Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2

Percentage Accurate: 55.3% → 81.9%

Time: 17.3s

Alternatives: 18

Speedup: 2.0×

Specification

Math

\[\begin{array}{l} \\ \frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (/
  (+ (* (+ (* (+ (* (+ (* x y) z) y) 27464.7644705) y) 230661.510616) y) t)
  (+ (* (+ (* (+ (* (+ y a) y) b) y) c) y) i)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return ((((((((x * y) + z) * y) + 27464.7644705) * y) + 230661.510616) * y) + t) / (((((((y + a) * y) + b) * y) + c) * y) + i);
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i)
    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), intent (in) :: i
    code = ((((((((x * y) + z) * y) + 27464.7644705d0) * y) + 230661.510616d0) * y) + t) / (((((((y + a) * y) + b) * y) + c) * y) + i)
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return ((((((((x * y) + z) * y) + 27464.7644705) * y) + 230661.510616) * y) + t) / (((((((y + a) * y) + b) * y) + c) * y) + i);
}

Python

def code(x, y, z, t, a, b, c, i):
	return ((((((((x * y) + z) * y) + 27464.7644705) * y) + 230661.510616) * y) + t) / (((((((y + a) * y) + b) * y) + c) * y) + i)

Julia

function code(x, y, z, t, a, b, c, i)
	return Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(x * y) + z) * y) + 27464.7644705) * y) + 230661.510616) * y) + t) / Float64(Float64(Float64(Float64(Float64(Float64(Float64(y + a) * y) + b) * y) + c) * y) + i))
end

MATLAB

function tmp = code(x, y, z, t, a, b, c, i)
	tmp = ((((((((x * y) + z) * y) + 27464.7644705) * y) + 230661.510616) * y) + t) / (((((((y + a) * y) + b) * y) + c) * y) + i);
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := N[(N[(N[(N[(N[(N[(N[(N[(N[(x * y), $MachinePrecision] + z), $MachinePrecision] * y), $MachinePrecision] + 27464.7644705), $MachinePrecision] * y), $MachinePrecision] + 230661.510616), $MachinePrecision] * y), $MachinePrecision] + t), $MachinePrecision] / N[(N[(N[(N[(N[(N[(N[(y + a), $MachinePrecision] * y), $MachinePrecision] + b), $MachinePrecision] * y), $MachinePrecision] + c), $MachinePrecision] * y), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision]

Initial Program: 55.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (/
  (+ (* (+ (* (+ (* (+ (* x y) z) y) 27464.7644705) y) 230661.510616) y) t)
  (+ (* (+ (* (+ (* (+ y a) y) b) y) c) y) i)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return ((((((((x * y) + z) * y) + 27464.7644705) * y) + 230661.510616) * y) + t) / (((((((y + a) * y) + b) * y) + c) * y) + i);
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i)
    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), intent (in) :: i
    code = ((((((((x * y) + z) * y) + 27464.7644705d0) * y) + 230661.510616d0) * y) + t) / (((((((y + a) * y) + b) * y) + c) * y) + i)
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return ((((((((x * y) + z) * y) + 27464.7644705) * y) + 230661.510616) * y) + t) / (((((((y + a) * y) + b) * y) + c) * y) + i);
}

Python

def code(x, y, z, t, a, b, c, i):
	return ((((((((x * y) + z) * y) + 27464.7644705) * y) + 230661.510616) * y) + t) / (((((((y + a) * y) + b) * y) + c) * y) + i)

Julia

function code(x, y, z, t, a, b, c, i)
	return Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(x * y) + z) * y) + 27464.7644705) * y) + 230661.510616) * y) + t) / Float64(Float64(Float64(Float64(Float64(Float64(Float64(y + a) * y) + b) * y) + c) * y) + i))
end

MATLAB

function tmp = code(x, y, z, t, a, b, c, i)
	tmp = ((((((((x * y) + z) * y) + 27464.7644705) * y) + 230661.510616) * y) + t) / (((((((y + a) * y) + b) * y) + c) * y) + i);
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := N[(N[(N[(N[(N[(N[(N[(N[(N[(x * y), $MachinePrecision] + z), $MachinePrecision] * y), $MachinePrecision] + 27464.7644705), $MachinePrecision] * y), $MachinePrecision] + 230661.510616), $MachinePrecision] * y), $MachinePrecision] + t), $MachinePrecision] / N[(N[(N[(N[(N[(N[(N[(y + a), $MachinePrecision] * y), $MachinePrecision] + b), $MachinePrecision] * y), $MachinePrecision] + c), $MachinePrecision] * y), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision]

Alternative 1: 81.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\frac{z}{y} + \left(\frac{27464.7644705}{y \cdot y} + x\right)\right) - \mathsf{fma}\left(a, \frac{z - a \cdot x}{y \cdot y} + \frac{x}{y}, \frac{b}{y} \cdot \frac{x}{y}\right)\\ \mathbf{if}\;y \leq -6.2 \cdot 10^{+47}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.12 \cdot 10^{+38}:\\ \;\;\;\;\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right) \cdot y, y, c \cdot y\right) + i}\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{+106}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(x, y, \frac{t}{{y}^{3}}\right) + \frac{230661.510616}{y \cdot y}\right) + \left(\frac{27464.7644705}{y} + z\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1
         (-
          (+ (/ z y) (+ (/ 27464.7644705 (* y y)) x))
          (fma a (+ (/ (- z (* a x)) (* y y)) (/ x y)) (* (/ b y) (/ x y))))))
   (if (<= y -6.2e+47)
     t_1
     (if (<= y 1.12e+38)
       (/
        (+
         (* (+ (* (+ (* (+ (* x y) z) y) 27464.7644705) y) 230661.510616) y)
         t)
        (+ (fma (* (fma (+ a y) y b) y) y (* c y)) i))
       (if (<= y 1.25e+106)
         (/
          (+
           (+ (fma x y (/ t (pow y 3.0))) (/ 230661.510616 (* y y)))
           (+ (/ 27464.7644705 y) z))
          a)
         t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = ((z / y) + ((27464.7644705 / (y * y)) + x)) - fma(a, (((z - (a * x)) / (y * y)) + (x / y)), ((b / y) * (x / y)));
	double tmp;
	if (y <= -6.2e+47) {
		tmp = t_1;
	} else if (y <= 1.12e+38) {
		tmp = ((((((((x * y) + z) * y) + 27464.7644705) * y) + 230661.510616) * y) + t) / (fma((fma((a + y), y, b) * y), y, (c * y)) + i);
	} else if (y <= 1.25e+106) {
		tmp = ((fma(x, y, (t / pow(y, 3.0))) + (230661.510616 / (y * y))) + ((27464.7644705 / y) + z)) / a;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(z / y) + Float64(Float64(27464.7644705 / Float64(y * y)) + x)) - fma(a, Float64(Float64(Float64(z - Float64(a * x)) / Float64(y * y)) + Float64(x / y)), Float64(Float64(b / y) * Float64(x / y))))
	tmp = 0.0
	if (y <= -6.2e+47)
		tmp = t_1;
	elseif (y <= 1.12e+38)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(x * y) + z) * y) + 27464.7644705) * y) + 230661.510616) * y) + t) / Float64(fma(Float64(fma(Float64(a + y), y, b) * y), y, Float64(c * y)) + i));
	elseif (y <= 1.25e+106)
		tmp = Float64(Float64(Float64(fma(x, y, Float64(t / (y ^ 3.0))) + Float64(230661.510616 / Float64(y * y))) + Float64(Float64(27464.7644705 / y) + z)) / a);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(z / y), $MachinePrecision] + N[(N[(27464.7644705 / N[(y * y), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision] - N[(a * N[(N[(N[(z - N[(a * x), $MachinePrecision]), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision] + N[(x / y), $MachinePrecision]), $MachinePrecision] + N[(N[(b / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -6.2e+47], t$95$1, If[LessEqual[y, 1.12e+38], N[(N[(N[(N[(N[(N[(N[(N[(N[(x * y), $MachinePrecision] + z), $MachinePrecision] * y), $MachinePrecision] + 27464.7644705), $MachinePrecision] * y), $MachinePrecision] + 230661.510616), $MachinePrecision] * y), $MachinePrecision] + t), $MachinePrecision] / N[(N[(N[(N[(N[(a + y), $MachinePrecision] * y + b), $MachinePrecision] * y), $MachinePrecision] * y + N[(c * y), $MachinePrecision]), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.25e+106], N[(N[(N[(N[(x * y + N[(t / N[Power[y, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(230661.510616 / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(27464.7644705 / y), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.2000000000000001e47 or 1.25e106 < y

   1. Initial program 1.6%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\left(x + \left(\frac{\frac{54929528941}{2000000}}{{y}^{2}} + \frac{z}{y}\right)\right) - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)} \]
   4. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(x + \left(\frac{\frac{54929528941}{2000000}}{{y}^{2}} + \frac{z}{y}\right)\right) - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)} \]

      2. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\left(x + \frac{\frac{54929528941}{2000000}}{{y}^{2}}\right) + \frac{z}{y}\right)} - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right) \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(x + \frac{\frac{54929528941}{2000000}}{{y}^{2}}\right) + \frac{z}{y}\right)} - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right) \]

      4. lower-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(x + \frac{\frac{54929528941}{2000000}}{{y}^{2}}\right)} + \frac{z}{y}\right) - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \left(\left(x + \color{blue}{\frac{\frac{54929528941}{2000000}}{{y}^{2}}}\right) + \frac{z}{y}\right) - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \left(\left(x + \frac{\frac{54929528941}{2000000}}{\color{blue}{y \cdot y}}\right) + \frac{z}{y}\right) - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\left(x + \frac{\frac{54929528941}{2000000}}{\color{blue}{y \cdot y}}\right) + \frac{z}{y}\right) - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \left(\left(x + \frac{\frac{54929528941}{2000000}}{y \cdot y}\right) + \color{blue}{\frac{z}{y}}\right) - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right) \]

      9. associate-+r+ N/A

         \[\leadsto \left(\left(x + \frac{\frac{54929528941}{2000000}}{y \cdot y}\right) + \frac{z}{y}\right) - \color{blue}{\left(\left(\frac{a \cdot x}{y} + \frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}}\right) + \frac{b \cdot x}{{y}^{2}}\right)} \]

      10. associate-/l* N/A

          \[\leadsto \left(\left(x + \frac{\frac{54929528941}{2000000}}{y \cdot y}\right) + \frac{z}{y}\right) - \left(\left(\color{blue}{a \cdot \frac{x}{y}} + \frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}}\right) + \frac{b \cdot x}{{y}^{2}}\right) \]

      11. associate-/l* N/A

          \[\leadsto \left(\left(x + \frac{\frac{54929528941}{2000000}}{y \cdot y}\right) + \frac{z}{y}\right) - \left(\left(a \cdot \frac{x}{y} + \color{blue}{a \cdot \frac{z - a \cdot x}{{y}^{2}}}\right) + \frac{b \cdot x}{{y}^{2}}\right) \]

      12. distribute-lft-out N/A

          \[\leadsto \left(\left(x + \frac{\frac{54929528941}{2000000}}{y \cdot y}\right) + \frac{z}{y}\right) - \left(\color{blue}{a \cdot \left(\frac{x}{y} + \frac{z - a \cdot x}{{y}^{2}}\right)} + \frac{b \cdot x}{{y}^{2}}\right) \]

      13. lower-fma.f64 N/A

          \[\leadsto \left(\left(x + \frac{\frac{54929528941}{2000000}}{y \cdot y}\right) + \frac{z}{y}\right) - \color{blue}{\mathsf{fma}\left(a, \frac{x}{y} + \frac{z - a \cdot x}{{y}^{2}}, \frac{b \cdot x}{{y}^{2}}\right)} \]

   5. Applied rewrites 68.5%

      \[\leadsto \color{blue}{\left(\left(x + \frac{27464.7644705}{y \cdot y}\right) + \frac{z}{y}\right) - \mathsf{fma}\left(a, \frac{x}{y} + \frac{z - a \cdot x}{y \cdot y}, \frac{x}{y} \cdot \frac{b}{y}\right)} \]

   if -6.2000000000000001e47 < y < 1.1199999999999999e38

   1. Initial program 94.6%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}{\color{blue}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y} + i} \]

      2. *-commutative N/A

         \[\leadsto \frac{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}{\color{blue}{y \cdot \left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right)} + i} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}{y \cdot \color{blue}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right)} + i} \]

