Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, D

Percentage Accurate: 59.5% → 97.0%

Time: 16.2s

Alternatives: 14

Speedup: 11.3×

Specification

Math

\[\begin{array}{l} \\ x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (+
  x
  (/
   (*
    y
    (+ (* (+ (* (+ (* (+ (* z 3.13060547623) 11.1667541262) z) t) z) a) z) b))
   (+
    (* (+ (* (+ (* (+ z 15.234687407) z) 31.4690115749) z) 11.9400905721) z)
    0.607771387771))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return x + ((y * ((((((((z * 3.13060547623) + 11.1667541262) * z) + t) * z) + a) * z) + b)) / (((((((z + 15.234687407) * z) + 31.4690115749) * z) + 11.9400905721) * z) + 0.607771387771));
}

Fortran

real(8) function code(x, y, z, t, a, b)
    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
    code = x + ((y * ((((((((z * 3.13060547623d0) + 11.1667541262d0) * z) + t) * z) + a) * z) + b)) / (((((((z + 15.234687407d0) * z) + 31.4690115749d0) * z) + 11.9400905721d0) * z) + 0.607771387771d0))
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return x + ((y * ((((((((z * 3.13060547623) + 11.1667541262) * z) + t) * z) + a) * z) + b)) / (((((((z + 15.234687407) * z) + 31.4690115749) * z) + 11.9400905721) * z) + 0.607771387771));
}

Python

def code(x, y, z, t, a, b):
	return x + ((y * ((((((((z * 3.13060547623) + 11.1667541262) * z) + t) * z) + a) * z) + b)) / (((((((z + 15.234687407) * z) + 31.4690115749) * z) + 11.9400905721) * z) + 0.607771387771))

Julia

function code(x, y, z, t, a, b)
	return Float64(x + Float64(Float64(y * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(z * 3.13060547623) + 11.1667541262) * z) + t) * z) + a) * z) + b)) / Float64(Float64(Float64(Float64(Float64(Float64(Float64(z + 15.234687407) * z) + 31.4690115749) * z) + 11.9400905721) * z) + 0.607771387771)))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = x + ((y * ((((((((z * 3.13060547623) + 11.1667541262) * z) + t) * z) + a) * z) + b)) / (((((((z + 15.234687407) * z) + 31.4690115749) * z) + 11.9400905721) * z) + 0.607771387771));
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(x + N[(N[(y * N[(N[(N[(N[(N[(N[(N[(N[(z * 3.13060547623), $MachinePrecision] + 11.1667541262), $MachinePrecision] * z), $MachinePrecision] + t), $MachinePrecision] * z), $MachinePrecision] + a), $MachinePrecision] * z), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(N[(N[(N[(z + 15.234687407), $MachinePrecision] * z), $MachinePrecision] + 31.4690115749), $MachinePrecision] * z), $MachinePrecision] + 11.9400905721), $MachinePrecision] * z), $MachinePrecision] + 0.607771387771), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 59.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (+
  x
  (/
   (*
    y
    (+ (* (+ (* (+ (* (+ (* z 3.13060547623) 11.1667541262) z) t) z) a) z) b))
   (+
    (* (+ (* (+ (* (+ z 15.234687407) z) 31.4690115749) z) 11.9400905721) z)
    0.607771387771))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return x + ((y * ((((((((z * 3.13060547623) + 11.1667541262) * z) + t) * z) + a) * z) + b)) / (((((((z + 15.234687407) * z) + 31.4690115749) * z) + 11.9400905721) * z) + 0.607771387771));
}

Fortran

real(8) function code(x, y, z, t, a, b)
    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
    code = x + ((y * ((((((((z * 3.13060547623d0) + 11.1667541262d0) * z) + t) * z) + a) * z) + b)) / (((((((z + 15.234687407d0) * z) + 31.4690115749d0) * z) + 11.9400905721d0) * z) + 0.607771387771d0))
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return x + ((y * ((((((((z * 3.13060547623) + 11.1667541262) * z) + t) * z) + a) * z) + b)) / (((((((z + 15.234687407) * z) + 31.4690115749) * z) + 11.9400905721) * z) + 0.607771387771));
}

Python

def code(x, y, z, t, a, b):
	return x + ((y * ((((((((z * 3.13060547623) + 11.1667541262) * z) + t) * z) + a) * z) + b)) / (((((((z + 15.234687407) * z) + 31.4690115749) * z) + 11.9400905721) * z) + 0.607771387771))

