <= 1e-141)
		tmp = (b + (9.0 * (x * y))) / (c_m * z);
	else
		tmp = a * ((-4.0 * (t / c_m)) + (b / (a * (c_m * z))));
	end
	tmp_2 = c_s * tmp;
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := N[(c$95$s * If[LessEqual[t, -3.1e-85], N[(t * N[(N[(-4.0 * N[(a / c$95$m), $MachinePrecision]), $MachinePrecision] + N[(b / N[(c$95$m * N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1e-141], N[(N[(b + N[(9.0 * N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision], N[(a * N[(N[(-4.0 * N[(t / c$95$m), $MachinePrecision]), $MachinePrecision] + N[(b / N[(a * N[(c$95$m * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < -3.1000000000000002e-85

   1. Initial program 70.1%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Step-by-step derivation

      1. +-commutative 70.1%

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)}}{z \cdot c} \]

      2. associate-+r- 70.1%

         \[\leadsto \frac{\color{blue}{\left(b + \left(x \cdot 9\right) \cdot y\right) - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}}{z \cdot c} \]

      3. *-commutative 70.1%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{a \cdot \left(\left(z \cdot 4\right) \cdot t\right)}}{z \cdot c} \]

      4. associate-*r* 78.1%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{\left(a \cdot \left(z \cdot 4\right)\right) \cdot t}}{z \cdot c} \]

      5. *-commutative 78.1%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{\left(\left(z \cdot 4\right) \cdot a\right)} \cdot t}{z \cdot c} \]

      6. associate-+r- 78.1%

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right)}}{z \cdot c} \]

      7. +-commutative 78.1%

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right) + b}}{z \cdot c} \]

      8. associate-*l* 78.1%

         \[\leadsto \frac{\left(\color{blue}{x \cdot \left(9 \cdot y\right)} - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right) + b}{z \cdot c} \]

      9. associate-*l* 74.7%

         \[\leadsto \frac{\left(x \cdot \left(9 \cdot y\right) - \color{blue}{\left(z \cdot 4\right) \cdot \left(a \cdot t\right)}\right) + b}{z \cdot c} \]

      10. *-commutative 74.7%

          \[\leadsto \frac{\left(x \cdot \left(9 \cdot y\right) - \left(z \cdot 4\right) \cdot \color{blue}{\left(t \cdot a\right)}\right) + b}{z \cdot c} \]

   3. Simplified 74.7%

      \[\leadsto \color{blue}{\frac{\left(x \cdot \left(9 \cdot y\right) - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 56.9%

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

   6. Taylor expanded in t around inf 75.0%

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

   if -3.1000000000000002e-85 < t < 1e-141

   1. Initial program 92.6%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Step-by-step derivation

      1. +-commutative 92.6%

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)}}{z \cdot c} \]

      2. associate-+r- 92.6%

         \[\leadsto \frac{\color{blue}{\left(b + \left(x \cdot 9\right) \cdot y\right) - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}}{z \cdot c} \]

      3. *-commutative 92.6%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{a \cdot \left(\left(z \cdot 4\right) \cdot t\right)}}{z \cdot c} \]

      4. associate-*r* 87.8%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{\left(a \cdot \left(z \cdot 4\right)\right) \cdot t}}{z \cdot c} \]

      5. *-commutative 87.8%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{\left(\left(z \cdot 4\right) \cdot a\right)} \cdot t}{z \cdot c} \]

      6. associate-+r- 87.8%

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right)}}{z \cdot c} \]

      7. +-commutative 87.8%

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right) + b}}{z \cdot c} \]

      8. associate-*l* 89.0%

         \[\leadsto \frac{\left(\color{blue}{x \cdot \left(9 \cdot y\right)} - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right) + b}{z \cdot c} \]

      9. associate-*l* 92.8%

         \[\leadsto \frac{\left(x \cdot \left(9 \cdot y\right) - \color{blue}{\left(z \cdot 4\right) \cdot \left(a \cdot t\right)}\right) + b}{z \cdot c} \]

      10. *-commutative 92.8%

          \[\leadsto \frac{\left(x \cdot \left(9 \cdot y\right) - \left(z \cdot 4\right) \cdot \color{blue}{\left(t \cdot a\right)}\right) + b}{z \cdot c} \]

   3. Simplified 92.8%

      \[\leadsto \color{blue}{\frac{\left(x \cdot \left(9 \cdot y\right) - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 85.7%

      \[\leadsto \frac{\color{blue}{9 \cdot \left(x \cdot y\right)} + b}{z \cdot c} \]

   if 1e-141 < t

   1. Initial program 78.1%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Step-by-step derivation

      1. +-commutative 78.1%

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)}}{z \cdot c} \]

      2. associate-+r- 78.1%

         \[\leadsto \frac{\color{blue}{\left(b + \left(x \cdot 9\right) \cdot y\right) - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}}{z \cdot c} \]

      3. *-commutative 78.1%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{a \cdot \left(\left(z \cdot 4\right) \cdot t\right)}}{z \cdot c} \]

      4. associate-*r* 77.2%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{\left(a \cdot \left(z \cdot 4\right)\right) \cdot t}}{z \cdot c} \]

      5. *-commutative 77.2%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{\left(\left(z \cdot 4\right) \cdot a\right)} \cdot t}{z \cdot c} \]

      6. associate-+r- 77.2%

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right)}}{z \cdot c} \]

      7. +-commutative 77.2%

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right) + b}}{z \cdot c} \]

      8. associate-*l* 77.2%

         \[\leadsto \frac{\left(\color{blue}{x \cdot \left(9 \cdot y\right)} - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right) + b}{z \cdot c} \]

      9. associate-*l* 79.4%

         \[\leadsto \frac{\left(x \cdot \left(9 \cdot y\right) - \color{blue}{\left(z \cdot 4\right) \cdot \left(a \cdot t\right)}\right) + b}{z \cdot c} \]

      10. *-commutative 79.4%

          \[\leadsto \frac{\left(x \cdot \left(9 \cdot y\right) - \left(z \cdot 4\right) \cdot \color{blue}{\left(t \cdot a\right)}\right) + b}{z \cdot c} \]

   3. Simplified 79.4%

      \[\leadsto \color{blue}{\frac{\left(x \cdot \left(9 \cdot y\right) - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 59.6%

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

   6. Taylor expanded in a around inf 66.9%

      \[\leadsto \color{blue}{a \cdot \left(-4 \cdot \frac{t}{c} + \frac{b}{a \cdot \left(c \cdot z\right)}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.1 \cdot 10^{-85}:\\ \;\;\;\;t \cdot \left(-4 \cdot \frac{a}{c} + \frac{b}{c \cdot \left(z \cdot t\right)}\right)\\ \mathbf{elif}\;t \leq 10^{-141}:\\ \;\;\;\;\frac{b + 9 \cdot \left(x \cdot y\right)}{c \cdot z}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(-4 \cdot \frac{t}{c} + \frac{b}{a \cdot \left(c \cdot z\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 48.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ c\_s \cdot \begin{array}{l} \mathbf{if}\;b \leq -7.6 \cdot 10^{-25}:\\ \;\;\;\;\frac{1}{\frac{z}{\frac{b}{c\_m}}}\\ \mathbf{elif}\;b \leq 3.7 \cdot 10^{-233}:\\ \;\;\;\;\frac{a}{c\_m} \cdot \left(t \cdot -4\right)\\ \mathbf{elif}\;b \leq 3.8 \cdot 10^{-30}:\\ \;\;\;\;\frac{x \cdot \frac{9 \cdot y}{c\_m}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{c\_m \cdot \frac{z}{b}}\\ \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (*
  c_s
  (if (<= b -7.6e-25)
    (/ 1.0 (/ z (/ b c_m)))
    (if (<= b 3.7e-233)
      (* (/ a c_m) (* t -4.0))
      (if (<= b 3.8e-30)
        (/ (* x (/ (* 9.0 y) c_m)) z)
        (/ 1.0 (* c_m (/ z b))))))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double tmp;
	if (b <= -7.6e-25) {
		tmp = 1.0 / (z / (b / c_m));
	} else if (b <= 3.7e-233) {
		tmp = (a / c_m) * (t * -4.0);
	} else if (b <= 3.8e-30) {
		tmp = (x * ((9.0 * y) / c_m)) / z;
	} else {
		tmp = 1.0 / (c_m * (z / b));
	}
	return c_s * tmp;
}

Fortran

c\_m = abs(c)
c\_s = copysign(1.0d0, c)
real(8) function code(c_s, x, y, z, t, a, b, c_m)
    real(8), intent (in) :: c_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c_m
    real(8) :: tmp
    if (b <= (-7.6d-25)) then
        tmp = 1.0d0 / (z / (b / c_m))
    else if (b <= 3.7d-233) then
        tmp = (a / c_m) * (t * (-4.0d0))
    else if (b <= 3.8d-30) then
        tmp = (x * ((9.0d0 * y) / c_m)) / z
    else
        tmp = 1.0d0 / (c_m * (z / b))
    end if
    code = c_s * tmp
end function

Java

c\_m = Math.abs(c);
c\_s = Math.copySign(1.0, c);
public static double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double tmp;
	if (b <= -7.6e-25) {
		tmp = 1.0 / (z / (b / c_m));
	} else if (b <= 3.7e-233) {
		tmp = (a / c_m) * (t * -4.0);
	} else if (b <= 3.8e-30) {
		tmp = (x * ((9.0 * y) / c_m)) / z;
	} else {
		tmp = 1.0 / (c_m * (z / b));
	}
	return c_s * tmp;
}

Python

c\_m = math.fabs(c)
c\_s = math.copysign(1.0, c)
def code(c_s, x, y, z, t, a, b, c_m):
	tmp = 0
	if b <= -7.6e-25:
		tmp = 1.0 / (z / (b / c_m))
	elif b <= 3.7e-233:
		tmp = (a / c_m) * (t * -4.0)
	elif b <= 3.8e-30:
		tmp = (x * ((9.0 * y) / c_m)) / z
	else:
		tmp = 1.0 / (c_m * (z / b))
	return c_s * tmp

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
function code(c_s, x, y, z, t, a, b, c_m)
	tmp = 0.0
	if (b <= -7.6e-25)
		tmp = Float64(1.0 / Float64(z / Float64(b / c_m)));
	elseif (b <= 3.7e-233)
		tmp = Float64(Float64(a / c_m) * Float64(t * -4.0));
	elseif (b <= 3.8e-30)
		tmp = Float64(Float64(x * Float64(Float64(9.0 * y) / c_m)) / z);
	else
		tmp = Float64(1.0 / Float64(c_m * Float64(z / b)));
	end
	return Float64(c_s * tmp)
end

MATLAB

c\_m = abs(c);
c\_s = sign(c) * abs(1.0);
function tmp_2 = code(c_s, x, y, z, t, a, b, c_m)
	tmp = 0.0;
	if (b <= -7.6e-25)
		tmp = 1.0 / (z / (b / c_m));
	elseif (b <= 3.7e-233)
		tmp = (a / c_m) * (t * -4.0);
	elseif (b <= 3.8e-30)
		tmp = (x * ((9.0 * y) / c_m)) / z;
	else
		tmp = 1.0 / (c_m * (z / b));
	end
	tmp_2 = c_s * tmp;
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := N[(c$95$s * If[LessEqual[b, -7.6e-25], N[(1.0 / N[(z / N[(b / c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 3.7e-233], N[(N[(a / c$95$m), $MachinePrecision] * N[(t * -4.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 3.8e-30], N[(N[(x * N[(N[(9.0 * y), $MachinePrecision] / c$95$m), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(1.0 / N[(c$95$m * N[(z / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if b < -7.5999999999999996e-25

   1. Initial program 82.4%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Step-by-step derivation

      1. +-commutative 82.4%

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)}}{z \cdot c} \]

      2. associate-+r- 82.4%

         \[\leadsto \frac{\color{blue}{\left(b + \left(x \cdot 9\right) \cdot y\right) - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}}{z \cdot c} \]

      3. *-commutative 82.4%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{a \cdot \left(\left(z \cdot 4\right) \cdot t\right)}}{z \cdot c} \]

      4. associate-*r* 83.9%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{\left(a \cdot \left(z \cdot 4\right)\right) \cdot t}}{z \cdot c} \]

      5. *-commutative 83.9%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{\left(\left(z \cdot 4\right) \cdot a\right)} \cdot t}{z \cdot c} \]

      6. associate-+r- 83.9%

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right)}}{z \cdot c} \]

      7. +-commutative 83.9%

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right) + b}}{z \cdot c} \]

      8. associate-*l* 83.9%

         \[\leadsto \frac{\left(\color{blue}{x \cdot \left(9 \cdot y\right)} - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right) + b}{z \cdot c} \]

      9. associate-*l* 81.0%

         \[\leadsto \frac{\left(x \cdot \left(9 \cdot y\right) - \color{blue}{\left(z \cdot 4\right) \cdot \left(a \cdot t\right)}\right) + b}{z \cdot c} \]

      10. *-commutative 81.0%

          \[\leadsto \frac{\left(x \cdot \left(9 \cdot y\right) - \left(z \cdot 4\right) \cdot \color{blue}{\left(t \cdot a\right)}\right) + b}{z \cdot c} \]

   3. Simplified 81.0%

      \[\leadsto \color{blue}{\frac{\left(x \cdot \left(9 \cdot y\right) - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 75.3%

      \[\leadsto \color{blue}{\frac{-4 \cdot \frac{a \cdot \left(t \cdot z\right)}{c} + \left(9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z}} \]
   6. Step-by-step derivation

      1. fma-define 75.3%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-4, \frac{a \cdot \left(t \cdot z\right)}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}}{z} \]

      2. *-commutative 75.3%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{\color{blue}{\left(t \cdot z\right) \cdot a}}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      3. *-commutative 75.3%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{\color{blue}{\left(z \cdot t\right)} \cdot a}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      4. associate-*r* 75.3%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{\color{blue}{z \cdot \left(t \cdot a\right)}}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      5. *-commutative 75.3%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{z \cdot \color{blue}{\left(a \cdot t\right)}}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      6. associate-/l* 75.4%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \color{blue}{z \cdot \frac{a \cdot t}{c}}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      7. associate-/l* 79.6%

         \[\leadsto \frac{\mathsf{fma}\left(-4, z \cdot \color{blue}{\left(a \cdot \frac{t}{c}\right)}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      8. fma-define 79.6%

         \[\leadsto \frac{\mathsf{fma}\left(-4, z \cdot \left(a \cdot \frac{t}{c}\right), \color{blue}{\mathsf{fma}\left(9, \frac{x \cdot y}{c}, \frac{b}{c}\right)}\right)}{z} \]

      9. associate-/l* 83.8%

         \[\leadsto \frac{\mathsf{fma}\left(-4, z \cdot \left(a \cdot \frac{t}{c}\right), \mathsf{fma}\left(9, \color{blue}{x \cdot \frac{y}{c}}, \frac{b}{c}\right)\right)}{z} \]

   7. Simplified 83.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-4, z \cdot \left(a \cdot \frac{t}{c}\right), \mathsf{fma}\left(9, x \cdot \frac{y}{c}, \frac{b}{c}\right)\right)}{z}} \]

   8. Taylor expanded in b around inf 62.2%

      \[\leadsto \frac{\color{blue}{\frac{b}{c}}}{z} \]
   9. Step-by-step derivation

      1. clear-num 62.3%

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

      2. inv-pow 62.3%

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

   10. Applied egg-rr 62.3%

       \[\leadsto \color{blue}{{\left(\frac{z}{\frac{b}{c}}\right)}^{-1}} \]
   11. Step-by-step derivation

       1. unpow-1 62.3%

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

   12. Simplified 62.3%

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

   if -7.5999999999999996e-25 < b < 3.6999999999999998e-233

   1. Initial program 74.6%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Step-by-step derivation

      1. +-commutative 74.6%

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)}}{z \cdot c} \]

      2. associate-+r- 74.6%

         \[\leadsto \frac{\color{blue}{\left(b + \left(x \cdot 9\right) \cdot y\right) - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}}{z \cdot c} \]

      3. *-commutative 74.6%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{a \cdot \left(\left(z \cdot 4\right) \cdot t\right)}}{z \cdot c} \]

      4. associate-*r* 73.4%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{\left(a \cdot \left(z \cdot 4\right)\right) \cdot t}}{z \cdot c} \]

      5. *-commutative 73.4%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{\left(\left(z \cdot 4\right) \cdot a\right)} \cdot t}{z \cdot c} \]

      6. associate-+r- 73.4%

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right)}}{z \cdot c} \]

      7. +-commutative 73.4%

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right) + b}}{z \cdot c} \]

      8. associate-*l* 73.4%

         \[\leadsto \frac{\left(\color{blue}{x \cdot \left(9 \cdot y\right)} - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right) + b}{z \cdot c} \]

      9. associate-*l* 77.4%

         \[\leadsto \frac{\left(x \cdot \left(9 \cdot y\right) - \color{blue}{\left(z \cdot 4\right) \cdot \left(a \cdot t\right)}\right) + b}{z \cdot c} \]

      10. *-commutative 77.4%

          \[\leadsto \frac{\left(x \cdot \left(9 \cdot y\right) - \left(z \cdot 4\right) \cdot \color{blue}{\left(t \cdot a\right)}\right) + b}{z \cdot c} \]

   3. Simplified 77.4%

      \[\leadsto \color{blue}{\frac{\left(x \cdot \left(9 \cdot y\right) - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 72.6%

      \[\leadsto \color{blue}{\frac{-4 \cdot \frac{a \cdot \left(t \cdot z\right)}{c} + \left(9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z}} \]
   6. Step-by-step derivation

      1. fma-define 72.6%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-4, \frac{a \cdot \left(t \cdot z\right)}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}}{z} \]

      2. *-commutative 72.6%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{\color{blue}{\left(t \cdot z\right) \cdot a}}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      3. *-commutative 72.6%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{\color{blue}{\left(z \cdot t\right)} \cdot a}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      4. associate-*r* 74.2%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{\color{blue}{z \cdot \left(t \cdot a\right)}}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      5. *-commutative 74.2%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{z \cdot \color{blue}{\left(a \cdot t\right)}}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      6. associate-/l* 79.5%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \color{blue}{z \cdot \frac{a \cdot t}{c}}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      7. associate-/l* 80.1%

         \[\leadsto \frac{\mathsf{fma}\left(-4, z \cdot \color{blue}{\left(a \cdot \frac{t}{c}\right)}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      8. fma-define 80.1%

         \[\leadsto \frac{\mathsf{fma}\left(-4, z \cdot \left(a \cdot \frac{t}{c}\right), \color{blue}{\mathsf{fma}\left(9, \frac{x \cdot y}{c}, \frac{b}{c}\right)}\right)}{z} \]

      9. associate-/l* 80.2%

         \[\leadsto \frac{\mathsf{fma}\left(-4, z \cdot \left(a \cdot \frac{t}{c}\right), \mathsf{fma}\left(9, \color{blue}{x \cdot \frac{y}{c}}, \frac{b}{c}\right)\right)}{z} \]

   7. Simplified 80.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-4, z \cdot \left(a \cdot \frac{t}{c}\right), \mathsf{fma}\left(9, x \cdot \frac{y}{c}, \frac{b}{c}\right)\right)}{z}} \]

   8. Taylor expanded in z around inf 49.2%

      \[\leadsto \color{blue}{-4 \cdot \frac{a \cdot t}{c}} \]
   9. Step-by-step derivation

      1. associate-*r/ 49.2%

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

      2. associate-*r* 49.2%

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

      3. associate-*l/ 56.8%

         \[\leadsto \color{blue}{\frac{-4 \cdot a}{c} \cdot t} \]

      4. associate-*r/ 56.8%

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

      5. *-commutative 56.8%

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

      6. associate-*l* 56.8%

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

   10. Simplified 56.8%

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

   if 3.6999999999999998e-233 < b < 3.8000000000000003e-30

   1. Initial program 82.9%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Step-by-step derivation

      1. +-commutative 82.9%

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)}}{z \cdot c} \]

      2. associate-+r- 82.9%

         \[\leadsto \frac{\color{blue}{\left(b + \left(x \cdot 9\right) \cdot y\right) - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}}{z \cdot c} \]

      3. *-commutative 82.9%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{a \cdot \left(\left(z \cdot 4\right) \cdot t\right)}}{z \cdot c} \]

      4. associate-*r* 83.1%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{\left(a \cdot \left(z \cdot 4\right)\right) \cdot t}}{z \cdot c} \]

      5. *-commutative 83.1%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{\left(\left(z \cdot 4\right) \cdot a\right)} \cdot t}{z \cdot c} \]

      6. associate-+r- 83.1%

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right)}}{z \cdot c} \]

      7. +-commutative 83.1%

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right) + b}}{z \cdot c} \]

      8. associate-*l* 83.2%

         \[\leadsto \frac{\left(\color{blue}{x \cdot \left(9 \cdot y\right)} - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right) + b}{z \cdot c} \]

      9. associate-*l* 83.2%

         \[\leadsto \frac{\left(x \cdot \left(9 \cdot y\right) - \color{blue}{\left(z \cdot 4\right) \cdot \left(a \cdot t\right)}\right) + b}{z \cdot c} \]

      10. *-commutative 83.2%

          \[\leadsto \frac{\left(x \cdot \left(9 \cdot y\right) - \left(z \cdot 4\right) \cdot \color{blue}{\left(t \cdot a\right)}\right) + b}{z \cdot c} \]

   3. Simplified 83.2%

      \[\leadsto \color{blue}{\frac{\left(x \cdot \left(9 \cdot y\right) - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 85.3%

      \[\leadsto \color{blue}{\frac{-4 \cdot \frac{a \cdot \left(t \cdot z\right)}{c} + \left(9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z}} \]
   6. Step-by-step derivation

      1. fma-define 85.3%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-4, \frac{a \cdot \left(t \cdot z\right)}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}}{z} \]

      2. *-commutative 85.3%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{\color{blue}{\left(t \cdot z\right) \cdot a}}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      3. *-commutative 85.3%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{\color{blue}{\left(z \cdot t\right)} \cdot a}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      4. associate-*r* 90.2%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{\color{blue}{z \cdot \left(t \cdot a\right)}}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      5. *-commutative 90.2%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{z \cdot \color{blue}{\left(a \cdot t\right)}}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      6. associate-/l* 87.7%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \color{blue}{z \cdot \frac{a \cdot t}{c}}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      7. associate-/l* 87.7%

         \[\leadsto \frac{\mathsf{fma}\left(-4, z \cdot \color{blue}{\left(a \cdot \frac{t}{c}\right)}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      8. fma-define 87.7%

         \[\leadsto \frac{\mathsf{fma}\left(-4, z \cdot \left(a \cdot \frac{t}{c}\right), \color{blue}{\mathsf{fma}\left(9, \frac{x \cdot y}{c}, \frac{b}{c}\right)}\right)}{z} \]

      9. associate-/l* 82.7%

         \[\leadsto \frac{\mathsf{fma}\left(-4, z \cdot \left(a \cdot \frac{t}{c}\right), \mathsf{fma}\left(9, \color{blue}{x \cdot \frac{y}{c}}, \frac{b}{c}\right)\right)}{z} \]

   7. Simplified 82.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-4, z \cdot \left(a \cdot \frac{t}{c}\right), \mathsf{fma}\left(9, x \cdot \frac{y}{c}, \frac{b}{c}\right)\right)}{z}} \]

   8. Taylor expanded in c around 0 85.3%

      \[\leadsto \frac{\color{blue}{\frac{b + \left(-4 \cdot \left(a \cdot \left(t \cdot z\right)\right) + 9 \cdot \left(x \cdot y\right)\right)}{c}}}{z} \]

   9. Taylor expanded in x around inf 59.1%

      \[\leadsto \frac{\color{blue}{9 \cdot \frac{x \cdot y}{c}}}{z} \]
   10. Step-by-step derivation

       1. *-commutative 59.1%

          \[\leadsto \frac{\color{blue}{\frac{x \cdot y}{c} \cdot 9}}{z} \]

       2. associate-/l* 59.0%

          \[\leadsto \frac{\color{blue}{\left(x \cdot \frac{y}{c}\right)} \cdot 9}{z} \]

       3. associate-*l* 59.0%

          \[\leadsto \frac{\color{blue}{x \cdot \left(\frac{y}{c} \cdot 9\right)}}{z} \]

       4. *-commutative 59.0%

          \[\leadsto \frac{x \cdot \color{blue}{\left(9 \cdot \frac{y}{c}\right)}}{z} \]

       5. associate-*r/ 59.1%

          \[\leadsto \frac{x \cdot \color{blue}{\frac{9 \cdot y}{c}}}{z} \]

   11. Simplified 59.1%

       \[\leadsto \frac{\color{blue}{x \cdot \frac{9 \cdot y}{c}}}{z} \]

   if 3.8000000000000003e-30 < b

   1. Initial program 81.9%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Step-by-step derivation

      1. +-commutative 81.9%

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)}}{z \cdot c} \]

      2. associate-+r- 81.9%

         \[\leadsto \frac{\color{blue}{\left(b + \left(x \cdot 9\right) \cdot y\right) - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}}{z \cdot c} \]

      3. *-commutative 81.9%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{a \cdot \left(\left(z \cdot 4\right) \cdot t\right)}}{z \cdot c} \]