      4. distribute-rgt-in N/A

         \[\leadsto \frac{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}{\color{blue}{\left(\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y\right) \cdot y + c \cdot y\right)} + i} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}{\color{blue}{\mathsf{fma}\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y, y, c \cdot y\right)} + i} \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}{\mathsf{fma}\left(\color{blue}{\left(\left(y + a\right) \cdot y + b\right)} \cdot y, y, c \cdot y\right) + i} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}{\mathsf{fma}\left(\left(\color{blue}{\left(y + a\right) \cdot y} + b\right) \cdot y, y, c \cdot y\right) + i} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(y + a, y, b\right)} \cdot y, y, c \cdot y\right) + i} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{y + a}, y, b\right) \cdot y, y, c \cdot y\right) + i} \]

      10. +-commutative N/A

          \[\leadsto \frac{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{a + y}, y, b\right) \cdot y, y, c \cdot y\right) + i} \]

      11. lower-+.f64 N/A

          \[\leadsto \frac{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{a + y}, y, b\right) \cdot y, y, c \cdot y\right) + i} \]

      12. lower-*.f64 94.6

          \[\leadsto \frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right) \cdot y, y, \color{blue}{c \cdot y}\right) + i} \]

   4. Applied rewrites 94.6%

      \[\leadsto \frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right) \cdot y, y, c \cdot y\right)} + i} \]

   if 1.1199999999999999e38 < y < 1.25e106

   1. Initial program 9.2%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{t}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)} + \frac{y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)} + \frac{t}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{y \cdot \frac{\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}} + \frac{t}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}, \frac{t}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}\right)} \]

   5. Applied rewrites 17.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, y, z\right), y, 27464.7644705\right), y, 230661.510616\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}, \frac{t}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}\right)} \]

   6. Taylor expanded in a around inf

      \[\leadsto \frac{z + \left(\frac{54929528941}{2000000} \cdot \frac{1}{y} + \left(\frac{28832688827}{125000} \cdot \frac{1}{{y}^{2}} + \left(x \cdot y + \frac{t}{{y}^{3}}\right)\right)\right)}{\color{blue}{a}} \]
   7. Step-by-step derivation

   8. Applied rewrites 68.3%

      \[\leadsto \frac{\left(z + \frac{27464.7644705}{y}\right) + \left(\frac{230661.510616}{y \cdot y} + \mathsf{fma}\left(x, y, \frac{t}{{y}^{3}}\right)\right)}{\color{blue}{a}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.2 \cdot 10^{+47}:\\ \;\;\;\;\left(\frac{z}{y} + \left(\frac{27464.7644705}{y \cdot y} + x\right)\right) - \mathsf{fma}\left(a, \frac{z - a \cdot x}{y \cdot y} + \frac{x}{y}, \frac{b}{y} \cdot \frac{x}{y}\right)\\ \mathbf{elif}\;y \leq 1.12 \cdot 10^{+38}:\\ \;\;\;\;\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right) \cdot y, y, c \cdot y\right) + i}\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{+106}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(x, y, \frac{t}{{y}^{3}}\right) + \frac{230661.510616}{y \cdot y}\right) + \left(\frac{27464.7644705}{y} + z\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z}{y} + \left(\frac{27464.7644705}{y \cdot y} + x\right)\right) - \mathsf{fma}\left(a, \frac{z - a \cdot x}{y \cdot y} + \frac{x}{y}, \frac{b}{y} \cdot \frac{x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 82.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)\\ \mathbf{if}\;\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(a + y\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, y, z\right), y, 27464.7644705\right), y, 230661.510616\right)}{t\_1}, \frac{t}{t\_1}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z - a \cdot x}{y} + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma (fma (fma (+ a y) y b) y c) y i)))
   (if (<=
        (/
         (+
          (* (+ (* (+ (* (+ (* x y) z) y) 27464.7644705) y) 230661.510616) y)
          t)
         (+ (* (+ (* (+ (* (+ a y) y) b) y) c) y) i))
        INFINITY)
     (fma
      y
      (/ (fma (fma (fma x y z) y 27464.7644705) y 230661.510616) t_1)
      (/ t t_1))
     (+ (/ (- z (* a x)) y) x))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma(fma(fma((a + y), y, b), y, c), y, i);
	double tmp;
	if ((((((((((x * y) + z) * y) + 27464.7644705) * y) + 230661.510616) * y) + t) / (((((((a + y) * y) + b) * y) + c) * y) + i)) <= ((double) INFINITY)) {
		tmp = fma(y, (fma(fma(fma(x, y, z), y, 27464.7644705), y, 230661.510616) / t_1), (t / t_1));
	} else {
		tmp = ((z - (a * x)) / y) + x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(fma(fma(Float64(a + y), y, b), y, c), y, i)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(x * y) + z) * y) + 27464.7644705) * y) + 230661.510616) * y) + t) / Float64(Float64(Float64(Float64(Float64(Float64(Float64(a + y) * y) + b) * y) + c) * y) + i)) <= Inf)
		tmp = fma(y, Float64(fma(fma(fma(x, y, z), y, 27464.7644705), y, 230661.510616) / t_1), Float64(t / t_1));
	else
		tmp = Float64(Float64(Float64(z - Float64(a * x)) / y) + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(N[(a + y), $MachinePrecision] * y + b), $MachinePrecision] * y + c), $MachinePrecision] * y + i), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(N[(N[(N[(N[(N[(x * y), $MachinePrecision] + z), $MachinePrecision] * y), $MachinePrecision] + 27464.7644705), $MachinePrecision] * y), $MachinePrecision] + 230661.510616), $MachinePrecision] * y), $MachinePrecision] + t), $MachinePrecision] / N[(N[(N[(N[(N[(N[(N[(a + y), $MachinePrecision] * y), $MachinePrecision] + b), $MachinePrecision] * y), $MachinePrecision] + c), $MachinePrecision] * y), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision], Infinity], N[(y * N[(N[(N[(N[(x * y + z), $MachinePrecision] * y + 27464.7644705), $MachinePrecision] * y + 230661.510616), $MachinePrecision] / t$95$1), $MachinePrecision] + N[(t / t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(N[(z - N[(a * x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 x y) z) y) #s(literal 54929528941/2000000 binary64)) y) #s(literal 28832688827/125000 binary64)) y) t) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 y a) y) b) y) c) y) i)) < +inf.0

   1. Initial program 90.4%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{t}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)} + \frac{y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)} + \frac{t}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{y \cdot \frac{\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}} + \frac{t}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}, \frac{t}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}\right)} \]

   5. Applied rewrites 90.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, y, z\right), y, 27464.7644705\right), y, 230661.510616\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}, \frac{t}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}\right)} \]

   if +inf.0 < (/.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 x y) z) y) #s(literal 54929528941/2000000 binary64)) y) #s(literal 28832688827/125000 binary64)) y) t) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 y a) y) b) y) c) y) i))

   1. Initial program 0.0%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{t}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)} + \frac{y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)} + \frac{t}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{y \cdot \frac{\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}} + \frac{t}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}, \frac{t}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}\right)} \]

   5. Applied rewrites 0.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, y, z\right), y, 27464.7644705\right), y, 230661.510616\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}, \frac{t}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}\right)} \]

   6. Taylor expanded in y around inf

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

      1. associate--l+ N/A

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

      2. div-sub N/A

         \[\leadsto x + \color{blue}{\frac{z - a \cdot x}{y}} \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{z - a \cdot x}{y}} \]

      4. lower-/.f64 N/A

         \[\leadsto x + \color{blue}{\frac{z - a \cdot x}{y}} \]

      5. lower--.f64 N/A

         \[\leadsto x + \frac{\color{blue}{z - a \cdot x}}{y} \]

      6. lower-*.f64 61.4

         \[\leadsto x + \frac{z - \color{blue}{a \cdot x}}{y} \]

   8. Applied rewrites 61.4%

      \[\leadsto \color{blue}{x + \frac{z - a \cdot x}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(a + y\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, y, z\right), y, 27464.7644705\right), y, 230661.510616\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}, \frac{t}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z - a \cdot x}{y} + x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(a + y\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}\\ \mathbf{if}\;t\_1 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{z - a \cdot x}{y} + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1
         (/
          (+
           (* (+ (* (+ (* (+ (* x y) z) y) 27464.7644705) y) 230661.510616) y)
           t)
          (+ (* (+ (* (+ (* (+ a y) y) b) y) c) y) i))))
   (if (<= t_1 INFINITY) t_1 (+ (/ (- z (* a x)) y) x))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = ((((((((x * y) + z) * y) + 27464.7644705) * y) + 230661.510616) * y) + t) / (((((((a + y) * y) + b) * y) + c) * y) + i);
	double tmp;
	if (t_1 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = ((z - (a * x)) / y) + x;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = ((((((((x * y) + z) * y) + 27464.7644705) * y) + 230661.510616) * y) + t) / (((((((a + y) * y) + b) * y) + c) * y) + i);
	double tmp;
	if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else {
		tmp = ((z - (a * x)) / y) + x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	t_1 = ((((((((x * y) + z) * y) + 27464.7644705) * y) + 230661.510616) * y) + t) / (((((((a + y) * y) + b) * y) + c) * y) + i)
	tmp = 0
	if t_1 <= math.inf:
		tmp = t_1
	else:
		tmp = ((z - (a * x)) / y) + x
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(x * y) + z) * y) + 27464.7644705) * y) + 230661.510616) * y) + t) / Float64(Float64(Float64(Float64(Float64(Float64(Float64(a + y) * y) + b) * y) + c) * y) + i))
	tmp = 0.0
	if (t_1 <= Inf)
		tmp = t_1;
	else
		tmp = Float64(Float64(Float64(z - Float64(a * x)) / y) + x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	t_1 = ((((((((x * y) + z) * y) + 27464.7644705) * y) + 230661.510616) * y) + t) / (((((((a + y) * y) + b) * y) + c) * y) + i);
	tmp = 0.0;
	if (t_1 <= Inf)
		tmp = t_1;
	else
		tmp = ((z - (a * x)) / y) + x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(N[(N[(N[(N[(N[(N[(x * y), $MachinePrecision] + z), $MachinePrecision] * y), $MachinePrecision] + 27464.7644705), $MachinePrecision] * y), $MachinePrecision] + 230661.510616), $MachinePrecision] * y), $MachinePrecision] + t), $MachinePrecision] / N[(N[(N[(N[(N[(N[(N[(a + y), $MachinePrecision] * y), $MachinePrecision] + b), $MachinePrecision] * y), $MachinePrecision] + c), $MachinePrecision] * y), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, Infinity], t$95$1, N[(N[(N[(z - N[(a * x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 x y) z) y) #s(literal 54929528941/2000000 binary64)) y) #s(literal 28832688827/125000 binary64)) y) t) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 y a) y) b) y) c) y) i)) < +inf.0

   1. Initial program 90.4%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   2. Add Preprocessing

   if +inf.0 < (/.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 x y) z) y) #s(literal 54929528941/2000000 binary64)) y) #s(literal 28832688827/125000 binary64)) y) t) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 y a) y) b) y) c) y) i))

   1. Initial program 0.0%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{t}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)} + \frac{y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)} + \frac{t}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{y \cdot \frac{\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}} + \frac{t}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}, \frac{t}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}\right)} \]

   5. Applied rewrites 0.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, y, z\right), y, 27464.7644705\right), y, 230661.510616\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}, \frac{t}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}\right)} \]

   6. Taylor expanded in y around inf

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

      1. associate--l+ N/A

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

      2. div-sub N/A

         \[\leadsto x + \color{blue}{\frac{z - a \cdot x}{y}} \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{z - a \cdot x}{y}} \]

      4. lower-/.f64 N/A

         \[\leadsto x + \color{blue}{\frac{z - a \cdot x}{y}} \]

      5. lower--.f64 N/A

         \[\leadsto x + \frac{\color{blue}{z - a \cdot x}}{y} \]

      6. lower-*.f64 61.4

         \[\leadsto x + \frac{z - \color{blue}{a \cdot x}}{y} \]

