Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2

Percentage Accurate: 56.7% → 85.7%

Time: 18.9s

Alternatives: 16

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: 56.7% 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: 85.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)\\ \mathbf{if}\;\frac{y \cdot \left(y \cdot \left(y \cdot \left(x \cdot y + z\right) + 27464.7644705\right) + 230661.510616\right) + t}{y \cdot \left(y \cdot \left(y \cdot \left(y + a\right) + b\right) + c\right) + i} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, z, 27464.7644705\right), 230661.510616\right)}{t\_1}, \mathsf{fma}\left(x, \left(y \cdot y\right) \cdot \frac{y \cdot y}{t\_1}, \frac{t}{t\_1}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{z \cdot \left(\frac{a}{y} + -1\right)}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma y (fma y (fma y (+ y a) b) c) i)))
   (if (<=
        (/
         (+
          (* y (+ (* y (+ (* y (+ (* x y) z)) 27464.7644705)) 230661.510616))
          t)
         (+ (* y (+ (* y (+ (* y (+ y a)) b)) c)) i))
        INFINITY)
     (fma
      y
      (/ (fma y (fma y z 27464.7644705) 230661.510616) t_1)
      (fma x (* (* y y) (/ (* y y) t_1)) (/ t t_1)))
     (- x (/ (* z (+ (/ a y) -1.0)) y)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma(y, fma(y, fma(y, (y + a), b), c), i);
	double tmp;
	if ((((y * ((y * ((y * ((x * y) + z)) + 27464.7644705)) + 230661.510616)) + t) / ((y * ((y * ((y * (y + a)) + b)) + c)) + i)) <= ((double) INFINITY)) {
		tmp = fma(y, (fma(y, fma(y, z, 27464.7644705), 230661.510616) / t_1), fma(x, ((y * y) * ((y * y) / t_1)), (t / t_1)));
	} else {
		tmp = x - ((z * ((a / y) + -1.0)) / y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(y, fma(y, fma(y, Float64(y + a), b), c), i)
	tmp = 0.0
	if (Float64(Float64(Float64(y * Float64(Float64(y * Float64(Float64(y * Float64(Float64(x * y) + z)) + 27464.7644705)) + 230661.510616)) + t) / Float64(Float64(y * Float64(Float64(y * Float64(Float64(y * Float64(y + a)) + b)) + c)) + i)) <= Inf)
		tmp = fma(y, Float64(fma(y, fma(y, z, 27464.7644705), 230661.510616) / t_1), fma(x, Float64(Float64(y * y) * Float64(Float64(y * y) / t_1)), Float64(t / t_1)));
	else
		tmp = Float64(x - Float64(Float64(z * Float64(Float64(a / y) + -1.0)) / y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(y * N[(y * N[(y * N[(y + a), $MachinePrecision] + b), $MachinePrecision] + c), $MachinePrecision] + i), $MachinePrecision]}, If[LessEqual[N[(N[(N[(y * N[(N[(y * N[(N[(y * N[(N[(x * y), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision] + 27464.7644705), $MachinePrecision]), $MachinePrecision] + 230661.510616), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision] / N[(N[(y * N[(N[(y * N[(N[(y * N[(y + a), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision]), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision], Infinity], N[(y * N[(N[(y * N[(y * z + 27464.7644705), $MachinePrecision] + 230661.510616), $MachinePrecision] / t$95$1), $MachinePrecision] + N[(x * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] + N[(t / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[(z * N[(N[(a / y), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $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 90.3%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

   5. Simplified 92.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, z, 27464.7644705\right), 230661.510616\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)}, \mathsf{fma}\left(x, \frac{{y}^{4}}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)}, \frac{t}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)}\right)\right)} \]
   6. Step-by-step derivation

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. pow2 N/A

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

      4. pow2 N/A

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

      5. associate-/l* N/A

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

      6. *-lowering-*.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. accelerator-lowering-fma.f64 N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. +-lowering-+.f64 92.8

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

   7. Applied egg-rr 92.8%

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

   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 y around -inf

      \[\leadsto \color{blue}{x + -1 \cdot \frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

         \[\leadsto \color{blue}{x - \frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \color{blue}{x - \frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]

   5. Simplified 60.6%

      \[\leadsto \color{blue}{x - \frac{\left(\left(0 - z\right) - \frac{27464.7644705 - \mathsf{fma}\left(\left(0 - z\right) + x \cdot a, 0 - a, x \cdot b\right)}{y}\right) + x \cdot a}{y}} \]

   6. Taylor expanded in z around inf

      \[\leadsto x - \frac{\color{blue}{z \cdot \left(\frac{a}{y} - 1\right)}}{y} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto x - \frac{\color{blue}{z \cdot \left(\frac{a}{y} - 1\right)}}{y} \]

      2. sub-neg N/A

         \[\leadsto x - \frac{z \cdot \color{blue}{\left(\frac{a}{y} + \left(\mathsf{neg}\left(1\right)\right)\right)}}{y} \]