      4. associate-*r* 84.5%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{\left(a \cdot \left(z \cdot 4\right)\right) \cdot t}}{z \cdot c} \]

      5. *-commutative 84.5%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{\left(\left(z \cdot 4\right) \cdot a\right)} \cdot t}{z \cdot c} \]

      6. associate-+r- 84.5%

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right)}}{z \cdot c} \]

      7. +-commutative 84.5%

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right) + b}}{z \cdot c} \]

      8. associate-*l* 85.8%

         \[\leadsto \frac{\left(\color{blue}{x \cdot \left(9 \cdot y\right)} - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right) + b}{z \cdot c} \]

      9. associate-*l* 87.2%

         \[\leadsto \frac{\left(x \cdot \left(9 \cdot y\right) - \color{blue}{\left(z \cdot 4\right) \cdot \left(a \cdot t\right)}\right) + b}{z \cdot c} \]

      10. *-commutative 87.2%

          \[\leadsto \frac{\left(x \cdot \left(9 \cdot y\right) - \left(z \cdot 4\right) \cdot \color{blue}{\left(t \cdot a\right)}\right) + b}{z \cdot c} \]

   3. Simplified 87.2%

      \[\leadsto \color{blue}{\frac{\left(x \cdot \left(9 \cdot y\right) - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around inf 68.7%

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

      1. *-commutative 68.7%

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

   7. Simplified 68.7%

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

      1. clear-num 68.6%

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

      2. inv-pow 68.6%

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

      3. *-commutative 68.6%

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

   9. Applied egg-rr 68.6%

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

       1. unpow-1 68.6%

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

       2. associate-/l* 69.2%

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

   11. Simplified 69.2%

       \[\leadsto \color{blue}{\frac{1}{c \cdot \frac{z}{b}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -7.6 \cdot 10^{-25}:\\ \;\;\;\;\frac{1}{\frac{z}{\frac{b}{c}}}\\ \mathbf{elif}\;b \leq 3.7 \cdot 10^{-233}:\\ \;\;\;\;\frac{a}{c} \cdot \left(t \cdot -4\right)\\ \mathbf{elif}\;b \leq 3.8 \cdot 10^{-30}:\\ \;\;\;\;\frac{x \cdot \frac{9 \cdot y}{c}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{c \cdot \frac{z}{b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 48.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ c\_s \cdot \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-24}:\\ \;\;\;\;\frac{1}{\frac{z}{\frac{b}{c\_m}}}\\ \mathbf{elif}\;b \leq 3.7 \cdot 10^{-233}:\\ \;\;\;\;\frac{a}{c\_m} \cdot \left(t \cdot -4\right)\\ \mathbf{elif}\;b \leq 2.4 \cdot 10^{-24}:\\ \;\;\;\;\frac{9 \cdot \left(x \cdot y\right)}{c\_m \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{c\_m \cdot \frac{z}{b}}\\ \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (*
  c_s
  (if (<= b -5e-24)
    (/ 1.0 (/ z (/ b c_m)))
    (if (<= b 3.7e-233)
      (* (/ a c_m) (* t -4.0))
      (if (<= b 2.4e-24)
        (/ (* 9.0 (* x y)) (* c_m z))
        (/ 1.0 (* c_m (/ z b))))))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double tmp;
	if (b <= -5e-24) {
		tmp = 1.0 / (z / (b / c_m));
	} else if (b <= 3.7e-233) {
		tmp = (a / c_m) * (t * -4.0);
	} else if (b <= 2.4e-24) {
		tmp = (9.0 * (x * y)) / (c_m * z);
	} else {
		tmp = 1.0 / (c_m * (z / b));
	}
	return c_s * tmp;
}

Fortran

c\_m = abs(c)
c\_s = copysign(1.0d0, c)
real(8) function code(c_s, x, y, z, t, a, b, c_m)
    real(8), intent (in) :: c_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c_m
    real(8) :: tmp
    if (b <= (-5d-24)) then
        tmp = 1.0d0 / (z / (b / c_m))
    else if (b <= 3.7d-233) then
        tmp = (a / c_m) * (t * (-4.0d0))
    else if (b <= 2.4d-24) then
        tmp = (9.0d0 * (x * y)) / (c_m * z)
    else
        tmp = 1.0d0 / (c_m * (z / b))
    end if
    code = c_s * tmp
end function

Java

c\_m = Math.abs(c);
c\_s = Math.copySign(1.0, c);
public static double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double tmp;
	if (b <= -5e-24) {
		tmp = 1.0 / (z / (b / c_m));
	} else if (b <= 3.7e-233) {
		tmp = (a / c_m) * (t * -4.0);
	} else if (b <= 2.4e-24) {
		tmp = (9.0 * (x * y)) / (c_m * z);
	} else {
		tmp = 1.0 / (c_m * (z / b));
	}
	return c_s * tmp;
}

Python

c\_m = math.fabs(c)
c\_s = math.copysign(1.0, c)
def code(c_s, x, y, z, t, a, b, c_m):
	tmp = 0
	if b <= -5e-24:
		tmp = 1.0 / (z / (b / c_m))
	elif b <= 3.7e-233:
		tmp = (a / c_m) * (t * -4.0)
	elif b <= 2.4e-24:
		tmp = (9.0 * (x * y)) / (c_m * z)
	else:
		tmp = 1.0 / (c_m * (z / b))
	return c_s * tmp

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
function code(c_s, x, y, z, t, a, b, c_m)
	tmp = 0.0
	if (b <= -5e-24)
		tmp = Float64(1.0 / Float64(z / Float64(b / c_m)));
	elseif (b <= 3.7e-233)
		tmp = Float64(Float64(a / c_m) * Float64(t * -4.0));
	elseif (b <= 2.4e-24)
		tmp = Float64(Float64(9.0 * Float64(x * y)) / Float64(c_m * z));
	else
		tmp = Float64(1.0 / Float64(c_m * Float64(z / b)));
	end
	return Float64(c_s * tmp)
end

MATLAB

c\_m = abs(c);
c\_s = sign(c) * abs(1.0);
function tmp_2 = code(c_s, x, y, z, t, a, b, c_m)
	tmp = 0.0;
	if (b <= -5e-24)
		tmp = 1.0 / (z / (b / c_m));
	elseif (b <= 3.7e-233)
		tmp = (a / c_m) * (t * -4.0);
	elseif (b <= 2.4e-24)
		tmp = (9.0 * (x * y)) / (c_m * z);
	else
		tmp = 1.0 / (c_m * (z / b));
	end
	tmp_2 = c_s * tmp;
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := N[(c$95$s * If[LessEqual[b, -5e-24], N[(1.0 / N[(z / N[(b / c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 3.7e-233], N[(N[(a / c$95$m), $MachinePrecision] * N[(t * -4.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.4e-24], N[(N[(9.0 * N[(x * y), $MachinePrecision]), $MachinePrecision] / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(c$95$m * N[(z / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if b < -4.9999999999999998e-24

   1. Initial program 82.4%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Step-by-step derivation

      1. +-commutative 82.4%

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)}}{z \cdot c} \]

      2. associate-+r- 82.4%

         \[\leadsto \frac{\color{blue}{\left(b + \left(x \cdot 9\right) \cdot y\right) - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}}{z \cdot c} \]

      3. *-commutative 82.4%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{a \cdot \left(\left(z \cdot 4\right) \cdot t\right)}}{z \cdot c} \]

      4. associate-*r* 83.9%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{\left(a \cdot \left(z \cdot 4\right)\right) \cdot t}}{z \cdot c} \]

      5. *-commutative 83.9%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{\left(\left(z \cdot 4\right) \cdot a\right)} \cdot t}{z \cdot c} \]

      6. associate-+r- 83.9%

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right)}}{z \cdot c} \]

      7. +-commutative 83.9%

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right) + b}}{z \cdot c} \]

      8. associate-*l* 83.9%

         \[\leadsto \frac{\left(\color{blue}{x \cdot \left(9 \cdot y\right)} - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right) + b}{z \cdot c} \]

      9. associate-*l* 81.0%

         \[\leadsto \frac{\left(x \cdot \left(9 \cdot y\right) - \color{blue}{\left(z \cdot 4\right) \cdot \left(a \cdot t\right)}\right) + b}{z \cdot c} \]

      10. *-commutative 81.0%

          \[\leadsto \frac{\left(x \cdot \left(9 \cdot y\right) - \left(z \cdot 4\right) \cdot \color{blue}{\left(t \cdot a\right)}\right) + b}{z \cdot c} \]

   3. Simplified 81.0%

      \[\leadsto \color{blue}{\frac{\left(x \cdot \left(9 \cdot y\right) - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 75.3%

      \[\leadsto \color{blue}{\frac{-4 \cdot \frac{a \cdot \left(t \cdot z\right)}{c} + \left(9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z}} \]
   6. Step-by-step derivation

      1. fma-define 75.3%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-4, \frac{a \cdot \left(t \cdot z\right)}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}}{z} \]

      2. *-commutative 75.3%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{\color{blue}{\left(t \cdot z\right) \cdot a}}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      3. *-commutative 75.3%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{\color{blue}{\left(z \cdot t\right)} \cdot a}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      4. associate-*r* 75.3%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{\color{blue}{z \cdot \left(t \cdot a\right)}}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      5. *-commutative 75.3%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{z \cdot \color{blue}{\left(a \cdot t\right)}}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      6. associate-/l* 75.4%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \color{blue}{z \cdot \frac{a \cdot t}{c}}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      7. associate-/l* 79.6%

         \[\leadsto \frac{\mathsf{fma}\left(-4, z \cdot \color{blue}{\left(a \cdot \frac{t}{c}\right)}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      8. fma-define 79.6%

         \[\leadsto \frac{\mathsf{fma}\left(-4, z \cdot \left(a \cdot \frac{t}{c}\right), \color{blue}{\mathsf{fma}\left(9, \frac{x \cdot y}{c}, \frac{b}{c}\right)}\right)}{z} \]

      9. associate-/l* 83.8%

         \[\leadsto \frac{\mathsf{fma}\left(-4, z \cdot \left(a \cdot \frac{t}{c}\right), \mathsf{fma}\left(9, \color{blue}{x \cdot \frac{y}{c}}, \frac{b}{c}\right)\right)}{z} \]

   7. Simplified 83.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-4, z \cdot \left(a \cdot \frac{t}{c}\right), \mathsf{fma}\left(9, x \cdot \frac{y}{c}, \frac{b}{c}\right)\right)}{z}} \]

   8. Taylor expanded in b around inf 62.2%

      \[\leadsto \frac{\color{blue}{\frac{b}{c}}}{z} \]
   9. Step-by-step derivation

      1. clear-num 62.3%

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

      2. inv-pow 62.3%

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

   10. Applied egg-rr 62.3%

       \[\leadsto \color{blue}{{\left(\frac{z}{\frac{b}{c}}\right)}^{-1}} \]
   11. Step-by-step derivation

       1. unpow-1 62.3%

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

   12. Simplified 62.3%

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

   if -4.9999999999999998e-24 < b < 3.6999999999999998e-233

   1. Initial program 74.6%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Step-by-step derivation

      1. +-commutative 74.6%

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)}}{z \cdot c} \]

      2. associate-+r- 74.6%

         \[\leadsto \frac{\color{blue}{\left(b + \left(x \cdot 9\right) \cdot y\right) - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}}{z \cdot c} \]

      3. *-commutative 74.6%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{a \cdot \left(\left(z \cdot 4\right) \cdot t\right)}}{z \cdot c} \]

      4. associate-*r* 73.4%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{\left(a \cdot \left(z \cdot 4\right)\right) \cdot t}}{z \cdot c} \]

      5. *-commutative 73.4%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{\left(\left(z \cdot 4\right) \cdot a\right)} \cdot t}{z \cdot c} \]

      6. associate-+r- 73.4%

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right)}}{z \cdot c} \]

      7. +-commutative 73.4%

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right) + b}}{z \cdot c} \]

      8. associate-*l* 73.4%

         \[\leadsto \frac{\left(\color{blue}{x \cdot \left(9 \cdot y\right)} - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right) + b}{z \cdot c} \]

      9. associate-*l* 77.4%

         \[\leadsto \frac{\left(x \cdot \left(9 \cdot y\right) - \color{blue}{\left(z \cdot 4\right) \cdot \left(a \cdot t\right)}\right) + b}{z \cdot c} \]

      10. *-commutative 77.4%

          \[\leadsto \frac{\left(x \cdot \left(9 \cdot y\right) - \left(z \cdot 4\right) \cdot \color{blue}{\left(t \cdot a\right)}\right) + b}{z \cdot c} \]

   3. Simplified 77.4%

      \[\leadsto \color{blue}{\frac{\left(x \cdot \left(9 \cdot y\right) - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 72.6%

      \[\leadsto \color{blue}{\frac{-4 \cdot \frac{a \cdot \left(t \cdot z\right)}{c} + \left(9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z}} \]
   6. Step-by-step derivation

      1. fma-define 72.6%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-4, \frac{a \cdot \left(t \cdot z\right)}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}}{z} \]

      2. *-commutative 72.6%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{\color{blue}{\left(t \cdot z\right) \cdot a}}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      3. *-commutative 72.6%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{\color{blue}{\left(z \cdot t\right)} \cdot a}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      4. associate-*r* 74.2%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{\color{blue}{z \cdot \left(t \cdot a\right)}}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      5. *-commutative 74.2%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{z \cdot \color{blue}{\left(a \cdot t\right)}}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      6. associate-/l* 79.5%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \color{blue}{z \cdot \frac{a \cdot t}{c}}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      7. associate-/l* 80.1%

         \[\leadsto \frac{\mathsf{fma}\left(-4, z \cdot \color{blue}{\left(a \cdot \frac{t}{c}\right)}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      8. fma-define 80.1%

         \[\leadsto \frac{\mathsf{fma}\left(-4, z \cdot \left(a \cdot \frac{t}{c}\right), \color{blue}{\mathsf{fma}\left(9, \frac{x \cdot y}{c}, \frac{b}{c}\right)}\right)}{z} \]

      9. associate-/l* 80.2%

         \[\leadsto \frac{\mathsf{fma}\left(-4, z \cdot \left(a \cdot \frac{t}{c}\right), \mathsf{fma}\left(9, \color{blue}{x \cdot \frac{y}{c}}, \frac{b}{c}\right)\right)}{z} \]

   7. Simplified 80.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-4, z \cdot \left(a \cdot \frac{t}{c}\right), \mathsf{fma}\left(9, x \cdot \frac{y}{c}, \frac{b}{c}\right)\right)}{z}} \]

   8. Taylor expanded in z around inf 49.2%

      \[\leadsto \color{blue}{-4 \cdot \frac{a \cdot t}{c}} \]
   9. Step-by-step derivation

      1. associate-*r/ 49.2%

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

      2. associate-*r* 49.2%

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

      3. associate-*l/ 56.8%

         \[\leadsto \color{blue}{\frac{-4 \cdot a}{c} \cdot t} \]

      4. associate-*r/ 56.8%

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

      5. *-commutative 56.8%

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

      6. associate-*l* 56.8%

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

   10. Simplified 56.8%

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

   if 3.6999999999999998e-233 < b < 2.3999999999999998e-24

   1. Initial program 82.9%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Step-by-step derivation

      1. +-commutative 82.9%

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)}}{z \cdot c} \]

      2. associate-+r- 82.9%

         \[\leadsto \frac{\color{blue}{\left(b + \left(x \cdot 9\right) \cdot y\right) - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}}{z \cdot c} \]

      3. *-commutative 82.9%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{a \cdot \left(\left(z \cdot 4\right) \cdot t\right)}}{z \cdot c} \]

      4. associate-*r* 83.1%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{\left(a \cdot \left(z \cdot 4\right)\right) \cdot t}}{z \cdot c} \]

      5. *-commutative 83.1%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{\left(\left(z \cdot 4\right) \cdot a\right)} \cdot t}{z \cdot c} \]

      6. associate-+r- 83.1%

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right)}}{z \cdot c} \]

      7. +-commutative 83.1%

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right) + b}}{z \cdot c} \]

      8. associate-*l* 83.2%

         \[\leadsto \frac{\left(\color{blue}{x \cdot \left(9 \cdot y\right)} - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right) + b}{z \cdot c} \]

      9. associate-*l* 83.2%

         \[\leadsto \frac{\left(x \cdot \left(9 \cdot y\right) - \color{blue}{\left(z \cdot 4\right) \cdot \left(a \cdot t\right)}\right) + b}{z \cdot c} \]

      10. *-commutative 83.2%

          \[\leadsto \frac{\left(x \cdot \left(9 \cdot y\right) - \left(z \cdot 4\right) \cdot \color{blue}{\left(t \cdot a\right)}\right) + b}{z \cdot c} \]

   3. Simplified 83.2%

      \[\leadsto \color{blue}{\frac{\left(x \cdot \left(9 \cdot y\right) - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 61.4%

      \[\leadsto \color{blue}{9 \cdot \frac{x \cdot y}{c \cdot z}} \]
   6. Step-by-step derivation

      1. associate-*r/ 61.4%

         \[\leadsto \color{blue}{\frac{9 \cdot \left(x \cdot y\right)}{c \cdot z}} \]

   7. Applied egg-rr 61.4%

      \[\leadsto \color{blue}{\frac{9 \cdot \left(x \cdot y\right)}{c \cdot z}} \]

   if 2.3999999999999998e-24 < b

   1. Initial program 81.9%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Step-by-step derivation

      1. +-commutative 81.9%

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)}}{z \cdot c} \]

      2. associate-+r- 81.9%

         \[\leadsto \frac{\color{blue}{\left(b + \left(x \cdot 9\right) \cdot y\right) - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}}{z \cdot c} \]

      3. *-commutative 81.9%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{a \cdot \left(\left(z \cdot 4\right) \cdot t\right)}}{z \cdot c} \]

      4. associate-*r* 84.5%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{\left(a \cdot \left(z \cdot 4\right)\right) \cdot t}}{z \cdot c} \]

      5. *-commutative 84.5%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{\left(\left(z \cdot 4\right) \cdot a\right)} \cdot t}{z \cdot c} \]

      6. associate-+r- 84.5%

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right)}}{z \cdot c} \]

      7. +-commutative 84.5%

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right) + b}}{z \cdot c} \]

      8. associate-*l* 85.8%

         \[\leadsto \frac{\left(\color{blue}{x \cdot \left(9 \cdot y\right)} - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right) + b}{z \cdot c} \]

      9. associate-*l* 87.2%

         \[\leadsto \frac{\left(x \cdot \left(9 \cdot y\right) - \color{blue}{\left(z \cdot 4\right) \cdot \left(a \cdot t\right)}\right) + b}{z \cdot c} \]

      10. *-commutative 87.2%

          \[\leadsto \frac{\left(x \cdot \left(9 \cdot y\right) - \left(z \cdot 4\right) \cdot \color{blue}{\left(t \cdot a\right)}\right) + b}{z \cdot c} \]

   3. Simplified 87.2%

      \[\leadsto \color{blue}{\frac{\left(x \cdot \left(9 \cdot y\right) - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around inf 68.7%

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

      1. *-commutative 68.7%

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

   7. Simplified 68.7%

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

      1. clear-num 68.6%

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

      2. inv-pow 68.6%

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

      3. *-commutative 68.6%

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

   9. Applied egg-rr 68.6%

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

       1. unpow-1 68.6%

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

       2. associate-/l* 69.2%

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

   11. Simplified 69.2%

       \[\leadsto \color{blue}{\frac{1}{c \cdot \frac{z}{b}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 62.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-24}:\\ \;\;\;\;\frac{1}{\frac{z}{\frac{b}{c}}}\\ \mathbf{elif}\;b \leq 3.7 \cdot 10^{-233}:\\ \;\;\;\;\frac{a}{c} \cdot \left(t \cdot -4\right)\\ \mathbf{elif}\;b \leq 2.4 \cdot 10^{-24}:\\ \;\;\;\;\frac{9 \cdot \left(x \cdot y\right)}{c \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{c \cdot \frac{z}{b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 48.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ c\_s \cdot \begin{array}{l} \mathbf{if}\;b \leq -2.9 \cdot 10^{-25}:\\ \;\;\;\;\frac{1}{\frac{z}{\frac{b}{c\_m}}}\\ \mathbf{elif}\;b \leq 4 \cdot 10^{-233}:\\ \;\;\;\;\frac{a}{c\_m} \cdot \left(t \cdot -4\right)\\ \mathbf{elif}\;b \leq 1.95 \cdot 10^{-25}:\\ \;\;\;\;9 \cdot \frac{x \cdot y}{c\_m \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{c\_m \cdot \frac{z}{b}}\\ \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (*
  c_s
  (if (<= b -2.9e-25)
    (/ 1.0 (/ z (/ b c_m)))
    (if (<= b 4e-233)
      (* (/ a c_m) (* t -4.0))
      (if (<= b 1.95e-25)
        (* 9.0 (/ (* x y) (* c_m z)))
        (/ 1.0 (* c_m (/ z b))))))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double tmp;
	if (b <= -2.9e-25) {
		tmp = 1.0 / (z / (b / c_m));
	} else if (b <= 4e-233) {
		tmp = (a / c_m) * (t * -4.0);
	} else if (b <= 1.95e-25) {
		tmp = 9.0 * ((x * y) / (c_m * z));
	} else {
		tmp = 1.0 / (c_m * (z / b));
	}
	return c_s * tmp;
}

Fortran

c\_m = abs(c)
c\_s = copysign(1.0d0, c)
real(8) function code(c_s, x, y, z, t, a, b, c_m)
    real(8), intent (in) :: c_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c_m
    real(8) :: tmp
    if (b <= (-2.9d-25)) then
        tmp = 1.0d0 / (z / (b / c_m))
    else if (b <= 4d-233) then
        tmp = (a / c_m) * (t * (-4.0d0))
    else if (b <= 1.95d-25) then
        tmp = 9.0d0 * ((x * y) / (c_m * z))
    else
        tmp = 1.0d0 / (c_m * (z / b))
    end if
    code = c_s * tmp
end function

Java

c\_m = Math.abs(c);
c\_s = Math.copySign(1.0, c);
public static double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double tmp;
	if (b <= -2.9e-25) {
		tmp = 1.0 / (z / (b / c_m));
	} else if (b <= 4e-233) {
		tmp = (a / c_m) * (t * -4.0);
	} else if (b <= 1.95e-25) {
		tmp = 9.0 * ((x * y) / (c_m * z));
	} else {
		tmp = 1.0 / (c_m * (z / b));
	}
	return c_s * tmp;
}

Python

c\_m = math.fabs(c)
c\_s = math.copysign(1.0, c)
def code(c_s, x, y, z, t, a, b, c_m):
	tmp = 0
	if b <= -2.9e-25:
		tmp = 1.0 / (z / (b / c_m))
	elif b <= 4e-233:
		tmp = (a / c_m) * (t * -4.0)
	elif b <= 1.95e-25:
		tmp = 9.0 * ((x * y) / (c_m * z))
	else:
		tmp = 1.0 / (c_m * (z / b))
	return c_s * tmp

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
function code(c_s, x, y, z, t, a, b, c_m)
	tmp = 0.0
	if (b <= -2.9e-25)
		tmp = Float64(1.0 / Float64(z / Float64(b / c_m)));
	elseif (b <= 4e-233)
		tmp = Float64(Float64(a / c_m) * Float64(t * -4.0));
	elseif (b <= 1.95e-25)
		tmp = Float64(9.0 * Float64(Float64(x * y) / Float64(c_m * z)));
	else
		tmp = Float64(1.0 / Float64(c_m * Float64(z / b)));
	end
	return Float64(c_s * tmp)
end

MATLAB

c\_m = abs(c);
c\_s = sign(c) * abs(1.0);
function tmp_2 = code(c_s, x, y, z, t, a, b, c_m)
	tmp = 0.0;
	if (b <= -2.9e-25)
		tmp = 1.0 / (z / (b / c_m));
	elseif (b <= 4e-233)
		tmp = (a / c_m) * (t * -4.0);
	elseif (b <= 1.95e-25)
		tmp = 9.0 * ((x * y) / (c_m * z));
	else
		tmp = 1.0 / (c_m * (z / b));
	end
	tmp_2 = c_s * tmp;
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := N[(c$95$s * If[LessEqual[b, -2.9e-25], N[(1.0 / N[(z / N[(b / c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 4e-233], N[(N[(a / c$95$m), $MachinePrecision] * N[(t * -4.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.95e-25], N[(9.0 * N[(N[(x * y), $MachinePrecision] / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(c$95$m * N[(z / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if b < -2.9000000000000001e-25

   1. Initial program 82.4%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Step-by-step derivation

      1. +-commutative 82.4%

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)}}{z \cdot c} \]

      2. associate-+r- 82.4%

         \[\leadsto \frac{\color{blue}{\left(b + \left(x \cdot 9\right) \cdot y\right) - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}}{z \cdot c} \]

      3. *-commutative 82.4%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{a \cdot \left(\left(z \cdot 4\right) \cdot t\right)}}{z \cdot c} \]

      4. associate-*r* 83.9%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{\left(a \cdot \left(z \cdot 4\right)\right) \cdot t}}{z \cdot c} \]

      5. *-commutative 83.9%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{\left(\left(z \cdot 4\right) \cdot a\right)} \cdot t}{z \cdot c} \]

      6. associate-+r- 83.9%

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right)}}{z \cdot c} \]