Julia

function code(x, y, z, t, a, b)
	return Float64(x + Float64(Float64(y * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(z * 3.13060547623) + 11.1667541262) * z) + t) * z) + a) * z) + b)) / Float64(Float64(Float64(Float64(Float64(Float64(Float64(z + 15.234687407) * z) + 31.4690115749) * z) + 11.9400905721) * z) + 0.607771387771)))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = x + ((y * ((((((((z * 3.13060547623) + 11.1667541262) * z) + t) * z) + a) * z) + b)) / (((((((z + 15.234687407) * z) + 31.4690115749) * z) + 11.9400905721) * z) + 0.607771387771));
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(x + N[(N[(y * N[(N[(N[(N[(N[(N[(N[(N[(z * 3.13060547623), $MachinePrecision] + 11.1667541262), $MachinePrecision] * z), $MachinePrecision] + t), $MachinePrecision] * z), $MachinePrecision] + a), $MachinePrecision] * z), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(N[(N[(N[(z + 15.234687407), $MachinePrecision] * z), $MachinePrecision] + 31.4690115749), $MachinePrecision] * z), $MachinePrecision] + 11.9400905721), $MachinePrecision] * z), $MachinePrecision] + 0.607771387771), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 97.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.4 \cdot 10^{+16}:\\ \;\;\;\;\mathsf{fma}\left(3.13060547623 - \frac{36.52704169880642 - \frac{457.9610022158428 + t}{z}}{z}, y, x\right)\\ \mathbf{elif}\;z \leq 2 \cdot 10^{+50}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(t, z, a\right), z, b\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(15.234687407 + z, z, 31.4690115749\right), z, 11.9400905721\right), z, 0.607771387771\right)}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(3.13060547623, y, x\right) - \frac{\frac{y}{z}}{z} \cdot \left(-t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -5.4e+16)
   (fma
    (- 3.13060547623 (/ (- 36.52704169880642 (/ (+ 457.9610022158428 t) z)) z))
    y
    x)
   (if (<= z 2e+50)
     (fma
      (/
       (fma (fma t z a) z b)
       (fma
        (fma (fma (+ 15.234687407 z) z 31.4690115749) z 11.9400905721)
        z
        0.607771387771))
      y
      x)
     (- (fma 3.13060547623 y x) (* (/ (/ y z) z) (- t))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -5.4e+16) {
		tmp = fma((3.13060547623 - ((36.52704169880642 - ((457.9610022158428 + t) / z)) / z)), y, x);
	} else if (z <= 2e+50) {
		tmp = fma((fma(fma(t, z, a), z, b) / fma(fma(fma((15.234687407 + z), z, 31.4690115749), z, 11.9400905721), z, 0.607771387771)), y, x);
	} else {
		tmp = fma(3.13060547623, y, x) - (((y / z) / z) * -t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -5.4e+16)
		tmp = fma(Float64(3.13060547623 - Float64(Float64(36.52704169880642 - Float64(Float64(457.9610022158428 + t) / z)) / z)), y, x);
	elseif (z <= 2e+50)
		tmp = fma(Float64(fma(fma(t, z, a), z, b) / fma(fma(fma(Float64(15.234687407 + z), z, 31.4690115749), z, 11.9400905721), z, 0.607771387771)), y, x);
	else
		tmp = Float64(fma(3.13060547623, y, x) - Float64(Float64(Float64(y / z) / z) * Float64(-t)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -5.4e+16], N[(N[(3.13060547623 - N[(N[(36.52704169880642 - N[(N[(457.9610022158428 + t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[z, 2e+50], N[(N[(N[(N[(t * z + a), $MachinePrecision] * z + b), $MachinePrecision] / N[(N[(N[(N[(15.234687407 + z), $MachinePrecision] * z + 31.4690115749), $MachinePrecision] * z + 11.9400905721), $MachinePrecision] * z + 0.607771387771), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision], N[(N[(3.13060547623 * y + x), $MachinePrecision] - N[(N[(N[(y / z), $MachinePrecision] / z), $MachinePrecision] * (-t)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.4e16

   1. Initial program 20.3%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto x + \frac{y \cdot \color{blue}{\left(a \cdot z\right)}}{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}} \]
   4. Step-by-step derivation

      1. lower-*.f64 43.1

         \[\leadsto x + \frac{y \cdot \color{blue}{\left(a \cdot z\right)}}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]

   5. Applied rewrites 43.1%

      \[\leadsto x + \frac{y \cdot \color{blue}{\left(a \cdot z\right)}}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{y \cdot \left(a \cdot z\right)}{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{y \cdot \left(a \cdot z\right)}{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}} + x} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y \cdot \left(a \cdot z\right)}{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}}} + x \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(a \cdot z\right)}}{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}} + x \]

      5. associate-/l* N/A

         \[\leadsto \color{blue}{y \cdot \frac{a \cdot z}{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}}} + x \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\frac{a \cdot z}{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}} \cdot y} + x \]

   7. Applied rewrites 43.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z \cdot a}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(15.234687407 + z, z, 31.4690115749\right), z, 11.9400905721\right), z, 0.607771387771\right)}, y, x\right)} \]

   8. Taylor expanded in z around -inf

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{313060547623}{100000000000} + -1 \cdot \frac{\frac{3652704169880641883561}{100000000000000000000} + -1 \cdot \frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z}}{z}}, y, x\right) \]

   10. Applied rewrites 97.0%

       \[\leadsto \mathsf{fma}\left(\color{blue}{3.13060547623 - \frac{36.52704169880642 - \frac{457.9610022158428 + t}{z}}{z}}, y, x\right) \]

   if -5.4e16 < z < 2.0000000000000002e50

   1. Initial program 98.4%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto x + \frac{y \cdot \color{blue}{\left(a \cdot z\right)}}{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}} \]
   4. Step-by-step derivation

      1. lower-*.f64 58.4

         \[\leadsto x + \frac{y \cdot \color{blue}{\left(a \cdot z\right)}}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]

   5. Applied rewrites 58.4%

      \[\leadsto x + \frac{y \cdot \color{blue}{\left(a \cdot z\right)}}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{y \cdot \left(a \cdot z\right)}{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{y \cdot \left(a \cdot z\right)}{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}} + x} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y \cdot \left(a \cdot z\right)}{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}}} + x \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(a \cdot z\right)}}{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}} + x \]

      5. associate-/l* N/A

         \[\leadsto \color{blue}{y \cdot \frac{a \cdot z}{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}}} + x \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\frac{a \cdot z}{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}} \cdot y} + x \]

   7. Applied rewrites 59.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z \cdot a}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(15.234687407 + z, z, 31.4690115749\right), z, 11.9400905721\right), z, 0.607771387771\right)}, y, x\right)} \]

   8. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{b + z \cdot \left(a + t \cdot z\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{15234687407}{1000000000} + z, z, \frac{314690115749}{10000000000}\right), z, \frac{119400905721}{10000000000}\right), z, \frac{607771387771}{1000000000000}\right)}, y, x\right) \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{z \cdot \left(a + t \cdot z\right) + b}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{15234687407}{1000000000} + z, z, \frac{314690115749}{10000000000}\right), z, \frac{119400905721}{10000000000}\right), z, \frac{607771387771}{1000000000000}\right)}, y, x\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\left(a + t \cdot z\right) \cdot z} + b}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{15234687407}{1000000000} + z, z, \frac{314690115749}{10000000000}\right), z, \frac{119400905721}{10000000000}\right), z, \frac{607771387771}{1000000000000}\right)}, y, x\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\mathsf{fma}\left(a + t \cdot z, z, b\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{15234687407}{1000000000} + z, z, \frac{314690115749}{10000000000}\right), z, \frac{119400905721}{10000000000}\right), z, \frac{607771387771}{1000000000000}\right)}, y, x\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(\color{blue}{t \cdot z + a}, z, b\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{15234687407}{1000000000} + z, z, \frac{314690115749}{10000000000}\right), z, \frac{119400905721}{10000000000}\right), z, \frac{607771387771}{1000000000000}\right)}, y, x\right) \]