      3. metadata-eval N/A

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

      4. +-lowering-+.f64 N/A

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

      5. /-lowering-/.f64 76.6

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

   8. Simplified 76.6%

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

4. Final simplification 86.2%

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

Alternative 2: 69.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)}\\ t_2 := \frac{y \cdot \left(y \cdot \left(y \cdot \left(x \cdot y + z\right) + 27464.7644705\right) + 230661.510616\right) + t}{y \cdot \left(y \cdot \left(y \cdot \left(y + a\right) + b\right) + c\right) + i}\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{-231}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{-87}:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(y, 27464.7644705, 230661.510616\right), t\right) \cdot \frac{1}{\mathsf{fma}\left(y, y \cdot b, i\right)}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;x - \frac{z \cdot \left(\frac{a}{y} + -1\right)}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (/ t (fma y (fma y (fma y (+ y a) b) c) i)))
        (t_2
         (/
          (+
           (* y (+ (* y (+ (* y (+ (* x y) z)) 27464.7644705)) 230661.510616))
           t)
          (+ (* y (+ (* y (+ (* y (+ y a)) b)) c)) i))))
   (if (<= t_2 -2e-231)
     t_1
     (if (<= t_2 1e-87)
       (*
        (fma y (fma y 27464.7644705 230661.510616) t)
        (/ 1.0 (fma y (* y b) i)))
       (if (<= t_2 INFINITY) t_1 (- x (/ (* z (+ (/ a y) -1.0)) y)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = t / fma(y, fma(y, fma(y, (y + a), b), c), i);
	double t_2 = ((y * ((y * ((y * ((x * y) + z)) + 27464.7644705)) + 230661.510616)) + t) / ((y * ((y * ((y * (y + a)) + b)) + c)) + i);
	double tmp;
	if (t_2 <= -2e-231) {
		tmp = t_1;
	} else if (t_2 <= 1e-87) {
		tmp = fma(y, fma(y, 27464.7644705, 230661.510616), t) * (1.0 / fma(y, (y * b), i));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = x - ((z * ((a / y) + -1.0)) / y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(t / fma(y, fma(y, fma(y, Float64(y + a), b), c), i))
	t_2 = Float64(Float64(Float64(y * Float64(Float64(y * Float64(Float64(y * Float64(Float64(x * y) + z)) + 27464.7644705)) + 230661.510616)) + t) / Float64(Float64(y * Float64(Float64(y * Float64(Float64(y * Float64(y + a)) + b)) + c)) + i))
	tmp = 0.0
	if (t_2 <= -2e-231)
		tmp = t_1;
	elseif (t_2 <= 1e-87)
		tmp = Float64(fma(y, fma(y, 27464.7644705, 230661.510616), t) * Float64(1.0 / fma(y, Float64(y * b), i)));
	elseif (t_2 <= Inf)
		tmp = t_1;
	else
		tmp = Float64(x - Float64(Float64(z * Float64(Float64(a / y) + -1.0)) / y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(t / N[(y * N[(y * N[(y * N[(y + a), $MachinePrecision] + b), $MachinePrecision] + c), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(y * N[(N[(y * N[(N[(y * N[(N[(x * y), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision] + 27464.7644705), $MachinePrecision]), $MachinePrecision] + 230661.510616), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision] / N[(N[(y * N[(N[(y * N[(N[(y * N[(y + a), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision]), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e-231], t$95$1, If[LessEqual[t$95$2, 1e-87], N[(N[(y * N[(y * 27464.7644705 + 230661.510616), $MachinePrecision] + t), $MachinePrecision] * N[(1.0 / N[(y * N[(y * b), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], t$95$1, N[(x - N[(N[(z * N[(N[(a / y), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 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)) < -2e-231 or 1.00000000000000002e-87 < (/.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 94.1%

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

   3. Taylor expanded in 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. /-lowering-/.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. accelerator-lowering-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. +-lowering-+.f64 69.2

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

   5. Simplified 69.2%

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

   if -2e-231 < (/.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)) < 1.00000000000000002e-87

   1. Initial program 76.7%

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

      1. div-inv N/A

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

      2. *-lowering-*.f64 N/A

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

      3. *-commutative N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. /-lowering-/.f64 N/A

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

   4. Applied egg-rr 76.7%

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

   5. Taylor expanded in b around inf

      \[\leadsto \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(x, y, z\right), \frac{54929528941}{2000000}\right), \frac{28832688827}{125000}\right), t\right) \cdot \frac{1}{\mathsf{fma}\left(y, \color{blue}{b \cdot y}, i\right)} \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 60.9

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

   7. Simplified 60.9%

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

   8. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{\frac{54929528941}{2000000}}, \frac{28832688827}{125000}\right), t\right) \cdot \frac{1}{\mathsf{fma}\left(y, b \cdot y, i\right)} \]
   9. Step-by-step derivation

   10. Simplified 58.2%

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

   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 y around -inf

      \[\leadsto \color{blue}{x + -1 \cdot \frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

         \[\leadsto \color{blue}{x - \frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \color{blue}{x - \frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]

   5. Simplified 60.6%

      \[\leadsto \color{blue}{x - \frac{\left(\left(0 - z\right) - \frac{27464.7644705 - \mathsf{fma}\left(\left(0 - z\right) + x \cdot a, 0 - a, x \cdot b\right)}{y}\right) + x \cdot a}{y}} \]

   6. Taylor expanded in z around inf

      \[\leadsto x - \frac{\color{blue}{z \cdot \left(\frac{a}{y} - 1\right)}}{y} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto x - \frac{\color{blue}{z \cdot \left(\frac{a}{y} - 1\right)}}{y} \]

      2. sub-neg N/A

         \[\leadsto x - \frac{z \cdot \color{blue}{\left(\frac{a}{y} + \left(\mathsf{neg}\left(1\right)\right)\right)}}{y} \]

      3. metadata-eval N/A

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

      4. +-lowering-+.f64 N/A

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

      5. /-lowering-/.f64 76.6

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

   8. Simplified 76.6%

      \[\leadsto x - \frac{\color{blue}{z \cdot \left(\frac{a}{y} + -1\right)}}{y} \]
3. Recombined 3 regimes into one program.