      7. +-commutative 83.9%

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right) + b}}{z \cdot c} \]

      8. associate-*l* 83.9%

         \[\leadsto \frac{\left(\color{blue}{x \cdot \left(9 \cdot y\right)} - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right) + b}{z \cdot c} \]

      9. associate-*l* 81.0%

         \[\leadsto \frac{\left(x \cdot \left(9 \cdot y\right) - \color{blue}{\left(z \cdot 4\right) \cdot \left(a \cdot t\right)}\right) + b}{z \cdot c} \]

      10. *-commutative 81.0%

          \[\leadsto \frac{\left(x \cdot \left(9 \cdot y\right) - \left(z \cdot 4\right) \cdot \color{blue}{\left(t \cdot a\right)}\right) + b}{z \cdot c} \]

   3. Simplified 81.0%

      \[\leadsto \color{blue}{\frac{\left(x \cdot \left(9 \cdot y\right) - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 75.3%

      \[\leadsto \color{blue}{\frac{-4 \cdot \frac{a \cdot \left(t \cdot z\right)}{c} + \left(9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z}} \]
   6. Step-by-step derivation

      1. fma-define 75.3%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-4, \frac{a \cdot \left(t \cdot z\right)}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}}{z} \]

      2. *-commutative 75.3%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{\color{blue}{\left(t \cdot z\right) \cdot a}}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      3. *-commutative 75.3%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{\color{blue}{\left(z \cdot t\right)} \cdot a}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      4. associate-*r* 75.3%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{\color{blue}{z \cdot \left(t \cdot a\right)}}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      5. *-commutative 75.3%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{z \cdot \color{blue}{\left(a \cdot t\right)}}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      6. associate-/l* 75.4%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \color{blue}{z \cdot \frac{a \cdot t}{c}}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      7. associate-/l* 79.6%

         \[\leadsto \frac{\mathsf{fma}\left(-4, z \cdot \color{blue}{\left(a \cdot \frac{t}{c}\right)}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      8. fma-define 79.6%

         \[\leadsto \frac{\mathsf{fma}\left(-4, z \cdot \left(a \cdot \frac{t}{c}\right), \color{blue}{\mathsf{fma}\left(9, \frac{x \cdot y}{c}, \frac{b}{c}\right)}\right)}{z} \]

      9. associate-/l* 83.8%

         \[\leadsto \frac{\mathsf{fma}\left(-4, z \cdot \left(a \cdot \frac{t}{c}\right), \mathsf{fma}\left(9, \color{blue}{x \cdot \frac{y}{c}}, \frac{b}{c}\right)\right)}{z} \]

   7. Simplified 83.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-4, z \cdot \left(a \cdot \frac{t}{c}\right), \mathsf{fma}\left(9, x \cdot \frac{y}{c}, \frac{b}{c}\right)\right)}{z}} \]

   8. Taylor expanded in b around inf 62.2%

      \[\leadsto \frac{\color{blue}{\frac{b}{c}}}{z} \]
   9. Step-by-step derivation

      1. clear-num 62.3%

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

      2. inv-pow 62.3%

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

   10. Applied egg-rr 62.3%

       \[\leadsto \color{blue}{{\left(\frac{z}{\frac{b}{c}}\right)}^{-1}} \]
   11. Step-by-step derivation

       1. unpow-1 62.3%

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

   12. Simplified 62.3%

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

   if -2.9000000000000001e-25 < b < 3.99999999999999983e-233

   1. Initial program 74.6%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Step-by-step derivation

      1. +-commutative 74.6%

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)}}{z \cdot c} \]

      2. associate-+r- 74.6%

         \[\leadsto \frac{\color{blue}{\left(b + \left(x \cdot 9\right) \cdot y\right) - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}}{z \cdot c} \]

      3. *-commutative 74.6%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{a \cdot \left(\left(z \cdot 4\right) \cdot t\right)}}{z \cdot c} \]

      4. associate-*r* 73.4%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{\left(a \cdot \left(z \cdot 4\right)\right) \cdot t}}{z \cdot c} \]

      5. *-commutative 73.4%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{\left(\left(z \cdot 4\right) \cdot a\right)} \cdot t}{z \cdot c} \]

      6. associate-+r- 73.4%

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right)}}{z \cdot c} \]

      7. +-commutative 73.4%

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right) + b}}{z \cdot c} \]

      8. associate-*l* 73.4%

         \[\leadsto \frac{\left(\color{blue}{x \cdot \left(9 \cdot y\right)} - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right) + b}{z \cdot c} \]

      9. associate-*l* 77.4%

         \[\leadsto \frac{\left(x \cdot \left(9 \cdot y\right) - \color{blue}{\left(z \cdot 4\right) \cdot \left(a \cdot t\right)}\right) + b}{z \cdot c} \]

      10. *-commutative 77.4%

          \[\leadsto \frac{\left(x \cdot \left(9 \cdot y\right) - \left(z \cdot 4\right) \cdot \color{blue}{\left(t \cdot a\right)}\right) + b}{z \cdot c} \]

   3. Simplified 77.4%

      \[\leadsto \color{blue}{\frac{\left(x \cdot \left(9 \cdot y\right) - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 72.6%

      \[\leadsto \color{blue}{\frac{-4 \cdot \frac{a \cdot \left(t \cdot z\right)}{c} + \left(9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z}} \]
   6. Step-by-step derivation

      1. fma-define 72.6%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-4, \frac{a \cdot \left(t \cdot z\right)}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}}{z} \]

      2. *-commutative 72.6%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{\color{blue}{\left(t \cdot z\right) \cdot a}}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      3. *-commutative 72.6%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{\color{blue}{\left(z \cdot t\right)} \cdot a}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      4. associate-*r* 74.2%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{\color{blue}{z \cdot \left(t \cdot a\right)}}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      5. *-commutative 74.2%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{z \cdot \color{blue}{\left(a \cdot t\right)}}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      6. associate-/l* 79.5%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \color{blue}{z \cdot \frac{a \cdot t}{c}}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      7. associate-/l* 80.1%

         \[\leadsto \frac{\mathsf{fma}\left(-4, z \cdot \color{blue}{\left(a \cdot \frac{t}{c}\right)}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      8. fma-define 80.1%

         \[\leadsto \frac{\mathsf{fma}\left(-4, z \cdot \left(a \cdot \frac{t}{c}\right), \color{blue}{\mathsf{fma}\left(9, \frac{x \cdot y}{c}, \frac{b}{c}\right)}\right)}{z} \]

      9. associate-/l* 80.2%

         \[\leadsto \frac{\mathsf{fma}\left(-4, z \cdot \left(a \cdot \frac{t}{c}\right), \mathsf{fma}\left(9, \color{blue}{x \cdot \frac{y}{c}}, \frac{b}{c}\right)\right)}{z} \]

   7. Simplified 80.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-4, z \cdot \left(a \cdot \frac{t}{c}\right), \mathsf{fma}\left(9, x \cdot \frac{y}{c}, \frac{b}{c}\right)\right)}{z}} \]

   8. Taylor expanded in z around inf 49.2%

      \[\leadsto \color{blue}{-4 \cdot \frac{a \cdot t}{c}} \]
   9. Step-by-step derivation

      1. associate-*r/ 49.2%

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

      2. associate-*r* 49.2%

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

      3. associate-*l/ 56.8%

         \[\leadsto \color{blue}{\frac{-4 \cdot a}{c} \cdot t} \]

      4. associate-*r/ 56.8%

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

      5. *-commutative 56.8%

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

      6. associate-*l* 56.8%

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

   10. Simplified 56.8%

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

   if 3.99999999999999983e-233 < b < 1.95e-25

   1. Initial program 82.9%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Step-by-step derivation

      1. +-commutative 82.9%

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)}}{z \cdot c} \]

      2. associate-+r- 82.9%

         \[\leadsto \frac{\color{blue}{\left(b + \left(x \cdot 9\right) \cdot y\right) - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}}{z \cdot c} \]

      3. *-commutative 82.9%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{a \cdot \left(\left(z \cdot 4\right) \cdot t\right)}}{z \cdot c} \]

      4. associate-*r* 83.1%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{\left(a \cdot \left(z \cdot 4\right)\right) \cdot t}}{z \cdot c} \]

      5. *-commutative 83.1%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{\left(\left(z \cdot 4\right) \cdot a\right)} \cdot t}{z \cdot c} \]

      6. associate-+r- 83.1%

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right)}}{z \cdot c} \]

      7. +-commutative 83.1%

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right) + b}}{z \cdot c} \]

      8. associate-*l* 83.2%

         \[\leadsto \frac{\left(\color{blue}{x \cdot \left(9 \cdot y\right)} - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right) + b}{z \cdot c} \]

      9. associate-*l* 83.2%

         \[\leadsto \frac{\left(x \cdot \left(9 \cdot y\right) - \color{blue}{\left(z \cdot 4\right) \cdot \left(a \cdot t\right)}\right) + b}{z \cdot c} \]

      10. *-commutative 83.2%

          \[\leadsto \frac{\left(x \cdot \left(9 \cdot y\right) - \left(z \cdot 4\right) \cdot \color{blue}{\left(t \cdot a\right)}\right) + b}{z \cdot c} \]

   3. Simplified 83.2%

      \[\leadsto \color{blue}{\frac{\left(x \cdot \left(9 \cdot y\right) - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 61.4%

      \[\leadsto \color{blue}{9 \cdot \frac{x \cdot y}{c \cdot z}} \]

   if 1.95e-25 < b

   1. Initial program 81.9%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Step-by-step derivation

      1. +-commutative 81.9%

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)}}{z \cdot c} \]

      2. associate-+r- 81.9%

         \[\leadsto \frac{\color{blue}{\left(b + \left(x \cdot 9\right) \cdot y\right) - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}}{z \cdot c} \]

      3. *-commutative 81.9%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{a \cdot \left(\left(z \cdot 4\right) \cdot t\right)}}{z \cdot c} \]

      4. associate-*r* 84.5%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{\left(a \cdot \left(z \cdot 4\right)\right) \cdot t}}{z \cdot c} \]

      5. *-commutative 84.5%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{\left(\left(z \cdot 4\right) \cdot a\right)} \cdot t}{z \cdot c} \]

      6. associate-+r- 84.5%

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right)}}{z \cdot c} \]

      7. +-commutative 84.5%

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right) + b}}{z \cdot c} \]

      8. associate-*l* 85.8%

         \[\leadsto \frac{\left(\color{blue}{x \cdot \left(9 \cdot y\right)} - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right) + b}{z \cdot c} \]

      9. associate-*l* 87.2%

         \[\leadsto \frac{\left(x \cdot \left(9 \cdot y\right) - \color{blue}{\left(z \cdot 4\right) \cdot \left(a \cdot t\right)}\right) + b}{z \cdot c} \]

      10. *-commutative 87.2%

          \[\leadsto \frac{\left(x \cdot \left(9 \cdot y\right) - \left(z \cdot 4\right) \cdot \color{blue}{\left(t \cdot a\right)}\right) + b}{z \cdot c} \]

   3. Simplified 87.2%

      \[\leadsto \color{blue}{\frac{\left(x \cdot \left(9 \cdot y\right) - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around inf 68.7%

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

      1. *-commutative 68.7%

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

   7. Simplified 68.7%

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

      1. clear-num 68.6%

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

      2. inv-pow 68.6%

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

      3. *-commutative 68.6%

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

   9. Applied egg-rr 68.6%

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

       1. unpow-1 68.6%

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

       2. associate-/l* 69.2%

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

   11. Simplified 69.2%

       \[\leadsto \color{blue}{\frac{1}{c \cdot \frac{z}{b}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 62.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.9 \cdot 10^{-25}:\\ \;\;\;\;\frac{1}{\frac{z}{\frac{b}{c}}}\\ \mathbf{elif}\;b \leq 4 \cdot 10^{-233}:\\ \;\;\;\;\frac{a}{c} \cdot \left(t \cdot -4\right)\\ \mathbf{elif}\;b \leq 1.95 \cdot 10^{-25}:\\ \;\;\;\;9 \cdot \frac{x \cdot y}{c \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{c \cdot \frac{z}{b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 48.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ c\_s \cdot \begin{array}{l} \mathbf{if}\;b \leq -1.18 \cdot 10^{-23}:\\ \;\;\;\;\frac{1}{\frac{z}{\frac{b}{c\_m}}}\\ \mathbf{elif}\;b \leq 3.8 \cdot 10^{-233}:\\ \;\;\;\;\frac{a}{c\_m} \cdot \left(t \cdot -4\right)\\ \mathbf{elif}\;b \leq 3.8 \cdot 10^{-30}:\\ \;\;\;\;9 \cdot \left(x \cdot \frac{y}{c\_m \cdot z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{c\_m \cdot \frac{z}{b}}\\ \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (*
  c_s
  (if (<= b -1.18e-23)
    (/ 1.0 (/ z (/ b c_m)))
    (if (<= b 3.8e-233)
      (* (/ a c_m) (* t -4.0))
      (if (<= b 3.8e-30)
        (* 9.0 (* x (/ y (* c_m z))))
        (/ 1.0 (* c_m (/ z b))))))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double tmp;
	if (b <= -1.18e-23) {
		tmp = 1.0 / (z / (b / c_m));
	} else if (b <= 3.8e-233) {
		tmp = (a / c_m) * (t * -4.0);
	} else if (b <= 3.8e-30) {
		tmp = 9.0 * (x * (y / (c_m * z)));
	} else {
		tmp = 1.0 / (c_m * (z / b));
	}
	return c_s * tmp;
}

Fortran

c\_m = abs(c)
c\_s = copysign(1.0d0, c)
real(8) function code(c_s, x, y, z, t, a, b, c_m)
    real(8), intent (in) :: c_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c_m
    real(8) :: tmp
    if (b <= (-1.18d-23)) then
        tmp = 1.0d0 / (z / (b / c_m))
    else if (b <= 3.8d-233) then
        tmp = (a / c_m) * (t * (-4.0d0))
    else if (b <= 3.8d-30) then
        tmp = 9.0d0 * (x * (y / (c_m * z)))
    else
        tmp = 1.0d0 / (c_m * (z / b))
    end if
    code = c_s * tmp
end function

Java

c\_m = Math.abs(c);
c\_s = Math.copySign(1.0, c);
public static double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double tmp;
	if (b <= -1.18e-23) {
		tmp = 1.0 / (z / (b / c_m));
	} else if (b <= 3.8e-233) {
		tmp = (a / c_m) * (t * -4.0);
	} else if (b <= 3.8e-30) {
		tmp = 9.0 * (x * (y / (c_m * z)));
	} else {
		tmp = 1.0 / (c_m * (z / b));
	}
	return c_s * tmp;
}

Python

c\_m = math.fabs(c)
c\_s = math.copysign(1.0, c)
def code(c_s, x, y, z, t, a, b, c_m):
	tmp = 0
	if b <= -1.18e-23:
		tmp = 1.0 / (z / (b / c_m))
	elif b <= 3.8e-233:
		tmp = (a / c_m) * (t * -4.0)
	elif b <= 3.8e-30:
		tmp = 9.0 * (x * (y / (c_m * z)))
	else:
		tmp = 1.0 / (c_m * (z / b))
	return c_s * tmp

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
function code(c_s, x, y, z, t, a, b, c_m)
	tmp = 0.0
	if (b <= -1.18e-23)
		tmp = Float64(1.0 / Float64(z / Float64(b / c_m)));
	elseif (b <= 3.8e-233)
		tmp = Float64(Float64(a / c_m) * Float64(t * -4.0));
	elseif (b <= 3.8e-30)
		tmp = Float64(9.0 * Float64(x * Float64(y / Float64(c_m * z))));
	else
		tmp = Float64(1.0 / Float64(c_m * Float64(z / b)));
	end
	return Float64(c_s * tmp)
end

MATLAB

c\_m = abs(c);
c\_s = sign(c) * abs(1.0);
function tmp_2 = code(c_s, x, y, z, t, a, b, c_m)
	tmp = 0.0;
	if (b <= -1.18e-23)
		tmp = 1.0 / (z / (b / c_m));
	elseif (b <= 3.8e-233)
		tmp = (a / c_m) * (t * -4.0);
	elseif (b <= 3.8e-30)
		tmp = 9.0 * (x * (y / (c_m * z)));
	else
		tmp = 1.0 / (c_m * (z / b));
	end
	tmp_2 = c_s * tmp;
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := N[(c$95$s * If[LessEqual[b, -1.18e-23], N[(1.0 / N[(z / N[(b / c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 3.8e-233], N[(N[(a / c$95$m), $MachinePrecision] * N[(t * -4.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 3.8e-30], N[(9.0 * N[(x * N[(y / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(c$95$m * N[(z / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if b < -1.18e-23

   1. Initial program 82.4%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Step-by-step derivation

      1. +-commutative 82.4%

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)}}{z \cdot c} \]

      2. associate-+r- 82.4%

         \[\leadsto \frac{\color{blue}{\left(b + \left(x \cdot 9\right) \cdot y\right) - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}}{z \cdot c} \]

      3. *-commutative 82.4%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{a \cdot \left(\left(z \cdot 4\right) \cdot t\right)}}{z \cdot c} \]

      4. associate-*r* 83.9%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{\left(a \cdot \left(z \cdot 4\right)\right) \cdot t}}{z \cdot c} \]

      5. *-commutative 83.9%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{\left(\left(z \cdot 4\right) \cdot a\right)} \cdot t}{z \cdot c} \]

      6. associate-+r- 83.9%

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right)}}{z \cdot c} \]

      7. +-commutative 83.9%

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right) + b}}{z \cdot c} \]

      8. associate-*l* 83.9%

         \[\leadsto \frac{\left(\color{blue}{x \cdot \left(9 \cdot y\right)} - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right) + b}{z \cdot c} \]

      9. associate-*l* 81.0%

         \[\leadsto \frac{\left(x \cdot \left(9 \cdot y\right) - \color{blue}{\left(z \cdot 4\right) \cdot \left(a \cdot t\right)}\right) + b}{z \cdot c} \]

      10. *-commutative 81.0%

          \[\leadsto \frac{\left(x \cdot \left(9 \cdot y\right) - \left(z \cdot 4\right) \cdot \color{blue}{\left(t \cdot a\right)}\right) + b}{z \cdot c} \]

   3. Simplified 81.0%

      \[\leadsto \color{blue}{\frac{\left(x \cdot \left(9 \cdot y\right) - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 75.3%

      \[\leadsto \color{blue}{\frac{-4 \cdot \frac{a \cdot \left(t \cdot z\right)}{c} + \left(9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z}} \]
   6. Step-by-step derivation

      1. fma-define 75.3%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-4, \frac{a \cdot \left(t \cdot z\right)}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}}{z} \]

      2. *-commutative 75.3%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{\color{blue}{\left(t \cdot z\right) \cdot a}}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      3. *-commutative 75.3%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{\color{blue}{\left(z \cdot t\right)} \cdot a}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      4. associate-*r* 75.3%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{\color{blue}{z \cdot \left(t \cdot a\right)}}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      5. *-commutative 75.3%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{z \cdot \color{blue}{\left(a \cdot t\right)}}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      6. associate-/l* 75.4%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \color{blue}{z \cdot \frac{a \cdot t}{c}}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      7. associate-/l* 79.6%

         \[\leadsto \frac{\mathsf{fma}\left(-4, z \cdot \color{blue}{\left(a \cdot \frac{t}{c}\right)}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      8. fma-define 79.6%

         \[\leadsto \frac{\mathsf{fma}\left(-4, z \cdot \left(a \cdot \frac{t}{c}\right), \color{blue}{\mathsf{fma}\left(9, \frac{x \cdot y}{c}, \frac{b}{c}\right)}\right)}{z} \]

      9. associate-/l* 83.8%

         \[\leadsto \frac{\mathsf{fma}\left(-4, z \cdot \left(a \cdot \frac{t}{c}\right), \mathsf{fma}\left(9, \color{blue}{x \cdot \frac{y}{c}}, \frac{b}{c}\right)\right)}{z} \]

   7. Simplified 83.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-4, z \cdot \left(a \cdot \frac{t}{c}\right), \mathsf{fma}\left(9, x \cdot \frac{y}{c}, \frac{b}{c}\right)\right)}{z}} \]

   8. Taylor expanded in b around inf 62.2%

      \[\leadsto \frac{\color{blue}{\frac{b}{c}}}{z} \]
   9. Step-by-step derivation

      1. clear-num 62.3%

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

      2. inv-pow 62.3%

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

   10. Applied egg-rr 62.3%

       \[\leadsto \color{blue}{{\left(\frac{z}{\frac{b}{c}}\right)}^{-1}} \]
   11. Step-by-step derivation

       1. unpow-1 62.3%

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

   12. Simplified 62.3%

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

   if -1.18e-23 < b < 3.8e-233

   1. Initial program 74.6%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Step-by-step derivation

      1. +-commutative 74.6%

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)}}{z \cdot c} \]

      2. associate-+r- 74.6%

         \[\leadsto \frac{\color{blue}{\left(b + \left(x \cdot 9\right) \cdot y\right) - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}}{z \cdot c} \]

      3. *-commutative 74.6%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{a \cdot \left(\left(z \cdot 4\right) \cdot t\right)}}{z \cdot c} \]

      4. associate-*r* 73.4%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{\left(a \cdot \left(z \cdot 4\right)\right) \cdot t}}{z \cdot c} \]

      5. *-commutative 73.4%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{\left(\left(z \cdot 4\right) \cdot a\right)} \cdot t}{z \cdot c} \]

      6. associate-+r- 73.4%

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right)}}{z \cdot c} \]

      7. +-commutative 73.4%

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right) + b}}{z \cdot c} \]

      8. associate-*l* 73.4%

         \[\leadsto \frac{\left(\color{blue}{x \cdot \left(9 \cdot y\right)} - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right) + b}{z \cdot c} \]

      9. associate-*l* 77.4%

         \[\leadsto \frac{\left(x \cdot \left(9 \cdot y\right) - \color{blue}{\left(z \cdot 4\right) \cdot \left(a \cdot t\right)}\right) + b}{z \cdot c} \]

      10. *-commutative 77.4%

          \[\leadsto \frac{\left(x \cdot \left(9 \cdot y\right) - \left(z \cdot 4\right) \cdot \color{blue}{\left(t \cdot a\right)}\right) + b}{z \cdot c} \]

   3. Simplified 77.4%

      \[\leadsto \color{blue}{\frac{\left(x \cdot \left(9 \cdot y\right) - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 72.6%

      \[\leadsto \color{blue}{\frac{-4 \cdot \frac{a \cdot \left(t \cdot z\right)}{c} + \left(9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z}} \]
   6. Step-by-step derivation

      1. fma-define 72.6%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-4, \frac{a \cdot \left(t \cdot z\right)}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}}{z} \]

      2. *-commutative 72.6%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{\color{blue}{\left(t \cdot z\right) \cdot a}}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      3. *-commutative 72.6%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{\color{blue}{\left(z \cdot t\right)} \cdot a}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      4. associate-*r* 74.2%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{\color{blue}{z \cdot \left(t \cdot a\right)}}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      5. *-commutative 74.2%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{z \cdot \color{blue}{\left(a \cdot t\right)}}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      6. associate-/l* 79.5%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \color{blue}{z \cdot \frac{a \cdot t}{c}}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      7. associate-/l* 80.1%

         \[\leadsto \frac{\mathsf{fma}\left(-4, z \cdot \color{blue}{\left(a \cdot \frac{t}{c}\right)}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      8. fma-define 80.1%

         \[\leadsto \frac{\mathsf{fma}\left(-4, z \cdot \left(a \cdot \frac{t}{c}\right), \color{blue}{\mathsf{fma}\left(9, \frac{x \cdot y}{c}, \frac{b}{c}\right)}\right)}{z} \]

      9. associate-/l* 80.2%

         \[\leadsto \frac{\mathsf{fma}\left(-4, z \cdot \left(a \cdot \frac{t}{c}\right), \mathsf{fma}\left(9, \color{blue}{x \cdot \frac{y}{c}}, \frac{b}{c}\right)\right)}{z} \]

   7. Simplified 80.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-4, z \cdot \left(a \cdot \frac{t}{c}\right), \mathsf{fma}\left(9, x \cdot \frac{y}{c}, \frac{b}{c}\right)\right)}{z}} \]

   8. Taylor expanded in z around inf 49.2%

      \[\leadsto \color{blue}{-4 \cdot \frac{a \cdot t}{c}} \]
   9. Step-by-step derivation

      1. associate-*r/ 49.2%

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

      2. associate-*r* 49.2%

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

      3. associate-*l/ 56.8%

         \[\leadsto \color{blue}{\frac{-4 \cdot a}{c} \cdot t} \]

      4. associate-*r/ 56.8%

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

      5. *-commutative 56.8%

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

      6. associate-*l* 56.8%

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

   10. Simplified 56.8%

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

   if 3.8e-233 < b < 3.8000000000000003e-30

   1. Initial program 82.9%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Step-by-step derivation

      1. +-commutative 82.9%

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)}}{z \cdot c} \]

      2. associate-+r- 82.9%

         \[\leadsto \frac{\color{blue}{\left(b + \left(x \cdot 9\right) \cdot y\right) - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}}{z \cdot c} \]

      3. *-commutative 82.9%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{a \cdot \left(\left(z \cdot 4\right) \cdot t\right)}}{z \cdot c} \]