   8. Applied rewrites 61.4%

      \[\leadsto \color{blue}{x + \frac{z - a \cdot x}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(a + y\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \leq \infty:\\ \;\;\;\;\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(a + y\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}\\ \mathbf{else}:\\ \;\;\;\;\frac{z - a \cdot x}{y} + x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(a + y\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \leq \infty:\\ \;\;\;\;\frac{--1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y, x, z\right), y, 27464.7644705\right), y, 230661.510616\right), y, t\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z - a \cdot x}{y} + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<=
      (/
       (+
        (* (+ (* (+ (* (+ (* x y) z) y) 27464.7644705) y) 230661.510616) y)
        t)
       (+ (* (+ (* (+ (* (+ a y) y) b) y) c) y) i))
      INFINITY)
   (*
    (/ (- -1.0) (fma (fma (fma (+ a y) y b) y c) y i))
    (fma (fma (fma (fma y x z) y 27464.7644705) y 230661.510616) y t))
   (+ (/ (- z (* a x)) y) x)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((((((((((x * y) + z) * y) + 27464.7644705) * y) + 230661.510616) * y) + t) / (((((((a + y) * y) + b) * y) + c) * y) + i)) <= ((double) INFINITY)) {
		tmp = (-(-1.0) / fma(fma(fma((a + y), y, b), y, c), y, i)) * fma(fma(fma(fma(y, x, z), y, 27464.7644705), y, 230661.510616), y, t);
	} else {
		tmp = ((z - (a * x)) / y) + x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(x * y) + z) * y) + 27464.7644705) * y) + 230661.510616) * y) + t) / Float64(Float64(Float64(Float64(Float64(Float64(Float64(a + y) * y) + b) * y) + c) * y) + i)) <= Inf)
		tmp = Float64(Float64(Float64(-(-1.0)) / fma(fma(fma(Float64(a + y), y, b), y, c), y, i)) * fma(fma(fma(fma(y, x, z), y, 27464.7644705), y, 230661.510616), y, t));
	else
		tmp = Float64(Float64(Float64(z - Float64(a * x)) / y) + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(N[(N[(N[(N[(N[(N[(N[(N[(x * y), $MachinePrecision] + z), $MachinePrecision] * y), $MachinePrecision] + 27464.7644705), $MachinePrecision] * y), $MachinePrecision] + 230661.510616), $MachinePrecision] * y), $MachinePrecision] + t), $MachinePrecision] / N[(N[(N[(N[(N[(N[(N[(a + y), $MachinePrecision] * y), $MachinePrecision] + b), $MachinePrecision] * y), $MachinePrecision] + c), $MachinePrecision] * y), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[((--1.0) / N[(N[(N[(N[(a + y), $MachinePrecision] * y + b), $MachinePrecision] * y + c), $MachinePrecision] * y + i), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(y * x + z), $MachinePrecision] * y + 27464.7644705), $MachinePrecision] * y + 230661.510616), $MachinePrecision] * y + t), $MachinePrecision]), $MachinePrecision], N[(N[(N[(z - N[(a * x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 x y) z) y) #s(literal 54929528941/2000000 binary64)) y) #s(literal 28832688827/125000 binary64)) y) t) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 y a) y) b) y) c) y) i)) < +inf.0

   1. Initial program 90.4%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}} \]

      2. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\left(\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t\right)\right)}{\mathsf{neg}\left(\left(\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i\right)\right)}} \]

      3. div-inv N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i\right)\right)}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i\right)\right)}} \]

   4. Applied rewrites 90.1%

      \[\leadsto \color{blue}{\left(-\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y, x, z\right), y, 27464.7644705\right), y, 230661.510616\right), y, t\right)\right) \cdot \frac{-1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}} \]

   if +inf.0 < (/.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 x y) z) y) #s(literal 54929528941/2000000 binary64)) y) #s(literal 28832688827/125000 binary64)) y) t) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 y a) y) b) y) c) y) i))

   1. Initial program 0.0%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{t}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)} + \frac{y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)} + \frac{t}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{y \cdot \frac{\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}} + \frac{t}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}, \frac{t}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}\right)} \]

   5. Applied rewrites 0.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, y, z\right), y, 27464.7644705\right), y, 230661.510616\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}, \frac{t}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}\right)} \]

   6. Taylor expanded in y around inf

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

      1. associate--l+ N/A

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

      2. div-sub N/A

         \[\leadsto x + \color{blue}{\frac{z - a \cdot x}{y}} \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{z - a \cdot x}{y}} \]

      4. lower-/.f64 N/A

         \[\leadsto x + \color{blue}{\frac{z - a \cdot x}{y}} \]

      5. lower--.f64 N/A

         \[\leadsto x + \frac{\color{blue}{z - a \cdot x}}{y} \]

      6. lower-*.f64 61.4

         \[\leadsto x + \frac{z - \color{blue}{a \cdot x}}{y} \]

   8. Applied rewrites 61.4%

      \[\leadsto \color{blue}{x + \frac{z - a \cdot x}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(a + y\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \leq \infty:\\ \;\;\;\;\frac{--1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y, x, z\right), y, 27464.7644705\right), y, 230661.510616\right), y, t\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z - a \cdot x}{y} + x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(a + y\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \leq \infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, y, z\right), y, 27464.7644705\right), y, 230661.510616\right), y, t\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y, y, b\right), y, c\right), y, i\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{z - a \cdot x}{y} + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<=
      (/
       (+
        (* (+ (* (+ (* (+ (* x y) z) y) 27464.7644705) y) 230661.510616) y)
        t)
       (+ (* (+ (* (+ (* (+ a y) y) b) y) c) y) i))
      INFINITY)
   (/
    (fma (fma (fma (fma x y z) y 27464.7644705) y 230661.510616) y t)
    (fma (fma (fma y y b) y c) y i))
   (+ (/ (- z (* a x)) y) x)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((((((((((x * y) + z) * y) + 27464.7644705) * y) + 230661.510616) * y) + t) / (((((((a + y) * y) + b) * y) + c) * y) + i)) <= ((double) INFINITY)) {
		tmp = fma(fma(fma(fma(x, y, z), y, 27464.7644705), y, 230661.510616), y, t) / fma(fma(fma(y, y, b), y, c), y, i);
	} else {
		tmp = ((z - (a * x)) / y) + x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(x * y) + z) * y) + 27464.7644705) * y) + 230661.510616) * y) + t) / Float64(Float64(Float64(Float64(Float64(Float64(Float64(a + y) * y) + b) * y) + c) * y) + i)) <= Inf)
		tmp = Float64(fma(fma(fma(fma(x, y, z), y, 27464.7644705), y, 230661.510616), y, t) / fma(fma(fma(y, y, b), y, c), y, i));
	else
		tmp = Float64(Float64(Float64(z - Float64(a * x)) / y) + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(N[(N[(N[(N[(N[(N[(N[(N[(x * y), $MachinePrecision] + z), $MachinePrecision] * y), $MachinePrecision] + 27464.7644705), $MachinePrecision] * y), $MachinePrecision] + 230661.510616), $MachinePrecision] * y), $MachinePrecision] + t), $MachinePrecision] / N[(N[(N[(N[(N[(N[(N[(a + y), $MachinePrecision] * y), $MachinePrecision] + b), $MachinePrecision] * y), $MachinePrecision] + c), $MachinePrecision] * y), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[(N[(N[(x * y + z), $MachinePrecision] * y + 27464.7644705), $MachinePrecision] * y + 230661.510616), $MachinePrecision] * y + t), $MachinePrecision] / N[(N[(N[(y * y + b), $MachinePrecision] * y + c), $MachinePrecision] * y + i), $MachinePrecision]), $MachinePrecision], N[(N[(N[(z - N[(a * x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 x y) z) y) #s(literal 54929528941/2000000 binary64)) y) #s(literal 28832688827/125000 binary64)) y) t) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 y a) y) b) y) c) y) i)) < +inf.0

   1. Initial program 90.4%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{i + y \cdot \left(c + y \cdot \left(b + {y}^{2}\right)\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{i + y \cdot \left(c + y \cdot \left(b + {y}^{2}\right)\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right) + t}}{i + y \cdot \left(c + y \cdot \left(b + {y}^{2}\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right) \cdot y} + t}{i + y \cdot \left(c + y \cdot \left(b + {y}^{2}\right)\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right), y, t\right)}}{i + y \cdot \left(c + y \cdot \left(b + {y}^{2}\right)\right)} \]

      5. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right) + \frac{28832688827}{125000}}, y, t\right)}{i + y \cdot \left(c + y \cdot \left(b + {y}^{2}\right)\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right) \cdot y} + \frac{28832688827}{125000}, y, t\right)}{i + y \cdot \left(c + y \cdot \left(b + {y}^{2}\right)\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right), y, \frac{28832688827}{125000}\right)}, y, t\right)}{i + y \cdot \left(c + y \cdot \left(b + {y}^{2}\right)\right)} \]

      8. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{y \cdot \left(z + x \cdot y\right) + \frac{54929528941}{2000000}}, y, \frac{28832688827}{125000}\right), y, t\right)}{i + y \cdot \left(c + y \cdot \left(b + {y}^{2}\right)\right)} \]

      9. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\left(z + x \cdot y\right) \cdot y} + \frac{54929528941}{2000000}, y, \frac{28832688827}{125000}\right), y, t\right)}{i + y \cdot \left(c + y \cdot \left(b + {y}^{2}\right)\right)} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(z + x \cdot y, y, \frac{54929528941}{2000000}\right)}, y, \frac{28832688827}{125000}\right), y, t\right)}{i + y \cdot \left(c + y \cdot \left(b + {y}^{2}\right)\right)} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{x \cdot y + z}, y, \frac{54929528941}{2000000}\right), y, \frac{28832688827}{125000}\right), y, t\right)}{i + y \cdot \left(c + y \cdot \left(b + {y}^{2}\right)\right)} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(x, y, z\right)}, y, \frac{54929528941}{2000000}\right), y, \frac{28832688827}{125000}\right), y, t\right)}{i + y \cdot \left(c + y \cdot \left(b + {y}^{2}\right)\right)} \]

      13. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, y, z\right), y, \frac{54929528941}{2000000}\right), y, \frac{28832688827}{125000}\right), y, t\right)}{\color{blue}{y \cdot \left(c + y \cdot \left(b + {y}^{2}\right)\right) + i}} \]

   5. Applied rewrites 85.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, y, z\right), y, 27464.7644705\right), y, 230661.510616\right), y, t\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y, y, b\right), y, c\right), y, i\right)}} \]

   if +inf.0 < (/.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 x y) z) y) #s(literal 54929528941/2000000 binary64)) y) #s(literal 28832688827/125000 binary64)) y) t) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 y a) y) b) y) c) y) i))

   1. Initial program 0.0%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{t}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)} + \frac{y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)} + \frac{t}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{y \cdot \frac{\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}} + \frac{t}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}, \frac{t}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}\right)} \]

   5. Applied rewrites 0.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, y, z\right), y, 27464.7644705\right), y, 230661.510616\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}, \frac{t}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}\right)} \]

   6. Taylor expanded in y around inf

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

      1. associate--l+ N/A

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

      2. div-sub N/A

         \[\leadsto x + \color{blue}{\frac{z - a \cdot x}{y}} \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{z - a \cdot x}{y}} \]

      4. lower-/.f64 N/A

         \[\leadsto x + \color{blue}{\frac{z - a \cdot x}{y}} \]

      5. lower--.f64 N/A

         \[\leadsto x + \frac{\color{blue}{z - a \cdot x}}{y} \]

      6. lower-*.f64 61.4

         \[\leadsto x + \frac{z - \color{blue}{a \cdot x}}{y} \]