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(y \cdot \left(y \cdot \left(x \cdot y + z\right) + 27464.7644705\right) + 230661.510616\right) + t}{y \cdot \left(y \cdot \left(y \cdot \left(y + a\right) + b\right) + c\right) + i} \leq -2 \cdot 10^{-231}:\\ \;\;\;\;\frac{t}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)}\\ \mathbf{elif}\;\frac{y \cdot \left(y \cdot \left(y \cdot \left(x \cdot y + z\right) + 27464.7644705\right) + 230661.510616\right) + t}{y \cdot \left(y \cdot \left(y \cdot \left(y + a\right) + b\right) + c\right) + i} \leq 10^{-87}:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(y, 27464.7644705, 230661.510616\right), t\right) \cdot \frac{1}{\mathsf{fma}\left(y, y \cdot b, i\right)}\\ \mathbf{elif}\;\frac{y \cdot \left(y \cdot \left(y \cdot \left(x \cdot y + z\right) + 27464.7644705\right) + 230661.510616\right) + t}{y \cdot \left(y \cdot \left(y \cdot \left(y + a\right) + b\right) + c\right) + i} \leq \infty:\\ \;\;\;\;\frac{t}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{z \cdot \left(\frac{a}{y} + -1\right)}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 69.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)}\\ t_2 := \frac{y \cdot \left(y \cdot \left(y \cdot \left(x \cdot y + z\right) + 27464.7644705\right) + 230661.510616\right) + t}{y \cdot \left(y \cdot \left(y \cdot \left(y + a\right) + b\right) + c\right) + i}\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{-231}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{-68}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, x, z\right), 27464.7644705\right), 230661.510616\right), t\right)}{i}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;x - \frac{z \cdot \left(\frac{a}{y} + -1\right)}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (/ t (fma y (fma y (fma y (+ y a) b) c) i)))
        (t_2
         (/
          (+
           (* y (+ (* y (+ (* y (+ (* x y) z)) 27464.7644705)) 230661.510616))
           t)
          (+ (* y (+ (* y (+ (* y (+ y a)) b)) c)) i))))
   (if (<= t_2 -2e-231)
     t_1
     (if (<= t_2 1e-68)
       (/ (fma y (fma y (fma y (fma y x z) 27464.7644705) 230661.510616) t) i)
       (if (<= t_2 INFINITY) t_1 (- x (/ (* z (+ (/ a y) -1.0)) y)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = t / fma(y, fma(y, fma(y, (y + a), b), c), i);
	double t_2 = ((y * ((y * ((y * ((x * y) + z)) + 27464.7644705)) + 230661.510616)) + t) / ((y * ((y * ((y * (y + a)) + b)) + c)) + i);
	double tmp;
	if (t_2 <= -2e-231) {
		tmp = t_1;
	} else if (t_2 <= 1e-68) {
		tmp = fma(y, fma(y, fma(y, fma(y, x, z), 27464.7644705), 230661.510616), t) / i;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = x - ((z * ((a / y) + -1.0)) / y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(t / fma(y, fma(y, fma(y, Float64(y + a), b), c), i))
	t_2 = Float64(Float64(Float64(y * Float64(Float64(y * Float64(Float64(y * Float64(Float64(x * y) + z)) + 27464.7644705)) + 230661.510616)) + t) / Float64(Float64(y * Float64(Float64(y * Float64(Float64(y * Float64(y + a)) + b)) + c)) + i))
	tmp = 0.0
	if (t_2 <= -2e-231)
		tmp = t_1;
	elseif (t_2 <= 1e-68)
		tmp = Float64(fma(y, fma(y, fma(y, fma(y, x, z), 27464.7644705), 230661.510616), t) / i);
	elseif (t_2 <= Inf)
		tmp = t_1;
	else
		tmp = Float64(x - Float64(Float64(z * Float64(Float64(a / y) + -1.0)) / y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(t / N[(y * N[(y * N[(y * N[(y + a), $MachinePrecision] + b), $MachinePrecision] + c), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(y * N[(N[(y * N[(N[(y * N[(N[(x * y), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision] + 27464.7644705), $MachinePrecision]), $MachinePrecision] + 230661.510616), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision] / N[(N[(y * N[(N[(y * N[(N[(y * N[(y + a), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision]), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e-231], t$95$1, If[LessEqual[t$95$2, 1e-68], N[(N[(y * N[(y * N[(y * N[(y * x + z), $MachinePrecision] + 27464.7644705), $MachinePrecision] + 230661.510616), $MachinePrecision] + t), $MachinePrecision] / i), $MachinePrecision], If[LessEqual[t$95$2, Infinity], t$95$1, N[(x - N[(N[(z * N[(N[(a / y), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 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)) < -2e-231 or 1.00000000000000007e-68 < (/.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 93.9%

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

   3. Taylor expanded in t around 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. /-lowering-/.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. accelerator-lowering-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. +-lowering-+.f64 70.1

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

   5. Simplified 70.1%

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

   if -2e-231 < (/.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)) < 1.00000000000000007e-68

   1. Initial program 78.6%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   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. /-lowering-/.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. accelerator-lowering-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. accelerator-lowering-fma.f64 55.7

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

   5. Simplified 55.7%

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

   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 y around -inf

      \[\leadsto \color{blue}{x + -1 \cdot \frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

         \[\leadsto \color{blue}{x - \frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \color{blue}{x - \frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]

   5. Simplified 60.6%

      \[\leadsto \color{blue}{x - \frac{\left(\left(0 - z\right) - \frac{27464.7644705 - \mathsf{fma}\left(\left(0 - z\right) + x \cdot a, 0 - a, x \cdot b\right)}{y}\right) + x \cdot a}{y}} \]

   6. Taylor expanded in z around inf

      \[\leadsto x - \frac{\color{blue}{z \cdot \left(\frac{a}{y} - 1\right)}}{y} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto x - \frac{\color{blue}{z \cdot \left(\frac{a}{y} - 1\right)}}{y} \]

      2. sub-neg N/A

         \[\leadsto x - \frac{z \cdot \color{blue}{\left(\frac{a}{y} + \left(\mathsf{neg}\left(1\right)\right)\right)}}{y} \]

      3. metadata-eval N/A

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

      4. +-lowering-+.f64 N/A

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

      5. /-lowering-/.f64 76.6

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

   8. Simplified 76.6%

      \[\leadsto x - \frac{\color{blue}{z \cdot \left(\frac{a}{y} + -1\right)}}{y} \]
3. Recombined 3 regimes into one program.