      4. associate-*r* 83.1%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{\left(a \cdot \left(z \cdot 4\right)\right) \cdot t}}{z \cdot c} \]

      5. *-commutative 83.1%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{\left(\left(z \cdot 4\right) \cdot a\right)} \cdot t}{z \cdot c} \]

      6. associate-+r- 83.1%

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right)}}{z \cdot c} \]

      7. +-commutative 83.1%

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right) + b}}{z \cdot c} \]

      8. associate-*l* 83.2%

         \[\leadsto \frac{\left(\color{blue}{x \cdot \left(9 \cdot y\right)} - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right) + b}{z \cdot c} \]

      9. associate-*l* 83.2%

         \[\leadsto \frac{\left(x \cdot \left(9 \cdot y\right) - \color{blue}{\left(z \cdot 4\right) \cdot \left(a \cdot t\right)}\right) + b}{z \cdot c} \]

      10. *-commutative 83.2%

          \[\leadsto \frac{\left(x \cdot \left(9 \cdot y\right) - \left(z \cdot 4\right) \cdot \color{blue}{\left(t \cdot a\right)}\right) + b}{z \cdot c} \]

   3. Simplified 83.2%

      \[\leadsto \color{blue}{\frac{\left(x \cdot \left(9 \cdot y\right) - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 61.4%

      \[\leadsto \color{blue}{9 \cdot \frac{x \cdot y}{c \cdot z}} \]
   6. Step-by-step derivation

      1. associate-/l* 56.5%

         \[\leadsto 9 \cdot \color{blue}{\left(x \cdot \frac{y}{c \cdot z}\right)} \]

      2. *-commutative 56.5%

         \[\leadsto 9 \cdot \left(x \cdot \frac{y}{\color{blue}{z \cdot c}}\right) \]

   7. Simplified 56.5%

      \[\leadsto \color{blue}{9 \cdot \left(x \cdot \frac{y}{z \cdot c}\right)} \]

   if 3.8000000000000003e-30 < b

   1. Initial program 81.9%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Step-by-step derivation

      1. +-commutative 81.9%

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)}}{z \cdot c} \]

      2. associate-+r- 81.9%

         \[\leadsto \frac{\color{blue}{\left(b + \left(x \cdot 9\right) \cdot y\right) - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}}{z \cdot c} \]

      3. *-commutative 81.9%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{a \cdot \left(\left(z \cdot 4\right) \cdot t\right)}}{z \cdot c} \]

      4. associate-*r* 84.5%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{\left(a \cdot \left(z \cdot 4\right)\right) \cdot t}}{z \cdot c} \]

      5. *-commutative 84.5%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{\left(\left(z \cdot 4\right) \cdot a\right)} \cdot t}{z \cdot c} \]

      6. associate-+r- 84.5%

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right)}}{z \cdot c} \]

      7. +-commutative 84.5%

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right) + b}}{z \cdot c} \]

      8. associate-*l* 85.8%

         \[\leadsto \frac{\left(\color{blue}{x \cdot \left(9 \cdot y\right)} - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right) + b}{z \cdot c} \]

      9. associate-*l* 87.2%

         \[\leadsto \frac{\left(x \cdot \left(9 \cdot y\right) - \color{blue}{\left(z \cdot 4\right) \cdot \left(a \cdot t\right)}\right) + b}{z \cdot c} \]

      10. *-commutative 87.2%

          \[\leadsto \frac{\left(x \cdot \left(9 \cdot y\right) - \left(z \cdot 4\right) \cdot \color{blue}{\left(t \cdot a\right)}\right) + b}{z \cdot c} \]

   3. Simplified 87.2%

      \[\leadsto \color{blue}{\frac{\left(x \cdot \left(9 \cdot y\right) - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around inf 68.7%

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

      1. *-commutative 68.7%

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

   7. Simplified 68.7%

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

      1. clear-num 68.6%

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

      2. inv-pow 68.6%

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

      3. *-commutative 68.6%

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

   9. Applied egg-rr 68.6%

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

       1. unpow-1 68.6%

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

       2. associate-/l* 69.2%

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

   11. Simplified 69.2%

       \[\leadsto \color{blue}{\frac{1}{c \cdot \frac{z}{b}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 61.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.18 \cdot 10^{-23}:\\ \;\;\;\;\frac{1}{\frac{z}{\frac{b}{c}}}\\ \mathbf{elif}\;b \leq 3.8 \cdot 10^{-233}:\\ \;\;\;\;\frac{a}{c} \cdot \left(t \cdot -4\right)\\ \mathbf{elif}\;b \leq 3.8 \cdot 10^{-30}:\\ \;\;\;\;9 \cdot \left(x \cdot \frac{y}{c \cdot z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{c \cdot \frac{z}{b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 65.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ c\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -5.8 \cdot 10^{+25}:\\ \;\;\;\;x \cdot \frac{\frac{9 \cdot y}{c\_m}}{z}\\ \mathbf{elif}\;y \leq 8.6 \cdot 10^{+69}:\\ \;\;\;\;\frac{\frac{b + -4 \cdot \left(t \cdot \left(z \cdot a\right)\right)}{c\_m}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + 9 \cdot \left(x \cdot y\right)}{c\_m \cdot z}\\ \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (*
  c_s
  (if (<= y -5.8e+25)
    (* x (/ (/ (* 9.0 y) c_m) z))
    (if (<= y 8.6e+69)
      (/ (/ (+ b (* -4.0 (* t (* z a)))) c_m) z)
      (/ (+ b (* 9.0 (* x y))) (* c_m z))))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double tmp;
	if (y <= -5.8e+25) {
		tmp = x * (((9.0 * y) / c_m) / z);
	} else if (y <= 8.6e+69) {
		tmp = ((b + (-4.0 * (t * (z * a)))) / c_m) / z;
	} else {
		tmp = (b + (9.0 * (x * y))) / (c_m * z);
	}
	return c_s * tmp;
}

Fortran

c\_m = abs(c)
c\_s = copysign(1.0d0, c)
real(8) function code(c_s, x, y, z, t, a, b, c_m)
    real(8), intent (in) :: c_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c_m
    real(8) :: tmp
    if (y <= (-5.8d+25)) then
        tmp = x * (((9.0d0 * y) / c_m) / z)
    else if (y <= 8.6d+69) then
        tmp = ((b + ((-4.0d0) * (t * (z * a)))) / c_m) / z
    else
        tmp = (b + (9.0d0 * (x * y))) / (c_m * z)
    end if
    code = c_s * tmp
end function

Java

c\_m = Math.abs(c);
c\_s = Math.copySign(1.0, c);
public static double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double tmp;
	if (y <= -5.8e+25) {
		tmp = x * (((9.0 * y) / c_m) / z);
	} else if (y <= 8.6e+69) {
		tmp = ((b + (-4.0 * (t * (z * a)))) / c_m) / z;
	} else {
		tmp = (b + (9.0 * (x * y))) / (c_m * z);
	}
	return c_s * tmp;
}

Python

c\_m = math.fabs(c)
c\_s = math.copysign(1.0, c)
def code(c_s, x, y, z, t, a, b, c_m):
	tmp = 0
	if y <= -5.8e+25:
		tmp = x * (((9.0 * y) / c_m) / z)
	elif y <= 8.6e+69:
		tmp = ((b + (-4.0 * (t * (z * a)))) / c_m) / z
	else:
		tmp = (b + (9.0 * (x * y))) / (c_m * z)
	return c_s * tmp

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
function code(c_s, x, y, z, t, a, b, c_m)
	tmp = 0.0
	if (y <= -5.8e+25)
		tmp = Float64(x * Float64(Float64(Float64(9.0 * y) / c_m) / z));
	elseif (y <= 8.6e+69)
		tmp = Float64(Float64(Float64(b + Float64(-4.0 * Float64(t * Float64(z * a)))) / c_m) / z);
	else
		tmp = Float64(Float64(b + Float64(9.0 * Float64(x * y))) / Float64(c_m * z));
	end
	return Float64(c_s * tmp)
end

MATLAB

c\_m = abs(c);
c\_s = sign(c) * abs(1.0);
function tmp_2 = code(c_s, x, y, z, t, a, b, c_m)
	tmp = 0.0;
	if (y <= -5.8e+25)
		tmp = x * (((9.0 * y) / c_m) / z);
	elseif (y <= 8.6e+69)
		tmp = ((b + (-4.0 * (t * (z * a)))) / c_m) / z;
	else
		tmp = (b + (9.0 * (x * y))) / (c_m * z);
	end
	tmp_2 = c_s * tmp;
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := N[(c$95$s * If[LessEqual[y, -5.8e+25], N[(x * N[(N[(N[(9.0 * y), $MachinePrecision] / c$95$m), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 8.6e+69], N[(N[(N[(b + N[(-4.0 * N[(t * N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c$95$m), $MachinePrecision] / z), $MachinePrecision], N[(N[(b + N[(9.0 * N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < -5.7999999999999998e25

   1. Initial program 64.6%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Step-by-step derivation

      1. +-commutative 64.6%

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)}}{z \cdot c} \]

      2. associate-+r- 64.6%

         \[\leadsto \frac{\color{blue}{\left(b + \left(x \cdot 9\right) \cdot y\right) - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}}{z \cdot c} \]

      3. *-commutative 64.6%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{a \cdot \left(\left(z \cdot 4\right) \cdot t\right)}}{z \cdot c} \]

      4. associate-*r* 64.7%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{\left(a \cdot \left(z \cdot 4\right)\right) \cdot t}}{z \cdot c} \]

      5. *-commutative 64.7%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{\left(\left(z \cdot 4\right) \cdot a\right)} \cdot t}{z \cdot c} \]

      6. associate-+r- 64.7%

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right)}}{z \cdot c} \]

      7. +-commutative 64.7%

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right) + b}}{z \cdot c} \]

      8. associate-*l* 64.7%

         \[\leadsto \frac{\left(\color{blue}{x \cdot \left(9 \cdot y\right)} - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right) + b}{z \cdot c} \]

      9. associate-*l* 66.4%

         \[\leadsto \frac{\left(x \cdot \left(9 \cdot y\right) - \color{blue}{\left(z \cdot 4\right) \cdot \left(a \cdot t\right)}\right) + b}{z \cdot c} \]

      10. *-commutative 66.4%

          \[\leadsto \frac{\left(x \cdot \left(9 \cdot y\right) - \left(z \cdot 4\right) \cdot \color{blue}{\left(t \cdot a\right)}\right) + b}{z \cdot c} \]

   3. Simplified 66.4%

      \[\leadsto \color{blue}{\frac{\left(x \cdot \left(9 \cdot y\right) - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 59.7%

      \[\leadsto \color{blue}{\frac{-4 \cdot \frac{a \cdot \left(t \cdot z\right)}{c} + \left(9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z}} \]
   6. Step-by-step derivation

      1. fma-define 59.7%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-4, \frac{a \cdot \left(t \cdot z\right)}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}}{z} \]

      2. *-commutative 59.7%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{\color{blue}{\left(t \cdot z\right) \cdot a}}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      3. *-commutative 59.7%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{\color{blue}{\left(z \cdot t\right)} \cdot a}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      4. associate-*r* 63.1%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{\color{blue}{z \cdot \left(t \cdot a\right)}}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      5. *-commutative 63.1%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{z \cdot \color{blue}{\left(a \cdot t\right)}}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      6. associate-/l* 63.1%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \color{blue}{z \cdot \frac{a \cdot t}{c}}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      7. associate-/l* 63.1%

         \[\leadsto \frac{\mathsf{fma}\left(-4, z \cdot \color{blue}{\left(a \cdot \frac{t}{c}\right)}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      8. fma-define 63.1%

         \[\leadsto \frac{\mathsf{fma}\left(-4, z \cdot \left(a \cdot \frac{t}{c}\right), \color{blue}{\mathsf{fma}\left(9, \frac{x \cdot y}{c}, \frac{b}{c}\right)}\right)}{z} \]

      9. associate-/l* 64.6%

         \[\leadsto \frac{\mathsf{fma}\left(-4, z \cdot \left(a \cdot \frac{t}{c}\right), \mathsf{fma}\left(9, \color{blue}{x \cdot \frac{y}{c}}, \frac{b}{c}\right)\right)}{z} \]

   7. Simplified 64.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-4, z \cdot \left(a \cdot \frac{t}{c}\right), \mathsf{fma}\left(9, x \cdot \frac{y}{c}, \frac{b}{c}\right)\right)}{z}} \]

   8. Taylor expanded in c around 0 70.1%

      \[\leadsto \frac{\color{blue}{\frac{b + \left(-4 \cdot \left(a \cdot \left(t \cdot z\right)\right) + 9 \cdot \left(x \cdot y\right)\right)}{c}}}{z} \]

   9. Taylor expanded in x around inf 40.4%

      \[\leadsto \color{blue}{9 \cdot \frac{x \cdot y}{c \cdot z}} \]
   10. Step-by-step derivation

       1. *-commutative 40.4%

          \[\leadsto \color{blue}{\frac{x \cdot y}{c \cdot z} \cdot 9} \]

       2. *-commutative 40.4%

          \[\leadsto \frac{x \cdot y}{\color{blue}{z \cdot c}} \cdot 9 \]

       3. associate-*r/ 49.4%

          \[\leadsto \color{blue}{\left(x \cdot \frac{y}{z \cdot c}\right)} \cdot 9 \]

       4. associate-*l* 49.4%

          \[\leadsto \color{blue}{x \cdot \left(\frac{y}{z \cdot c} \cdot 9\right)} \]

       5. *-commutative 49.4%

          \[\leadsto x \cdot \left(\frac{y}{\color{blue}{c \cdot z}} \cdot 9\right) \]

       6. *-commutative 49.4%

          \[\leadsto x \cdot \color{blue}{\left(9 \cdot \frac{y}{c \cdot z}\right)} \]

       7. associate-/r* 52.8%

          \[\leadsto x \cdot \left(9 \cdot \color{blue}{\frac{\frac{y}{c}}{z}}\right) \]

       8. associate-*r/ 52.8%

          \[\leadsto x \cdot \color{blue}{\frac{9 \cdot \frac{y}{c}}{z}} \]

       9. associate-*r/ 52.8%

          \[\leadsto x \cdot \frac{\color{blue}{\frac{9 \cdot y}{c}}}{z} \]

   11. Simplified 52.8%

       \[\leadsto \color{blue}{x \cdot \frac{\frac{9 \cdot y}{c}}{z}} \]

   if -5.7999999999999998e25 < y < 8.59999999999999986e69

   1. Initial program 84.1%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Step-by-step derivation

      1. +-commutative 84.1%

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)}}{z \cdot c} \]

      2. associate-+r- 84.1%

         \[\leadsto \frac{\color{blue}{\left(b + \left(x \cdot 9\right) \cdot y\right) - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}}{z \cdot c} \]

      3. *-commutative 84.1%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{a \cdot \left(\left(z \cdot 4\right) \cdot t\right)}}{z \cdot c} \]

      4. associate-*r* 84.2%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{\left(a \cdot \left(z \cdot 4\right)\right) \cdot t}}{z \cdot c} \]

      5. *-commutative 84.2%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{\left(\left(z \cdot 4\right) \cdot a\right)} \cdot t}{z \cdot c} \]

      6. associate-+r- 84.2%

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right)}}{z \cdot c} \]

      7. +-commutative 84.2%

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right) + b}}{z \cdot c} \]

      8. associate-*l* 84.9%

         \[\leadsto \frac{\left(\color{blue}{x \cdot \left(9 \cdot y\right)} - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right) + b}{z \cdot c} \]

      9. associate-*l* 85.6%

         \[\leadsto \frac{\left(x \cdot \left(9 \cdot y\right) - \color{blue}{\left(z \cdot 4\right) \cdot \left(a \cdot t\right)}\right) + b}{z \cdot c} \]

      10. *-commutative 85.6%

          \[\leadsto \frac{\left(x \cdot \left(9 \cdot y\right) - \left(z \cdot 4\right) \cdot \color{blue}{\left(t \cdot a\right)}\right) + b}{z \cdot c} \]

   3. Simplified 85.6%

      \[\leadsto \color{blue}{\frac{\left(x \cdot \left(9 \cdot y\right) - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 80.4%

      \[\leadsto \color{blue}{\frac{-4 \cdot \frac{a \cdot \left(t \cdot z\right)}{c} + \left(9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z}} \]
   6. Step-by-step derivation

      1. fma-define 80.4%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-4, \frac{a \cdot \left(t \cdot z\right)}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}}{z} \]

      2. *-commutative 80.4%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{\color{blue}{\left(t \cdot z\right) \cdot a}}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      3. *-commutative 80.4%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{\color{blue}{\left(z \cdot t\right)} \cdot a}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      4. associate-*r* 81.9%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{\color{blue}{z \cdot \left(t \cdot a\right)}}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      5. *-commutative 81.9%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{z \cdot \color{blue}{\left(a \cdot t\right)}}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      6. associate-/l* 84.6%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \color{blue}{z \cdot \frac{a \cdot t}{c}}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      7. associate-/l* 84.7%

         \[\leadsto \frac{\mathsf{fma}\left(-4, z \cdot \color{blue}{\left(a \cdot \frac{t}{c}\right)}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      8. fma-define 84.7%

         \[\leadsto \frac{\mathsf{fma}\left(-4, z \cdot \left(a \cdot \frac{t}{c}\right), \color{blue}{\mathsf{fma}\left(9, \frac{x \cdot y}{c}, \frac{b}{c}\right)}\right)}{z} \]

      9. associate-/l* 84.1%

         \[\leadsto \frac{\mathsf{fma}\left(-4, z \cdot \left(a \cdot \frac{t}{c}\right), \mathsf{fma}\left(9, \color{blue}{x \cdot \frac{y}{c}}, \frac{b}{c}\right)\right)}{z} \]

   7. Simplified 84.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-4, z \cdot \left(a \cdot \frac{t}{c}\right), \mathsf{fma}\left(9, x \cdot \frac{y}{c}, \frac{b}{c}\right)\right)}{z}} \]

   8. Taylor expanded in c around 0 83.1%

      \[\leadsto \frac{\color{blue}{\frac{b + \left(-4 \cdot \left(a \cdot \left(t \cdot z\right)\right) + 9 \cdot \left(x \cdot y\right)\right)}{c}}}{z} \]

   9. Taylor expanded in a around inf 70.8%

      \[\leadsto \frac{\frac{b + \color{blue}{-4 \cdot \left(a \cdot \left(t \cdot z\right)\right)}}{c}}{z} \]
   10. Step-by-step derivation

       1. *-commutative 70.8%

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

       2. associate-*l* 72.2%

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

       3. *-commutative 72.2%

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

   11. Simplified 72.2%

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

   if 8.59999999999999986e69 < y

   1. Initial program 86.3%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Step-by-step derivation

      1. +-commutative 86.3%

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)}}{z \cdot c} \]

      2. associate-+r- 86.3%

         \[\leadsto \frac{\color{blue}{\left(b + \left(x \cdot 9\right) \cdot y\right) - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}}{z \cdot c} \]

      3. *-commutative 86.3%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{a \cdot \left(\left(z \cdot 4\right) \cdot t\right)}}{z \cdot c} \]

      4. associate-*r* 90.1%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{\left(a \cdot \left(z \cdot 4\right)\right) \cdot t}}{z \cdot c} \]

      5. *-commutative 90.1%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{\left(\left(z \cdot 4\right) \cdot a\right)} \cdot t}{z \cdot c} \]

      6. associate-+r- 90.1%

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right)}}{z \cdot c} \]

      7. +-commutative 90.1%

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right) + b}}{z \cdot c} \]

      8. associate-*l* 90.1%

         \[\leadsto \frac{\left(\color{blue}{x \cdot \left(9 \cdot y\right)} - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right) + b}{z \cdot c} \]

      9. associate-*l* 90.1%

         \[\leadsto \frac{\left(x \cdot \left(9 \cdot y\right) - \color{blue}{\left(z \cdot 4\right) \cdot \left(a \cdot t\right)}\right) + b}{z \cdot c} \]

      10. *-commutative 90.1%

          \[\leadsto \frac{\left(x \cdot \left(9 \cdot y\right) - \left(z \cdot 4\right) \cdot \color{blue}{\left(t \cdot a\right)}\right) + b}{z \cdot c} \]

   3. Simplified 90.1%

      \[\leadsto \color{blue}{\frac{\left(x \cdot \left(9 \cdot y\right) - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 84.0%

      \[\leadsto \frac{\color{blue}{9 \cdot \left(x \cdot y\right)} + b}{z \cdot c} \]
3. Recombined 3 regimes into one program.

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.8 \cdot 10^{+25}:\\ \;\;\;\;x \cdot \frac{\frac{9 \cdot y}{c}}{z}\\ \mathbf{elif}\;y \leq 8.6 \cdot 10^{+69}:\\ \;\;\;\;\frac{\frac{b + -4 \cdot \left(t \cdot \left(z \cdot a\right)\right)}{c}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + 9 \cdot \left(x \cdot y\right)}{c \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 65.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ c\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -8.5 \cdot 10^{-16}:\\ \;\;\;\;x \cdot \frac{\frac{9 \cdot y}{c\_m}}{z}\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{+22}:\\ \;\;\;\;\frac{b - 4 \cdot \left(a \cdot \left(z \cdot t\right)\right)}{c\_m \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + 9 \cdot \left(x \cdot y\right)}{c\_m \cdot z}\\ \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (*
  c_s
  (if (<= y -8.5e-16)
    (* x (/ (/ (* 9.0 y) c_m) z))
    (if (<= y 3.6e+22)
      (/ (- b (* 4.0 (* a (* z t)))) (* c_m z))
      (/ (+ b (* 9.0 (* x y))) (* c_m z))))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double tmp;
	if (y <= -8.5e-16) {
		tmp = x * (((9.0 * y) / c_m) / z);
	} else if (y <= 3.6e+22) {
		tmp = (b - (4.0 * (a * (z * t)))) / (c_m * z);
	} else {
		tmp = (b + (9.0 * (x * y))) / (c_m * z);
	}
	return c_s * tmp;
}

Fortran

c\_m = abs(c)
c\_s = copysign(1.0d0, c)
real(8) function code(c_s, x, y, z, t, a, b, c_m)
    real(8), intent (in) :: c_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c_m
    real(8) :: tmp
    if (y <= (-8.5d-16)) then
        tmp = x * (((9.0d0 * y) / c_m) / z)
    else if (y <= 3.6d+22) then
        tmp = (b - (4.0d0 * (a * (z * t)))) / (c_m * z)
    else
        tmp = (b + (9.0d0 * (x * y))) / (c_m * z)
    end if
    code = c_s * tmp
end function

Java

c\_m = Math.abs(c);
c\_s = Math.copySign(1.0, c);
public static double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double tmp;
	if (y <= -8.5e-16) {
		tmp = x * (((9.0 * y) / c_m) / z);
	} else if (y <= 3.6e+22) {
		tmp = (b - (4.0 * (a * (z * t)))) / (c_m * z);
	} else {
		tmp = (b + (9.0 * (x * y))) / (c_m * z);
	}
	return c_s * tmp;
}

Python

c\_m = math.fabs(c)
c\_s = math.copysign(1.0, c)
def code(c_s, x, y, z, t, a, b, c_m):
	tmp = 0
	if y <= -8.5e-16:
		tmp = x * (((9.0 * y) / c_m) / z)
	elif y <= 3.6e+22:
		tmp = (b - (4.0 * (a * (z * t)))) / (c_m * z)
	else:
		tmp = (b + (9.0 * (x * y))) / (c_m * z)
	return c_s * tmp

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
function code(c_s, x, y, z, t, a, b, c_m)
	tmp = 0.0
	if (y <= -8.5e-16)
		tmp = Float64(x * Float64(Float64(Float64(9.0 * y) / c_m) / z));
	elseif (y <= 3.6e+22)
		tmp = Float64(Float64(b - Float64(4.0 * Float64(a * Float64(z * t)))) / Float64(c_m * z));
	else
		tmp = Float64(Float64(b + Float64(9.0 * Float64(x * y))) / Float64(c_m * z));
	end
	return Float64(c_s * tmp)
end

MATLAB

c\_m = abs(c);
c\_s = sign(c) * abs(1.0);
function tmp_2 = code(c_s, x, y, z, t, a, b, c_m)
	tmp = 0.0;
	if (y <= -8.5e-16)
		tmp = x * (((9.0 * y) / c_m) / z);
	elseif (y <= 3.6e+22)
		tmp = (b - (4.0 * (a * (z * t)))) / (c_m * z);
	else
		tmp = (b + (9.0 * (x * y))) / (c_m * z);
	end
	tmp_2 = c_s * tmp;
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := N[(c$95$s * If[LessEqual[y, -8.5e-16], N[(x * N[(N[(N[(9.0 * y), $MachinePrecision] / c$95$m), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.6e+22], N[(N[(b - N[(4.0 * N[(a * N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision], N[(N[(b + N[(9.0 * N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < -8.5000000000000001e-16

   1. Initial program 67.4%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Step-by-step derivation

      1. +-commutative 67.4%

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)}}{z \cdot c} \]

      2. associate-+r- 67.4%

         \[\leadsto \frac{\color{blue}{\left(b + \left(x \cdot 9\right) \cdot y\right) - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}}{z \cdot c} \]