   8. Applied rewrites 61.4%

      \[\leadsto \color{blue}{x + \frac{z - a \cdot x}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(a + y\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \leq \infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, y, z\right), y, 27464.7644705\right), y, 230661.510616\right), y, t\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y, y, b\right), y, c\right), y, i\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{z - a \cdot x}{y} + x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\frac{z}{y} + \left(\frac{27464.7644705}{y \cdot y} + x\right)\right) - \mathsf{fma}\left(a, \frac{z - a \cdot x}{y \cdot y} + \frac{x}{y}, \frac{b}{y} \cdot \frac{x}{y}\right)\\ \mathbf{if}\;y \leq -6.2 \cdot 10^{+47}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{+25}:\\ \;\;\;\;\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right) \cdot y, y, c \cdot y\right) + i}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1
         (-
          (+ (/ z y) (+ (/ 27464.7644705 (* y y)) x))
          (fma a (+ (/ (- z (* a x)) (* y y)) (/ x y)) (* (/ b y) (/ x y))))))
   (if (<= y -6.2e+47)
     t_1
     (if (<= y 2.5e+25)
       (/
        (+
         (* (+ (* (+ (* (+ (* x y) z) y) 27464.7644705) y) 230661.510616) y)
         t)
        (+ (fma (* (fma (+ a y) y b) y) y (* c y)) i))
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = ((z / y) + ((27464.7644705 / (y * y)) + x)) - fma(a, (((z - (a * x)) / (y * y)) + (x / y)), ((b / y) * (x / y)));
	double tmp;
	if (y <= -6.2e+47) {
		tmp = t_1;
	} else if (y <= 2.5e+25) {
		tmp = ((((((((x * y) + z) * y) + 27464.7644705) * y) + 230661.510616) * y) + t) / (fma((fma((a + y), y, b) * y), y, (c * y)) + i);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(z / y) + Float64(Float64(27464.7644705 / Float64(y * y)) + x)) - fma(a, Float64(Float64(Float64(z - Float64(a * x)) / Float64(y * y)) + Float64(x / y)), Float64(Float64(b / y) * Float64(x / y))))
	tmp = 0.0
	if (y <= -6.2e+47)
		tmp = t_1;
	elseif (y <= 2.5e+25)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(x * y) + z) * y) + 27464.7644705) * y) + 230661.510616) * y) + t) / Float64(fma(Float64(fma(Float64(a + y), y, b) * y), y, Float64(c * y)) + i));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(z / y), $MachinePrecision] + N[(N[(27464.7644705 / N[(y * y), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision] - N[(a * N[(N[(N[(z - N[(a * x), $MachinePrecision]), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision] + N[(x / y), $MachinePrecision]), $MachinePrecision] + N[(N[(b / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -6.2e+47], t$95$1, If[LessEqual[y, 2.5e+25], N[(N[(N[(N[(N[(N[(N[(N[(N[(x * y), $MachinePrecision] + z), $MachinePrecision] * y), $MachinePrecision] + 27464.7644705), $MachinePrecision] * y), $MachinePrecision] + 230661.510616), $MachinePrecision] * y), $MachinePrecision] + t), $MachinePrecision] / N[(N[(N[(N[(N[(a + y), $MachinePrecision] * y + b), $MachinePrecision] * y), $MachinePrecision] * y + N[(c * y), $MachinePrecision]), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -6.2000000000000001e47 or 2.50000000000000012e25 < y

   1. Initial program 3.4%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\left(x + \left(\frac{\frac{54929528941}{2000000}}{{y}^{2}} + \frac{z}{y}\right)\right) - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)} \]
   4. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(x + \left(\frac{\frac{54929528941}{2000000}}{{y}^{2}} + \frac{z}{y}\right)\right) - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)} \]

      2. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\left(x + \frac{\frac{54929528941}{2000000}}{{y}^{2}}\right) + \frac{z}{y}\right)} - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right) \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(x + \frac{\frac{54929528941}{2000000}}{{y}^{2}}\right) + \frac{z}{y}\right)} - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right) \]

      4. lower-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(x + \frac{\frac{54929528941}{2000000}}{{y}^{2}}\right)} + \frac{z}{y}\right) - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \left(\left(x + \color{blue}{\frac{\frac{54929528941}{2000000}}{{y}^{2}}}\right) + \frac{z}{y}\right) - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \left(\left(x + \frac{\frac{54929528941}{2000000}}{\color{blue}{y \cdot y}}\right) + \frac{z}{y}\right) - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\left(x + \frac{\frac{54929528941}{2000000}}{\color{blue}{y \cdot y}}\right) + \frac{z}{y}\right) - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \left(\left(x + \frac{\frac{54929528941}{2000000}}{y \cdot y}\right) + \color{blue}{\frac{z}{y}}\right) - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right) \]

      9. associate-+r+ N/A

         \[\leadsto \left(\left(x + \frac{\frac{54929528941}{2000000}}{y \cdot y}\right) + \frac{z}{y}\right) - \color{blue}{\left(\left(\frac{a \cdot x}{y} + \frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}}\right) + \frac{b \cdot x}{{y}^{2}}\right)} \]

      10. associate-/l* N/A

          \[\leadsto \left(\left(x + \frac{\frac{54929528941}{2000000}}{y \cdot y}\right) + \frac{z}{y}\right) - \left(\left(\color{blue}{a \cdot \frac{x}{y}} + \frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}}\right) + \frac{b \cdot x}{{y}^{2}}\right) \]

      11. associate-/l* N/A

          \[\leadsto \left(\left(x + \frac{\frac{54929528941}{2000000}}{y \cdot y}\right) + \frac{z}{y}\right) - \left(\left(a \cdot \frac{x}{y} + \color{blue}{a \cdot \frac{z - a \cdot x}{{y}^{2}}}\right) + \frac{b \cdot x}{{y}^{2}}\right) \]

      12. distribute-lft-out N/A

          \[\leadsto \left(\left(x + \frac{\frac{54929528941}{2000000}}{y \cdot y}\right) + \frac{z}{y}\right) - \left(\color{blue}{a \cdot \left(\frac{x}{y} + \frac{z - a \cdot x}{{y}^{2}}\right)} + \frac{b \cdot x}{{y}^{2}}\right) \]

      13. lower-fma.f64 N/A

          \[\leadsto \left(\left(x + \frac{\frac{54929528941}{2000000}}{y \cdot y}\right) + \frac{z}{y}\right) - \color{blue}{\mathsf{fma}\left(a, \frac{x}{y} + \frac{z - a \cdot x}{{y}^{2}}, \frac{b \cdot x}{{y}^{2}}\right)} \]

   5. Applied rewrites 62.6%

      \[\leadsto \color{blue}{\left(\left(x + \frac{27464.7644705}{y \cdot y}\right) + \frac{z}{y}\right) - \mathsf{fma}\left(a, \frac{x}{y} + \frac{z - a \cdot x}{y \cdot y}, \frac{x}{y} \cdot \frac{b}{y}\right)} \]

   if -6.2000000000000001e47 < y < 2.50000000000000012e25

   1. Initial program 95.2%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}{\color{blue}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y} + i} \]

      2. *-commutative N/A

         \[\leadsto \frac{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}{\color{blue}{y \cdot \left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right)} + i} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}{y \cdot \color{blue}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right)} + i} \]

      4. distribute-rgt-in N/A

         \[\leadsto \frac{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}{\color{blue}{\left(\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y\right) \cdot y + c \cdot y\right)} + i} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}{\color{blue}{\mathsf{fma}\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y, y, c \cdot y\right)} + i} \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}{\mathsf{fma}\left(\color{blue}{\left(\left(y + a\right) \cdot y + b\right)} \cdot y, y, c \cdot y\right) + i} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}{\mathsf{fma}\left(\left(\color{blue}{\left(y + a\right) \cdot y} + b\right) \cdot y, y, c \cdot y\right) + i} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(y + a, y, b\right)} \cdot y, y, c \cdot y\right) + i} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{y + a}, y, b\right) \cdot y, y, c \cdot y\right) + i} \]

      10. +-commutative N/A

          \[\leadsto \frac{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{a + y}, y, b\right) \cdot y, y, c \cdot y\right) + i} \]

      11. lower-+.f64 N/A

          \[\leadsto \frac{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{a + y}, y, b\right) \cdot y, y, c \cdot y\right) + i} \]

      12. lower-*.f64 95.2

          \[\leadsto \frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right) \cdot y, y, \color{blue}{c \cdot y}\right) + i} \]

   4. Applied rewrites 95.2%

      \[\leadsto \frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right) \cdot y, y, c \cdot y\right)} + i} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.2 \cdot 10^{+47}:\\ \;\;\;\;\left(\frac{z}{y} + \left(\frac{27464.7644705}{y \cdot y} + x\right)\right) - \mathsf{fma}\left(a, \frac{z - a \cdot x}{y \cdot y} + \frac{x}{y}, \frac{b}{y} \cdot \frac{x}{y}\right)\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{+25}:\\ \;\;\;\;\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right) \cdot y, y, c \cdot y\right) + i}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z}{y} + \left(\frac{27464.7644705}{y \cdot y} + x\right)\right) - \mathsf{fma}\left(a, \frac{z - a \cdot x}{y \cdot y} + \frac{x}{y}, \frac{b}{y} \cdot \frac{x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 73.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - a \cdot x}{y} + x\\ \mathbf{if}\;y \leq -510000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 4100:\\ \;\;\;\;\frac{\mathsf{fma}\left(230661.510616, y, t\right)}{\left(\left(\left(a + y\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}\\ \mathbf{elif}\;y \leq 8 \cdot 10^{+98}:\\ \;\;\;\;\frac{\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616}{\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ (/ (- z (* a x)) y) x)))
   (if (<= y -510000000.0)
     t_1
     (if (<= y 4100.0)
       (/ (fma 230661.510616 y t) (+ (* (+ (* (+ (* (+ a y) y) b) y) c) y) i))
       (if (<= y 8e+98)
         (/
          (+ (* (+ (* (+ (* x y) z) y) 27464.7644705) y) 230661.510616)
          (fma (fma (+ a y) y b) y c))
         t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = ((z - (a * x)) / y) + x;
	double tmp;
	if (y <= -510000000.0) {
		tmp = t_1;
	} else if (y <= 4100.0) {
		tmp = fma(230661.510616, y, t) / (((((((a + y) * y) + b) * y) + c) * y) + i);
	} else if (y <= 8e+98) {
		tmp = ((((((x * y) + z) * y) + 27464.7644705) * y) + 230661.510616) / fma(fma((a + y), y, b), y, c);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(z - Float64(a * x)) / y) + x)
	tmp = 0.0
	if (y <= -510000000.0)
		tmp = t_1;
	elseif (y <= 4100.0)
		tmp = Float64(fma(230661.510616, y, t) / Float64(Float64(Float64(Float64(Float64(Float64(Float64(a + y) * y) + b) * y) + c) * y) + i));
	elseif (y <= 8e+98)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(x * y) + z) * y) + 27464.7644705) * y) + 230661.510616) / fma(fma(Float64(a + y), y, b), y, c));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(z - N[(a * x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[y, -510000000.0], t$95$1, If[LessEqual[y, 4100.0], N[(N[(230661.510616 * y + t), $MachinePrecision] / N[(N[(N[(N[(N[(N[(N[(a + y), $MachinePrecision] * y), $MachinePrecision] + b), $MachinePrecision] * y), $MachinePrecision] + c), $MachinePrecision] * y), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 8e+98], N[(N[(N[(N[(N[(N[(N[(x * y), $MachinePrecision] + z), $MachinePrecision] * y), $MachinePrecision] + 27464.7644705), $MachinePrecision] * y), $MachinePrecision] + 230661.510616), $MachinePrecision] / N[(N[(N[(a + y), $MachinePrecision] * y + b), $MachinePrecision] * y + c), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if y < -5.1e8 or 7.99999999999999998e98 < y

   1. Initial program 6.3%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{t}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)} + \frac{y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)} + \frac{t}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{y \cdot \frac{\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}} + \frac{t}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}, \frac{t}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}\right)} \]

   5. Applied rewrites 6.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, y, z\right), y, 27464.7644705\right), y, 230661.510616\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}, \frac{t}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}\right)} \]

   6. Taylor expanded in y around inf

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

      1. associate--l+ N/A

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

      2. div-sub N/A

         \[\leadsto x + \color{blue}{\frac{z - a \cdot x}{y}} \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{z - a \cdot x}{y}} \]

      4. lower-/.f64 N/A

         \[\leadsto x + \color{blue}{\frac{z - a \cdot x}{y}} \]

      5. lower--.f64 N/A

         \[\leadsto x + \frac{\color{blue}{z - a \cdot x}}{y} \]

      6. lower-*.f64 60.4

         \[\leadsto x + \frac{z - \color{blue}{a \cdot x}}{y} \]

   8. Applied rewrites 60.4%

      \[\leadsto \color{blue}{x + \frac{z - a \cdot x}{y}} \]

   if -5.1e8 < y < 4100

   1. Initial program 99.8%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{t + \frac{28832688827}{125000} \cdot y}}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{28832688827}{125000} \cdot y + t}}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      2. lower-fma.f64 91.7

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(230661.510616, y, t\right)}}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