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(y \cdot \left(y \cdot \left(x \cdot y + z\right) + 27464.7644705\right) + 230661.510616\right) + t}{y \cdot \left(y \cdot \left(y \cdot \left(y + a\right) + b\right) + c\right) + i} \leq -2 \cdot 10^{-231}:\\ \;\;\;\;\frac{t}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)}\\ \mathbf{elif}\;\frac{y \cdot \left(y \cdot \left(y \cdot \left(x \cdot y + z\right) + 27464.7644705\right) + 230661.510616\right) + t}{y \cdot \left(y \cdot \left(y \cdot \left(y + a\right) + b\right) + c\right) + i} \leq 10^{-68}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, x, z\right), 27464.7644705\right), 230661.510616\right), t\right)}{i}\\ \mathbf{elif}\;\frac{y \cdot \left(y \cdot \left(y \cdot \left(x \cdot y + z\right) + 27464.7644705\right) + 230661.510616\right) + t}{y \cdot \left(y \cdot \left(y \cdot \left(y + a\right) + b\right) + c\right) + i} \leq \infty:\\ \;\;\;\;\frac{t}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{z \cdot \left(\frac{a}{y} + -1\right)}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)\\ \mathbf{if}\;\frac{y \cdot \left(y \cdot \left(y \cdot \left(x \cdot y + z\right) + 27464.7644705\right) + 230661.510616\right) + t}{y \cdot \left(y \cdot \left(y \cdot \left(y + a\right) + b\right) + c\right) + i} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, x, z\right), 27464.7644705\right), 230661.510616\right)}{t\_1}, \frac{t}{t\_1}\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{z \cdot \left(\frac{a}{y} + -1\right)}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma y (fma y (fma y (+ y a) b) c) i)))
   (if (<=
        (/
         (+
          (* y (+ (* y (+ (* y (+ (* x y) z)) 27464.7644705)) 230661.510616))
          t)
         (+ (* y (+ (* y (+ (* y (+ y a)) b)) c)) i))
        INFINITY)
     (fma
      y
      (/ (fma y (fma y (fma y x z) 27464.7644705) 230661.510616) t_1)
      (/ t t_1))
     (- x (/ (* z (+ (/ a y) -1.0)) y)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma(y, fma(y, fma(y, (y + a), b), c), i);
	double tmp;
	if ((((y * ((y * ((y * ((x * y) + z)) + 27464.7644705)) + 230661.510616)) + t) / ((y * ((y * ((y * (y + a)) + b)) + c)) + i)) <= ((double) INFINITY)) {
		tmp = fma(y, (fma(y, fma(y, fma(y, x, z), 27464.7644705), 230661.510616) / t_1), (t / t_1));
	} else {
		tmp = x - ((z * ((a / y) + -1.0)) / y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(y, fma(y, fma(y, Float64(y + a), b), c), i)
	tmp = 0.0
	if (Float64(Float64(Float64(y * Float64(Float64(y * Float64(Float64(y * Float64(Float64(x * y) + z)) + 27464.7644705)) + 230661.510616)) + t) / Float64(Float64(y * Float64(Float64(y * Float64(Float64(y * Float64(y + a)) + b)) + c)) + i)) <= Inf)
		tmp = fma(y, Float64(fma(y, fma(y, fma(y, x, z), 27464.7644705), 230661.510616) / t_1), Float64(t / t_1));
	else
		tmp = Float64(x - Float64(Float64(z * Float64(Float64(a / y) + -1.0)) / y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(y * N[(y * N[(y * N[(y + a), $MachinePrecision] + b), $MachinePrecision] + c), $MachinePrecision] + i), $MachinePrecision]}, If[LessEqual[N[(N[(N[(y * N[(N[(y * N[(N[(y * N[(N[(x * y), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision] + 27464.7644705), $MachinePrecision]), $MachinePrecision] + 230661.510616), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision] / N[(N[(y * N[(N[(y * N[(N[(y * N[(y + a), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision]), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision], Infinity], N[(y * N[(N[(y * N[(y * N[(y * x + z), $MachinePrecision] + 27464.7644705), $MachinePrecision] + 230661.510616), $MachinePrecision] / t$95$1), $MachinePrecision] + N[(t / t$95$1), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[(z * N[(N[(a / y), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $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 90.3%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   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. accelerator-lowering-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. Simplified 91.0%

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

   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 y around -inf

      \[\leadsto \color{blue}{x + -1 \cdot \frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

         \[\leadsto \color{blue}{x - \frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \color{blue}{x - \frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]

   5. Simplified 60.6%

      \[\leadsto \color{blue}{x - \frac{\left(\left(0 - z\right) - \frac{27464.7644705 - \mathsf{fma}\left(\left(0 - z\right) + x \cdot a, 0 - a, x \cdot b\right)}{y}\right) + x \cdot a}{y}} \]

   6. Taylor expanded in z around inf

      \[\leadsto x - \frac{\color{blue}{z \cdot \left(\frac{a}{y} - 1\right)}}{y} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto x - \frac{\color{blue}{z \cdot \left(\frac{a}{y} - 1\right)}}{y} \]

      2. sub-neg N/A

         \[\leadsto x - \frac{z \cdot \color{blue}{\left(\frac{a}{y} + \left(\mathsf{neg}\left(1\right)\right)\right)}}{y} \]