      3. *-commutative 67.4%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{a \cdot \left(\left(z \cdot 4\right) \cdot t\right)}}{z \cdot c} \]

      4. associate-*r* 68.9%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{\left(a \cdot \left(z \cdot 4\right)\right) \cdot t}}{z \cdot c} \]

      5. *-commutative 68.9%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{\left(\left(z \cdot 4\right) \cdot a\right)} \cdot t}{z \cdot c} \]

      6. associate-+r- 68.9%

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right)}}{z \cdot c} \]

      7. +-commutative 68.9%

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right) + b}}{z \cdot c} \]

      8. associate-*l* 68.9%

         \[\leadsto \frac{\left(\color{blue}{x \cdot \left(9 \cdot y\right)} - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right) + b}{z \cdot c} \]

      9. associate-*l* 70.3%

         \[\leadsto \frac{\left(x \cdot \left(9 \cdot y\right) - \color{blue}{\left(z \cdot 4\right) \cdot \left(a \cdot t\right)}\right) + b}{z \cdot c} \]

      10. *-commutative 70.3%

          \[\leadsto \frac{\left(x \cdot \left(9 \cdot y\right) - \left(z \cdot 4\right) \cdot \color{blue}{\left(t \cdot a\right)}\right) + b}{z \cdot c} \]

   3. Simplified 70.3%

      \[\leadsto \color{blue}{\frac{\left(x \cdot \left(9 \cdot y\right) - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 63.3%

      \[\leadsto \color{blue}{\frac{-4 \cdot \frac{a \cdot \left(t \cdot z\right)}{c} + \left(9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z}} \]
   6. Step-by-step derivation

      1. fma-define 63.3%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-4, \frac{a \cdot \left(t \cdot z\right)}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}}{z} \]

      2. *-commutative 63.3%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{\color{blue}{\left(t \cdot z\right) \cdot a}}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      3. *-commutative 63.3%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{\color{blue}{\left(z \cdot t\right)} \cdot a}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      4. associate-*r* 67.6%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{\color{blue}{z \cdot \left(t \cdot a\right)}}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      5. *-commutative 67.6%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{z \cdot \color{blue}{\left(a \cdot t\right)}}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      6. associate-/l* 67.6%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \color{blue}{z \cdot \frac{a \cdot t}{c}}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      7. associate-/l* 66.2%

         \[\leadsto \frac{\mathsf{fma}\left(-4, z \cdot \color{blue}{\left(a \cdot \frac{t}{c}\right)}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      8. fma-define 66.2%

         \[\leadsto \frac{\mathsf{fma}\left(-4, z \cdot \left(a \cdot \frac{t}{c}\right), \color{blue}{\mathsf{fma}\left(9, \frac{x \cdot y}{c}, \frac{b}{c}\right)}\right)}{z} \]

      9. associate-/l* 67.4%

         \[\leadsto \frac{\mathsf{fma}\left(-4, z \cdot \left(a \cdot \frac{t}{c}\right), \mathsf{fma}\left(9, \color{blue}{x \cdot \frac{y}{c}}, \frac{b}{c}\right)\right)}{z} \]

   7. Simplified 67.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-4, z \cdot \left(a \cdot \frac{t}{c}\right), \mathsf{fma}\left(9, x \cdot \frac{y}{c}, \frac{b}{c}\right)\right)}{z}} \]

   8. Taylor expanded in c around 0 72.1%

      \[\leadsto \frac{\color{blue}{\frac{b + \left(-4 \cdot \left(a \cdot \left(t \cdot z\right)\right) + 9 \cdot \left(x \cdot y\right)\right)}{c}}}{z} \]

   9. Taylor expanded in x around inf 41.4%

      \[\leadsto \color{blue}{9 \cdot \frac{x \cdot y}{c \cdot z}} \]
   10. Step-by-step derivation

       1. *-commutative 41.4%

          \[\leadsto \color{blue}{\frac{x \cdot y}{c \cdot z} \cdot 9} \]

       2. *-commutative 41.4%

          \[\leadsto \frac{x \cdot y}{\color{blue}{z \cdot c}} \cdot 9 \]

       3. associate-*r/ 49.0%

          \[\leadsto \color{blue}{\left(x \cdot \frac{y}{z \cdot c}\right)} \cdot 9 \]

       4. associate-*l* 49.0%

          \[\leadsto \color{blue}{x \cdot \left(\frac{y}{z \cdot c} \cdot 9\right)} \]

       5. *-commutative 49.0%

          \[\leadsto x \cdot \left(\frac{y}{\color{blue}{c \cdot z}} \cdot 9\right) \]

       6. *-commutative 49.0%

          \[\leadsto x \cdot \color{blue}{\left(9 \cdot \frac{y}{c \cdot z}\right)} \]

       7. associate-/r* 51.8%

          \[\leadsto x \cdot \left(9 \cdot \color{blue}{\frac{\frac{y}{c}}{z}}\right) \]

       8. associate-*r/ 51.8%

          \[\leadsto x \cdot \color{blue}{\frac{9 \cdot \frac{y}{c}}{z}} \]

       9. associate-*r/ 51.8%

          \[\leadsto x \cdot \frac{\color{blue}{\frac{9 \cdot y}{c}}}{z} \]

   11. Simplified 51.8%

       \[\leadsto \color{blue}{x \cdot \frac{\frac{9 \cdot y}{c}}{z}} \]

   if -8.5000000000000001e-16 < y < 3.6e22

   1. Initial program 84.0%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Step-by-step derivation

      1. +-commutative 84.0%

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)}}{z \cdot c} \]

      2. associate-+r- 84.0%

         \[\leadsto \frac{\color{blue}{\left(b + \left(x \cdot 9\right) \cdot y\right) - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}}{z \cdot c} \]

      3. *-commutative 84.0%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{a \cdot \left(\left(z \cdot 4\right) \cdot t\right)}}{z \cdot c} \]

      4. associate-*r* 82.7%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{\left(a \cdot \left(z \cdot 4\right)\right) \cdot t}}{z \cdot c} \]

      5. *-commutative 82.7%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{\left(\left(z \cdot 4\right) \cdot a\right)} \cdot t}{z \cdot c} \]

      6. associate-+r- 82.7%

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right)}}{z \cdot c} \]

      7. +-commutative 82.7%

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right) + b}}{z \cdot c} \]

      8. associate-*l* 83.4%

         \[\leadsto \frac{\left(\color{blue}{x \cdot \left(9 \cdot y\right)} - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right) + b}{z \cdot c} \]

      9. associate-*l* 84.3%

         \[\leadsto \frac{\left(x \cdot \left(9 \cdot y\right) - \color{blue}{\left(z \cdot 4\right) \cdot \left(a \cdot t\right)}\right) + b}{z \cdot c} \]

      10. *-commutative 84.3%

          \[\leadsto \frac{\left(x \cdot \left(9 \cdot y\right) - \left(z \cdot 4\right) \cdot \color{blue}{\left(t \cdot a\right)}\right) + b}{z \cdot c} \]

   3. Simplified 84.3%

      \[\leadsto \color{blue}{\frac{\left(x \cdot \left(9 \cdot y\right) - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 74.9%

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

   if 3.6e22 < y

   1. Initial program 86.5%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Step-by-step derivation

      1. +-commutative 86.5%

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)}}{z \cdot c} \]

      2. associate-+r- 86.5%

         \[\leadsto \frac{\color{blue}{\left(b + \left(x \cdot 9\right) \cdot y\right) - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}}{z \cdot c} \]

      3. *-commutative 86.5%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{a \cdot \left(\left(z \cdot 4\right) \cdot t\right)}}{z \cdot c} \]

      4. associate-*r* 91.4%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{\left(a \cdot \left(z \cdot 4\right)\right) \cdot t}}{z \cdot c} \]

      5. *-commutative 91.4%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{\left(\left(z \cdot 4\right) \cdot a\right)} \cdot t}{z \cdot c} \]

      6. associate-+r- 91.4%

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right)}}{z \cdot c} \]

      7. +-commutative 91.4%

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right) + b}}{z \cdot c} \]

      8. associate-*l* 91.4%

         \[\leadsto \frac{\left(\color{blue}{x \cdot \left(9 \cdot y\right)} - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right) + b}{z \cdot c} \]

      9. associate-*l* 91.4%

         \[\leadsto \frac{\left(x \cdot \left(9 \cdot y\right) - \color{blue}{\left(z \cdot 4\right) \cdot \left(a \cdot t\right)}\right) + b}{z \cdot c} \]

      10. *-commutative 91.4%

          \[\leadsto \frac{\left(x \cdot \left(9 \cdot y\right) - \left(z \cdot 4\right) \cdot \color{blue}{\left(t \cdot a\right)}\right) + b}{z \cdot c} \]

   3. Simplified 91.4%

      \[\leadsto \color{blue}{\frac{\left(x \cdot \left(9 \cdot y\right) - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 84.5%

      \[\leadsto \frac{\color{blue}{9 \cdot \left(x \cdot y\right)} + b}{z \cdot c} \]
3. Recombined 3 regimes into one program.

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.5 \cdot 10^{-16}:\\ \;\;\;\;x \cdot \frac{\frac{9 \cdot y}{c}}{z}\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{+22}:\\ \;\;\;\;\frac{b - 4 \cdot \left(a \cdot \left(z \cdot t\right)\right)}{c \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + 9 \cdot \left(x \cdot y\right)}{c \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 64.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ c\_s \cdot \begin{array}{l} \mathbf{if}\;a \leq -1.1 \cdot 10^{-25}:\\ \;\;\;\;a \cdot \frac{t \cdot -4}{c\_m}\\ \mathbf{elif}\;a \leq 1.6 \cdot 10^{+151}:\\ \;\;\;\;\frac{x \cdot \left(9 \cdot y\right) + b}{c\_m \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c\_m} \cdot \left(t \cdot -4\right)\\ \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (*
  c_s
  (if (<= a -1.1e-25)
    (* a (/ (* t -4.0) c_m))
    (if (<= a 1.6e+151)
      (/ (+ (* x (* 9.0 y)) b) (* c_m z))
      (* (/ a c_m) (* t -4.0))))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double tmp;
	if (a <= -1.1e-25) {
		tmp = a * ((t * -4.0) / c_m);
	} else if (a <= 1.6e+151) {
		tmp = ((x * (9.0 * y)) + b) / (c_m * z);
	} else {
		tmp = (a / c_m) * (t * -4.0);
	}
	return c_s * tmp;
}

Fortran

c\_m = abs(c)
c\_s = copysign(1.0d0, c)
real(8) function code(c_s, x, y, z, t, a, b, c_m)
    real(8), intent (in) :: c_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c_m
    real(8) :: tmp
    if (a <= (-1.1d-25)) then
        tmp = a * ((t * (-4.0d0)) / c_m)
    else if (a <= 1.6d+151) then
        tmp = ((x * (9.0d0 * y)) + b) / (c_m * z)
    else
        tmp = (a / c_m) * (t * (-4.0d0))
    end if
    code = c_s * tmp
end function

Java

c\_m = Math.abs(c);
c\_s = Math.copySign(1.0, c);
public static double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double tmp;
	if (a <= -1.1e-25) {
		tmp = a * ((t * -4.0) / c_m);
	} else if (a <= 1.6e+151) {
		tmp = ((x * (9.0 * y)) + b) / (c_m * z);
	} else {
		tmp = (a / c_m) * (t * -4.0);
	}
	return c_s * tmp;
}

Python

c\_m = math.fabs(c)
c\_s = math.copysign(1.0, c)
def code(c_s, x, y, z, t, a, b, c_m):
	tmp = 0
	if a <= -1.1e-25:
		tmp = a * ((t * -4.0) / c_m)
	elif a <= 1.6e+151:
		tmp = ((x * (9.0 * y)) + b) / (c_m * z)
	else:
		tmp = (a / c_m) * (t * -4.0)
	return c_s * tmp

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
function code(c_s, x, y, z, t, a, b, c_m)
	tmp = 0.0
	if (a <= -1.1e-25)
		tmp = Float64(a * Float64(Float64(t * -4.0) / c_m));
	elseif (a <= 1.6e+151)
		tmp = Float64(Float64(Float64(x * Float64(9.0 * y)) + b) / Float64(c_m * z));
	else
		tmp = Float64(Float64(a / c_m) * Float64(t * -4.0));
	end
	return Float64(c_s * tmp)
end

MATLAB

c\_m = abs(c);
c\_s = sign(c) * abs(1.0);
function tmp_2 = code(c_s, x, y, z, t, a, b, c_m)
	tmp = 0.0;
	if (a <= -1.1e-25)
		tmp = a * ((t * -4.0) / c_m);
	elseif (a <= 1.6e+151)
		tmp = ((x * (9.0 * y)) + b) / (c_m * z);
	else
		tmp = (a / c_m) * (t * -4.0);
	end
	tmp_2 = c_s * tmp;
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := N[(c$95$s * If[LessEqual[a, -1.1e-25], N[(a * N[(N[(t * -4.0), $MachinePrecision] / c$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1.6e+151], N[(N[(N[(x * N[(9.0 * y), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision] / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision], N[(N[(a / c$95$m), $MachinePrecision] * N[(t * -4.0), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if a < -1.1000000000000001e-25

   1. Initial program 82.1%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Step-by-step derivation

      1. +-commutative 82.1%

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)}}{z \cdot c} \]

      2. associate-+r- 82.1%

         \[\leadsto \frac{\color{blue}{\left(b + \left(x \cdot 9\right) \cdot y\right) - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}}{z \cdot c} \]

      3. *-commutative 82.1%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{a \cdot \left(\left(z \cdot 4\right) \cdot t\right)}}{z \cdot c} \]

      4. associate-*r* 77.3%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{\left(a \cdot \left(z \cdot 4\right)\right) \cdot t}}{z \cdot c} \]

      5. *-commutative 77.3%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{\left(\left(z \cdot 4\right) \cdot a\right)} \cdot t}{z \cdot c} \]

      6. associate-+r- 77.3%

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right)}}{z \cdot c} \]

      7. +-commutative 77.3%

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right) + b}}{z \cdot c} \]

      8. associate-*l* 77.3%

         \[\leadsto \frac{\left(\color{blue}{x \cdot \left(9 \cdot y\right)} - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right) + b}{z \cdot c} \]

      9. associate-*l* 80.5%

         \[\leadsto \frac{\left(x \cdot \left(9 \cdot y\right) - \color{blue}{\left(z \cdot 4\right) \cdot \left(a \cdot t\right)}\right) + b}{z \cdot c} \]

      10. *-commutative 80.5%

          \[\leadsto \frac{\left(x \cdot \left(9 \cdot y\right) - \left(z \cdot 4\right) \cdot \color{blue}{\left(t \cdot a\right)}\right) + b}{z \cdot c} \]

   3. Simplified 80.5%

      \[\leadsto \color{blue}{\frac{\left(x \cdot \left(9 \cdot y\right) - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 47.6%

      \[\leadsto \color{blue}{-4 \cdot \frac{a \cdot t}{c}} \]
   6. Step-by-step derivation

      1. associate-/l* 50.3%

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

      2. associate-*r* 50.3%

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

      3. *-commutative 50.3%

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

      4. associate-*r* 50.3%

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

      5. *-commutative 50.3%

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

      6. associate-*l/ 50.3%

         \[\leadsto a \cdot \color{blue}{\frac{t \cdot -4}{c}} \]

   7. Simplified 50.3%

      \[\leadsto \color{blue}{a \cdot \frac{t \cdot -4}{c}} \]

   if -1.1000000000000001e-25 < a < 1.59999999999999997e151

   1. Initial program 80.9%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Step-by-step derivation

      1. +-commutative 80.9%

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)}}{z \cdot c} \]

      2. associate-+r- 80.9%

         \[\leadsto \frac{\color{blue}{\left(b + \left(x \cdot 9\right) \cdot y\right) - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}}{z \cdot c} \]

      3. *-commutative 80.9%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{a \cdot \left(\left(z \cdot 4\right) \cdot t\right)}}{z \cdot c} \]

      4. associate-*r* 85.0%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{\left(a \cdot \left(z \cdot 4\right)\right) \cdot t}}{z \cdot c} \]

      5. *-commutative 85.0%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{\left(\left(z \cdot 4\right) \cdot a\right)} \cdot t}{z \cdot c} \]

      6. associate-+r- 85.0%

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right)}}{z \cdot c} \]

      7. +-commutative 85.0%

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right) + b}}{z \cdot c} \]

      8. associate-*l* 85.0%

         \[\leadsto \frac{\left(\color{blue}{x \cdot \left(9 \cdot y\right)} - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right) + b}{z \cdot c} \]

      9. associate-*l* 84.5%

         \[\leadsto \frac{\left(x \cdot \left(9 \cdot y\right) - \color{blue}{\left(z \cdot 4\right) \cdot \left(a \cdot t\right)}\right) + b}{z \cdot c} \]

      10. *-commutative 84.5%

          \[\leadsto \frac{\left(x \cdot \left(9 \cdot y\right) - \left(z \cdot 4\right) \cdot \color{blue}{\left(t \cdot a\right)}\right) + b}{z \cdot c} \]

   3. Simplified 84.5%

      \[\leadsto \color{blue}{\frac{\left(x \cdot \left(9 \cdot y\right) - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 73.6%

      \[\leadsto \frac{\color{blue}{9 \cdot \left(x \cdot y\right)} + b}{z \cdot c} \]
   6. Step-by-step derivation

      1. associate-*r* 73.7%

         \[\leadsto \frac{\color{blue}{\left(9 \cdot x\right) \cdot y} + b}{z \cdot c} \]

      2. *-commutative 73.7%

         \[\leadsto \frac{\color{blue}{\left(x \cdot 9\right)} \cdot y + b}{z \cdot c} \]

      3. associate-*r* 73.7%

         \[\leadsto \frac{\color{blue}{x \cdot \left(9 \cdot y\right)} + b}{z \cdot c} \]

   7. Simplified 73.7%

      \[\leadsto \frac{\color{blue}{x \cdot \left(9 \cdot y\right)} + b}{z \cdot c} \]

   if 1.59999999999999997e151 < a

   1. Initial program 70.0%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Step-by-step derivation

      1. +-commutative 70.0%

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)}}{z \cdot c} \]

      2. associate-+r- 70.0%

         \[\leadsto \frac{\color{blue}{\left(b + \left(x \cdot 9\right) \cdot y\right) - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}}{z \cdot c} \]

      3. *-commutative 70.0%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{a \cdot \left(\left(z \cdot 4\right) \cdot t\right)}}{z \cdot c} \]

      4. associate-*r* 62.4%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{\left(a \cdot \left(z \cdot 4\right)\right) \cdot t}}{z \cdot c} \]

      5. *-commutative 62.4%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{\left(\left(z \cdot 4\right) \cdot a\right)} \cdot t}{z \cdot c} \]

      6. associate-+r- 62.4%

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right)}}{z \cdot c} \]

      7. +-commutative 62.4%

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right) + b}}{z \cdot c} \]

      8. associate-*l* 66.3%

         \[\leadsto \frac{\left(\color{blue}{x \cdot \left(9 \cdot y\right)} - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right) + b}{z \cdot c} \]

      9. associate-*l* 70.2%

         \[\leadsto \frac{\left(x \cdot \left(9 \cdot y\right) - \color{blue}{\left(z \cdot 4\right) \cdot \left(a \cdot t\right)}\right) + b}{z \cdot c} \]

      10. *-commutative 70.2%

          \[\leadsto \frac{\left(x \cdot \left(9 \cdot y\right) - \left(z \cdot 4\right) \cdot \color{blue}{\left(t \cdot a\right)}\right) + b}{z \cdot c} \]

   3. Simplified 70.2%

      \[\leadsto \color{blue}{\frac{\left(x \cdot \left(9 \cdot y\right) - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 66.7%

      \[\leadsto \color{blue}{\frac{-4 \cdot \frac{a \cdot \left(t \cdot z\right)}{c} + \left(9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z}} \]
   6. Step-by-step derivation

      1. fma-define 66.7%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-4, \frac{a \cdot \left(t \cdot z\right)}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}}{z} \]

      2. *-commutative 66.7%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{\color{blue}{\left(t \cdot z\right) \cdot a}}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      3. *-commutative 66.7%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{\color{blue}{\left(z \cdot t\right)} \cdot a}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      4. associate-*r* 63.1%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{\color{blue}{z \cdot \left(t \cdot a\right)}}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      5. *-commutative 63.1%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{z \cdot \color{blue}{\left(a \cdot t\right)}}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      6. associate-/l* 66.8%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \color{blue}{z \cdot \frac{a \cdot t}{c}}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      7. associate-/l* 70.2%

         \[\leadsto \frac{\mathsf{fma}\left(-4, z \cdot \color{blue}{\left(a \cdot \frac{t}{c}\right)}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      8. fma-define 70.2%

         \[\leadsto \frac{\mathsf{fma}\left(-4, z \cdot \left(a \cdot \frac{t}{c}\right), \color{blue}{\mathsf{fma}\left(9, \frac{x \cdot y}{c}, \frac{b}{c}\right)}\right)}{z} \]

      9. associate-/l* 70.2%

         \[\leadsto \frac{\mathsf{fma}\left(-4, z \cdot \left(a \cdot \frac{t}{c}\right), \mathsf{fma}\left(9, \color{blue}{x \cdot \frac{y}{c}}, \frac{b}{c}\right)\right)}{z} \]

   7. Simplified 70.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-4, z \cdot \left(a \cdot \frac{t}{c}\right), \mathsf{fma}\left(9, x \cdot \frac{y}{c}, \frac{b}{c}\right)\right)}{z}} \]

   8. Taylor expanded in z around inf 45.2%

      \[\leadsto \color{blue}{-4 \cdot \frac{a \cdot t}{c}} \]
   9. Step-by-step derivation

      1. associate-*r/ 45.2%

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

      2. associate-*r* 45.2%

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

      3. associate-*l/ 62.6%

         \[\leadsto \color{blue}{\frac{-4 \cdot a}{c} \cdot t} \]

      4. associate-*r/ 66.1%

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

      5. *-commutative 66.1%

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

      6. associate-*l* 66.1%

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

   10. Simplified 66.1%

       \[\leadsto \color{blue}{\frac{a}{c} \cdot \left(-4 \cdot t\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.1 \cdot 10^{-25}:\\ \;\;\;\;a \cdot \frac{t \cdot -4}{c}\\ \mathbf{elif}\;a \leq 1.6 \cdot 10^{+151}:\\ \;\;\;\;\frac{x \cdot \left(9 \cdot y\right) + b}{c \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} \cdot \left(t \cdot -4\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 64.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ c\_s \cdot \begin{array}{l} \mathbf{if}\;a \leq -1 \cdot 10^{-25}:\\ \;\;\;\;a \cdot \frac{t \cdot -4}{c\_m}\\ \mathbf{elif}\;a \leq 2.1 \cdot 10^{+148}:\\ \;\;\;\;\frac{b + 9 \cdot \left(x \cdot y\right)}{c\_m \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c\_m} \cdot \left(t \cdot -4\right)\\ \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (*
  c_s
  (if (<= a -1e-25)
    (* a (/ (* t -4.0) c_m))
    (if (<= a 2.1e+148)
      (/ (+ b (* 9.0 (* x y))) (* c_m z))
      (* (/ a c_m) (* t -4.0))))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double tmp;
	if (a <= -1e-25) {
		tmp = a * ((t * -4.0) / c_m);
	} else if (a <= 2.1e+148) {
		tmp = (b + (9.0 * (x * y))) / (c_m * z);
	} else {
		tmp = (a / c_m) * (t * -4.0);
	}
	return c_s * tmp;
}

Fortran

c\_m = abs(c)
c\_s = copysign(1.0d0, c)
real(8) function code(c_s, x, y, z, t, a, b, c_m)
    real(8), intent (in) :: c_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c_m
    real(8) :: tmp
    if (a <= (-1d-25)) then
        tmp = a * ((t * (-4.0d0)) / c_m)
    else if (a <= 2.1d+148) then
        tmp = (b + (9.0d0 * (x * y))) / (c_m * z)
    else
        tmp = (a / c_m) * (t * (-4.0d0))
    end if
    code = c_s * tmp
end function

Java

c\_m = Math.abs(c);
c\_s = Math.copySign(1.0, c);
public static double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double tmp;
	if (a <= -1e-25) {
		tmp = a * ((t * -4.0) / c_m);
	} else if (a <= 2.1e+148) {
		tmp = (b + (9.0 * (x * y))) / (c_m * z);
	} else {
		tmp = (a / c_m) * (t * -4.0);
	}
	return c_s * tmp;
}

Python

c\_m = math.fabs(c)
c\_s = math.copysign(1.0, c)
def code(c_s, x, y, z, t, a, b, c_m):
	tmp = 0
	if a <= -1e-25:
		tmp = a * ((t * -4.0) / c_m)
	elif a <= 2.1e+148:
		tmp = (b + (9.0 * (x * y))) / (c_m * z)
	else:
		tmp = (a / c_m) * (t * -4.0)
	return c_s * tmp

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
function code(c_s, x, y, z, t, a, b, c_m)
	tmp = 0.0
	if (a <= -1e-25)
		tmp = Float64(a * Float64(Float64(t * -4.0) / c_m));
	elseif (a <= 2.1e+148)
		tmp = Float64(Float64(b + Float64(9.0 * Float64(x * y))) / Float64(c_m * z));
	else
		tmp = Float64(Float64(a / c_m) * Float64(t * -4.0));
	end
	return Float64(c_s * tmp)
end