      5. lower-fma.f64 99.0

         \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(t, z, a\right)}, z, b\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(15.234687407 + z, z, 31.4690115749\right), z, 11.9400905721\right), z, 0.607771387771\right)}, y, x\right) \]

   10. Applied rewrites 99.0%

       \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(t, z, a\right), z, b\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(15.234687407 + z, z, 31.4690115749\right), z, 11.9400905721\right), z, 0.607771387771\right)}, y, x\right) \]

   if 2.0000000000000002e50 < z

   1. Initial program 3.7%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
   2. Add Preprocessing

   3. Taylor expanded in z around -inf

      \[\leadsto \color{blue}{x + \left(-1 \cdot \frac{\left(\frac{-55833770631}{5000000000} \cdot y + -1 \cdot \frac{t \cdot y - \left(\frac{-15234687407}{1000000000} \cdot \left(\frac{-55833770631}{5000000000} \cdot y - \frac{-4769379582500641883561}{100000000000000000000} \cdot y\right) + \frac{98517059967927196814627}{1000000000000000000000} \cdot y\right)}{z}\right) - \frac{-4769379582500641883561}{100000000000000000000} \cdot y}{z} + \frac{313060547623}{100000000000} \cdot y\right)} \]

   5. Applied rewrites 84.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(3.13060547623, y, x\right) - \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(t, y, \mathsf{fma}\left(-98.5170599679272, y, y \cdot 556.47806218377\right)\right)}{z}, -1, 36.52704169880642 \cdot y\right)}{z}} \]

   6. Taylor expanded in t around inf

      \[\leadsto \mathsf{fma}\left(\frac{313060547623}{100000000000}, y, x\right) - -1 \cdot \color{blue}{\frac{t \cdot y}{{z}^{2}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 99.9%

      \[\leadsto \mathsf{fma}\left(3.13060547623, y, x\right) - \left(-t\right) \cdot \color{blue}{\frac{\frac{y}{z}}{z}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.4 \cdot 10^{+16}:\\ \;\;\;\;\mathsf{fma}\left(3.13060547623 - \frac{36.52704169880642 - \frac{457.9610022158428 + t}{z}}{z}, y, x\right)\\ \mathbf{elif}\;z \leq 2 \cdot 10^{+50}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(t, z, a\right), z, b\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(15.234687407 + z, z, 31.4690115749\right), z, 11.9400905721\right), z, 0.607771387771\right)}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(3.13060547623, y, x\right) - \frac{\frac{y}{z}}{z} \cdot \left(-t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 70.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(1.6453555072203998 \cdot y\right) \cdot b\\ t_2 := \frac{\left(b + \left(a + \left(t + \left(11.1667541262 + 3.13060547623 \cdot z\right) \cdot z\right) \cdot z\right) \cdot z\right) \cdot y}{0.607771387771 + \left(11.9400905721 + \left(31.4690115749 + \left(15.234687407 + z\right) \cdot z\right) \cdot z\right) \cdot z}\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+20}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+183}:\\ \;\;\;\;\mathsf{fma}\left(3.13060547623, y, x\right)\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(3.13060547623, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (* 1.6453555072203998 y) b))
        (t_2
         (/
          (*
           (+
            b
            (*
             (+ a (* (+ t (* (+ 11.1667541262 (* 3.13060547623 z)) z)) z))
             z))
           y)
          (+
           0.607771387771
           (*
            (+ 11.9400905721 (* (+ 31.4690115749 (* (+ 15.234687407 z) z)) z))
            z)))))
   (if (<= t_2 -1e+20)
     t_1
     (if (<= t_2 2e+183)
       (fma 3.13060547623 y x)
       (if (<= t_2 INFINITY) t_1 (fma 3.13060547623 y x))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (1.6453555072203998 * y) * b;
	double t_2 = ((b + ((a + ((t + ((11.1667541262 + (3.13060547623 * z)) * z)) * z)) * z)) * y) / (0.607771387771 + ((11.9400905721 + ((31.4690115749 + ((15.234687407 + z) * z)) * z)) * z));
	double tmp;
	if (t_2 <= -1e+20) {
		tmp = t_1;
	} else if (t_2 <= 2e+183) {
		tmp = fma(3.13060547623, y, x);
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = fma(3.13060547623, y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(1.6453555072203998 * y) * b)
	t_2 = Float64(Float64(Float64(b + Float64(Float64(a + Float64(Float64(t + Float64(Float64(11.1667541262 + Float64(3.13060547623 * z)) * z)) * z)) * z)) * y) / Float64(0.607771387771 + Float64(Float64(11.9400905721 + Float64(Float64(31.4690115749 + Float64(Float64(15.234687407 + z) * z)) * z)) * z)))
	tmp = 0.0
	if (t_2 <= -1e+20)
		tmp = t_1;
	elseif (t_2 <= 2e+183)
		tmp = fma(3.13060547623, y, x);
	elseif (t_2 <= Inf)
		tmp = t_1;
	else
		tmp = fma(3.13060547623, y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(1.6453555072203998 * y), $MachinePrecision] * b), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(b + N[(N[(a + N[(N[(t + N[(N[(11.1667541262 + N[(3.13060547623 * z), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision] / N[(0.607771387771 + N[(N[(11.9400905721 + N[(N[(31.4690115749 + N[(N[(15.234687407 + z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+20], t$95$1, If[LessEqual[t$95$2, 2e+183], N[(3.13060547623 * y + x), $MachinePrecision], If[LessEqual[t$95$2, Infinity], t$95$1, N[(3.13060547623 * y + x), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 313060547623/100000000000 binary64)) #s(literal 55833770631/5000000000 binary64)) z) t) z) a) z) b)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 z #s(literal 15234687407/1000000000 binary64)) z) #s(literal 314690115749/10000000000 binary64)) z) #s(literal 119400905721/10000000000 binary64)) z) #s(literal 607771387771/1000000000000 binary64))) < -1e20 or 1.99999999999999989e183 < (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 313060547623/100000000000 binary64)) #s(literal 55833770631/5000000000 binary64)) z) t) z) a) z) b)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 z #s(literal 15234687407/1000000000 binary64)) z) #s(literal 314690115749/10000000000 binary64)) z) #s(literal 119400905721/10000000000 binary64)) z) #s(literal 607771387771/1000000000000 binary64))) < +inf.0