      3. metadata-eval N/A

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

      4. +-lowering-+.f64 N/A

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

      5. /-lowering-/.f64 76.6

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

   8. Simplified 76.6%

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

4. Final simplification 85.1%

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

Alternative 5: 84.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(y \cdot \left(y \cdot \left(y + a\right) + b\right) + c\right) + i\\ \mathbf{if}\;\frac{y \cdot \left(y \cdot \left(y \cdot \left(x \cdot y + z\right) + 27464.7644705\right) + 230661.510616\right) + t}{t\_1} \leq \infty:\\ \;\;\;\;\frac{t + \mathsf{fma}\left(y \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(x, y, z\right), 27464.7644705\right), y, y \cdot 230661.510616\right)}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{z \cdot \left(\frac{a}{y} + -1\right)}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ (* y (+ (* y (+ (* y (+ y a)) b)) c)) i)))
   (if (<=
        (/
         (+
          (* y (+ (* y (+ (* y (+ (* x y) z)) 27464.7644705)) 230661.510616))
          t)
         t_1)
        INFINITY)
     (/
      (+ t (fma (* y (fma y (fma x y z) 27464.7644705)) y (* y 230661.510616)))
      t_1)
     (- x (/ (* z (+ (/ a y) -1.0)) y)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (y * ((y * ((y * (y + a)) + b)) + c)) + i;
	double tmp;
	if ((((y * ((y * ((y * ((x * y) + z)) + 27464.7644705)) + 230661.510616)) + t) / t_1) <= ((double) INFINITY)) {
		tmp = (t + fma((y * fma(y, fma(x, y, z), 27464.7644705)), y, (y * 230661.510616))) / t_1;
	} else {
		tmp = x - ((z * ((a / y) + -1.0)) / y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(y * Float64(Float64(y * Float64(Float64(y * Float64(y + a)) + b)) + c)) + i)
	tmp = 0.0
	if (Float64(Float64(Float64(y * Float64(Float64(y * Float64(Float64(y * Float64(Float64(x * y) + z)) + 27464.7644705)) + 230661.510616)) + t) / t_1) <= Inf)
		tmp = Float64(Float64(t + fma(Float64(y * fma(y, fma(x, y, z), 27464.7644705)), y, Float64(y * 230661.510616))) / t_1);
	else
		tmp = Float64(x - Float64(Float64(z * Float64(Float64(a / y) + -1.0)) / y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(y * N[(N[(y * N[(N[(y * N[(y + a), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision]), $MachinePrecision] + i), $MachinePrecision]}, If[LessEqual[N[(N[(N[(y * N[(N[(y * N[(N[(y * N[(N[(x * y), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision] + 27464.7644705), $MachinePrecision]), $MachinePrecision] + 230661.510616), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision] / t$95$1), $MachinePrecision], Infinity], N[(N[(t + N[(N[(y * N[(y * N[(x * y + z), $MachinePrecision] + 27464.7644705), $MachinePrecision]), $MachinePrecision] * y + N[(y * 230661.510616), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], N[(x - N[(N[(z * N[(N[(a / y), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $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 90.3%

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

      1. *-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. *-lowering-*.f64 90.3

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

   4. Applied egg-rr 90.3%

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

   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 y around -inf

      \[\leadsto \color{blue}{x + -1 \cdot \frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

         \[\leadsto \color{blue}{x - \frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \color{blue}{x - \frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]

   5. Simplified 60.6%

      \[\leadsto \color{blue}{x - \frac{\left(\left(0 - z\right) - \frac{27464.7644705 - \mathsf{fma}\left(\left(0 - z\right) + x \cdot a, 0 - a, x \cdot b\right)}{y}\right) + x \cdot a}{y}} \]

   6. Taylor expanded in z around inf

      \[\leadsto x - \frac{\color{blue}{z \cdot \left(\frac{a}{y} - 1\right)}}{y} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto x - \frac{\color{blue}{z \cdot \left(\frac{a}{y} - 1\right)}}{y} \]

      2. sub-neg N/A

         \[\leadsto x - \frac{z \cdot \color{blue}{\left(\frac{a}{y} + \left(\mathsf{neg}\left(1\right)\right)\right)}}{y} \]

      3. metadata-eval N/A

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

      4. +-lowering-+.f64 N/A

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

      5. /-lowering-/.f64 76.6

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

   8. Simplified 76.6%

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

4. Final simplification 84.7%

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

Alternative 6: 84.5% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(N[(N[(y * N[(N[(y * N[(N[(y * N[(N[(x * y), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision] + 27464.7644705), $MachinePrecision]), $MachinePrecision] + 230661.510616), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision] / N[(N[(y * N[(N[(y * N[(N[(y * N[(y + a), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision]), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(y * N[(y * N[(y * N[(x * y + z), $MachinePrecision] + 27464.7644705), $MachinePrecision] + 230661.510616), $MachinePrecision] + t), $MachinePrecision] * N[(1.0 / N[(y * N[(y * N[(y * N[(y + a), $MachinePrecision] + b), $MachinePrecision] + c), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[(z * N[(N[(a / y), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $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 90.3%

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

      1. div-inv N/A

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

      2. *-lowering-*.f64 N/A

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

      3. *-commutative N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. /-lowering-/.f64 N/A

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

   4. Applied egg-rr 90.2%

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

   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 y around -inf

      \[\leadsto \color{blue}{x + -1 \cdot \frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

         \[\leadsto \color{blue}{x - \frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \color{blue}{x - \frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]

   5. Simplified 60.6%

      \[\leadsto \color{blue}{x - \frac{\left(\left(0 - z\right) - \frac{27464.7644705 - \mathsf{fma}\left(\left(0 - z\right) + x \cdot a, 0 - a, x \cdot b\right)}{y}\right) + x \cdot a}{y}} \]

   6. Taylor expanded in z around inf

      \[\leadsto x - \frac{\color{blue}{z \cdot \left(\frac{a}{y} - 1\right)}}{y} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto x - \frac{\color{blue}{z \cdot \left(\frac{a}{y} - 1\right)}}{y} \]

      2. sub-neg N/A

         \[\leadsto x - \frac{z \cdot \color{blue}{\left(\frac{a}{y} + \left(\mathsf{neg}\left(1\right)\right)\right)}}{y} \]

      3. metadata-eval N/A

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

      4. +-lowering-+.f64 N/A

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

      5. /-lowering-/.f64 76.6

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

   8. Simplified 76.6%

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

4. Final simplification 84.7%

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

Alternative 7: 81.0% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(N[(N[(y * N[(N[(y * N[(N[(y * N[(N[(x * y), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision] + 27464.7644705), $MachinePrecision]), $MachinePrecision] + 230661.510616), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision] / N[(N[(y * N[(N[(y * N[(N[(y * N[(y + a), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision]), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(y * N[(y * N[(y * z + 27464.7644705), $MachinePrecision] + 230661.510616), $MachinePrecision] + t), $MachinePrecision] / N[(y * N[(y * N[(y * N[(y + a), $MachinePrecision] + b), $MachinePrecision] + c), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[(z * N[(N[(a / y), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $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 90.3%

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

   3. Taylor expanded in 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. /-lowering-/.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. accelerator-lowering-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. accelerator-lowering-fma.f64 N/A

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

      12. +-commutative N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. +-commutative N/A