MATLAB

c\_m = abs(c);
c\_s = sign(c) * abs(1.0);
function tmp_2 = code(c_s, x, y, z, t, a, b, c_m)
	tmp = 0.0;
	if (a <= -1e-25)
		tmp = a * ((t * -4.0) / c_m);
	elseif (a <= 2.1e+148)
		tmp = (b + (9.0 * (x * y))) / (c_m * z);
	else
		tmp = (a / c_m) * (t * -4.0);
	end
	tmp_2 = c_s * tmp;
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := N[(c$95$s * If[LessEqual[a, -1e-25], N[(a * N[(N[(t * -4.0), $MachinePrecision] / c$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 2.1e+148], N[(N[(b + N[(9.0 * N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision], N[(N[(a / c$95$m), $MachinePrecision] * N[(t * -4.0), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if a < -1.00000000000000004e-25

   1. Initial program 82.1%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Step-by-step derivation

      1. +-commutative 82.1%

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)}}{z \cdot c} \]

      2. associate-+r- 82.1%

         \[\leadsto \frac{\color{blue}{\left(b + \left(x \cdot 9\right) \cdot y\right) - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}}{z \cdot c} \]

      3. *-commutative 82.1%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{a \cdot \left(\left(z \cdot 4\right) \cdot t\right)}}{z \cdot c} \]

      4. associate-*r* 77.3%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{\left(a \cdot \left(z \cdot 4\right)\right) \cdot t}}{z \cdot c} \]

      5. *-commutative 77.3%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{\left(\left(z \cdot 4\right) \cdot a\right)} \cdot t}{z \cdot c} \]

      6. associate-+r- 77.3%

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right)}}{z \cdot c} \]

      7. +-commutative 77.3%

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right) + b}}{z \cdot c} \]

      8. associate-*l* 77.3%

         \[\leadsto \frac{\left(\color{blue}{x \cdot \left(9 \cdot y\right)} - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right) + b}{z \cdot c} \]

      9. associate-*l* 80.5%

         \[\leadsto \frac{\left(x \cdot \left(9 \cdot y\right) - \color{blue}{\left(z \cdot 4\right) \cdot \left(a \cdot t\right)}\right) + b}{z \cdot c} \]

      10. *-commutative 80.5%

          \[\leadsto \frac{\left(x \cdot \left(9 \cdot y\right) - \left(z \cdot 4\right) \cdot \color{blue}{\left(t \cdot a\right)}\right) + b}{z \cdot c} \]

   3. Simplified 80.5%

      \[\leadsto \color{blue}{\frac{\left(x \cdot \left(9 \cdot y\right) - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 47.6%

      \[\leadsto \color{blue}{-4 \cdot \frac{a \cdot t}{c}} \]
   6. Step-by-step derivation

      1. associate-/l* 50.3%

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

      2. associate-*r* 50.3%

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

      3. *-commutative 50.3%

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

      4. associate-*r* 50.3%

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

      5. *-commutative 50.3%

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

      6. associate-*l/ 50.3%

         \[\leadsto a \cdot \color{blue}{\frac{t \cdot -4}{c}} \]

   7. Simplified 50.3%

      \[\leadsto \color{blue}{a \cdot \frac{t \cdot -4}{c}} \]

   if -1.00000000000000004e-25 < a < 2.09999999999999999e148

   1. Initial program 80.9%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Step-by-step derivation

      1. +-commutative 80.9%

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)}}{z \cdot c} \]

      2. associate-+r- 80.9%

         \[\leadsto \frac{\color{blue}{\left(b + \left(x \cdot 9\right) \cdot y\right) - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}}{z \cdot c} \]

      3. *-commutative 80.9%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{a \cdot \left(\left(z \cdot 4\right) \cdot t\right)}}{z \cdot c} \]

      4. associate-*r* 85.0%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{\left(a \cdot \left(z \cdot 4\right)\right) \cdot t}}{z \cdot c} \]

      5. *-commutative 85.0%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{\left(\left(z \cdot 4\right) \cdot a\right)} \cdot t}{z \cdot c} \]

      6. associate-+r- 85.0%

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right)}}{z \cdot c} \]

      7. +-commutative 85.0%

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right) + b}}{z \cdot c} \]

      8. associate-*l* 85.0%

         \[\leadsto \frac{\left(\color{blue}{x \cdot \left(9 \cdot y\right)} - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right) + b}{z \cdot c} \]

      9. associate-*l* 84.5%

         \[\leadsto \frac{\left(x \cdot \left(9 \cdot y\right) - \color{blue}{\left(z \cdot 4\right) \cdot \left(a \cdot t\right)}\right) + b}{z \cdot c} \]

      10. *-commutative 84.5%

          \[\leadsto \frac{\left(x \cdot \left(9 \cdot y\right) - \left(z \cdot 4\right) \cdot \color{blue}{\left(t \cdot a\right)}\right) + b}{z \cdot c} \]

   3. Simplified 84.5%

      \[\leadsto \color{blue}{\frac{\left(x \cdot \left(9 \cdot y\right) - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 73.6%

      \[\leadsto \frac{\color{blue}{9 \cdot \left(x \cdot y\right)} + b}{z \cdot c} \]

   if 2.09999999999999999e148 < a

   1. Initial program 70.0%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Step-by-step derivation

      1. +-commutative 70.0%

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)}}{z \cdot c} \]

      2. associate-+r- 70.0%

         \[\leadsto \frac{\color{blue}{\left(b + \left(x \cdot 9\right) \cdot y\right) - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}}{z \cdot c} \]

      3. *-commutative 70.0%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{a \cdot \left(\left(z \cdot 4\right) \cdot t\right)}}{z \cdot c} \]

      4. associate-*r* 62.4%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{\left(a \cdot \left(z \cdot 4\right)\right) \cdot t}}{z \cdot c} \]

      5. *-commutative 62.4%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{\left(\left(z \cdot 4\right) \cdot a\right)} \cdot t}{z \cdot c} \]

      6. associate-+r- 62.4%

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right)}}{z \cdot c} \]

      7. +-commutative 62.4%

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right) + b}}{z \cdot c} \]

      8. associate-*l* 66.3%

         \[\leadsto \frac{\left(\color{blue}{x \cdot \left(9 \cdot y\right)} - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right) + b}{z \cdot c} \]

      9. associate-*l* 70.2%

         \[\leadsto \frac{\left(x \cdot \left(9 \cdot y\right) - \color{blue}{\left(z \cdot 4\right) \cdot \left(a \cdot t\right)}\right) + b}{z \cdot c} \]

      10. *-commutative 70.2%

          \[\leadsto \frac{\left(x \cdot \left(9 \cdot y\right) - \left(z \cdot 4\right) \cdot \color{blue}{\left(t \cdot a\right)}\right) + b}{z \cdot c} \]

   3. Simplified 70.2%

      \[\leadsto \color{blue}{\frac{\left(x \cdot \left(9 \cdot y\right) - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 66.7%

      \[\leadsto \color{blue}{\frac{-4 \cdot \frac{a \cdot \left(t \cdot z\right)}{c} + \left(9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z}} \]
   6. Step-by-step derivation

      1. fma-define 66.7%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-4, \frac{a \cdot \left(t \cdot z\right)}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}}{z} \]

      2. *-commutative 66.7%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{\color{blue}{\left(t \cdot z\right) \cdot a}}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      3. *-commutative 66.7%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{\color{blue}{\left(z \cdot t\right)} \cdot a}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      4. associate-*r* 63.1%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{\color{blue}{z \cdot \left(t \cdot a\right)}}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      5. *-commutative 63.1%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{z \cdot \color{blue}{\left(a \cdot t\right)}}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      6. associate-/l* 66.8%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \color{blue}{z \cdot \frac{a \cdot t}{c}}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      7. associate-/l* 70.2%

         \[\leadsto \frac{\mathsf{fma}\left(-4, z \cdot \color{blue}{\left(a \cdot \frac{t}{c}\right)}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      8. fma-define 70.2%

         \[\leadsto \frac{\mathsf{fma}\left(-4, z \cdot \left(a \cdot \frac{t}{c}\right), \color{blue}{\mathsf{fma}\left(9, \frac{x \cdot y}{c}, \frac{b}{c}\right)}\right)}{z} \]

      9. associate-/l* 70.2%

         \[\leadsto \frac{\mathsf{fma}\left(-4, z \cdot \left(a \cdot \frac{t}{c}\right), \mathsf{fma}\left(9, \color{blue}{x \cdot \frac{y}{c}}, \frac{b}{c}\right)\right)}{z} \]

   7. Simplified 70.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-4, z \cdot \left(a \cdot \frac{t}{c}\right), \mathsf{fma}\left(9, x \cdot \frac{y}{c}, \frac{b}{c}\right)\right)}{z}} \]

   8. Taylor expanded in z around inf 45.2%

      \[\leadsto \color{blue}{-4 \cdot \frac{a \cdot t}{c}} \]
   9. Step-by-step derivation

      1. associate-*r/ 45.2%

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

      2. associate-*r* 45.2%

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

      3. associate-*l/ 62.6%

         \[\leadsto \color{blue}{\frac{-4 \cdot a}{c} \cdot t} \]

      4. associate-*r/ 66.1%

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

      5. *-commutative 66.1%

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

      6. associate-*l* 66.1%

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

   10. Simplified 66.1%

       \[\leadsto \color{blue}{\frac{a}{c} \cdot \left(-4 \cdot t\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1 \cdot 10^{-25}:\\ \;\;\;\;a \cdot \frac{t \cdot -4}{c}\\ \mathbf{elif}\;a \leq 2.1 \cdot 10^{+148}:\\ \;\;\;\;\frac{b + 9 \cdot \left(x \cdot y\right)}{c \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} \cdot \left(t \cdot -4\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 48.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ c\_s \cdot \begin{array}{l} \mathbf{if}\;b \leq -4.9 \cdot 10^{-24}:\\ \;\;\;\;\frac{1}{\frac{z}{\frac{b}{c\_m}}}\\ \mathbf{elif}\;b \leq 2.8 \cdot 10^{-54}:\\ \;\;\;\;\frac{a}{c\_m} \cdot \left(t \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c\_m \cdot z}\\ \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (*
  c_s
  (if (<= b -4.9e-24)
    (/ 1.0 (/ z (/ b c_m)))
    (if (<= b 2.8e-54) (* (/ a c_m) (* t -4.0)) (/ b (* c_m z))))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double tmp;
	if (b <= -4.9e-24) {
		tmp = 1.0 / (z / (b / c_m));
	} else if (b <= 2.8e-54) {
		tmp = (a / c_m) * (t * -4.0);
	} else {
		tmp = b / (c_m * z);
	}
	return c_s * tmp;
}

Fortran

c\_m = abs(c)
c\_s = copysign(1.0d0, c)
real(8) function code(c_s, x, y, z, t, a, b, c_m)
    real(8), intent (in) :: c_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c_m
    real(8) :: tmp
    if (b <= (-4.9d-24)) then
        tmp = 1.0d0 / (z / (b / c_m))
    else if (b <= 2.8d-54) then
        tmp = (a / c_m) * (t * (-4.0d0))
    else
        tmp = b / (c_m * z)
    end if
    code = c_s * tmp
end function

Java

c\_m = Math.abs(c);
c\_s = Math.copySign(1.0, c);
public static double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double tmp;
	if (b <= -4.9e-24) {
		tmp = 1.0 / (z / (b / c_m));
	} else if (b <= 2.8e-54) {
		tmp = (a / c_m) * (t * -4.0);
	} else {
		tmp = b / (c_m * z);
	}
	return c_s * tmp;
}

Python

c\_m = math.fabs(c)
c\_s = math.copysign(1.0, c)
def code(c_s, x, y, z, t, a, b, c_m):
	tmp = 0
	if b <= -4.9e-24:
		tmp = 1.0 / (z / (b / c_m))
	elif b <= 2.8e-54:
		tmp = (a / c_m) * (t * -4.0)
	else:
		tmp = b / (c_m * z)
	return c_s * tmp

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
function code(c_s, x, y, z, t, a, b, c_m)
	tmp = 0.0
	if (b <= -4.9e-24)
		tmp = Float64(1.0 / Float64(z / Float64(b / c_m)));
	elseif (b <= 2.8e-54)
		tmp = Float64(Float64(a / c_m) * Float64(t * -4.0));
	else
		tmp = Float64(b / Float64(c_m * z));
	end
	return Float64(c_s * tmp)
end

MATLAB

c\_m = abs(c);
c\_s = sign(c) * abs(1.0);
function tmp_2 = code(c_s, x, y, z, t, a, b, c_m)
	tmp = 0.0;
	if (b <= -4.9e-24)
		tmp = 1.0 / (z / (b / c_m));
	elseif (b <= 2.8e-54)
		tmp = (a / c_m) * (t * -4.0);
	else
		tmp = b / (c_m * z);
	end
	tmp_2 = c_s * tmp;
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := N[(c$95$s * If[LessEqual[b, -4.9e-24], N[(1.0 / N[(z / N[(b / c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.8e-54], N[(N[(a / c$95$m), $MachinePrecision] * N[(t * -4.0), $MachinePrecision]), $MachinePrecision], N[(b / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if b < -4.9000000000000001e-24

   1. Initial program 82.4%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Step-by-step derivation

      1. +-commutative 82.4%

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)}}{z \cdot c} \]

      2. associate-+r- 82.4%

         \[\leadsto \frac{\color{blue}{\left(b + \left(x \cdot 9\right) \cdot y\right) - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}}{z \cdot c} \]

      3. *-commutative 82.4%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{a \cdot \left(\left(z \cdot 4\right) \cdot t\right)}}{z \cdot c} \]

      4. associate-*r* 83.9%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{\left(a \cdot \left(z \cdot 4\right)\right) \cdot t}}{z \cdot c} \]

      5. *-commutative 83.9%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{\left(\left(z \cdot 4\right) \cdot a\right)} \cdot t}{z \cdot c} \]

      6. associate-+r- 83.9%

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right)}}{z \cdot c} \]

      7. +-commutative 83.9%

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right) + b}}{z \cdot c} \]

      8. associate-*l* 83.9%

         \[\leadsto \frac{\left(\color{blue}{x \cdot \left(9 \cdot y\right)} - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right) + b}{z \cdot c} \]

      9. associate-*l* 81.0%

         \[\leadsto \frac{\left(x \cdot \left(9 \cdot y\right) - \color{blue}{\left(z \cdot 4\right) \cdot \left(a \cdot t\right)}\right) + b}{z \cdot c} \]

      10. *-commutative 81.0%

          \[\leadsto \frac{\left(x \cdot \left(9 \cdot y\right) - \left(z \cdot 4\right) \cdot \color{blue}{\left(t \cdot a\right)}\right) + b}{z \cdot c} \]

   3. Simplified 81.0%

      \[\leadsto \color{blue}{\frac{\left(x \cdot \left(9 \cdot y\right) - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 75.3%

      \[\leadsto \color{blue}{\frac{-4 \cdot \frac{a \cdot \left(t \cdot z\right)}{c} + \left(9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z}} \]
   6. Step-by-step derivation

      1. fma-define 75.3%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-4, \frac{a \cdot \left(t \cdot z\right)}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}}{z} \]

      2. *-commutative 75.3%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{\color{blue}{\left(t \cdot z\right) \cdot a}}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      3. *-commutative 75.3%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{\color{blue}{\left(z \cdot t\right)} \cdot a}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      4. associate-*r* 75.3%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{\color{blue}{z \cdot \left(t \cdot a\right)}}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      5. *-commutative 75.3%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{z \cdot \color{blue}{\left(a \cdot t\right)}}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      6. associate-/l* 75.4%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \color{blue}{z \cdot \frac{a \cdot t}{c}}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      7. associate-/l* 79.6%

         \[\leadsto \frac{\mathsf{fma}\left(-4, z \cdot \color{blue}{\left(a \cdot \frac{t}{c}\right)}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      8. fma-define 79.6%

         \[\leadsto \frac{\mathsf{fma}\left(-4, z \cdot \left(a \cdot \frac{t}{c}\right), \color{blue}{\mathsf{fma}\left(9, \frac{x \cdot y}{c}, \frac{b}{c}\right)}\right)}{z} \]

      9. associate-/l* 83.8%

         \[\leadsto \frac{\mathsf{fma}\left(-4, z \cdot \left(a \cdot \frac{t}{c}\right), \mathsf{fma}\left(9, \color{blue}{x \cdot \frac{y}{c}}, \frac{b}{c}\right)\right)}{z} \]

   7. Simplified 83.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-4, z \cdot \left(a \cdot \frac{t}{c}\right), \mathsf{fma}\left(9, x \cdot \frac{y}{c}, \frac{b}{c}\right)\right)}{z}} \]

   8. Taylor expanded in b around inf 62.2%

      \[\leadsto \frac{\color{blue}{\frac{b}{c}}}{z} \]
   9. Step-by-step derivation

      1. clear-num 62.3%

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

      2. inv-pow 62.3%

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

   10. Applied egg-rr 62.3%

       \[\leadsto \color{blue}{{\left(\frac{z}{\frac{b}{c}}\right)}^{-1}} \]
   11. Step-by-step derivation

       1. unpow-1 62.3%

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

   12. Simplified 62.3%

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

   if -4.9000000000000001e-24 < b < 2.8000000000000002e-54

   1. Initial program 76.5%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Step-by-step derivation

      1. +-commutative 76.5%

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)}}{z \cdot c} \]

      2. associate-+r- 76.5%

         \[\leadsto \frac{\color{blue}{\left(b + \left(x \cdot 9\right) \cdot y\right) - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}}{z \cdot c} \]

      3. *-commutative 76.5%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{a \cdot \left(\left(z \cdot 4\right) \cdot t\right)}}{z \cdot c} \]

      4. associate-*r* 75.8%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{\left(a \cdot \left(z \cdot 4\right)\right) \cdot t}}{z \cdot c} \]

      5. *-commutative 75.8%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{\left(\left(z \cdot 4\right) \cdot a\right)} \cdot t}{z \cdot c} \]

      6. associate-+r- 75.8%

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right)}}{z \cdot c} \]

      7. +-commutative 75.8%

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right) + b}}{z \cdot c} \]

      8. associate-*l* 75.8%

         \[\leadsto \frac{\left(\color{blue}{x \cdot \left(9 \cdot y\right)} - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right) + b}{z \cdot c} \]

      9. associate-*l* 78.5%

         \[\leadsto \frac{\left(x \cdot \left(9 \cdot y\right) - \color{blue}{\left(z \cdot 4\right) \cdot \left(a \cdot t\right)}\right) + b}{z \cdot c} \]

      10. *-commutative 78.5%

          \[\leadsto \frac{\left(x \cdot \left(9 \cdot y\right) - \left(z \cdot 4\right) \cdot \color{blue}{\left(t \cdot a\right)}\right) + b}{z \cdot c} \]

   3. Simplified 78.5%

      \[\leadsto \color{blue}{\frac{\left(x \cdot \left(9 \cdot y\right) - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 76.0%

      \[\leadsto \color{blue}{\frac{-4 \cdot \frac{a \cdot \left(t \cdot z\right)}{c} + \left(9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z}} \]
   6. Step-by-step derivation

      1. fma-define 76.0%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-4, \frac{a \cdot \left(t \cdot z\right)}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}}{z} \]

      2. *-commutative 76.0%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{\color{blue}{\left(t \cdot z\right) \cdot a}}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      3. *-commutative 76.0%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{\color{blue}{\left(z \cdot t\right)} \cdot a}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      4. associate-*r* 78.9%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{\color{blue}{z \cdot \left(t \cdot a\right)}}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      5. *-commutative 78.9%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{z \cdot \color{blue}{\left(a \cdot t\right)}}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      6. associate-/l* 82.5%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \color{blue}{z \cdot \frac{a \cdot t}{c}}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      7. associate-/l* 82.9%

         \[\leadsto \frac{\mathsf{fma}\left(-4, z \cdot \color{blue}{\left(a \cdot \frac{t}{c}\right)}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      8. fma-define 82.9%

         \[\leadsto \frac{\mathsf{fma}\left(-4, z \cdot \left(a \cdot \frac{t}{c}\right), \color{blue}{\mathsf{fma}\left(9, \frac{x \cdot y}{c}, \frac{b}{c}\right)}\right)}{z} \]

      9. associate-/l* 82.0%

         \[\leadsto \frac{\mathsf{fma}\left(-4, z \cdot \left(a \cdot \frac{t}{c}\right), \mathsf{fma}\left(9, \color{blue}{x \cdot \frac{y}{c}}, \frac{b}{c}\right)\right)}{z} \]

   7. Simplified 82.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-4, z \cdot \left(a \cdot \frac{t}{c}\right), \mathsf{fma}\left(9, x \cdot \frac{y}{c}, \frac{b}{c}\right)\right)}{z}} \]

   8. Taylor expanded in z around inf 48.3%

      \[\leadsto \color{blue}{-4 \cdot \frac{a \cdot t}{c}} \]
   9. Step-by-step derivation

      1. associate-*r/ 48.3%

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

      2. associate-*r* 48.3%

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

      3. associate-*l/ 52.7%

         \[\leadsto \color{blue}{\frac{-4 \cdot a}{c} \cdot t} \]

      4. associate-*r/ 52.7%

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

      5. *-commutative 52.7%

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

      6. associate-*l* 52.7%

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

   10. Simplified 52.7%

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

   if 2.8000000000000002e-54 < b

   1. Initial program 83.1%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Step-by-step derivation

      1. +-commutative 83.1%

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)}}{z \cdot c} \]

      2. associate-+r- 83.1%

         \[\leadsto \frac{\color{blue}{\left(b + \left(x \cdot 9\right) \cdot y\right) - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}}{z \cdot c} \]

      3. *-commutative 83.1%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{a \cdot \left(\left(z \cdot 4\right) \cdot t\right)}}{z \cdot c} \]

      4. associate-*r* 85.5%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{\left(a \cdot \left(z \cdot 4\right)\right) \cdot t}}{z \cdot c} \]

      5. *-commutative 85.5%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{\left(\left(z \cdot 4\right) \cdot a\right)} \cdot t}{z \cdot c} \]

      6. associate-+r- 85.5%

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right)}}{z \cdot c} \]

      7. +-commutative 85.5%

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right) + b}}{z \cdot c} \]

      8. associate-*l* 86.7%

         \[\leadsto \frac{\left(\color{blue}{x \cdot \left(9 \cdot y\right)} - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right) + b}{z \cdot c} \]

      9. associate-*l* 88.0%

         \[\leadsto \frac{\left(x \cdot \left(9 \cdot y\right) - \color{blue}{\left(z \cdot 4\right) \cdot \left(a \cdot t\right)}\right) + b}{z \cdot c} \]

      10. *-commutative 88.0%

          \[\leadsto \frac{\left(x \cdot \left(9 \cdot y\right) - \left(z \cdot 4\right) \cdot \color{blue}{\left(t \cdot a\right)}\right) + b}{z \cdot c} \]

   3. Simplified 88.0%

      \[\leadsto \color{blue}{\frac{\left(x \cdot \left(9 \cdot y\right) - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around inf 68.3%

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

      1. *-commutative 68.3%

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

   7. Simplified 68.3%

      \[\leadsto \color{blue}{\frac{b}{z \cdot c}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 60.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.9 \cdot 10^{-24}:\\ \;\;\;\;\frac{1}{\frac{z}{\frac{b}{c}}}\\ \mathbf{elif}\;b \leq 2.8 \cdot 10^{-54}:\\ \;\;\;\;\frac{a}{c} \cdot \left(t \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 48.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ c\_s \cdot \begin{array}{l} \mathbf{if}\;b \leq -3.6 \cdot 10^{-25}:\\ \;\;\;\;\frac{\frac{b}{c\_m}}{z}\\ \mathbf{elif}\;b \leq 1.15 \cdot 10^{-53}:\\ \;\;\;\;\frac{a}{c\_m} \cdot \left(t \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c\_m \cdot z}\\ \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (*
  c_s
  (if (<= b -3.6e-25)
    (/ (/ b c_m) z)
    (if (<= b 1.15e-53) (* (/ a c_m) (* t -4.0)) (/ b (* c_m z))))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double tmp;
	if (b <= -3.6e-25) {
		tmp = (b / c_m) / z;
	} else if (b <= 1.15e-53) {
		tmp = (a / c_m) * (t * -4.0);
	} else {
		tmp = b / (c_m * z);
	}
	return c_s * tmp;
}

Fortran

c\_m = abs(c)
c\_s = copysign(1.0d0, c)
real(8) function code(c_s, x, y, z, t, a, b, c_m)
    real(8), intent (in) :: c_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c_m
    real(8) :: tmp
    if (b <= (-3.6d-25)) then
        tmp = (b / c_m) / z
    else if (b <= 1.15d-53) then
        tmp = (a / c_m) * (t * (-4.0d0))
    else
        tmp = b / (c_m * z)
    end if
    code = c_s * tmp
end function

Java

c\_m = Math.abs(c);
c\_s = Math.copySign(1.0, c);
public static double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double tmp;
	if (b <= -3.6e-25) {
		tmp = (b / c_m) / z;
	} else if (b <= 1.15e-53) {
		tmp = (a / c_m) * (t * -4.0);
	} else {
		tmp = b / (c_m * z);
	}
	return c_s * tmp;
}