   5. Applied rewrites 91.7%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(230661.510616, y, t\right)}}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

   if 4100 < y < 7.99999999999999998e98

   1. Initial program 36.1%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   2. Add Preprocessing

   3. Taylor expanded in i around 0

      \[\leadsto \color{blue}{\frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{y}}{c + y \cdot \left(b + y \cdot \left(a + y\right)\right)}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{y}}{c + y \cdot \left(b + y \cdot \left(a + y\right)\right)}} \]

   5. Applied rewrites 40.1%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, y, z\right), y, 27464.7644705\right), y, 230661.510616\right), y, t\right)}{y}}{\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right)}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \frac{\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(a + y, y, b\right)}, y, c\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 49.9%

      \[\leadsto \frac{230661.510616 + y \cdot \left(27464.7644705 + y \cdot \left(z + x \cdot y\right)\right)}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(a + y, y, b\right)}, y, c\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -510000000:\\ \;\;\;\;\frac{z - a \cdot x}{y} + x\\ \mathbf{elif}\;y \leq 4100:\\ \;\;\;\;\frac{\mathsf{fma}\left(230661.510616, y, t\right)}{\left(\left(\left(a + y\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}\\ \mathbf{elif}\;y \leq 8 \cdot 10^{+98}:\\ \;\;\;\;\frac{\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616}{\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{z - a \cdot x}{y} + x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 77.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - a \cdot x}{y} + x\\ \mathbf{if}\;y \leq -1.52 \cdot 10^{+47}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 6000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, y, 27464.7644705\right), y, 230661.510616\right), y, t\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ (/ (- z (* a x)) y) x)))
   (if (<= y -1.52e+47)
     t_1
     (if (<= y 6000000.0)
       (/
        (fma (fma (fma z y 27464.7644705) y 230661.510616) y t)
        (fma (fma (fma (+ a y) y b) y c) y i))
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = ((z - (a * x)) / y) + x;
	double tmp;
	if (y <= -1.52e+47) {
		tmp = t_1;
	} else if (y <= 6000000.0) {
		tmp = fma(fma(fma(z, y, 27464.7644705), y, 230661.510616), y, t) / fma(fma(fma((a + y), y, b), y, c), y, i);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(z - Float64(a * x)) / y) + x)
	tmp = 0.0
	if (y <= -1.52e+47)
		tmp = t_1;
	elseif (y <= 6000000.0)
		tmp = Float64(fma(fma(fma(z, y, 27464.7644705), y, 230661.510616), y, t) / fma(fma(fma(Float64(a + y), y, b), y, c), y, i));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(z - N[(a * x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[y, -1.52e+47], t$95$1, If[LessEqual[y, 6000000.0], N[(N[(N[(N[(z * y + 27464.7644705), $MachinePrecision] * y + 230661.510616), $MachinePrecision] * y + t), $MachinePrecision] / N[(N[(N[(N[(a + y), $MachinePrecision] * y + b), $MachinePrecision] * y + c), $MachinePrecision] * y + i), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.52e47 or 6e6 < y

   1. Initial program 6.8%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{t}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)} + \frac{y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)} + \frac{t}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{y \cdot \frac{\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}} + \frac{t}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}, \frac{t}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}\right)} \]

   5. Applied rewrites 8.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, y, z\right), y, 27464.7644705\right), y, 230661.510616\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}, \frac{t}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}\right)} \]

   6. Taylor expanded in y around inf

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

      1. associate--l+ N/A

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

      2. div-sub N/A

         \[\leadsto x + \color{blue}{\frac{z - a \cdot x}{y}} \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{z - a \cdot x}{y}} \]

      4. lower-/.f64 N/A

         \[\leadsto x + \color{blue}{\frac{z - a \cdot x}{y}} \]

      5. lower--.f64 N/A

         \[\leadsto x + \frac{\color{blue}{z - a \cdot x}}{y} \]

      6. lower-*.f64 59.1

         \[\leadsto x + \frac{z - \color{blue}{a \cdot x}}{y} \]

   8. Applied rewrites 59.1%

      \[\leadsto \color{blue}{x + \frac{z - a \cdot x}{y}} \]

   if -1.52e47 < y < 6e6

   1. Initial program 97.0%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot z\right)\right)}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot z\right)\right)}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot z\right)\right) + t}}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot z\right)\right) \cdot y} + t}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot z\right), y, t\right)}}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)} \]

      5. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{y \cdot \left(\frac{54929528941}{2000000} + y \cdot z\right) + \frac{28832688827}{125000}}, y, t\right)}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(\frac{54929528941}{2000000} + y \cdot z\right) \cdot y} + \frac{28832688827}{125000}, y, t\right)}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{54929528941}{2000000} + y \cdot z, y, \frac{28832688827}{125000}\right)}, y, t\right)}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)} \]

      8. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{y \cdot z + \frac{54929528941}{2000000}}, y, \frac{28832688827}{125000}\right), y, t\right)}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)} \]

      9. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{z \cdot y} + \frac{54929528941}{2000000}, y, \frac{28832688827}{125000}\right), y, t\right)}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(z, y, \frac{54929528941}{2000000}\right)}, y, \frac{28832688827}{125000}\right), y, t\right)}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, y, \frac{54929528941}{2000000}\right), y, \frac{28832688827}{125000}\right), y, t\right)}{\color{blue}{y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right) + i}} \]

      12. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, y, \frac{54929528941}{2000000}\right), y, \frac{28832688827}{125000}\right), y, t\right)}{\color{blue}{\left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right) \cdot y} + i} \]

      13. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, y, \frac{54929528941}{2000000}\right), y, \frac{28832688827}{125000}\right), y, t\right)}{\color{blue}{\mathsf{fma}\left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right), y, i\right)}} \]

   5. Applied rewrites 91.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, y, 27464.7644705\right), y, 230661.510616\right), y, t\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.52 \cdot 10^{+47}:\\ \;\;\;\;\frac{z - a \cdot x}{y} + x\\ \mathbf{elif}\;y \leq 6000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, y, 27464.7644705\right), y, 230661.510616\right), y, t\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{z - a \cdot x}{y} + x\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 72.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - a \cdot x}{y} + x\\ \mathbf{if}\;y \leq -510000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 6000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(230661.510616, y, t\right)}{\left(\left(\left(a + y\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ (/ (- z (* a x)) y) x)))
   (if (<= y -510000000.0)
     t_1
     (if (<= y 6000000.0)
       (/ (fma 230661.510616 y t) (+ (* (+ (* (+ (* (+ a y) y) b) y) c) y) i))
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = ((z - (a * x)) / y) + x;
	double tmp;
	if (y <= -510000000.0) {
		tmp = t_1;
	} else if (y <= 6000000.0) {
		tmp = fma(230661.510616, y, t) / (((((((a + y) * y) + b) * y) + c) * y) + i);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(z - Float64(a * x)) / y) + x)
	tmp = 0.0
	if (y <= -510000000.0)
		tmp = t_1;
	elseif (y <= 6000000.0)
		tmp = Float64(fma(230661.510616, y, t) / Float64(Float64(Float64(Float64(Float64(Float64(Float64(a + y) * y) + b) * y) + c) * y) + i));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(z - N[(a * x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[y, -510000000.0], t$95$1, If[LessEqual[y, 6000000.0], N[(N[(230661.510616 * y + t), $MachinePrecision] / N[(N[(N[(N[(N[(N[(N[(a + y), $MachinePrecision] * y), $MachinePrecision] + b), $MachinePrecision] * y), $MachinePrecision] + c), $MachinePrecision] * y), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -5.1e8 or 6e6 < y

   1. Initial program 10.4%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{t}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)} + \frac{y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)} + \frac{t}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{y \cdot \frac{\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}} + \frac{t}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}, \frac{t}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}\right)} \]

   5. Applied rewrites 12.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, y, z\right), y, 27464.7644705\right), y, 230661.510616\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}, \frac{t}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}\right)} \]

   6. Taylor expanded in y around inf

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

      1. associate--l+ N/A

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

      2. div-sub N/A

         \[\leadsto x + \color{blue}{\frac{z - a \cdot x}{y}} \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{z - a \cdot x}{y}} \]

      4. lower-/.f64 N/A

         \[\leadsto x + \color{blue}{\frac{z - a \cdot x}{y}} \]

      5. lower--.f64 N/A

         \[\leadsto x + \frac{\color{blue}{z - a \cdot x}}{y} \]

      6. lower-*.f64 55.2

         \[\leadsto x + \frac{z - \color{blue}{a \cdot x}}{y} \]

   8. Applied rewrites 55.2%

      \[\leadsto \color{blue}{x + \frac{z - a \cdot x}{y}} \]

   if -5.1e8 < y < 6e6

   1. Initial program 99.8%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{t + \frac{28832688827}{125000} \cdot y}}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{28832688827}{125000} \cdot y + t}}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      2. lower-fma.f64 91.1

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(230661.510616, y, t\right)}}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

   5. Applied rewrites 91.1%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(230661.510616, y, t\right)}}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -510000000:\\ \;\;\;\;\frac{z - a \cdot x}{y} + x\\ \mathbf{elif}\;y \leq 6000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(230661.510616, y, t\right)}{\left(\left(\left(a + y\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}\\ \mathbf{else}:\\ \;\;\;\;\frac{z - a \cdot x}{y} + x\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 65.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - a \cdot x}{y} + x\\ \mathbf{if}\;y \leq -1.5 \cdot 10^{+19}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -2 \cdot 10^{-39}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, y, z\right), y, 27464.7644705\right), y, 230661.510616\right), y, t\right)}{i}\\ \mathbf{elif}\;y \leq 6000000:\\ \;\;\;\;\frac{t}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ (/ (- z (* a x)) y) x)))
   (if (<= y -1.5e+19)
     t_1
     (if (<= y -2e-39)
       (/ (fma (fma (fma (fma x y z) y 27464.7644705) y 230661.510616) y t) i)
       (if (<= y 6000000.0)
         (/ t (fma (fma (fma (+ a y) y b) y c) y i))
         t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = ((z - (a * x)) / y) + x;
	double tmp;
	if (y <= -1.5e+19) {
		tmp = t_1;
	} else if (y <= -2e-39) {
		tmp = fma(fma(fma(fma(x, y, z), y, 27464.7644705), y, 230661.510616), y, t) / i;
	} else if (y <= 6000000.0) {
		tmp = t / fma(fma(fma((a + y), y, b), y, c), y, i);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(z - Float64(a * x)) / y) + x)
	tmp = 0.0
	if (y <= -1.5e+19)
		tmp = t_1;
	elseif (y <= -2e-39)
		tmp = Float64(fma(fma(fma(fma(x, y, z), y, 27464.7644705), y, 230661.510616), y, t) / i);
	elseif (y <= 6000000.0)
		tmp = Float64(t / fma(fma(fma(Float64(a + y), y, b), y, c), y, i));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(z - N[(a * x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[y, -1.5e+19], t$95$1, If[LessEqual[y, -2e-39], N[(N[(N[(N[(N[(x * y + z), $MachinePrecision] * y + 27464.7644705), $MachinePrecision] * y + 230661.510616), $MachinePrecision] * y + t), $MachinePrecision] / i), $MachinePrecision], If[LessEqual[y, 6000000.0], N[(t / N[(N[(N[(N[(a + y), $MachinePrecision] * y + b), $MachinePrecision] * y + c), $MachinePrecision] * y + i), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.5e19 or 6e6 < y

   1. Initial program 8.9%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{t}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)} + \frac{y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)} + \frac{t}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{y \cdot \frac{\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}} + \frac{t}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}, \frac{t}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}\right)} \]

   5. Applied rewrites 10.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, y, z\right), y, 27464.7644705\right), y, 230661.510616\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}, \frac{t}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}\right)} \]

   6. Taylor expanded in y around inf

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

      1. associate--l+ N/A

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

      2. div-sub N/A

         \[\leadsto x + \color{blue}{\frac{z - a \cdot x}{y}} \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{z - a \cdot x}{y}} \]

      4. lower-/.f64 N/A

         \[\leadsto x + \color{blue}{\frac{z - a \cdot x}{y}} \]

      5. lower--.f64 N/A

         \[\leadsto x + \frac{\color{blue}{z - a \cdot x}}{y} \]

      6. lower-*.f64 56.0

         \[\leadsto x + \frac{z - \color{blue}{a \cdot x}}{y} \]