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

      15. +-lowering-+.f64 85.9

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

   5. Simplified 85.9%

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

   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 y around -inf

      \[\leadsto \color{blue}{x + -1 \cdot \frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

         \[\leadsto \color{blue}{x - \frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \color{blue}{x - \frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]

   5. Simplified 60.6%

      \[\leadsto \color{blue}{x - \frac{\left(\left(0 - z\right) - \frac{27464.7644705 - \mathsf{fma}\left(\left(0 - z\right) + x \cdot a, 0 - a, x \cdot b\right)}{y}\right) + x \cdot a}{y}} \]

   6. Taylor expanded in z around inf

      \[\leadsto x - \frac{\color{blue}{z \cdot \left(\frac{a}{y} - 1\right)}}{y} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto x - \frac{\color{blue}{z \cdot \left(\frac{a}{y} - 1\right)}}{y} \]

      2. sub-neg N/A

         \[\leadsto x - \frac{z \cdot \color{blue}{\left(\frac{a}{y} + \left(\mathsf{neg}\left(1\right)\right)\right)}}{y} \]

      3. metadata-eval N/A

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

      4. +-lowering-+.f64 N/A

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

      5. /-lowering-/.f64 76.6

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

   8. Simplified 76.6%

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

4. Final simplification 82.1%

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

Alternative 8: 76.0% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(N[(N[(y * N[(N[(y * N[(N[(y * N[(N[(x * y), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision] + 27464.7644705), $MachinePrecision]), $MachinePrecision] + 230661.510616), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision] / N[(N[(y * N[(N[(y * N[(N[(y * N[(y + a), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision]), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(1.0 / N[(y * N[(y * N[(y * N[(y + a), $MachinePrecision] + b), $MachinePrecision] + c), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision] * N[(y * N[(y * 27464.7644705 + 230661.510616), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[(z * N[(N[(a / y), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $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 90.3%

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

      1. div-inv N/A

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

      2. *-lowering-*.f64 N/A

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

      3. *-commutative N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. /-lowering-/.f64 N/A

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

   4. Applied egg-rr 90.2%

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

   5. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{\frac{54929528941}{2000000}}, \frac{28832688827}{125000}\right), t\right) \cdot \frac{1}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)} \]
   6. Step-by-step derivation

   7. Simplified 75.1%

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

   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 y around -inf

      \[\leadsto \color{blue}{x + -1 \cdot \frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

         \[\leadsto \color{blue}{x - \frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \color{blue}{x - \frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]

   5. Simplified 60.6%

      \[\leadsto \color{blue}{x - \frac{\left(\left(0 - z\right) - \frac{27464.7644705 - \mathsf{fma}\left(\left(0 - z\right) + x \cdot a, 0 - a, x \cdot b\right)}{y}\right) + x \cdot a}{y}} \]

   6. Taylor expanded in z around inf

      \[\leadsto x - \frac{\color{blue}{z \cdot \left(\frac{a}{y} - 1\right)}}{y} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto x - \frac{\color{blue}{z \cdot \left(\frac{a}{y} - 1\right)}}{y} \]

      2. sub-neg N/A

         \[\leadsto x - \frac{z \cdot \color{blue}{\left(\frac{a}{y} + \left(\mathsf{neg}\left(1\right)\right)\right)}}{y} \]

      3. metadata-eval N/A

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

      4. +-lowering-+.f64 N/A

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

      5. /-lowering-/.f64 76.6

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

   8. Simplified 76.6%

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

4. Final simplification 75.7%

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

Alternative 9: 75.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(y \cdot \left(y \cdot \left(y + a\right) + b\right) + c\right) + i\\ \mathbf{if}\;\frac{y \cdot \left(y \cdot \left(y \cdot \left(x \cdot y + z\right) + 27464.7644705\right) + 230661.510616\right) + t}{t\_1} \leq \infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, 230661.510616, t\right)}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{z \cdot \left(\frac{a}{y} + -1\right)}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ (* y (+ (* y (+ (* y (+ y a)) b)) c)) i)))
   (if (<=
        (/
         (+
          (* y (+ (* y (+ (* y (+ (* x y) z)) 27464.7644705)) 230661.510616))
          t)
         t_1)
        INFINITY)
     (/ (fma y 230661.510616 t) t_1)
     (- x (/ (* z (+ (/ a y) -1.0)) y)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (y * ((y * ((y * (y + a)) + b)) + c)) + i;
	double tmp;
	if ((((y * ((y * ((y * ((x * y) + z)) + 27464.7644705)) + 230661.510616)) + t) / t_1) <= ((double) INFINITY)) {
		tmp = fma(y, 230661.510616, t) / t_1;
	} else {
		tmp = x - ((z * ((a / y) + -1.0)) / y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(y * Float64(Float64(y * Float64(Float64(y * Float64(y + a)) + b)) + c)) + i)
	tmp = 0.0
	if (Float64(Float64(Float64(y * Float64(Float64(y * Float64(Float64(y * Float64(Float64(x * y) + z)) + 27464.7644705)) + 230661.510616)) + t) / t_1) <= Inf)
		tmp = Float64(fma(y, 230661.510616, t) / t_1);
	else
		tmp = Float64(x - Float64(Float64(z * Float64(Float64(a / y) + -1.0)) / y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(y * N[(N[(y * N[(N[(y * N[(y + a), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision]), $MachinePrecision] + i), $MachinePrecision]}, If[LessEqual[N[(N[(N[(y * N[(N[(y * N[(N[(y * N[(N[(x * y), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision] + 27464.7644705), $MachinePrecision]), $MachinePrecision] + 230661.510616), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision] / t$95$1), $MachinePrecision], Infinity], N[(N[(y * 230661.510616 + t), $MachinePrecision] / t$95$1), $MachinePrecision], N[(x - N[(N[(z * N[(N[(a / y), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $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 90.3%

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

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{t + \frac{28832688827}{125000} \cdot y}}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   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. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 72.9

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

   5. Simplified 72.9%

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

   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 y around -inf

      \[\leadsto \color{blue}{x + -1 \cdot \frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

         \[\leadsto \color{blue}{x - \frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \color{blue}{x - \frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]