   1. Initial program 87.2%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{b \cdot y}{\frac{607771387771}{1000000000000} + z \cdot \left(\frac{119400905721}{10000000000} + z \cdot \left(\frac{314690115749}{10000000000} + z \cdot \left(\frac{15234687407}{1000000000} + z\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{b}{\frac{607771387771}{1000000000000} + z \cdot \left(\frac{119400905721}{10000000000} + z \cdot \left(\frac{314690115749}{10000000000} + z \cdot \left(\frac{15234687407}{1000000000} + z\right)\right)\right)} \cdot y} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{b}{\frac{607771387771}{1000000000000} + z \cdot \left(\frac{119400905721}{10000000000} + z \cdot \left(\frac{314690115749}{10000000000} + z \cdot \left(\frac{15234687407}{1000000000} + z\right)\right)\right)} \cdot y} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{b}{\frac{607771387771}{1000000000000} + z \cdot \left(\frac{119400905721}{10000000000} + z \cdot \left(\frac{314690115749}{10000000000} + z \cdot \left(\frac{15234687407}{1000000000} + z\right)\right)\right)}} \cdot y \]

      4. +-commutative N/A

         \[\leadsto \frac{b}{\color{blue}{z \cdot \left(\frac{119400905721}{10000000000} + z \cdot \left(\frac{314690115749}{10000000000} + z \cdot \left(\frac{15234687407}{1000000000} + z\right)\right)\right) + \frac{607771387771}{1000000000000}}} \cdot y \]

      5. *-commutative N/A

         \[\leadsto \frac{b}{\color{blue}{\left(\frac{119400905721}{10000000000} + z \cdot \left(\frac{314690115749}{10000000000} + z \cdot \left(\frac{15234687407}{1000000000} + z\right)\right)\right) \cdot z} + \frac{607771387771}{1000000000000}} \cdot y \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{b}{\color{blue}{\mathsf{fma}\left(\frac{119400905721}{10000000000} + z \cdot \left(\frac{314690115749}{10000000000} + z \cdot \left(\frac{15234687407}{1000000000} + z\right)\right), z, \frac{607771387771}{1000000000000}\right)}} \cdot y \]

      7. +-commutative N/A

         \[\leadsto \frac{b}{\mathsf{fma}\left(\color{blue}{z \cdot \left(\frac{314690115749}{10000000000} + z \cdot \left(\frac{15234687407}{1000000000} + z\right)\right) + \frac{119400905721}{10000000000}}, z, \frac{607771387771}{1000000000000}\right)} \cdot y \]

      8. *-commutative N/A

         \[\leadsto \frac{b}{\mathsf{fma}\left(\color{blue}{\left(\frac{314690115749}{10000000000} + z \cdot \left(\frac{15234687407}{1000000000} + z\right)\right) \cdot z} + \frac{119400905721}{10000000000}, z, \frac{607771387771}{1000000000000}\right)} \cdot y \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{b}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{314690115749}{10000000000} + z \cdot \left(\frac{15234687407}{1000000000} + z\right), z, \frac{119400905721}{10000000000}\right)}, z, \frac{607771387771}{1000000000000}\right)} \cdot y \]

      10. +-commutative N/A

          \[\leadsto \frac{b}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{z \cdot \left(\frac{15234687407}{1000000000} + z\right) + \frac{314690115749}{10000000000}}, z, \frac{119400905721}{10000000000}\right), z, \frac{607771387771}{1000000000000}\right)} \cdot y \]

      11. *-commutative N/A

          \[\leadsto \frac{b}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\left(\frac{15234687407}{1000000000} + z\right) \cdot z} + \frac{314690115749}{10000000000}, z, \frac{119400905721}{10000000000}\right), z, \frac{607771387771}{1000000000000}\right)} \cdot y \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{b}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{15234687407}{1000000000} + z, z, \frac{314690115749}{10000000000}\right)}, z, \frac{119400905721}{10000000000}\right), z, \frac{607771387771}{1000000000000}\right)} \cdot y \]

      13. lower-+.f64 49.5

          \[\leadsto \frac{b}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{15.234687407 + z}, z, 31.4690115749\right), z, 11.9400905721\right), z, 0.607771387771\right)} \cdot y \]

   5. Applied rewrites 49.5%

      \[\leadsto \color{blue}{\frac{b}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(15.234687407 + z, z, 31.4690115749\right), z, 11.9400905721\right), z, 0.607771387771\right)} \cdot y} \]

   6. Taylor expanded in z around 0

      \[\leadsto \frac{-11940090572100000000000000}{369386059793087248348441} \cdot \left(b \cdot \left(y \cdot z\right)\right) + \color{blue}{\frac{1000000000000}{607771387771} \cdot \left(b \cdot y\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 48.2%

      \[\leadsto \mathsf{fma}\left(-32.324150453290734 \cdot \left(z \cdot y\right), \color{blue}{b}, 1.6453555072203998 \cdot \left(b \cdot y\right)\right) \]

   9. Taylor expanded in z around 0

      \[\leadsto \frac{1000000000000}{607771387771} \cdot \color{blue}{\left(b \cdot y\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 48.5%

       \[\leadsto \left(1.6453555072203998 \cdot y\right) \cdot \color{blue}{b} \]

   if -1e20 < (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 313060547623/100000000000 binary64)) #s(literal 55833770631/5000000000 binary64)) z) t) z) a) z) b)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 z #s(literal 15234687407/1000000000 binary64)) z) #s(literal 314690115749/10000000000 binary64)) z) #s(literal 119400905721/10000000000 binary64)) z) #s(literal 607771387771/1000000000000 binary64))) < 1.99999999999999989e183 or +inf.0 < (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 313060547623/100000000000 binary64)) #s(literal 55833770631/5000000000 binary64)) z) t) z) a) z) b)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 z #s(literal 15234687407/1000000000 binary64)) z) #s(literal 314690115749/10000000000 binary64)) z) #s(literal 119400905721/10000000000 binary64)) z) #s(literal 607771387771/1000000000000 binary64)))