Python

c\_m = math.fabs(c)
c\_s = math.copysign(1.0, c)
def code(c_s, x, y, z, t, a, b, c_m):
	tmp = 0
	if b <= -3.6e-25:
		tmp = (b / c_m) / z
	elif b <= 1.15e-53:
		tmp = (a / c_m) * (t * -4.0)
	else:
		tmp = b / (c_m * z)
	return c_s * tmp

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
function code(c_s, x, y, z, t, a, b, c_m)
	tmp = 0.0
	if (b <= -3.6e-25)
		tmp = Float64(Float64(b / c_m) / z);
	elseif (b <= 1.15e-53)
		tmp = Float64(Float64(a / c_m) * Float64(t * -4.0));
	else
		tmp = Float64(b / Float64(c_m * z));
	end
	return Float64(c_s * tmp)
end

MATLAB

c\_m = abs(c);
c\_s = sign(c) * abs(1.0);
function tmp_2 = code(c_s, x, y, z, t, a, b, c_m)
	tmp = 0.0;
	if (b <= -3.6e-25)
		tmp = (b / c_m) / z;
	elseif (b <= 1.15e-53)
		tmp = (a / c_m) * (t * -4.0);
	else
		tmp = b / (c_m * z);
	end
	tmp_2 = c_s * tmp;
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := N[(c$95$s * If[LessEqual[b, -3.6e-25], N[(N[(b / c$95$m), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[b, 1.15e-53], N[(N[(a / c$95$m), $MachinePrecision] * N[(t * -4.0), $MachinePrecision]), $MachinePrecision], N[(b / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if b < -3.5999999999999999e-25

   1. Initial program 82.4%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Step-by-step derivation

      1. +-commutative 82.4%

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)}}{z \cdot c} \]

      2. associate-+r- 82.4%

         \[\leadsto \frac{\color{blue}{\left(b + \left(x \cdot 9\right) \cdot y\right) - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}}{z \cdot c} \]

      3. *-commutative 82.4%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{a \cdot \left(\left(z \cdot 4\right) \cdot t\right)}}{z \cdot c} \]

      4. associate-*r* 83.9%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{\left(a \cdot \left(z \cdot 4\right)\right) \cdot t}}{z \cdot c} \]

      5. *-commutative 83.9%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{\left(\left(z \cdot 4\right) \cdot a\right)} \cdot t}{z \cdot c} \]

      6. associate-+r- 83.9%

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right)}}{z \cdot c} \]

      7. +-commutative 83.9%

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right) + b}}{z \cdot c} \]

      8. associate-*l* 83.9%

         \[\leadsto \frac{\left(\color{blue}{x \cdot \left(9 \cdot y\right)} - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right) + b}{z \cdot c} \]

      9. associate-*l* 81.0%

         \[\leadsto \frac{\left(x \cdot \left(9 \cdot y\right) - \color{blue}{\left(z \cdot 4\right) \cdot \left(a \cdot t\right)}\right) + b}{z \cdot c} \]

      10. *-commutative 81.0%

          \[\leadsto \frac{\left(x \cdot \left(9 \cdot y\right) - \left(z \cdot 4\right) \cdot \color{blue}{\left(t \cdot a\right)}\right) + b}{z \cdot c} \]

   3. Simplified 81.0%

      \[\leadsto \color{blue}{\frac{\left(x \cdot \left(9 \cdot y\right) - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 75.3%

      \[\leadsto \color{blue}{\frac{-4 \cdot \frac{a \cdot \left(t \cdot z\right)}{c} + \left(9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z}} \]
   6. Step-by-step derivation

      1. fma-define 75.3%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-4, \frac{a \cdot \left(t \cdot z\right)}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}}{z} \]

      2. *-commutative 75.3%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{\color{blue}{\left(t \cdot z\right) \cdot a}}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      3. *-commutative 75.3%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{\color{blue}{\left(z \cdot t\right)} \cdot a}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      4. associate-*r* 75.3%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{\color{blue}{z \cdot \left(t \cdot a\right)}}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      5. *-commutative 75.3%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{z \cdot \color{blue}{\left(a \cdot t\right)}}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      6. associate-/l* 75.4%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \color{blue}{z \cdot \frac{a \cdot t}{c}}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      7. associate-/l* 79.6%

         \[\leadsto \frac{\mathsf{fma}\left(-4, z \cdot \color{blue}{\left(a \cdot \frac{t}{c}\right)}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      8. fma-define 79.6%

         \[\leadsto \frac{\mathsf{fma}\left(-4, z \cdot \left(a \cdot \frac{t}{c}\right), \color{blue}{\mathsf{fma}\left(9, \frac{x \cdot y}{c}, \frac{b}{c}\right)}\right)}{z} \]

      9. associate-/l* 83.8%

         \[\leadsto \frac{\mathsf{fma}\left(-4, z \cdot \left(a \cdot \frac{t}{c}\right), \mathsf{fma}\left(9, \color{blue}{x \cdot \frac{y}{c}}, \frac{b}{c}\right)\right)}{z} \]

   7. Simplified 83.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-4, z \cdot \left(a \cdot \frac{t}{c}\right), \mathsf{fma}\left(9, x \cdot \frac{y}{c}, \frac{b}{c}\right)\right)}{z}} \]

   8. Taylor expanded in b around inf 62.2%

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

   if -3.5999999999999999e-25 < b < 1.1500000000000001e-53

   1. Initial program 76.5%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Step-by-step derivation

      1. +-commutative 76.5%

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)}}{z \cdot c} \]

      2. associate-+r- 76.5%

         \[\leadsto \frac{\color{blue}{\left(b + \left(x \cdot 9\right) \cdot y\right) - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}}{z \cdot c} \]

      3. *-commutative 76.5%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{a \cdot \left(\left(z \cdot 4\right) \cdot t\right)}}{z \cdot c} \]

      4. associate-*r* 75.8%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{\left(a \cdot \left(z \cdot 4\right)\right) \cdot t}}{z \cdot c} \]

      5. *-commutative 75.8%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{\left(\left(z \cdot 4\right) \cdot a\right)} \cdot t}{z \cdot c} \]

      6. associate-+r- 75.8%

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right)}}{z \cdot c} \]

      7. +-commutative 75.8%

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right) + b}}{z \cdot c} \]

      8. associate-*l* 75.8%

         \[\leadsto \frac{\left(\color{blue}{x \cdot \left(9 \cdot y\right)} - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right) + b}{z \cdot c} \]

      9. associate-*l* 78.5%

         \[\leadsto \frac{\left(x \cdot \left(9 \cdot y\right) - \color{blue}{\left(z \cdot 4\right) \cdot \left(a \cdot t\right)}\right) + b}{z \cdot c} \]

      10. *-commutative 78.5%

          \[\leadsto \frac{\left(x \cdot \left(9 \cdot y\right) - \left(z \cdot 4\right) \cdot \color{blue}{\left(t \cdot a\right)}\right) + b}{z \cdot c} \]

   3. Simplified 78.5%

      \[\leadsto \color{blue}{\frac{\left(x \cdot \left(9 \cdot y\right) - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 76.0%

      \[\leadsto \color{blue}{\frac{-4 \cdot \frac{a \cdot \left(t \cdot z\right)}{c} + \left(9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z}} \]
   6. Step-by-step derivation

      1. fma-define 76.0%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-4, \frac{a \cdot \left(t \cdot z\right)}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}}{z} \]

      2. *-commutative 76.0%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{\color{blue}{\left(t \cdot z\right) \cdot a}}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      3. *-commutative 76.0%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{\color{blue}{\left(z \cdot t\right)} \cdot a}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      4. associate-*r* 78.9%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{\color{blue}{z \cdot \left(t \cdot a\right)}}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      5. *-commutative 78.9%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{z \cdot \color{blue}{\left(a \cdot t\right)}}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      6. associate-/l* 82.5%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \color{blue}{z \cdot \frac{a \cdot t}{c}}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      7. associate-/l* 82.9%

         \[\leadsto \frac{\mathsf{fma}\left(-4, z \cdot \color{blue}{\left(a \cdot \frac{t}{c}\right)}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      8. fma-define 82.9%

         \[\leadsto \frac{\mathsf{fma}\left(-4, z \cdot \left(a \cdot \frac{t}{c}\right), \color{blue}{\mathsf{fma}\left(9, \frac{x \cdot y}{c}, \frac{b}{c}\right)}\right)}{z} \]

      9. associate-/l* 82.0%

         \[\leadsto \frac{\mathsf{fma}\left(-4, z \cdot \left(a \cdot \frac{t}{c}\right), \mathsf{fma}\left(9, \color{blue}{x \cdot \frac{y}{c}}, \frac{b}{c}\right)\right)}{z} \]

   7. Simplified 82.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-4, z \cdot \left(a \cdot \frac{t}{c}\right), \mathsf{fma}\left(9, x \cdot \frac{y}{c}, \frac{b}{c}\right)\right)}{z}} \]

   8. Taylor expanded in z around inf 48.3%

      \[\leadsto \color{blue}{-4 \cdot \frac{a \cdot t}{c}} \]
   9. Step-by-step derivation

      1. associate-*r/ 48.3%

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

      2. associate-*r* 48.3%

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

      3. associate-*l/ 52.7%

         \[\leadsto \color{blue}{\frac{-4 \cdot a}{c} \cdot t} \]

      4. associate-*r/ 52.7%

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

      5. *-commutative 52.7%

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

      6. associate-*l* 52.7%

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

   10. Simplified 52.7%

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

   if 1.1500000000000001e-53 < b

   1. Initial program 83.1%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Step-by-step derivation

      1. +-commutative 83.1%

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)}}{z \cdot c} \]

      2. associate-+r- 83.1%

         \[\leadsto \frac{\color{blue}{\left(b + \left(x \cdot 9\right) \cdot y\right) - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}}{z \cdot c} \]

      3. *-commutative 83.1%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{a \cdot \left(\left(z \cdot 4\right) \cdot t\right)}}{z \cdot c} \]

      4. associate-*r* 85.5%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{\left(a \cdot \left(z \cdot 4\right)\right) \cdot t}}{z \cdot c} \]

      5. *-commutative 85.5%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{\left(\left(z \cdot 4\right) \cdot a\right)} \cdot t}{z \cdot c} \]

      6. associate-+r- 85.5%

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right)}}{z \cdot c} \]

      7. +-commutative 85.5%

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right) + b}}{z \cdot c} \]

      8. associate-*l* 86.7%

         \[\leadsto \frac{\left(\color{blue}{x \cdot \left(9 \cdot y\right)} - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right) + b}{z \cdot c} \]

      9. associate-*l* 88.0%

         \[\leadsto \frac{\left(x \cdot \left(9 \cdot y\right) - \color{blue}{\left(z \cdot 4\right) \cdot \left(a \cdot t\right)}\right) + b}{z \cdot c} \]

      10. *-commutative 88.0%

          \[\leadsto \frac{\left(x \cdot \left(9 \cdot y\right) - \left(z \cdot 4\right) \cdot \color{blue}{\left(t \cdot a\right)}\right) + b}{z \cdot c} \]

   3. Simplified 88.0%

      \[\leadsto \color{blue}{\frac{\left(x \cdot \left(9 \cdot y\right) - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around inf 68.3%

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

      1. *-commutative 68.3%

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

   7. Simplified 68.3%

      \[\leadsto \color{blue}{\frac{b}{z \cdot c}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 60.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.6 \cdot 10^{-25}:\\ \;\;\;\;\frac{\frac{b}{c}}{z}\\ \mathbf{elif}\;b \leq 1.15 \cdot 10^{-53}:\\ \;\;\;\;\frac{a}{c} \cdot \left(t \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 48.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ c\_s \cdot \begin{array}{l} \mathbf{if}\;b \leq -2.2 \cdot 10^{-26}:\\ \;\;\;\;\frac{\frac{b}{c\_m}}{z}\\ \mathbf{elif}\;b \leq 4.2 \cdot 10^{-54}:\\ \;\;\;\;a \cdot \frac{t \cdot -4}{c\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c\_m \cdot z}\\ \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (*
  c_s
  (if (<= b -2.2e-26)
    (/ (/ b c_m) z)
    (if (<= b 4.2e-54) (* a (/ (* t -4.0) c_m)) (/ b (* c_m z))))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double tmp;
	if (b <= -2.2e-26) {
		tmp = (b / c_m) / z;
	} else if (b <= 4.2e-54) {
		tmp = a * ((t * -4.0) / c_m);
	} else {
		tmp = b / (c_m * z);
	}
	return c_s * tmp;
}

Fortran

c\_m = abs(c)
c\_s = copysign(1.0d0, c)
real(8) function code(c_s, x, y, z, t, a, b, c_m)
    real(8), intent (in) :: c_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c_m
    real(8) :: tmp
    if (b <= (-2.2d-26)) then
        tmp = (b / c_m) / z
    else if (b <= 4.2d-54) then
        tmp = a * ((t * (-4.0d0)) / c_m)
    else
        tmp = b / (c_m * z)
    end if
    code = c_s * tmp
end function

Java

c\_m = Math.abs(c);
c\_s = Math.copySign(1.0, c);
public static double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double tmp;
	if (b <= -2.2e-26) {
		tmp = (b / c_m) / z;
	} else if (b <= 4.2e-54) {
		tmp = a * ((t * -4.0) / c_m);
	} else {
		tmp = b / (c_m * z);
	}
	return c_s * tmp;
}

Python

c\_m = math.fabs(c)
c\_s = math.copysign(1.0, c)
def code(c_s, x, y, z, t, a, b, c_m):
	tmp = 0
	if b <= -2.2e-26:
		tmp = (b / c_m) / z
	elif b <= 4.2e-54:
		tmp = a * ((t * -4.0) / c_m)
	else:
		tmp = b / (c_m * z)
	return c_s * tmp

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
function code(c_s, x, y, z, t, a, b, c_m)
	tmp = 0.0
	if (b <= -2.2e-26)
		tmp = Float64(Float64(b / c_m) / z);
	elseif (b <= 4.2e-54)
		tmp = Float64(a * Float64(Float64(t * -4.0) / c_m));
	else
		tmp = Float64(b / Float64(c_m * z));
	end
	return Float64(c_s * tmp)
end

MATLAB

c\_m = abs(c);
c\_s = sign(c) * abs(1.0);
function tmp_2 = code(c_s, x, y, z, t, a, b, c_m)
	tmp = 0.0;
	if (b <= -2.2e-26)
		tmp = (b / c_m) / z;
	elseif (b <= 4.2e-54)
		tmp = a * ((t * -4.0) / c_m);
	else
		tmp = b / (c_m * z);
	end
	tmp_2 = c_s * tmp;
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := N[(c$95$s * If[LessEqual[b, -2.2e-26], N[(N[(b / c$95$m), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[b, 4.2e-54], N[(a * N[(N[(t * -4.0), $MachinePrecision] / c$95$m), $MachinePrecision]), $MachinePrecision], N[(b / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if b < -2.2000000000000001e-26

   1. Initial program 82.4%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Step-by-step derivation

      1. +-commutative 82.4%

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)}}{z \cdot c} \]

      2. associate-+r- 82.4%

         \[\leadsto \frac{\color{blue}{\left(b + \left(x \cdot 9\right) \cdot y\right) - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}}{z \cdot c} \]

      3. *-commutative 82.4%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{a \cdot \left(\left(z \cdot 4\right) \cdot t\right)}}{z \cdot c} \]

      4. associate-*r* 83.9%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{\left(a \cdot \left(z \cdot 4\right)\right) \cdot t}}{z \cdot c} \]

      5. *-commutative 83.9%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{\left(\left(z \cdot 4\right) \cdot a\right)} \cdot t}{z \cdot c} \]

      6. associate-+r- 83.9%

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right)}}{z \cdot c} \]

      7. +-commutative 83.9%

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right) + b}}{z \cdot c} \]

      8. associate-*l* 83.9%

         \[\leadsto \frac{\left(\color{blue}{x \cdot \left(9 \cdot y\right)} - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right) + b}{z \cdot c} \]

      9. associate-*l* 81.0%

         \[\leadsto \frac{\left(x \cdot \left(9 \cdot y\right) - \color{blue}{\left(z \cdot 4\right) \cdot \left(a \cdot t\right)}\right) + b}{z \cdot c} \]

      10. *-commutative 81.0%

          \[\leadsto \frac{\left(x \cdot \left(9 \cdot y\right) - \left(z \cdot 4\right) \cdot \color{blue}{\left(t \cdot a\right)}\right) + b}{z \cdot c} \]

   3. Simplified 81.0%

      \[\leadsto \color{blue}{\frac{\left(x \cdot \left(9 \cdot y\right) - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 75.3%

      \[\leadsto \color{blue}{\frac{-4 \cdot \frac{a \cdot \left(t \cdot z\right)}{c} + \left(9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z}} \]
   6. Step-by-step derivation

      1. fma-define 75.3%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-4, \frac{a \cdot \left(t \cdot z\right)}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}}{z} \]

      2. *-commutative 75.3%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{\color{blue}{\left(t \cdot z\right) \cdot a}}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      3. *-commutative 75.3%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{\color{blue}{\left(z \cdot t\right)} \cdot a}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      4. associate-*r* 75.3%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{\color{blue}{z \cdot \left(t \cdot a\right)}}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      5. *-commutative 75.3%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{z \cdot \color{blue}{\left(a \cdot t\right)}}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      6. associate-/l* 75.4%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \color{blue}{z \cdot \frac{a \cdot t}{c}}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      7. associate-/l* 79.6%

         \[\leadsto \frac{\mathsf{fma}\left(-4, z \cdot \color{blue}{\left(a \cdot \frac{t}{c}\right)}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      8. fma-define 79.6%

         \[\leadsto \frac{\mathsf{fma}\left(-4, z \cdot \left(a \cdot \frac{t}{c}\right), \color{blue}{\mathsf{fma}\left(9, \frac{x \cdot y}{c}, \frac{b}{c}\right)}\right)}{z} \]

      9. associate-/l* 83.8%

         \[\leadsto \frac{\mathsf{fma}\left(-4, z \cdot \left(a \cdot \frac{t}{c}\right), \mathsf{fma}\left(9, \color{blue}{x \cdot \frac{y}{c}}, \frac{b}{c}\right)\right)}{z} \]

   7. Simplified 83.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-4, z \cdot \left(a \cdot \frac{t}{c}\right), \mathsf{fma}\left(9, x \cdot \frac{y}{c}, \frac{b}{c}\right)\right)}{z}} \]

   8. Taylor expanded in b around inf 62.2%

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

   if -2.2000000000000001e-26 < b < 4.2e-54

   1. Initial program 76.5%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Step-by-step derivation

      1. +-commutative 76.5%

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)}}{z \cdot c} \]

      2. associate-+r- 76.5%

         \[\leadsto \frac{\color{blue}{\left(b + \left(x \cdot 9\right) \cdot y\right) - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}}{z \cdot c} \]

      3. *-commutative 76.5%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{a \cdot \left(\left(z \cdot 4\right) \cdot t\right)}}{z \cdot c} \]

      4. associate-*r* 75.8%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{\left(a \cdot \left(z \cdot 4\right)\right) \cdot t}}{z \cdot c} \]

      5. *-commutative 75.8%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{\left(\left(z \cdot 4\right) \cdot a\right)} \cdot t}{z \cdot c} \]

      6. associate-+r- 75.8%

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right)}}{z \cdot c} \]

      7. +-commutative 75.8%

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right) + b}}{z \cdot c} \]

      8. associate-*l* 75.8%

         \[\leadsto \frac{\left(\color{blue}{x \cdot \left(9 \cdot y\right)} - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right) + b}{z \cdot c} \]

      9. associate-*l* 78.5%

         \[\leadsto \frac{\left(x \cdot \left(9 \cdot y\right) - \color{blue}{\left(z \cdot 4\right) \cdot \left(a \cdot t\right)}\right) + b}{z \cdot c} \]

      10. *-commutative 78.5%

          \[\leadsto \frac{\left(x \cdot \left(9 \cdot y\right) - \left(z \cdot 4\right) \cdot \color{blue}{\left(t \cdot a\right)}\right) + b}{z \cdot c} \]

   3. Simplified 78.5%

      \[\leadsto \color{blue}{\frac{\left(x \cdot \left(9 \cdot y\right) - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 48.3%

      \[\leadsto \color{blue}{-4 \cdot \frac{a \cdot t}{c}} \]
   6. Step-by-step derivation

      1. associate-/l* 52.2%

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

      2. associate-*r* 52.2%

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

      3. *-commutative 52.2%

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

      4. associate-*r* 52.2%

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

      5. *-commutative 52.2%

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

      6. associate-*l/ 52.2%

         \[\leadsto a \cdot \color{blue}{\frac{t \cdot -4}{c}} \]

   7. Simplified 52.2%

      \[\leadsto \color{blue}{a \cdot \frac{t \cdot -4}{c}} \]

   if 4.2e-54 < b

   1. Initial program 83.1%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Step-by-step derivation

      1. +-commutative 83.1%

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)}}{z \cdot c} \]

      2. associate-+r- 83.1%

         \[\leadsto \frac{\color{blue}{\left(b + \left(x \cdot 9\right) \cdot y\right) - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}}{z \cdot c} \]

      3. *-commutative 83.1%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{a \cdot \left(\left(z \cdot 4\right) \cdot t\right)}}{z \cdot c} \]

      4. associate-*r* 85.5%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{\left(a \cdot \left(z \cdot 4\right)\right) \cdot t}}{z \cdot c} \]

      5. *-commutative 85.5%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{\left(\left(z \cdot 4\right) \cdot a\right)} \cdot t}{z \cdot c} \]

      6. associate-+r- 85.5%

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right)}}{z \cdot c} \]

      7. +-commutative 85.5%

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right) + b}}{z \cdot c} \]

      8. associate-*l* 86.7%

         \[\leadsto \frac{\left(\color{blue}{x \cdot \left(9 \cdot y\right)} - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right) + b}{z \cdot c} \]

      9. associate-*l* 88.0%

         \[\leadsto \frac{\left(x \cdot \left(9 \cdot y\right) - \color{blue}{\left(z \cdot 4\right) \cdot \left(a \cdot t\right)}\right) + b}{z \cdot c} \]

      10. *-commutative 88.0%

          \[\leadsto \frac{\left(x \cdot \left(9 \cdot y\right) - \left(z \cdot 4\right) \cdot \color{blue}{\left(t \cdot a\right)}\right) + b}{z \cdot c} \]

   3. Simplified 88.0%

      \[\leadsto \color{blue}{\frac{\left(x \cdot \left(9 \cdot y\right) - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around inf 68.3%

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

      1. *-commutative 68.3%

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

   7. Simplified 68.3%

      \[\leadsto \color{blue}{\frac{b}{z \cdot c}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 59.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.2 \cdot 10^{-26}:\\ \;\;\;\;\frac{\frac{b}{c}}{z}\\ \mathbf{elif}\;b \leq 4.2 \cdot 10^{-54}:\\ \;\;\;\;a \cdot \frac{t \cdot -4}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 48.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ c\_s \cdot \begin{array}{l} \mathbf{if}\;b \leq -1.92 \cdot 10^{-25}:\\ \;\;\;\;\frac{\frac{b}{c\_m}}{z}\\ \mathbf{elif}\;b \leq 6.5 \cdot 10^{-54}:\\ \;\;\;\;a \cdot \left(t \cdot \frac{-4}{c\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c\_m \cdot z}\\ \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (*
  c_s
  (if (<= b -1.92e-25)
    (/ (/ b c_m) z)
    (if (<= b 6.5e-54) (* a (* t (/ -4.0 c_m))) (/ b (* c_m z))))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double tmp;
	if (b <= -1.92e-25) {
		tmp = (b / c_m) / z;
	} else if (b <= 6.5e-54) {
		tmp = a * (t * (-4.0 / c_m));
	} else {
		tmp = b / (c_m * z);
	}
	return c_s * tmp;
}

Fortran

c\_m = abs(c)
c\_s = copysign(1.0d0, c)
real(8) function code(c_s, x, y, z, t, a, b, c_m)
    real(8), intent (in) :: c_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c_m
    real(8) :: tmp
    if (b <= (-1.92d-25)) then
        tmp = (b / c_m) / z
    else if (b <= 6.5d-54) then
        tmp = a * (t * ((-4.0d0) / c_m))
    else
        tmp = b / (c_m * z)
    end if
    code = c_s * tmp
end function

Java

c\_m = Math.abs(c);
c\_s = Math.copySign(1.0, c);
public static double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double tmp;
	if (b <= -1.92e-25) {
		tmp = (b / c_m) / z;
	} else if (b <= 6.5e-54) {
		tmp = a * (t * (-4.0 / c_m));
	} else {
		tmp = b / (c_m * z);
	}
	return c_s * tmp;
}

Python

c\_m = math.fabs(c)
c\_s = math.copysign(1.0, c)
def code(c_s, x, y, z, t, a, b, c_m):
	tmp = 0
	if b <= -1.92e-25:
		tmp = (b / c_m) / z
	elif b <= 6.5e-54:
		tmp = a * (t * (-4.0 / c_m))
	else:
		tmp = b / (c_m * z)
	return c_s * tmp