   5. Simplified 60.6%

      \[\leadsto \color{blue}{x - \frac{\left(\left(0 - z\right) - \frac{27464.7644705 - \mathsf{fma}\left(\left(0 - z\right) + x \cdot a, 0 - a, x \cdot b\right)}{y}\right) + x \cdot a}{y}} \]

   6. Taylor expanded in z around inf

      \[\leadsto x - \frac{\color{blue}{z \cdot \left(\frac{a}{y} - 1\right)}}{y} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto x - \frac{\color{blue}{z \cdot \left(\frac{a}{y} - 1\right)}}{y} \]

      2. sub-neg N/A

         \[\leadsto x - \frac{z \cdot \color{blue}{\left(\frac{a}{y} + \left(\mathsf{neg}\left(1\right)\right)\right)}}{y} \]

      3. metadata-eval N/A

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

      4. +-lowering-+.f64 N/A

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

      5. /-lowering-/.f64 76.6

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

   8. Simplified 76.6%

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

4. Final simplification 74.4%

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

Alternative 10: 75.0% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(N[(N[(y * N[(N[(y * N[(N[(y * N[(N[(x * y), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision] + 27464.7644705), $MachinePrecision]), $MachinePrecision] + 230661.510616), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision] / N[(N[(y * N[(N[(y * N[(N[(y * N[(y + a), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision]), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(1.0 / N[(y * N[(y * N[(y * N[(y + a), $MachinePrecision] + b), $MachinePrecision] + c), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision] * N[(y * 230661.510616 + t), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[(z * N[(N[(a / y), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $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 90.3%

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

      1. div-inv N/A

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

      2. *-lowering-*.f64 N/A

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

      3. *-commutative N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. /-lowering-/.f64 N/A

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

   4. Applied egg-rr 90.2%

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

   5. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{28832688827}{125000}}, t\right) \cdot \frac{1}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)} \]
   6. Step-by-step derivation

   7. Simplified 72.8%

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

   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 y around -inf

      \[\leadsto \color{blue}{x + -1 \cdot \frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

         \[\leadsto \color{blue}{x - \frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \color{blue}{x - \frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]

   5. Simplified 60.6%

      \[\leadsto \color{blue}{x - \frac{\left(\left(0 - z\right) - \frac{27464.7644705 - \mathsf{fma}\left(\left(0 - z\right) + x \cdot a, 0 - a, x \cdot b\right)}{y}\right) + x \cdot a}{y}} \]

   6. Taylor expanded in z around inf

      \[\leadsto x - \frac{\color{blue}{z \cdot \left(\frac{a}{y} - 1\right)}}{y} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto x - \frac{\color{blue}{z \cdot \left(\frac{a}{y} - 1\right)}}{y} \]

      2. sub-neg N/A

         \[\leadsto x - \frac{z \cdot \color{blue}{\left(\frac{a}{y} + \left(\mathsf{neg}\left(1\right)\right)\right)}}{y} \]

      3. metadata-eval N/A

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

      4. +-lowering-+.f64 N/A

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

      5. /-lowering-/.f64 76.6

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

   8. Simplified 76.6%

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

4. Final simplification 74.3%

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

Alternative 11: 68.3% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(N[(N[(y * N[(N[(y * N[(N[(y * N[(N[(x * y), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision] + 27464.7644705), $MachinePrecision]), $MachinePrecision] + 230661.510616), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision] / N[(N[(y * N[(N[(y * N[(N[(y * N[(y + a), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision]), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision], Infinity], N[(t / N[(y * N[(y * N[(y * N[(y + a), $MachinePrecision] + b), $MachinePrecision] + c), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[(z * N[(N[(a / y), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $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 90.3%

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

   3. Taylor expanded in t around 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. /-lowering-/.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. accelerator-lowering-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. +-lowering-+.f64 61.0

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

   5. Simplified 61.0%

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

   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 y around -inf

      \[\leadsto \color{blue}{x + -1 \cdot \frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

         \[\leadsto \color{blue}{x - \frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \color{blue}{x - \frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]

   5. Simplified 60.6%

      \[\leadsto \color{blue}{x - \frac{\left(\left(0 - z\right) - \frac{27464.7644705 - \mathsf{fma}\left(\left(0 - z\right) + x \cdot a, 0 - a, x \cdot b\right)}{y}\right) + x \cdot a}{y}} \]

   6. Taylor expanded in z around inf

      \[\leadsto x - \frac{\color{blue}{z \cdot \left(\frac{a}{y} - 1\right)}}{y} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto x - \frac{\color{blue}{z \cdot \left(\frac{a}{y} - 1\right)}}{y} \]

      2. sub-neg N/A

         \[\leadsto x - \frac{z \cdot \color{blue}{\left(\frac{a}{y} + \left(\mathsf{neg}\left(1\right)\right)\right)}}{y} \]

      3. metadata-eval N/A

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

      4. +-lowering-+.f64 N/A

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

      5. /-lowering-/.f64 76.6

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

   8. Simplified 76.6%

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

4. Final simplification 67.3%

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

Alternative 12: 66.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(y \cdot \left(y \cdot \left(x \cdot y + z\right) + 27464.7644705\right) + 230661.510616\right) + t}{y \cdot \left(y \cdot \left(y \cdot \left(y + a\right) + b\right) + c\right) + i} \leq \infty:\\ \;\;\;\;\frac{t}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\mathsf{fma}\left(a, x, 0 - z\right)}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<=
      (/
       (+
        (* y (+ (* y (+ (* y (+ (* x y) z)) 27464.7644705)) 230661.510616))
        t)
       (+ (* y (+ (* y (+ (* y (+ y a)) b)) c)) i))
      INFINITY)
   (/ t (fma y (fma y (fma y (+ y a) b) c) i))
   (- x (/ (fma a x (- 0.0 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 * ((y * ((y * ((x * y) + z)) + 27464.7644705)) + 230661.510616)) + t) / ((y * ((y * ((y * (y + a)) + b)) + c)) + i)) <= ((double) INFINITY)) {
		tmp = t / fma(y, fma(y, fma(y, (y + a), b), c), i);
	} else {
		tmp = x - (fma(a, x, (0.0 - z)) / y);
	}
	return tmp;
}