   1. Initial program 47.1%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{x + \frac{313060547623}{100000000000} \cdot y} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{313060547623}{100000000000} \cdot y + x} \]

      2. lower-fma.f64 79.9

         \[\leadsto \color{blue}{\mathsf{fma}\left(3.13060547623, y, x\right)} \]

   5. Applied rewrites 79.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(3.13060547623, y, x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(b + \left(a + \left(t + \left(11.1667541262 + 3.13060547623 \cdot z\right) \cdot z\right) \cdot z\right) \cdot z\right) \cdot y}{0.607771387771 + \left(11.9400905721 + \left(31.4690115749 + \left(15.234687407 + z\right) \cdot z\right) \cdot z\right) \cdot z} \leq -1 \cdot 10^{+20}:\\ \;\;\;\;\left(1.6453555072203998 \cdot y\right) \cdot b\\ \mathbf{elif}\;\frac{\left(b + \left(a + \left(t + \left(11.1667541262 + 3.13060547623 \cdot z\right) \cdot z\right) \cdot z\right) \cdot z\right) \cdot y}{0.607771387771 + \left(11.9400905721 + \left(31.4690115749 + \left(15.234687407 + z\right) \cdot z\right) \cdot z\right) \cdot z} \leq 2 \cdot 10^{+183}:\\ \;\;\;\;\mathsf{fma}\left(3.13060547623, y, x\right)\\ \mathbf{elif}\;\frac{\left(b + \left(a + \left(t + \left(11.1667541262 + 3.13060547623 \cdot z\right) \cdot z\right) \cdot z\right) \cdot z\right) \cdot y}{0.607771387771 + \left(11.9400905721 + \left(31.4690115749 + \left(15.234687407 + z\right) \cdot z\right) \cdot z\right) \cdot z} \leq \infty:\\ \;\;\;\;\left(1.6453555072203998 \cdot y\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(3.13060547623, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 70.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(b \cdot y\right) \cdot 1.6453555072203998\\ t_2 := \frac{\left(b + \left(a + \left(t + \left(11.1667541262 + 3.13060547623 \cdot z\right) \cdot z\right) \cdot z\right) \cdot z\right) \cdot y}{0.607771387771 + \left(11.9400905721 + \left(31.4690115749 + \left(15.234687407 + z\right) \cdot z\right) \cdot z\right) \cdot z}\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+20}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+183}:\\ \;\;\;\;\mathsf{fma}\left(3.13060547623, y, x\right)\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(3.13060547623, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (* b y) 1.6453555072203998))
        (t_2
         (/
          (*
           (+
            b
            (*
             (+ a (* (+ t (* (+ 11.1667541262 (* 3.13060547623 z)) z)) z))
             z))
           y)
          (+
           0.607771387771
           (*
            (+ 11.9400905721 (* (+ 31.4690115749 (* (+ 15.234687407 z) z)) z))
            z)))))
   (if (<= t_2 -1e+20)
     t_1
     (if (<= t_2 2e+183)
       (fma 3.13060547623 y x)
       (if (<= t_2 INFINITY) t_1 (fma 3.13060547623 y x))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (b * y) * 1.6453555072203998;
	double t_2 = ((b + ((a + ((t + ((11.1667541262 + (3.13060547623 * z)) * z)) * z)) * z)) * y) / (0.607771387771 + ((11.9400905721 + ((31.4690115749 + ((15.234687407 + z) * z)) * z)) * z));
	double tmp;
	if (t_2 <= -1e+20) {
		tmp = t_1;
	} else if (t_2 <= 2e+183) {
		tmp = fma(3.13060547623, y, x);
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = fma(3.13060547623, y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(b * y) * 1.6453555072203998)
	t_2 = Float64(Float64(Float64(b + Float64(Float64(a + Float64(Float64(t + Float64(Float64(11.1667541262 + Float64(3.13060547623 * z)) * z)) * z)) * z)) * y) / Float64(0.607771387771 + Float64(Float64(11.9400905721 + Float64(Float64(31.4690115749 + Float64(Float64(15.234687407 + z) * z)) * z)) * z)))
	tmp = 0.0
	if (t_2 <= -1e+20)
		tmp = t_1;
	elseif (t_2 <= 2e+183)
		tmp = fma(3.13060547623, y, x);
	elseif (t_2 <= Inf)
		tmp = t_1;
	else
		tmp = fma(3.13060547623, y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(b * y), $MachinePrecision] * 1.6453555072203998), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(b + N[(N[(a + N[(N[(t + N[(N[(11.1667541262 + N[(3.13060547623 * z), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision] / N[(0.607771387771 + N[(N[(11.9400905721 + N[(N[(31.4690115749 + N[(N[(15.234687407 + z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+20], t$95$1, If[LessEqual[t$95$2, 2e+183], N[(3.13060547623 * y + x), $MachinePrecision], If[LessEqual[t$95$2, Infinity], t$95$1, N[(3.13060547623 * y + x), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 313060547623/100000000000 binary64)) #s(literal 55833770631/5000000000 binary64)) z) t) z) a) z) b)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 z #s(literal 15234687407/1000000000 binary64)) z) #s(literal 314690115749/10000000000 binary64)) z) #s(literal 119400905721/10000000000 binary64)) z) #s(literal 607771387771/1000000000000 binary64))) < -1e20 or 1.99999999999999989e183 < (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 313060547623/100000000000 binary64)) #s(literal 55833770631/5000000000 binary64)) z) t) z) a) z) b)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 z #s(literal 15234687407/1000000000 binary64)) z) #s(literal 314690115749/10000000000 binary64)) z) #s(literal 119400905721/10000000000 binary64)) z) #s(literal 607771387771/1000000000000 binary64))) < +inf.0