   8. Applied rewrites 56.0%

      \[\leadsto \color{blue}{x + \frac{z - a \cdot x}{y}} \]

   if -1.5e19 < y < -1.99999999999999986e-39

   1. Initial program 99.6%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   2. Add Preprocessing

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{i}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{i}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right) + t}}{i} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right) \cdot y} + t}{i} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right), y, t\right)}}{i} \]

      5. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right) + \frac{28832688827}{125000}}, y, t\right)}{i} \]

      6. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right) \cdot y} + \frac{28832688827}{125000}, y, t\right)}{i} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right), y, \frac{28832688827}{125000}\right)}, y, t\right)}{i} \]

      8. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{y \cdot \left(z + x \cdot y\right) + \frac{54929528941}{2000000}}, y, \frac{28832688827}{125000}\right), y, t\right)}{i} \]

      9. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\left(z + x \cdot y\right) \cdot y} + \frac{54929528941}{2000000}, y, \frac{28832688827}{125000}\right), y, t\right)}{i} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(z + x \cdot y, y, \frac{54929528941}{2000000}\right)}, y, \frac{28832688827}{125000}\right), y, t\right)}{i} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{x \cdot y + z}, y, \frac{54929528941}{2000000}\right), y, \frac{28832688827}{125000}\right), y, t\right)}{i} \]

      12. lower-fma.f64 58.3

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(x, y, z\right)}, y, 27464.7644705\right), y, 230661.510616\right), y, t\right)}{i} \]

   5. Applied rewrites 58.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, y, z\right), y, 27464.7644705\right), y, 230661.510616\right), y, t\right)}{i}} \]

   if -1.99999999999999986e-39 < y < 6e6

   1. Initial program 99.8%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{t}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{t}{\color{blue}{y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right) + i}} \]

      3. *-commutative N/A

         \[\leadsto \frac{t}{\color{blue}{\left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right) \cdot y} + i} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{t}{\color{blue}{\mathsf{fma}\left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right), y, i\right)}} \]

      5. +-commutative N/A

         \[\leadsto \frac{t}{\mathsf{fma}\left(\color{blue}{y \cdot \left(b + y \cdot \left(a + y\right)\right) + c}, y, i\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{t}{\mathsf{fma}\left(\color{blue}{\left(b + y \cdot \left(a + y\right)\right) \cdot y} + c, y, i\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{t}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(b + y \cdot \left(a + y\right), y, c\right)}, y, i\right)} \]

      8. +-commutative N/A

         \[\leadsto \frac{t}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{y \cdot \left(a + y\right) + b}, y, c\right), y, i\right)} \]

      9. *-commutative N/A

         \[\leadsto \frac{t}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\left(a + y\right) \cdot y} + b, y, c\right), y, i\right)} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{t}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(a + y, y, b\right)}, y, c\right), y, i\right)} \]

      11. lower-+.f64 85.3

          \[\leadsto \frac{t}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{a + y}, y, b\right), y, c\right), y, i\right)} \]

   5. Applied rewrites 85.3%

      \[\leadsto \color{blue}{\frac{t}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.5 \cdot 10^{+19}:\\ \;\;\;\;\frac{z - a \cdot x}{y} + x\\ \mathbf{elif}\;y \leq -2 \cdot 10^{-39}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, y, z\right), y, 27464.7644705\right), y, 230661.510616\right), y, t\right)}{i}\\ \mathbf{elif}\;y \leq 6000000:\\ \;\;\;\;\frac{t}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{z - a \cdot x}{y} + x\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 65.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - a \cdot x}{y} + x\\ \mathbf{if}\;y \leq -510000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 6000000:\\ \;\;\;\;\frac{t}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ (/ (- z (* a x)) y) x)))
   (if (<= y -510000000.0)
     t_1
     (if (<= y 6000000.0) (/ t (fma (fma (fma (+ a y) y b) y c) y i)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = ((z - (a * x)) / y) + x;
	double tmp;
	if (y <= -510000000.0) {
		tmp = t_1;
	} else if (y <= 6000000.0) {
		tmp = t / fma(fma(fma((a + y), y, b), y, c), y, i);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(z - Float64(a * x)) / y) + x)
	tmp = 0.0
	if (y <= -510000000.0)
		tmp = t_1;
	elseif (y <= 6000000.0)
		tmp = Float64(t / fma(fma(fma(Float64(a + y), y, b), y, c), y, i));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(z - N[(a * x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[y, -510000000.0], t$95$1, If[LessEqual[y, 6000000.0], N[(t / N[(N[(N[(N[(a + y), $MachinePrecision] * y + b), $MachinePrecision] * y + c), $MachinePrecision] * y + i), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -5.1e8 or 6e6 < y

   1. Initial program 10.4%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{t}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)} + \frac{y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)} + \frac{t}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{y \cdot \frac{\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}} + \frac{t}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}, \frac{t}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}\right)} \]

   5. Applied rewrites 12.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, y, z\right), y, 27464.7644705\right), y, 230661.510616\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}, \frac{t}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}\right)} \]

   6. Taylor expanded in y around inf

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

      1. associate--l+ N/A

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

      2. div-sub N/A

         \[\leadsto x + \color{blue}{\frac{z - a \cdot x}{y}} \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{z - a \cdot x}{y}} \]

      4. lower-/.f64 N/A

         \[\leadsto x + \color{blue}{\frac{z - a \cdot x}{y}} \]

      5. lower--.f64 N/A

         \[\leadsto x + \frac{\color{blue}{z - a \cdot x}}{y} \]

      6. lower-*.f64 55.2

         \[\leadsto x + \frac{z - \color{blue}{a \cdot x}}{y} \]

   8. Applied rewrites 55.2%

      \[\leadsto \color{blue}{x + \frac{z - a \cdot x}{y}} \]

   if -5.1e8 < y < 6e6

   1. Initial program 99.8%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{t}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{t}{\color{blue}{y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right) + i}} \]

      3. *-commutative N/A

         \[\leadsto \frac{t}{\color{blue}{\left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right) \cdot y} + i} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{t}{\color{blue}{\mathsf{fma}\left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right), y, i\right)}} \]

      5. +-commutative N/A

         \[\leadsto \frac{t}{\mathsf{fma}\left(\color{blue}{y \cdot \left(b + y \cdot \left(a + y\right)\right) + c}, y, i\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{t}{\mathsf{fma}\left(\color{blue}{\left(b + y \cdot \left(a + y\right)\right) \cdot y} + c, y, i\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{t}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(b + y \cdot \left(a + y\right), y, c\right)}, y, i\right)} \]

      8. +-commutative N/A

         \[\leadsto \frac{t}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{y \cdot \left(a + y\right) + b}, y, c\right), y, i\right)} \]

      9. *-commutative N/A

         \[\leadsto \frac{t}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\left(a + y\right) \cdot y} + b, y, c\right), y, i\right)} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{t}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(a + y, y, b\right)}, y, c\right), y, i\right)} \]

      11. lower-+.f64 80.0

          \[\leadsto \frac{t}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{a + y}, y, b\right), y, c\right), y, i\right)} \]

   5. Applied rewrites 80.0%

      \[\leadsto \color{blue}{\frac{t}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -510000000:\\ \;\;\;\;\frac{z - a \cdot x}{y} + x\\ \mathbf{elif}\;y \leq 6000000:\\ \;\;\;\;\frac{t}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{z - a \cdot x}{y} + x\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 64.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - a \cdot x}{y} + x\\ \mathbf{if}\;y \leq -510000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 6000000:\\ \;\;\;\;\frac{t}{\mathsf{fma}\left(b \cdot y + c, y, i\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ (/ (- z (* a x)) y) x)))
   (if (<= y -510000000.0)
     t_1
     (if (<= y 6000000.0) (/ t (fma (+ (* b y) c) y i)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = ((z - (a * x)) / y) + x;
	double tmp;
	if (y <= -510000000.0) {
		tmp = t_1;
	} else if (y <= 6000000.0) {
		tmp = t / fma(((b * y) + c), y, i);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(z - Float64(a * x)) / y) + x)
	tmp = 0.0
	if (y <= -510000000.0)
		tmp = t_1;
	elseif (y <= 6000000.0)
		tmp = Float64(t / fma(Float64(Float64(b * y) + c), y, i));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(z - N[(a * x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[y, -510000000.0], t$95$1, If[LessEqual[y, 6000000.0], N[(t / N[(N[(N[(b * y), $MachinePrecision] + c), $MachinePrecision] * y + i), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -5.1e8 or 6e6 < y

   1. Initial program 10.4%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{t}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)} + \frac{y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)} + \frac{t}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{y \cdot \frac{\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}} + \frac{t}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}, \frac{t}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}\right)} \]

   5. Applied rewrites 12.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, y, z\right), y, 27464.7644705\right), y, 230661.510616\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}, \frac{t}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}\right)} \]

   6. Taylor expanded in y around inf

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

      1. associate--l+ N/A

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

      2. div-sub N/A

         \[\leadsto x + \color{blue}{\frac{z - a \cdot x}{y}} \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{z - a \cdot x}{y}} \]

      4. lower-/.f64 N/A

         \[\leadsto x + \color{blue}{\frac{z - a \cdot x}{y}} \]

      5. lower--.f64 N/A

         \[\leadsto x + \frac{\color{blue}{z - a \cdot x}}{y} \]

      6. lower-*.f64 55.2

         \[\leadsto x + \frac{z - \color{blue}{a \cdot x}}{y} \]

   8. Applied rewrites 55.2%

      \[\leadsto \color{blue}{x + \frac{z - a \cdot x}{y}} \]

   if -5.1e8 < y < 6e6

   1. Initial program 99.8%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{t}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{t}{\color{blue}{y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right) + i}} \]

      3. *-commutative N/A

         \[\leadsto \frac{t}{\color{blue}{\left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right) \cdot y} + i} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{t}{\color{blue}{\mathsf{fma}\left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right), y, i\right)}} \]

      5. +-commutative N/A

         \[\leadsto \frac{t}{\mathsf{fma}\left(\color{blue}{y \cdot \left(b + y \cdot \left(a + y\right)\right) + c}, y, i\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{t}{\mathsf{fma}\left(\color{blue}{\left(b + y \cdot \left(a + y\right)\right) \cdot y} + c, y, i\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{t}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(b + y \cdot \left(a + y\right), y, c\right)}, y, i\right)} \]

      8. +-commutative N/A

         \[\leadsto \frac{t}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{y \cdot \left(a + y\right) + b}, y, c\right), y, i\right)} \]

      9. *-commutative N/A

         \[\leadsto \frac{t}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\left(a + y\right) \cdot y} + b, y, c\right), y, i\right)} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{t}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(a + y, y, b\right)}, y, c\right), y, i\right)} \]

      11. lower-+.f64 80.0

          \[\leadsto \frac{t}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{a + y}, y, b\right), y, c\right), y, i\right)} \]

   5. Applied rewrites 80.0%

      \[\leadsto \color{blue}{\frac{t}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}} \]

   6. Taylor expanded in y around 0

      \[\leadsto \frac{t}{\mathsf{fma}\left(c + b \cdot y, y, i\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 78.6%

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

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -510000000:\\ \;\;\;\;\frac{z - a \cdot x}{y} + x\\ \mathbf{elif}\;y \leq 6000000:\\ \;\;\;\;\frac{t}{\mathsf{fma}\left(b \cdot y + c, y, i\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{z - a \cdot x}{y} + x\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 62.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - a \cdot x}{y} + x\\ \mathbf{if}\;y \leq -1000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 26000:\\ \;\;\;\;\frac{t}{c \cdot y + i}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ (/ (- z (* a x)) y) x)))
   (if (<= y -1000000.0) t_1 (if (<= y 26000.0) (/ t (+ (* c y) i)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = ((z - (a * x)) / y) + x;
	double tmp;
	if (y <= -1000000.0) {
		tmp = t_1;
	} else if (y <= 26000.0) {
		tmp = t / ((c * y) + i);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i)
    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), intent (in) :: i
    real(8) :: t_1
    real(8) :: tmp
    t_1 = ((z - (a * x)) / y) + x
    if (y <= (-1000000.0d0)) then
        tmp = t_1
    else if (y <= 26000.0d0) then
        tmp = t / ((c * y) + i)
    else
        tmp = t_1
    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 i) {
	double t_1 = ((z - (a * x)) / y) + x;
	double tmp;
	if (y <= -1000000.0) {
		tmp = t_1;
	} else if (y <= 26000.0) {
		tmp = t / ((c * y) + i);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	t_1 = ((z - (a * x)) / y) + x
	tmp = 0
	if y <= -1000000.0:
		tmp = t_1
	elif y <= 26000.0:
		tmp = t / ((c * y) + i)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(z - Float64(a * x)) / y) + x)
	tmp = 0.0
	if (y <= -1000000.0)
		tmp = t_1;
	elseif (y <= 26000.0)
		tmp = Float64(t / Float64(Float64(c * y) + i));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	t_1 = ((z - (a * x)) / y) + x;
	tmp = 0.0;
	if (y <= -1000000.0)
		tmp = t_1;
	elseif (y <= 26000.0)
		tmp = t / ((c * y) + i);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(z - N[(a * x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[y, -1000000.0], t$95$1, If[LessEqual[y, 26000.0], N[(t / N[(N[(c * y), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -1e6 or 26000 < y