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
function code(c_s, x, y, z, t, a, b, c_m)
	tmp = 0.0
	if (b <= -1.92e-25)
		tmp = Float64(Float64(b / c_m) / z);
	elseif (b <= 6.5e-54)
		tmp = Float64(a * Float64(t * Float64(-4.0 / c_m)));
	else
		tmp = Float64(b / Float64(c_m * z));
	end
	return Float64(c_s * tmp)
end

MATLAB

c\_m = abs(c);
c\_s = sign(c) * abs(1.0);
function tmp_2 = code(c_s, x, y, z, t, a, b, c_m)
	tmp = 0.0;
	if (b <= -1.92e-25)
		tmp = (b / c_m) / z;
	elseif (b <= 6.5e-54)
		tmp = a * (t * (-4.0 / c_m));
	else
		tmp = b / (c_m * z);
	end
	tmp_2 = c_s * tmp;
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := N[(c$95$s * If[LessEqual[b, -1.92e-25], N[(N[(b / c$95$m), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[b, 6.5e-54], N[(a * N[(t * N[(-4.0 / c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(b / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if b < -1.9200000000000001e-25

   1. Initial program 82.4%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Step-by-step derivation

      1. +-commutative 82.4%

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)}}{z \cdot c} \]

      2. associate-+r- 82.4%

         \[\leadsto \frac{\color{blue}{\left(b + \left(x \cdot 9\right) \cdot y\right) - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}}{z \cdot c} \]

      3. *-commutative 82.4%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{a \cdot \left(\left(z \cdot 4\right) \cdot t\right)}}{z \cdot c} \]

      4. associate-*r* 83.9%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{\left(a \cdot \left(z \cdot 4\right)\right) \cdot t}}{z \cdot c} \]

      5. *-commutative 83.9%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{\left(\left(z \cdot 4\right) \cdot a\right)} \cdot t}{z \cdot c} \]

      6. associate-+r- 83.9%

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right)}}{z \cdot c} \]

      7. +-commutative 83.9%

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right) + b}}{z \cdot c} \]

      8. associate-*l* 83.9%

         \[\leadsto \frac{\left(\color{blue}{x \cdot \left(9 \cdot y\right)} - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right) + b}{z \cdot c} \]

      9. associate-*l* 81.0%

         \[\leadsto \frac{\left(x \cdot \left(9 \cdot y\right) - \color{blue}{\left(z \cdot 4\right) \cdot \left(a \cdot t\right)}\right) + b}{z \cdot c} \]

      10. *-commutative 81.0%

          \[\leadsto \frac{\left(x \cdot \left(9 \cdot y\right) - \left(z \cdot 4\right) \cdot \color{blue}{\left(t \cdot a\right)}\right) + b}{z \cdot c} \]

   3. Simplified 81.0%

      \[\leadsto \color{blue}{\frac{\left(x \cdot \left(9 \cdot y\right) - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 75.3%

      \[\leadsto \color{blue}{\frac{-4 \cdot \frac{a \cdot \left(t \cdot z\right)}{c} + \left(9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z}} \]
   6. Step-by-step derivation

      1. fma-define 75.3%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-4, \frac{a \cdot \left(t \cdot z\right)}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}}{z} \]

      2. *-commutative 75.3%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{\color{blue}{\left(t \cdot z\right) \cdot a}}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      3. *-commutative 75.3%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{\color{blue}{\left(z \cdot t\right)} \cdot a}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      4. associate-*r* 75.3%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{\color{blue}{z \cdot \left(t \cdot a\right)}}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      5. *-commutative 75.3%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{z \cdot \color{blue}{\left(a \cdot t\right)}}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      6. associate-/l* 75.4%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \color{blue}{z \cdot \frac{a \cdot t}{c}}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      7. associate-/l* 79.6%

         \[\leadsto \frac{\mathsf{fma}\left(-4, z \cdot \color{blue}{\left(a \cdot \frac{t}{c}\right)}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      8. fma-define 79.6%

         \[\leadsto \frac{\mathsf{fma}\left(-4, z \cdot \left(a \cdot \frac{t}{c}\right), \color{blue}{\mathsf{fma}\left(9, \frac{x \cdot y}{c}, \frac{b}{c}\right)}\right)}{z} \]

      9. associate-/l* 83.8%

         \[\leadsto \frac{\mathsf{fma}\left(-4, z \cdot \left(a \cdot \frac{t}{c}\right), \mathsf{fma}\left(9, \color{blue}{x \cdot \frac{y}{c}}, \frac{b}{c}\right)\right)}{z} \]

   7. Simplified 83.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-4, z \cdot \left(a \cdot \frac{t}{c}\right), \mathsf{fma}\left(9, x \cdot \frac{y}{c}, \frac{b}{c}\right)\right)}{z}} \]

   8. Taylor expanded in b around inf 62.2%

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

   if -1.9200000000000001e-25 < b < 6.49999999999999991e-54

   1. Initial program 76.5%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Step-by-step derivation

      1. +-commutative 76.5%

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)}}{z \cdot c} \]

      2. associate-+r- 76.5%

         \[\leadsto \frac{\color{blue}{\left(b + \left(x \cdot 9\right) \cdot y\right) - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}}{z \cdot c} \]

      3. *-commutative 76.5%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{a \cdot \left(\left(z \cdot 4\right) \cdot t\right)}}{z \cdot c} \]

      4. associate-*r* 75.8%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{\left(a \cdot \left(z \cdot 4\right)\right) \cdot t}}{z \cdot c} \]

      5. *-commutative 75.8%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{\left(\left(z \cdot 4\right) \cdot a\right)} \cdot t}{z \cdot c} \]

      6. associate-+r- 75.8%

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right)}}{z \cdot c} \]

      7. +-commutative 75.8%

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right) + b}}{z \cdot c} \]

      8. associate-*l* 75.8%

         \[\leadsto \frac{\left(\color{blue}{x \cdot \left(9 \cdot y\right)} - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right) + b}{z \cdot c} \]

      9. associate-*l* 78.5%

         \[\leadsto \frac{\left(x \cdot \left(9 \cdot y\right) - \color{blue}{\left(z \cdot 4\right) \cdot \left(a \cdot t\right)}\right) + b}{z \cdot c} \]

      10. *-commutative 78.5%

          \[\leadsto \frac{\left(x \cdot \left(9 \cdot y\right) - \left(z \cdot 4\right) \cdot \color{blue}{\left(t \cdot a\right)}\right) + b}{z \cdot c} \]

   3. Simplified 78.5%

      \[\leadsto \color{blue}{\frac{\left(x \cdot \left(9 \cdot y\right) - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 76.0%

      \[\leadsto \color{blue}{\frac{-4 \cdot \frac{a \cdot \left(t \cdot z\right)}{c} + \left(9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z}} \]
   6. Step-by-step derivation

      1. fma-define 76.0%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-4, \frac{a \cdot \left(t \cdot z\right)}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}}{z} \]

      2. *-commutative 76.0%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{\color{blue}{\left(t \cdot z\right) \cdot a}}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      3. *-commutative 76.0%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{\color{blue}{\left(z \cdot t\right)} \cdot a}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      4. associate-*r* 78.9%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{\color{blue}{z \cdot \left(t \cdot a\right)}}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      5. *-commutative 78.9%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{z \cdot \color{blue}{\left(a \cdot t\right)}}{c}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      6. associate-/l* 82.5%

         \[\leadsto \frac{\mathsf{fma}\left(-4, \color{blue}{z \cdot \frac{a \cdot t}{c}}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      7. associate-/l* 82.9%

         \[\leadsto \frac{\mathsf{fma}\left(-4, z \cdot \color{blue}{\left(a \cdot \frac{t}{c}\right)}, 9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}\right)}{z} \]

      8. fma-define 82.9%

         \[\leadsto \frac{\mathsf{fma}\left(-4, z \cdot \left(a \cdot \frac{t}{c}\right), \color{blue}{\mathsf{fma}\left(9, \frac{x \cdot y}{c}, \frac{b}{c}\right)}\right)}{z} \]

      9. associate-/l* 82.0%

         \[\leadsto \frac{\mathsf{fma}\left(-4, z \cdot \left(a \cdot \frac{t}{c}\right), \mathsf{fma}\left(9, \color{blue}{x \cdot \frac{y}{c}}, \frac{b}{c}\right)\right)}{z} \]

   7. Simplified 82.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-4, z \cdot \left(a \cdot \frac{t}{c}\right), \mathsf{fma}\left(9, x \cdot \frac{y}{c}, \frac{b}{c}\right)\right)}{z}} \]

   8. Taylor expanded in c around 0 77.0%

      \[\leadsto \frac{\color{blue}{\frac{b + \left(-4 \cdot \left(a \cdot \left(t \cdot z\right)\right) + 9 \cdot \left(x \cdot y\right)\right)}{c}}}{z} \]

   9. Taylor expanded in t around inf 75.2%

      \[\leadsto \frac{\frac{b + \color{blue}{t \cdot \left(-4 \cdot \left(a \cdot z\right) + 9 \cdot \frac{x \cdot y}{t}\right)}}{c}}{z} \]

   10. Taylor expanded in t around inf 48.3%

       \[\leadsto \color{blue}{-4 \cdot \frac{a \cdot t}{c}} \]
   11. Step-by-step derivation

       1. associate-*r/ 48.3%

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

       2. *-commutative 48.3%

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

       3. associate-*r/ 48.2%

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

       4. associate-*r* 52.2%

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

   12. Simplified 52.2%

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

   if 6.49999999999999991e-54 < b

   1. Initial program 83.1%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Step-by-step derivation

      1. +-commutative 83.1%

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)}}{z \cdot c} \]

      2. associate-+r- 83.1%

         \[\leadsto \frac{\color{blue}{\left(b + \left(x \cdot 9\right) \cdot y\right) - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}}{z \cdot c} \]

      3. *-commutative 83.1%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{a \cdot \left(\left(z \cdot 4\right) \cdot t\right)}}{z \cdot c} \]

      4. associate-*r* 85.5%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{\left(a \cdot \left(z \cdot 4\right)\right) \cdot t}}{z \cdot c} \]

      5. *-commutative 85.5%

         \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{\left(\left(z \cdot 4\right) \cdot a\right)} \cdot t}{z \cdot c} \]

      6. associate-+r- 85.5%

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right)}}{z \cdot c} \]

      7. +-commutative 85.5%

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right) + b}}{z \cdot c} \]

      8. associate-*l* 86.7%

         \[\leadsto \frac{\left(\color{blue}{x \cdot \left(9 \cdot y\right)} - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right) + b}{z \cdot c} \]

      9. associate-*l* 88.0%

         \[\leadsto \frac{\left(x \cdot \left(9 \cdot y\right) - \color{blue}{\left(z \cdot 4\right) \cdot \left(a \cdot t\right)}\right) + b}{z \cdot c} \]

      10. *-commutative 88.0%

          \[\leadsto \frac{\left(x \cdot \left(9 \cdot y\right) - \left(z \cdot 4\right) \cdot \color{blue}{\left(t \cdot a\right)}\right) + b}{z \cdot c} \]

   3. Simplified 88.0%

      \[\leadsto \color{blue}{\frac{\left(x \cdot \left(9 \cdot y\right) - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around inf 68.3%

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

      1. *-commutative 68.3%

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

   7. Simplified 68.3%

      \[\leadsto \color{blue}{\frac{b}{z \cdot c}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 59.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.92 \cdot 10^{-25}:\\ \;\;\;\;\frac{\frac{b}{c}}{z}\\ \mathbf{elif}\;b \leq 6.5 \cdot 10^{-54}:\\ \;\;\;\;a \cdot \left(t \cdot \frac{-4}{c}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 34.9% accurate, 3.8× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ c\_s \cdot \frac{b}{c\_m \cdot z} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
(FPCore (c_s x y z t a b c_m) :precision binary64 (* c_s (/ b (* c_m z))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	return c_s * (b / (c_m * z));
}

Fortran

c\_m = abs(c)
c\_s = copysign(1.0d0, c)
real(8) function code(c_s, x, y, z, t, a, b, c_m)
    real(8), intent (in) :: c_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c_m
    code = c_s * (b / (c_m * z))
end function

Java

c\_m = Math.abs(c);
c\_s = Math.copySign(1.0, c);
public static double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	return c_s * (b / (c_m * z));
}

Python

c\_m = math.fabs(c)
c\_s = math.copysign(1.0, c)
def code(c_s, x, y, z, t, a, b, c_m):
	return c_s * (b / (c_m * z))

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
function code(c_s, x, y, z, t, a, b, c_m)
	return Float64(c_s * Float64(b / Float64(c_m * z)))
end

MATLAB

c\_m = abs(c);
c\_s = sign(c) * abs(1.0);
function tmp = code(c_s, x, y, z, t, a, b, c_m)
	tmp = c_s * (b / (c_m * z));
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := N[(c$95$s * N[(b / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.1%

   \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
2. Step-by-step derivation

   1. +-commutative 80.1%

      \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)}}{z \cdot c} \]

   2. associate-+r- 80.1%

      \[\leadsto \frac{\color{blue}{\left(b + \left(x \cdot 9\right) \cdot y\right) - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}}{z \cdot c} \]

   3. *-commutative 80.1%

      \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{a \cdot \left(\left(z \cdot 4\right) \cdot t\right)}}{z \cdot c} \]

   4. associate-*r* 80.9%

      \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{\left(a \cdot \left(z \cdot 4\right)\right) \cdot t}}{z \cdot c} \]

   5. *-commutative 80.9%

      \[\leadsto \frac{\left(b + \left(x \cdot 9\right) \cdot y\right) - \color{blue}{\left(\left(z \cdot 4\right) \cdot a\right)} \cdot t}{z \cdot c} \]

   6. associate-+r- 80.9%

      \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right)}}{z \cdot c} \]

   7. +-commutative 80.9%

      \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right) + b}}{z \cdot c} \]

   8. associate-*l* 81.3%

      \[\leadsto \frac{\left(\color{blue}{x \cdot \left(9 \cdot y\right)} - \left(\left(z \cdot 4\right) \cdot a\right) \cdot t\right) + b}{z \cdot c} \]

   9. associate-*l* 82.1%

      \[\leadsto \frac{\left(x \cdot \left(9 \cdot y\right) - \color{blue}{\left(z \cdot 4\right) \cdot \left(a \cdot t\right)}\right) + b}{z \cdot c} \]

   10. *-commutative 82.1%

       \[\leadsto \frac{\left(x \cdot \left(9 \cdot y\right) - \left(z \cdot 4\right) \cdot \color{blue}{\left(t \cdot a\right)}\right) + b}{z \cdot c} \]

3. Simplified 82.1%

   \[\leadsto \color{blue}{\frac{\left(x \cdot \left(9 \cdot y\right) - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}} \]
4. Add Preprocessing

5. Taylor expanded in b around inf 45.3%

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

   1. *-commutative 45.3%

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

7. Simplified 45.3%

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

8. Final simplification 45.3%

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

Developer Target 1: 79.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{b}{c \cdot z}\\ t_2 := 4 \cdot \frac{a \cdot t}{c}\\ t_3 := \left(x \cdot 9\right) \cdot y\\ t_4 := \left(t\_3 - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b\\ t_5 := \frac{t\_4}{z \cdot c}\\ t_6 := \frac{\left(t\_3 - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{if}\;t\_5 < -1.100156740804105 \cdot 10^{-171}:\\ \;\;\;\;t\_6\\ \mathbf{elif}\;t\_5 < 0:\\ \;\;\;\;\frac{\frac{t\_4}{z}}{c}\\ \mathbf{elif}\;t\_5 < 1.1708877911747488 \cdot 10^{-53}:\\ \;\;\;\;t\_6\\ \mathbf{elif}\;t\_5 < 2.876823679546137 \cdot 10^{+130}:\\ \;\;\;\;\left(\left(9 \cdot \frac{y}{c}\right) \cdot \frac{x}{z} + t\_1\right) - t\_2\\ \mathbf{elif}\;t\_5 < 1.3838515042456319 \cdot 10^{+158}:\\ \;\;\;\;t\_6\\ \mathbf{else}:\\ \;\;\;\;\left(9 \cdot \left(\frac{y}{c \cdot z} \cdot x\right) + t\_1\right) - t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ b (* c z)))
        (t_2 (* 4.0 (/ (* a t) c)))
        (t_3 (* (* x 9.0) y))
        (t_4 (+ (- t_3 (* (* (* z 4.0) t) a)) b))
        (t_5 (/ t_4 (* z c)))
        (t_6 (/ (+ (- t_3 (* (* z 4.0) (* t a))) b) (* z c))))
   (if (< t_5 -1.100156740804105e-171)
     t_6
     (if (< t_5 0.0)
       (/ (/ t_4 z) c)
       (if (< t_5 1.1708877911747488e-53)
         t_6
         (if (< t_5 2.876823679546137e+130)
           (- (+ (* (* 9.0 (/ y c)) (/ x z)) t_1) t_2)
           (if (< t_5 1.3838515042456319e+158)
             t_6
             (- (+ (* 9.0 (* (/ y (* c z)) x)) t_1) t_2))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = b / (c * z);
	double t_2 = 4.0 * ((a * t) / c);
	double t_3 = (x * 9.0) * y;
	double t_4 = (t_3 - (((z * 4.0) * t) * a)) + b;
	double t_5 = t_4 / (z * c);
	double t_6 = ((t_3 - ((z * 4.0) * (t * a))) + b) / (z * c);
	double tmp;
	if (t_5 < -1.100156740804105e-171) {
		tmp = t_6;
	} else if (t_5 < 0.0) {
		tmp = (t_4 / z) / c;
	} else if (t_5 < 1.1708877911747488e-53) {
		tmp = t_6;
	} else if (t_5 < 2.876823679546137e+130) {
		tmp = (((9.0 * (y / c)) * (x / z)) + t_1) - t_2;
	} else if (t_5 < 1.3838515042456319e+158) {
		tmp = t_6;
	} else {
		tmp = ((9.0 * ((y / (c * z)) * x)) + t_1) - t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: t_6
    real(8) :: tmp
    t_1 = b / (c * z)
    t_2 = 4.0d0 * ((a * t) / c)
    t_3 = (x * 9.0d0) * y
    t_4 = (t_3 - (((z * 4.0d0) * t) * a)) + b
    t_5 = t_4 / (z * c)
    t_6 = ((t_3 - ((z * 4.0d0) * (t * a))) + b) / (z * c)
    if (t_5 < (-1.100156740804105d-171)) then
        tmp = t_6
    else if (t_5 < 0.0d0) then
        tmp = (t_4 / z) / c
    else if (t_5 < 1.1708877911747488d-53) then
        tmp = t_6
    else if (t_5 < 2.876823679546137d+130) then
        tmp = (((9.0d0 * (y / c)) * (x / z)) + t_1) - t_2
    else if (t_5 < 1.3838515042456319d+158) then
        tmp = t_6
    else
        tmp = ((9.0d0 * ((y / (c * z)) * x)) + t_1) - t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = b / (c * z);
	double t_2 = 4.0 * ((a * t) / c);
	double t_3 = (x * 9.0) * y;
	double t_4 = (t_3 - (((z * 4.0) * t) * a)) + b;
	double t_5 = t_4 / (z * c);
	double t_6 = ((t_3 - ((z * 4.0) * (t * a))) + b) / (z * c);
	double tmp;
	if (t_5 < -1.100156740804105e-171) {
		tmp = t_6;
	} else if (t_5 < 0.0) {
		tmp = (t_4 / z) / c;
	} else if (t_5 < 1.1708877911747488e-53) {
		tmp = t_6;
	} else if (t_5 < 2.876823679546137e+130) {
		tmp = (((9.0 * (y / c)) * (x / z)) + t_1) - t_2;
	} else if (t_5 < 1.3838515042456319e+158) {
		tmp = t_6;
	} else {
		tmp = ((9.0 * ((y / (c * z)) * x)) + t_1) - t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	t_1 = b / (c * z)
	t_2 = 4.0 * ((a * t) / c)
	t_3 = (x * 9.0) * y
	t_4 = (t_3 - (((z * 4.0) * t) * a)) + b
	t_5 = t_4 / (z * c)
	t_6 = ((t_3 - ((z * 4.0) * (t * a))) + b) / (z * c)
	tmp = 0
	if t_5 < -1.100156740804105e-171:
		tmp = t_6
	elif t_5 < 0.0:
		tmp = (t_4 / z) / c
	elif t_5 < 1.1708877911747488e-53:
		tmp = t_6
	elif t_5 < 2.876823679546137e+130:
		tmp = (((9.0 * (y / c)) * (x / z)) + t_1) - t_2
	elif t_5 < 1.3838515042456319e+158:
		tmp = t_6
	else:
		tmp = ((9.0 * ((y / (c * z)) * x)) + t_1) - t_2
	return tmp

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(b / Float64(c * z))
	t_2 = Float64(4.0 * Float64(Float64(a * t) / c))
	t_3 = Float64(Float64(x * 9.0) * y)
	t_4 = Float64(Float64(t_3 - Float64(Float64(Float64(z * 4.0) * t) * a)) + b)
	t_5 = Float64(t_4 / Float64(z * c))
	t_6 = Float64(Float64(Float64(t_3 - Float64(Float64(z * 4.0) * Float64(t * a))) + b) / Float64(z * c))
	tmp = 0.0
	if (t_5 < -1.100156740804105e-171)
		tmp = t_6;
	elseif (t_5 < 0.0)
		tmp = Float64(Float64(t_4 / z) / c);
	elseif (t_5 < 1.1708877911747488e-53)
		tmp = t_6;
	elseif (t_5 < 2.876823679546137e+130)
		tmp = Float64(Float64(Float64(Float64(9.0 * Float64(y / c)) * Float64(x / z)) + t_1) - t_2);
	elseif (t_5 < 1.3838515042456319e+158)
		tmp = t_6;
	else
		tmp = Float64(Float64(Float64(9.0 * Float64(Float64(y / Float64(c * z)) * x)) + t_1) - t_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c)
	t_1 = b / (c * z);
	t_2 = 4.0 * ((a * t) / c);
	t_3 = (x * 9.0) * y;
	t_4 = (t_3 - (((z * 4.0) * t) * a)) + b;
	t_5 = t_4 / (z * c);
	t_6 = ((t_3 - ((z * 4.0) * (t * a))) + b) / (z * c);
	tmp = 0.0;
	if (t_5 < -1.100156740804105e-171)
		tmp = t_6;
	elseif (t_5 < 0.0)
		tmp = (t_4 / z) / c;
	elseif (t_5 < 1.1708877911747488e-53)
		tmp = t_6;
	elseif (t_5 < 2.876823679546137e+130)
		tmp = (((9.0 * (y / c)) * (x / z)) + t_1) - t_2;
	elseif (t_5 < 1.3838515042456319e+158)
		tmp = t_6;
	else
		tmp = ((9.0 * ((y / (c * z)) * x)) + t_1) - t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(b / N[(c * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(4.0 * N[(N[(a * t), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x * 9.0), $MachinePrecision] * y), $MachinePrecision]}, Block[{t$95$4 = N[(N[(t$95$3 - N[(N[(N[(z * 4.0), $MachinePrecision] * t), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$4 / N[(z * c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(N[(N[(t$95$3 - N[(N[(z * 4.0), $MachinePrecision] * N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision] / N[(z * c), $MachinePrecision]), $MachinePrecision]}, If[Less[t$95$5, -1.100156740804105e-171], t$95$6, If[Less[t$95$5, 0.0], N[(N[(t$95$4 / z), $MachinePrecision] / c), $MachinePrecision], If[Less[t$95$5, 1.1708877911747488e-53], t$95$6, If[Less[t$95$5, 2.876823679546137e+130], N[(N[(N[(N[(9.0 * N[(y / c), $MachinePrecision]), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] - t$95$2), $MachinePrecision], If[Less[t$95$5, 1.3838515042456319e+158], t$95$6, N[(N[(N[(9.0 * N[(N[(y / N[(c * z), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] - t$95$2), $MachinePrecision]]]]]]]]]]]]

Reproduce

herbie shell --seed 2024135 
(FPCore (x y z t a b c)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, J"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< (/ (+ (- (* (* x 9) y) (* (* (* z 4) t) a)) b) (* z c)) -220031348160821/200000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (/ (+ (- (* (* x 9) y) (* (* z 4) (* t a))) b) (* z c)) (if (< (/ (+ (- (* (* x 9) y) (* (* (* z 4) t) a)) b) (* z c)) 0) (/ (/ (+ (- (* (* x 9) y) (* (* (* z 4) t) a)) b) z) c) (if (< (/ (+ (- (* (* x 9) y) (* (* (* z 4) t) a)) b) (* z c)) 365902434742109/31250000000000000000000000000000000000000000000000000000000000000000) (/ (+ (- (* (* x 9) y) (* (* z 4) (* t a))) b) (* z c)) (if (< (/ (+ (- (* (* x 9) y) (* (* (* z 4) t) a)) b) (* z c)) 28768236795461370000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (- (+ (* (* 9 (/ y c)) (/ x z)) (/ b (* c z))) (* 4 (/ (* a t) c))) (if (< (/ (+ (- (* (* x 9) y) (* (* (* z 4) t) a)) b) (* z c)) 138385150424563190000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (/ (+ (- (* (* x 9) y) (* (* z 4) (* t a))) b) (* z c)) (- (+ (* 9 (* (/ y (* c z)) x)) (/ b (* c z))) (* 4 (/ (* a t) c)))))))))

  (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)))