   1. Initial program 87.2%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{b \cdot y}{\frac{607771387771}{1000000000000} + z \cdot \left(\frac{119400905721}{10000000000} + z \cdot \left(\frac{314690115749}{10000000000} + z \cdot \left(\frac{15234687407}{1000000000} + z\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{b}{\frac{607771387771}{1000000000000} + z \cdot \left(\frac{119400905721}{10000000000} + z \cdot \left(\frac{314690115749}{10000000000} + z \cdot \left(\frac{15234687407}{1000000000} + z\right)\right)\right)} \cdot y} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{b}{\frac{607771387771}{1000000000000} + z \cdot \left(\frac{119400905721}{10000000000} + z \cdot \left(\frac{314690115749}{10000000000} + z \cdot \left(\frac{15234687407}{1000000000} + z\right)\right)\right)} \cdot y} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{b}{\frac{607771387771}{1000000000000} + z \cdot \left(\frac{119400905721}{10000000000} + z \cdot \left(\frac{314690115749}{10000000000} + z \cdot \left(\frac{15234687407}{1000000000} + z\right)\right)\right)}} \cdot y \]

      4. +-commutative N/A

         \[\leadsto \frac{b}{\color{blue}{z \cdot \left(\frac{119400905721}{10000000000} + z \cdot \left(\frac{314690115749}{10000000000} + z \cdot \left(\frac{15234687407}{1000000000} + z\right)\right)\right) + \frac{607771387771}{1000000000000}}} \cdot y \]

      5. *-commutative N/A

         \[\leadsto \frac{b}{\color{blue}{\left(\frac{119400905721}{10000000000} + z \cdot \left(\frac{314690115749}{10000000000} + z \cdot \left(\frac{15234687407}{1000000000} + z\right)\right)\right) \cdot z} + \frac{607771387771}{1000000000000}} \cdot y \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{b}{\color{blue}{\mathsf{fma}\left(\frac{119400905721}{10000000000} + z \cdot \left(\frac{314690115749}{10000000000} + z \cdot \left(\frac{15234687407}{1000000000} + z\right)\right), z, \frac{607771387771}{1000000000000}\right)}} \cdot y \]

      7. +-commutative N/A

         \[\leadsto \frac{b}{\mathsf{fma}\left(\color{blue}{z \cdot \left(\frac{314690115749}{10000000000} + z \cdot \left(\frac{15234687407}{1000000000} + z\right)\right) + \frac{119400905721}{10000000000}}, z, \frac{607771387771}{1000000000000}\right)} \cdot y \]

      8. *-commutative N/A

         \[\leadsto \frac{b}{\mathsf{fma}\left(\color{blue}{\left(\frac{314690115749}{10000000000} + z \cdot \left(\frac{15234687407}{1000000000} + z\right)\right) \cdot z} + \frac{119400905721}{10000000000}, z, \frac{607771387771}{1000000000000}\right)} \cdot y \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{b}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{314690115749}{10000000000} + z \cdot \left(\frac{15234687407}{1000000000} + z\right), z, \frac{119400905721}{10000000000}\right)}, z, \frac{607771387771}{1000000000000}\right)} \cdot y \]

      10. +-commutative N/A

          \[\leadsto \frac{b}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{z \cdot \left(\frac{15234687407}{1000000000} + z\right) + \frac{314690115749}{10000000000}}, z, \frac{119400905721}{10000000000}\right), z, \frac{607771387771}{1000000000000}\right)} \cdot y \]

      11. *-commutative N/A

          \[\leadsto \frac{b}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\left(\frac{15234687407}{1000000000} + z\right) \cdot z} + \frac{314690115749}{10000000000}, z, \frac{119400905721}{10000000000}\right), z, \frac{607771387771}{1000000000000}\right)} \cdot y \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{b}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{15234687407}{1000000000} + z, z, \frac{314690115749}{10000000000}\right)}, z, \frac{119400905721}{10000000000}\right), z, \frac{607771387771}{1000000000000}\right)} \cdot y \]

      13. lower-+.f64 49.5

          \[\leadsto \frac{b}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{15.234687407 + z}, z, 31.4690115749\right), z, 11.9400905721\right), z, 0.607771387771\right)} \cdot y \]

   5. Applied rewrites 49.5%

      \[\leadsto \color{blue}{\frac{b}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(15.234687407 + z, z, 31.4690115749\right), z, 11.9400905721\right), z, 0.607771387771\right)} \cdot y} \]

   6. Taylor expanded in z around 0

      \[\leadsto \frac{1000000000000}{607771387771} \cdot \color{blue}{\left(b \cdot y\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 48.5%

      \[\leadsto 1.6453555072203998 \cdot \color{blue}{\left(b \cdot y\right)} \]

   if -1e20 < (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 313060547623/100000000000 binary64)) #s(literal 55833770631/5000000000 binary64)) z) t) z) a) z) b)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 z #s(literal 15234687407/1000000000 binary64)) z) #s(literal 314690115749/10000000000 binary64)) z) #s(literal 119400905721/10000000000 binary64)) z) #s(literal 607771387771/1000000000000 binary64))) < 1.99999999999999989e183 or +inf.0 < (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 313060547623/100000000000 binary64)) #s(literal 55833770631/5000000000 binary64)) z) t) z) a) z) b)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 z #s(literal 15234687407/1000000000 binary64)) z) #s(literal 314690115749/10000000000 binary64)) z) #s(literal 119400905721/10000000000 binary64)) z) #s(literal 607771387771/1000000000000 binary64)))

   1. Initial program 47.1%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{x + \frac{313060547623}{100000000000} \cdot y} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{313060547623}{100000000000} \cdot y + x} \]

      2. lower-fma.f64 79.9

         \[\leadsto \color{blue}{\mathsf{fma}\left(3.13060547623, y, x\right)} \]