   1. Initial program 10.4%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{t}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)} + \frac{y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)} + \frac{t}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{y \cdot \frac{\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}} + \frac{t}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}, \frac{t}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}\right)} \]

   5. Applied rewrites 12.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, y, z\right), y, 27464.7644705\right), y, 230661.510616\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}, \frac{t}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}\right)} \]

   6. Taylor expanded in y around inf

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

      1. associate--l+ N/A

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

      2. div-sub N/A

         \[\leadsto x + \color{blue}{\frac{z - a \cdot x}{y}} \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{z - a \cdot x}{y}} \]

      4. lower-/.f64 N/A

         \[\leadsto x + \color{blue}{\frac{z - a \cdot x}{y}} \]

      5. lower--.f64 N/A

         \[\leadsto x + \frac{\color{blue}{z - a \cdot x}}{y} \]

      6. lower-*.f64 55.2

         \[\leadsto x + \frac{z - \color{blue}{a \cdot x}}{y} \]

   8. Applied rewrites 55.2%

      \[\leadsto \color{blue}{x + \frac{z - a \cdot x}{y}} \]

   if -1e6 < y < 26000

   1. Initial program 99.8%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{t}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{t}{\color{blue}{y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right) + i}} \]

      3. *-commutative N/A

         \[\leadsto \frac{t}{\color{blue}{\left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right) \cdot y} + i} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{t}{\color{blue}{\mathsf{fma}\left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right), y, i\right)}} \]

      5. +-commutative N/A

         \[\leadsto \frac{t}{\mathsf{fma}\left(\color{blue}{y \cdot \left(b + y \cdot \left(a + y\right)\right) + c}, y, i\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{t}{\mathsf{fma}\left(\color{blue}{\left(b + y \cdot \left(a + y\right)\right) \cdot y} + c, y, i\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{t}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(b + y \cdot \left(a + y\right), y, c\right)}, y, i\right)} \]

      8. +-commutative N/A

         \[\leadsto \frac{t}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{y \cdot \left(a + y\right) + b}, y, c\right), y, i\right)} \]

      9. *-commutative N/A

         \[\leadsto \frac{t}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\left(a + y\right) \cdot y} + b, y, c\right), y, i\right)} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{t}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(a + y, y, b\right)}, y, c\right), y, i\right)} \]

      11. lower-+.f64 80.0

          \[\leadsto \frac{t}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{a + y}, y, b\right), y, c\right), y, i\right)} \]

   5. Applied rewrites 80.0%

      \[\leadsto \color{blue}{\frac{t}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}} \]

   6. Taylor expanded in y around 0

      \[\leadsto \frac{t}{i + \color{blue}{c \cdot y}} \]
   7. Step-by-step derivation

   8. Applied rewrites 72.2%

      \[\leadsto \frac{t}{i + \color{blue}{c \cdot y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1000000:\\ \;\;\;\;\frac{z - a \cdot x}{y} + x\\ \mathbf{elif}\;y \leq 26000:\\ \;\;\;\;\frac{t}{c \cdot y + i}\\ \mathbf{else}:\\ \;\;\;\;\frac{z - a \cdot x}{y} + x\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 39.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.65 \cdot 10^{+22}:\\ \;\;\;\;\frac{t}{c \cdot y + i}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= y 1.65e+22) (/ t (+ (* c y) i)) (/ z y)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (y <= 1.65e+22) {
		tmp = t / ((c * y) + i);
	} else {
		tmp = z / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i)
    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), intent (in) :: i
    real(8) :: tmp
    if (y <= 1.65d+22) then
        tmp = t / ((c * y) + i)
    else
        tmp = z / y
    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 i) {
	double tmp;
	if (y <= 1.65e+22) {
		tmp = t / ((c * y) + i);
	} else {
		tmp = z / y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if y <= 1.65e+22:
		tmp = t / ((c * y) + i)
	else:
		tmp = z / y
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (y <= 1.65e+22)
		tmp = Float64(t / Float64(Float64(c * y) + i));
	else
		tmp = Float64(z / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	tmp = 0.0;
	if (y <= 1.65e+22)
		tmp = t / ((c * y) + i);
	else
		tmp = z / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[y, 1.65e+22], N[(t / N[(N[(c * y), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision], N[(z / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.6499999999999999e22

   1. Initial program 74.7%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{t}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{t}{\color{blue}{y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right) + i}} \]

      3. *-commutative N/A

         \[\leadsto \frac{t}{\color{blue}{\left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right) \cdot y} + i} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{t}{\color{blue}{\mathsf{fma}\left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right), y, i\right)}} \]

      5. +-commutative N/A

         \[\leadsto \frac{t}{\mathsf{fma}\left(\color{blue}{y \cdot \left(b + y \cdot \left(a + y\right)\right) + c}, y, i\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{t}{\mathsf{fma}\left(\color{blue}{\left(b + y \cdot \left(a + y\right)\right) \cdot y} + c, y, i\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{t}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(b + y \cdot \left(a + y\right), y, c\right)}, y, i\right)} \]

      8. +-commutative N/A

         \[\leadsto \frac{t}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{y \cdot \left(a + y\right) + b}, y, c\right), y, i\right)} \]

      9. *-commutative N/A

         \[\leadsto \frac{t}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\left(a + y\right) \cdot y} + b, y, c\right), y, i\right)} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{t}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(a + y, y, b\right)}, y, c\right), y, i\right)} \]

      11. lower-+.f64 57.2

          \[\leadsto \frac{t}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{a + y}, y, b\right), y, c\right), y, i\right)} \]

   5. Applied rewrites 57.2%

      \[\leadsto \color{blue}{\frac{t}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}} \]

   6. Taylor expanded in y around 0

      \[\leadsto \frac{t}{i + \color{blue}{c \cdot y}} \]
   7. Step-by-step derivation

   8. Applied rewrites 51.8%

      \[\leadsto \frac{t}{i + \color{blue}{c \cdot y}} \]

   if 1.6499999999999999e22 < y

   1. Initial program 6.4%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

         \[\leadsto {y}^{3} \cdot \frac{z}{\color{blue}{\mathsf{fma}\left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right), y, i\right)}} \]

      8. +-commutative N/A

         \[\leadsto {y}^{3} \cdot \frac{z}{\mathsf{fma}\left(\color{blue}{y \cdot \left(b + y \cdot \left(a + y\right)\right) + c}, y, i\right)} \]

      9. *-commutative N/A

         \[\leadsto {y}^{3} \cdot \frac{z}{\mathsf{fma}\left(\color{blue}{\left(b + y \cdot \left(a + y\right)\right) \cdot y} + c, y, i\right)} \]

      10. lower-fma.f64 N/A

          \[\leadsto {y}^{3} \cdot \frac{z}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(b + y \cdot \left(a + y\right), y, c\right)}, y, i\right)} \]

      11. +-commutative N/A

          \[\leadsto {y}^{3} \cdot \frac{z}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{y \cdot \left(a + y\right) + b}, y, c\right), y, i\right)} \]

      12. *-commutative N/A

          \[\leadsto {y}^{3} \cdot \frac{z}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\left(a + y\right) \cdot y} + b, y, c\right), y, i\right)} \]

      13. lower-fma.f64 N/A

          \[\leadsto {y}^{3} \cdot \frac{z}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(a + y, y, b\right)}, y, c\right), y, i\right)} \]

      14. lower-+.f64 5.0

          \[\leadsto {y}^{3} \cdot \frac{z}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{a + y}, y, b\right), y, c\right), y, i\right)} \]

   5. Applied rewrites 5.0%

      \[\leadsto \color{blue}{{y}^{3} \cdot \frac{z}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}} \]

   6. Taylor expanded in y around inf

      \[\leadsto \frac{z}{\color{blue}{y}} \]
   7. Step-by-step derivation

   8. Applied rewrites 19.9%

      \[\leadsto \frac{z}{\color{blue}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 43.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.65 \cdot 10^{+22}:\\ \;\;\;\;\frac{t}{c \cdot y + i}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 13.6% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.2 \cdot 10^{+100}:\\ \;\;\;\;\frac{z}{a}\\ \mathbf{elif}\;a \leq 3 \cdot 10^{+103}:\\ \;\;\;\;\frac{z}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= a -2.2e+100) (/ z a) (if (<= a 3e+103) (/ z y) (/ z a))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (a <= -2.2e+100) {
		tmp = z / a;
	} else if (a <= 3e+103) {
		tmp = z / y;
	} else {
		tmp = z / a;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i)
    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), intent (in) :: i
    real(8) :: tmp
    if (a <= (-2.2d+100)) then
        tmp = z / a
    else if (a <= 3d+103) then
        tmp = z / y
    else
        tmp = z / a
    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 i) {
	double tmp;
	if (a <= -2.2e+100) {
		tmp = z / a;
	} else if (a <= 3e+103) {
		tmp = z / y;
	} else {
		tmp = z / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if a <= -2.2e+100:
		tmp = z / a
	elif a <= 3e+103:
		tmp = z / y
	else:
		tmp = z / a
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (a <= -2.2e+100)
		tmp = Float64(z / a);
	elseif (a <= 3e+103)
		tmp = Float64(z / y);
	else
		tmp = Float64(z / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	tmp = 0.0;
	if (a <= -2.2e+100)
		tmp = z / a;
	elseif (a <= 3e+103)
		tmp = z / y;
	else
		tmp = z / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[a, -2.2e+100], N[(z / a), $MachinePrecision], If[LessEqual[a, 3e+103], N[(z / y), $MachinePrecision], N[(z / a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -2.2000000000000001e100 or 3e103 < a

   1. Initial program 54.5%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

         \[\leadsto {y}^{3} \cdot \frac{z}{\color{blue}{\mathsf{fma}\left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right), y, i\right)}} \]

      8. +-commutative N/A

         \[\leadsto {y}^{3} \cdot \frac{z}{\mathsf{fma}\left(\color{blue}{y \cdot \left(b + y \cdot \left(a + y\right)\right) + c}, y, i\right)} \]

      9. *-commutative N/A

         \[\leadsto {y}^{3} \cdot \frac{z}{\mathsf{fma}\left(\color{blue}{\left(b + y \cdot \left(a + y\right)\right) \cdot y} + c, y, i\right)} \]

      10. lower-fma.f64 N/A

          \[\leadsto {y}^{3} \cdot \frac{z}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(b + y \cdot \left(a + y\right), y, c\right)}, y, i\right)} \]

      11. +-commutative N/A

          \[\leadsto {y}^{3} \cdot \frac{z}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{y \cdot \left(a + y\right) + b}, y, c\right), y, i\right)} \]

      12. *-commutative N/A

          \[\leadsto {y}^{3} \cdot \frac{z}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\left(a + y\right) \cdot y} + b, y, c\right), y, i\right)} \]

      13. lower-fma.f64 N/A

          \[\leadsto {y}^{3} \cdot \frac{z}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(a + y, y, b\right)}, y, c\right), y, i\right)} \]

      14. lower-+.f64 14.0

          \[\leadsto {y}^{3} \cdot \frac{z}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{a + y}, y, b\right), y, c\right), y, i\right)} \]

   5. Applied rewrites 14.0%

      \[\leadsto \color{blue}{{y}^{3} \cdot \frac{z}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}} \]

   6. Taylor expanded in a around inf

      \[\leadsto \frac{z}{\color{blue}{a}} \]
   7. Step-by-step derivation

   8. Applied rewrites 14.4%

      \[\leadsto \frac{z}{\color{blue}{a}} \]

   if -2.2000000000000001e100 < a < 3e103

   1. Initial program 58.1%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

         \[\leadsto {y}^{3} \cdot \frac{z}{\color{blue}{\mathsf{fma}\left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right), y, i\right)}} \]