Julia

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

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(N[(N[(y * N[(N[(y * N[(N[(y * N[(N[(x * y), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision] + 27464.7644705), $MachinePrecision]), $MachinePrecision] + 230661.510616), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision] / N[(N[(y * N[(N[(y * N[(N[(y * N[(y + a), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision]), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision], Infinity], N[(t / N[(y * N[(y * N[(y * N[(y + a), $MachinePrecision] + b), $MachinePrecision] + c), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[(a * x + N[(0.0 - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $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 90.3%

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

   3. Taylor expanded in t around 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. /-lowering-/.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. accelerator-lowering-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. +-lowering-+.f64 61.0

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

   5. Simplified 61.0%

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

   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 y around -inf

      \[\leadsto \color{blue}{x + -1 \cdot \frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

         \[\leadsto \color{blue}{x - \frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \color{blue}{x - \frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]

   5. Simplified 60.6%

      \[\leadsto \color{blue}{x - \frac{\left(\left(0 - z\right) - \frac{27464.7644705 - \mathsf{fma}\left(\left(0 - z\right) + x \cdot a, 0 - a, x \cdot b\right)}{y}\right) + x \cdot a}{y}} \]

   6. Taylor expanded in y around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

         \[\leadsto x - \frac{a \cdot x + \color{blue}{-1 \cdot z}}{y} \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto x - \frac{\color{blue}{\mathsf{fma}\left(a, x, -1 \cdot z\right)}}{y} \]

      4. mul-1-neg N/A

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

      5. neg-lowering-neg.f64 70.6

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

   8. Simplified 70.6%

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

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(y \cdot \left(y \cdot \left(x \cdot y + z\right) + 27464.7644705\right) + 230661.510616\right) + t}{y \cdot \left(y \cdot \left(y \cdot \left(y + a\right) + b\right) + c\right) + i} \leq \infty:\\ \;\;\;\;\frac{t}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\mathsf{fma}\left(a, x, 0 - z\right)}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 57.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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 90.3%

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

   3. Taylor expanded in t around 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. /-lowering-/.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. accelerator-lowering-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. +-lowering-+.f64 61.0

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

   5. Simplified 61.0%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 53.9

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

   8. Simplified 53.9%

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

   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 y around inf

      \[\leadsto \color{blue}{x} \]
   4. Step-by-step derivation

   5. Simplified 65.4%

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

4. Final simplification 58.6%

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

Alternative 14: 50.2% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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 90.3%

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

   3. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 40.0

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

   5. Simplified 40.0%

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

   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 y around inf

      \[\leadsto \color{blue}{x} \]
   4. Step-by-step derivation

   5. Simplified 65.4%

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

4. Final simplification 50.3%

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

Alternative 15: 63.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \frac{\mathsf{fma}\left(a, x, 0 - z\right)}{y}\\ \mathbf{if}\;y \leq -6 \cdot 10^{+14}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 2.2:\\ \;\;\;\;\frac{t}{\mathsf{fma}\left(y, c, i\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (- x (/ (fma a x (- 0.0 z)) y))))
   (if (<= y -6e+14) t_1 (if (<= y 2.2) (/ t (fma y c 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 = x - (fma(a, x, (0.0 - z)) / y);
	double tmp;
	if (y <= -6e+14) {
		tmp = t_1;
	} else if (y <= 2.2) {
		tmp = t / fma(y, c, i);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(x - Float64(fma(a, x, Float64(0.0 - z)) / y))
	tmp = 0.0
	if (y <= -6e+14)
		tmp = t_1;
	elseif (y <= 2.2)
		tmp = Float64(t / fma(y, c, i));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(x - N[(N[(a * x + N[(0.0 - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -6e+14], t$95$1, If[LessEqual[y, 2.2], N[(t / N[(y * c + i), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -6e14 or 2.2000000000000002 < y

   1. Initial program 14.2%

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

   3. Taylor expanded in y around -inf

      \[\leadsto \color{blue}{x + -1 \cdot \frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

         \[\leadsto \color{blue}{x - \frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \color{blue}{x - \frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]

   5. Simplified 52.4%

      \[\leadsto \color{blue}{x - \frac{\left(\left(0 - z\right) - \frac{27464.7644705 - \mathsf{fma}\left(\left(0 - z\right) + x \cdot a, 0 - a, x \cdot b\right)}{y}\right) + x \cdot a}{y}} \]

   6. Taylor expanded in y around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

         \[\leadsto x - \frac{a \cdot x + \color{blue}{-1 \cdot z}}{y} \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto x - \frac{\color{blue}{\mathsf{fma}\left(a, x, -1 \cdot z\right)}}{y} \]

      4. mul-1-neg N/A

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

      5. neg-lowering-neg.f64 58.0

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

   8. Simplified 58.0%

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

   if -6e14 < y < 2.2000000000000002

   1. Initial program 99.7%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   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. /-lowering-/.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. accelerator-lowering-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. +-lowering-+.f64 73.2

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

   5. Simplified 73.2%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 67.7

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

   8. Simplified 67.7%

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

4. Final simplification 62.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6 \cdot 10^{+14}:\\ \;\;\;\;x - \frac{\mathsf{fma}\left(a, x, 0 - z\right)}{y}\\ \mathbf{elif}\;y \leq 2.2:\\ \;\;\;\;\frac{t}{\mathsf{fma}\left(y, c, i\right)}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\mathsf{fma}\left(a, x, 0 - z\right)}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 25.3% accurate, 71.0× speedup

Math

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

FPCore

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

C

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

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
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;
}

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := x

Derivation

1. Initial program 53.6%

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

3. Taylor expanded in y around inf

   \[\leadsto \color{blue}{x} \]
4. Step-by-step derivation

5. Simplified 30.2%

   \[\leadsto \color{blue}{x} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024195 
(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)))