   5. Applied rewrites 79.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(3.13060547623, y, x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(b + \left(a + \left(t + \left(11.1667541262 + 3.13060547623 \cdot z\right) \cdot z\right) \cdot z\right) \cdot z\right) \cdot y}{0.607771387771 + \left(11.9400905721 + \left(31.4690115749 + \left(15.234687407 + z\right) \cdot z\right) \cdot z\right) \cdot z} \leq -1 \cdot 10^{+20}:\\ \;\;\;\;\left(b \cdot y\right) \cdot 1.6453555072203998\\ \mathbf{elif}\;\frac{\left(b + \left(a + \left(t + \left(11.1667541262 + 3.13060547623 \cdot z\right) \cdot z\right) \cdot z\right) \cdot z\right) \cdot y}{0.607771387771 + \left(11.9400905721 + \left(31.4690115749 + \left(15.234687407 + z\right) \cdot z\right) \cdot z\right) \cdot z} \leq 2 \cdot 10^{+183}:\\ \;\;\;\;\mathsf{fma}\left(3.13060547623, y, x\right)\\ \mathbf{elif}\;\frac{\left(b + \left(a + \left(t + \left(11.1667541262 + 3.13060547623 \cdot z\right) \cdot z\right) \cdot z\right) \cdot z\right) \cdot y}{0.607771387771 + \left(11.9400905721 + \left(31.4690115749 + \left(15.234687407 + z\right) \cdot z\right) \cdot z\right) \cdot z} \leq \infty:\\ \;\;\;\;\left(b \cdot y\right) \cdot 1.6453555072203998\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(3.13060547623, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(15.234687407 + z, z, 31.4690115749\right), z, 11.9400905721\right), z, 0.607771387771\right)\\ \mathbf{if}\;\frac{\left(b + \left(a + \left(t + \left(11.1667541262 + 3.13060547623 \cdot z\right) \cdot z\right) \cdot z\right) \cdot z\right) \cdot y}{0.607771387771 + \left(11.9400905721 + \left(31.4690115749 + \left(15.234687407 + z\right) \cdot z\right) \cdot z\right) \cdot z} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(3.13060547623, z, 11.1667541262\right), z, t\right), z, a\right), \frac{z}{t\_1}, \frac{b}{t\_1}\right), y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(3.13060547623, y, x\right) - \frac{\frac{y}{z}}{z} \cdot \left(-t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1
         (fma
          (fma (fma (+ 15.234687407 z) z 31.4690115749) z 11.9400905721)
          z
          0.607771387771)))
   (if (<=
        (/
         (*
          (+
           b
           (* (+ a (* (+ t (* (+ 11.1667541262 (* 3.13060547623 z)) z)) z)) z))
          y)
         (+
          0.607771387771
          (*
           (+ 11.9400905721 (* (+ 31.4690115749 (* (+ 15.234687407 z) z)) z))
           z)))
        INFINITY)
     (fma
      (fma
       (fma (fma (fma 3.13060547623 z 11.1667541262) z t) z a)
       (/ z t_1)
       (/ b t_1))
      y
      x)
     (- (fma 3.13060547623 y x) (* (/ (/ y z) z) (- t))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(fma(fma((15.234687407 + z), z, 31.4690115749), z, 11.9400905721), z, 0.607771387771);
	double tmp;
	if ((((b + ((a + ((t + ((11.1667541262 + (3.13060547623 * z)) * z)) * z)) * z)) * y) / (0.607771387771 + ((11.9400905721 + ((31.4690115749 + ((15.234687407 + z) * z)) * z)) * z))) <= ((double) INFINITY)) {
		tmp = fma(fma(fma(fma(fma(3.13060547623, z, 11.1667541262), z, t), z, a), (z / t_1), (b / t_1)), y, x);
	} else {
		tmp = fma(3.13060547623, y, x) - (((y / z) / z) * -t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(fma(fma(Float64(15.234687407 + z), z, 31.4690115749), z, 11.9400905721), z, 0.607771387771)
	tmp = 0.0
	if (Float64(Float64(Float64(b + Float64(Float64(a + Float64(Float64(t + Float64(Float64(11.1667541262 + Float64(3.13060547623 * z)) * z)) * z)) * z)) * y) / Float64(0.607771387771 + Float64(Float64(11.9400905721 + Float64(Float64(31.4690115749 + Float64(Float64(15.234687407 + z) * z)) * z)) * z))) <= Inf)
		tmp = fma(fma(fma(fma(fma(3.13060547623, z, 11.1667541262), z, t), z, a), Float64(z / t_1), Float64(b / t_1)), y, x);
	else
		tmp = Float64(fma(3.13060547623, y, x) - Float64(Float64(Float64(y / z) / z) * Float64(-t)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(N[(15.234687407 + z), $MachinePrecision] * z + 31.4690115749), $MachinePrecision] * z + 11.9400905721), $MachinePrecision] * z + 0.607771387771), $MachinePrecision]}, If[LessEqual[N[(N[(N[(b + N[(N[(a + N[(N[(t + N[(N[(11.1667541262 + N[(3.13060547623 * z), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision] / N[(0.607771387771 + N[(N[(11.9400905721 + N[(N[(31.4690115749 + N[(N[(15.234687407 + z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[(N[(N[(3.13060547623 * z + 11.1667541262), $MachinePrecision] * z + t), $MachinePrecision] * z + a), $MachinePrecision] * N[(z / t$95$1), $MachinePrecision] + N[(b / t$95$1), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision], N[(N[(3.13060547623 * y + x), $MachinePrecision] - N[(N[(N[(y / z), $MachinePrecision] / z), $MachinePrecision] * (-t)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 313060547623/100000000000 binary64)) #s(literal 55833770631/5000000000 binary64)) z) t) z) a) z) b)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 z #s(literal 15234687407/1000000000 binary64)) z) #s(literal 314690115749/10000000000 binary64)) z) #s(literal 119400905721/10000000000 binary64)) z) #s(literal 607771387771/1000000000000 binary64))) < +inf.0

   1. Initial program 93.9%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0

      \[\leadsto \color{blue}{x + \left(\frac{b \cdot y}{\frac{607771387771}{1000000000000} + z \cdot \left(\frac{119400905721}{10000000000} + z \cdot \left(\frac{314690115749}{10000000000} + z \cdot \left(\frac{15234687407}{1000000000} + z\right)\right)\right)} + \frac{y \cdot \left(z \cdot \left(a + z \cdot \left(t + z \cdot \left(\frac{55833770631}{5000000000} + \frac{313060547623}{100000000000} \cdot z\right)\right)\right)\right)}{\frac{607771387771}{1000000000000} + z \cdot \left(\frac{119400905721}{10000000000} + z \cdot \left(\frac{314690115749}{10000000000} + z \cdot \left(\frac{15234687407}{1000000000} + z\right)\right)\right)}\right)} \]