      8. +-commutative N/A

         \[\leadsto {y}^{3} \cdot \frac{z}{\mathsf{fma}\left(\color{blue}{y \cdot \left(b + y \cdot \left(a + y\right)\right) + c}, y, i\right)} \]

      9. *-commutative N/A

         \[\leadsto {y}^{3} \cdot \frac{z}{\mathsf{fma}\left(\color{blue}{\left(b + y \cdot \left(a + y\right)\right) \cdot y} + c, y, i\right)} \]

      10. lower-fma.f64 N/A

          \[\leadsto {y}^{3} \cdot \frac{z}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(b + y \cdot \left(a + y\right), y, c\right)}, y, i\right)} \]

      11. +-commutative N/A

          \[\leadsto {y}^{3} \cdot \frac{z}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{y \cdot \left(a + y\right) + b}, y, c\right), y, i\right)} \]

      12. *-commutative N/A

          \[\leadsto {y}^{3} \cdot \frac{z}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\left(a + y\right) \cdot y} + b, y, c\right), y, i\right)} \]

      13. lower-fma.f64 N/A

          \[\leadsto {y}^{3} \cdot \frac{z}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(a + y, y, b\right)}, y, c\right), y, i\right)} \]

      14. lower-+.f64 9.5

          \[\leadsto {y}^{3} \cdot \frac{z}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{a + y}, y, b\right), y, c\right), y, i\right)} \]

   5. Applied rewrites 9.5%

      \[\leadsto \color{blue}{{y}^{3} \cdot \frac{z}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}} \]

   6. Taylor expanded in y around inf

      \[\leadsto \frac{z}{\color{blue}{y}} \]
   7. Step-by-step derivation

   8. Applied rewrites 11.9%

      \[\leadsto \frac{z}{\color{blue}{y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 34.2% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.85 \cdot 10^{+18}:\\ \;\;\;\;\frac{\mathsf{fma}\left(230661.510616, y, t\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= y 2.85e+18) (/ (fma 230661.510616 y t) i) (/ z y)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (y <= 2.85e+18) {
		tmp = fma(230661.510616, y, t) / i;
	} else {
		tmp = z / y;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (y <= 2.85e+18)
		tmp = Float64(fma(230661.510616, y, t) / i);
	else
		tmp = Float64(z / y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[y, 2.85e+18], N[(N[(230661.510616 * y + t), $MachinePrecision] / i), $MachinePrecision], N[(z / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.85e18

   1. Initial program 75.1%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   2. Add Preprocessing

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{i}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{i}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right) + t}}{i} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right) \cdot y} + t}{i} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right), y, t\right)}}{i} \]

      5. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right) + \frac{28832688827}{125000}}, y, t\right)}{i} \]

      6. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right) \cdot y} + \frac{28832688827}{125000}, y, t\right)}{i} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right), y, \frac{28832688827}{125000}\right)}, y, t\right)}{i} \]

      8. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{y \cdot \left(z + x \cdot y\right) + \frac{54929528941}{2000000}}, y, \frac{28832688827}{125000}\right), y, t\right)}{i} \]

      9. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\left(z + x \cdot y\right) \cdot y} + \frac{54929528941}{2000000}, y, \frac{28832688827}{125000}\right), y, t\right)}{i} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(z + x \cdot y, y, \frac{54929528941}{2000000}\right)}, y, \frac{28832688827}{125000}\right), y, t\right)}{i} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{x \cdot y + z}, y, \frac{54929528941}{2000000}\right), y, \frac{28832688827}{125000}\right), y, t\right)}{i} \]

      12. lower-fma.f64 47.2

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(x, y, z\right)}, y, 27464.7644705\right), y, 230661.510616\right), y, t\right)}{i} \]

   5. Applied rewrites 47.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, y, z\right), y, 27464.7644705\right), y, 230661.510616\right), y, t\right)}{i}} \]

   6. Taylor expanded in y around 0

      \[\leadsto \frac{\mathsf{fma}\left(\frac{28832688827}{125000}, y, t\right)}{i} \]
   7. Step-by-step derivation

   8. Applied rewrites 44.3%

      \[\leadsto \frac{\mathsf{fma}\left(230661.510616, y, t\right)}{i} \]

   if 2.85e18 < y

   1. Initial program 6.3%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

         \[\leadsto {y}^{3} \cdot \frac{z}{\color{blue}{\mathsf{fma}\left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right), y, i\right)}} \]

      8. +-commutative N/A

         \[\leadsto {y}^{3} \cdot \frac{z}{\mathsf{fma}\left(\color{blue}{y \cdot \left(b + y \cdot \left(a + y\right)\right) + c}, y, i\right)} \]

      9. *-commutative N/A

         \[\leadsto {y}^{3} \cdot \frac{z}{\mathsf{fma}\left(\color{blue}{\left(b + y \cdot \left(a + y\right)\right) \cdot y} + c, y, i\right)} \]

      10. lower-fma.f64 N/A

          \[\leadsto {y}^{3} \cdot \frac{z}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(b + y \cdot \left(a + y\right), y, c\right)}, y, i\right)} \]

      11. +-commutative N/A

          \[\leadsto {y}^{3} \cdot \frac{z}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{y \cdot \left(a + y\right) + b}, y, c\right), y, i\right)} \]

      12. *-commutative N/A

          \[\leadsto {y}^{3} \cdot \frac{z}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\left(a + y\right) \cdot y} + b, y, c\right), y, i\right)} \]

      13. lower-fma.f64 N/A

          \[\leadsto {y}^{3} \cdot \frac{z}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(a + y, y, b\right)}, y, c\right), y, i\right)} \]

      14. lower-+.f64 5.0

          \[\leadsto {y}^{3} \cdot \frac{z}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{a + y}, y, b\right), y, c\right), y, i\right)} \]

   5. Applied rewrites 5.0%

      \[\leadsto \color{blue}{{y}^{3} \cdot \frac{z}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}} \]

   6. Taylor expanded in y around inf

      \[\leadsto \frac{z}{\color{blue}{y}} \]
   7. Step-by-step derivation

   8. Applied rewrites 19.7%

      \[\leadsto \frac{z}{\color{blue}{y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 31.1% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 19000000000:\\ \;\;\;\;\frac{t}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= y 19000000000.0) (/ t i) (/ z y)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (y <= 19000000000.0) {
		tmp = t / i;
	} else {
		tmp = z / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i)
    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), intent (in) :: i
    real(8) :: tmp
    if (y <= 19000000000.0d0) then
        tmp = t / i
    else
        tmp = z / y
    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 i) {
	double tmp;
	if (y <= 19000000000.0) {
		tmp = t / i;
	} else {
		tmp = z / y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if y <= 19000000000.0:
		tmp = t / i
	else:
		tmp = z / y
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (y <= 19000000000.0)
		tmp = Float64(t / i);
	else
		tmp = Float64(z / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	tmp = 0.0;
	if (y <= 19000000000.0)
		tmp = t / i;
	else
		tmp = z / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[y, 19000000000.0], N[(t / i), $MachinePrecision], N[(z / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.9e10

   1. Initial program 75.3%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{t}{i}} \]
   4. Step-by-step derivation

      1. lower-/.f64 41.8

         \[\leadsto \color{blue}{\frac{t}{i}} \]

   5. Applied rewrites 41.8%

      \[\leadsto \color{blue}{\frac{t}{i}} \]

   if 1.9e10 < y

   1. Initial program 8.8%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

         \[\leadsto {y}^{3} \cdot \frac{z}{\color{blue}{\mathsf{fma}\left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right), y, i\right)}} \]

      8. +-commutative N/A

         \[\leadsto {y}^{3} \cdot \frac{z}{\mathsf{fma}\left(\color{blue}{y \cdot \left(b + y \cdot \left(a + y\right)\right) + c}, y, i\right)} \]

      9. *-commutative N/A

         \[\leadsto {y}^{3} \cdot \frac{z}{\mathsf{fma}\left(\color{blue}{\left(b + y \cdot \left(a + y\right)\right) \cdot y} + c, y, i\right)} \]

      10. lower-fma.f64 N/A

          \[\leadsto {y}^{3} \cdot \frac{z}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(b + y \cdot \left(a + y\right), y, c\right)}, y, i\right)} \]

      11. +-commutative N/A

          \[\leadsto {y}^{3} \cdot \frac{z}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{y \cdot \left(a + y\right) + b}, y, c\right), y, i\right)} \]

      12. *-commutative N/A

          \[\leadsto {y}^{3} \cdot \frac{z}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\left(a + y\right) \cdot y} + b, y, c\right), y, i\right)} \]

      13. lower-fma.f64 N/A

          \[\leadsto {y}^{3} \cdot \frac{z}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(a + y, y, b\right)}, y, c\right), y, i\right)} \]

      14. lower-+.f64 4.9

          \[\leadsto {y}^{3} \cdot \frac{z}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{a + y}, y, b\right), y, c\right), y, i\right)} \]

   5. Applied rewrites 4.9%

      \[\leadsto \color{blue}{{y}^{3} \cdot \frac{z}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}} \]

   6. Taylor expanded in y around inf

      \[\leadsto \frac{z}{\color{blue}{y}} \]
   7. Step-by-step derivation

   8. Applied rewrites 19.0%

      \[\leadsto \frac{z}{\color{blue}{y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 8.0% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \frac{z}{a} \end{array} \]

FPCore

(FPCore (x y z t a b c i) :precision binary64 (/ z a))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return z / a;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i)
    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), intent (in) :: i
    code = z / a
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return z / a;
}

Python

def code(x, y, z, t, a, b, c, i):
	return z / a

Julia

function code(x, y, z, t, a, b, c, i)
	return Float64(z / a)
end

MATLAB

function tmp = code(x, y, z, t, a, b, c, i)
	tmp = z / a;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := N[(z / a), $MachinePrecision]

Derivation

1. Initial program 56.8%

   \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
2. Add Preprocessing

3. Taylor expanded in z around inf

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

   1. associate-/l* N/A

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

   2. lower-*.f64 N/A

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

   3. lower-pow.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. +-commutative N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

      \[\leadsto {y}^{3} \cdot \frac{z}{\color{blue}{\mathsf{fma}\left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right), y, i\right)}} \]

   8. +-commutative N/A

      \[\leadsto {y}^{3} \cdot \frac{z}{\mathsf{fma}\left(\color{blue}{y \cdot \left(b + y \cdot \left(a + y\right)\right) + c}, y, i\right)} \]

   9. *-commutative N/A

      \[\leadsto {y}^{3} \cdot \frac{z}{\mathsf{fma}\left(\color{blue}{\left(b + y \cdot \left(a + y\right)\right) \cdot y} + c, y, i\right)} \]

   10. lower-fma.f64 N/A

       \[\leadsto {y}^{3} \cdot \frac{z}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(b + y \cdot \left(a + y\right), y, c\right)}, y, i\right)} \]

   11. +-commutative N/A

       \[\leadsto {y}^{3} \cdot \frac{z}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{y \cdot \left(a + y\right) + b}, y, c\right), y, i\right)} \]

   12. *-commutative N/A

       \[\leadsto {y}^{3} \cdot \frac{z}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\left(a + y\right) \cdot y} + b, y, c\right), y, i\right)} \]

   13. lower-fma.f64 N/A

       \[\leadsto {y}^{3} \cdot \frac{z}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(a + y, y, b\right)}, y, c\right), y, i\right)} \]

   14. lower-+.f64 11.1

       \[\leadsto {y}^{3} \cdot \frac{z}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{a + y}, y, b\right), y, c\right), y, i\right)} \]

5. Applied rewrites 11.1%

   \[\leadsto \color{blue}{{y}^{3} \cdot \frac{z}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}} \]

6. Taylor expanded in a around inf

   \[\leadsto \frac{z}{\color{blue}{a}} \]
7. Step-by-step derivation

8. Applied rewrites 7.4%

   \[\leadsto \frac{z}{\color{blue}{a}} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024268 
(FPCore (x y z t a b c i)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2"
  :precision binary64
  (/ (+ (* (+ (* (+ (* (+ (* x y) z) y) 27464.7644705) y) 230661.510616) y) t) (+ (* (+ (* (+ (* (+ y a) y) b) y) c) y) i)))