   5. Applied rewrites 97.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(3.13060547623, z, 11.1667541262\right), z, t\right), z, a\right), \frac{z}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(15.234687407 + z, z, 31.4690115749\right), z, 11.9400905721\right), z, 0.607771387771\right)}, \frac{b}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(15.234687407 + z, z, 31.4690115749\right), z, 11.9400905721\right), z, 0.607771387771\right)}\right), y, x\right)} \]

   if +inf.0 < (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 313060547623/100000000000 binary64)) #s(literal 55833770631/5000000000 binary64)) z) t) z) a) z) b)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 z #s(literal 15234687407/1000000000 binary64)) z) #s(literal 314690115749/10000000000 binary64)) z) #s(literal 119400905721/10000000000 binary64)) z) #s(literal 607771387771/1000000000000 binary64)))

   1. Initial program 0.0%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
   2. Add Preprocessing

   3. Taylor expanded in z around -inf

      \[\leadsto \color{blue}{x + \left(-1 \cdot \frac{\left(\frac{-55833770631}{5000000000} \cdot y + -1 \cdot \frac{t \cdot y - \left(\frac{-15234687407}{1000000000} \cdot \left(\frac{-55833770631}{5000000000} \cdot y - \frac{-4769379582500641883561}{100000000000000000000} \cdot y\right) + \frac{98517059967927196814627}{1000000000000000000000} \cdot y\right)}{z}\right) - \frac{-4769379582500641883561}{100000000000000000000} \cdot y}{z} + \frac{313060547623}{100000000000} \cdot y\right)} \]

   5. Applied rewrites 82.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(3.13060547623, y, x\right) - \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(t, y, \mathsf{fma}\left(-98.5170599679272, y, y \cdot 556.47806218377\right)\right)}{z}, -1, 36.52704169880642 \cdot y\right)}{z}} \]

   6. Taylor expanded in t around inf

      \[\leadsto \mathsf{fma}\left(\frac{313060547623}{100000000000}, y, x\right) - -1 \cdot \color{blue}{\frac{t \cdot y}{{z}^{2}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 99.9%

      \[\leadsto \mathsf{fma}\left(3.13060547623, y, x\right) - \left(-t\right) \cdot \color{blue}{\frac{\frac{y}{z}}{z}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(b + \left(a + \left(t + \left(11.1667541262 + 3.13060547623 \cdot z\right) \cdot z\right) \cdot z\right) \cdot z\right) \cdot y}{0.607771387771 + \left(11.9400905721 + \left(31.4690115749 + \left(15.234687407 + z\right) \cdot z\right) \cdot z\right) \cdot z} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(3.13060547623, z, 11.1667541262\right), z, t\right), z, a\right), \frac{z}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(15.234687407 + z, z, 31.4690115749\right), z, 11.9400905721\right), z, 0.607771387771\right)}, \frac{b}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(15.234687407 + z, z, 31.4690115749\right), z, 11.9400905721\right), z, 0.607771387771\right)}\right), y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(3.13060547623, y, x\right) - \frac{\frac{y}{z}}{z} \cdot \left(-t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(b + \left(a + \left(t + \left(11.1667541262 + 3.13060547623 \cdot z\right) \cdot z\right) \cdot z\right) \cdot z\right) \cdot y}{0.607771387771 + \left(11.9400905721 + \left(31.4690115749 + \left(15.234687407 + z\right) \cdot z\right) \cdot z\right) \cdot z} \leq \infty:\\ \;\;\;\;x - \frac{-1}{\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(15.234687407 + z, z, 31.4690115749\right), z, 11.9400905721\right), z, 0.607771387771\right)}{y}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(3.13060547623, z, 11.1667541262\right), z, t\right), z, a\right), z, b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(3.13060547623, y, x\right) - \frac{\frac{y}{z}}{z} \cdot \left(-t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<=
      (/
       (*
        (+
         b
         (* (+ a (* (+ t (* (+ 11.1667541262 (* 3.13060547623 z)) z)) z)) z))
        y)
       (+
        0.607771387771
        (*
         (+ 11.9400905721 (* (+ 31.4690115749 (* (+ 15.234687407 z) z)) z))
         z)))
      INFINITY)
   (-
    x
    (/
     -1.0
     (/
      (/
       (fma
        (fma (fma (+ 15.234687407 z) z 31.4690115749) z 11.9400905721)
        z
        0.607771387771)
       y)
      (fma (fma (fma (fma 3.13060547623 z 11.1667541262) z t) z a) z b))))
   (- (fma 3.13060547623 y x) (* (/ (/ y z) z) (- t)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((((b + ((a + ((t + ((11.1667541262 + (3.13060547623 * z)) * z)) * z)) * z)) * y) / (0.607771387771 + ((11.9400905721 + ((31.4690115749 + ((15.234687407 + z) * z)) * z)) * z))) <= ((double) INFINITY)) {
		tmp = x - (-1.0 / ((fma(fma(fma((15.234687407 + z), z, 31.4690115749), z, 11.9400905721), z, 0.607771387771) / y) / fma(fma(fma(fma(3.13060547623, z, 11.1667541262), z, t), z, a), z, b)));
	} else {
		tmp = fma(3.13060547623, y, x) - (((y / z) / z) * -t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(Float64(Float64(b + Float64(Float64(a + Float64(Float64(t + Float64(Float64(11.1667541262 + Float64(3.13060547623 * z)) * z)) * z)) * z)) * y) / Float64(0.607771387771 + Float64(Float64(11.9400905721 + Float64(Float64(31.4690115749 + Float64(Float64(15.234687407 + z) * z)) * z)) * z))) <= Inf)
		tmp = Float64(x - Float64(-1.0 / Float64(Float64(fma(fma(fma(Float64(15.234687407 + z), z, 31.4690115749), z, 11.9400905721), z, 0.607771387771) / y) / fma(fma(fma(fma(3.13060547623, z, 11.1667541262), z, t), z, a), z, b))));
	else
		tmp = Float64(fma(3.13060547623, y, x) - Float64(Float64(Float64(y / z) / z) * Float64(-t)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(N[(N[(b + N[(N[(a + N[(N[(t + N[(N[(11.1667541262 + N[(3.13060547623 * z), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision] / N[(0.607771387771 + N[(N[(11.9400905721 + N[(N[(31.4690115749 + N[(N[(15.234687407 + z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(x - N[(-1.0 / N[(N[(N[(N[(N[(N[(15.234687407 + z), $MachinePrecision] * z + 31.4690115749), $MachinePrecision] * z + 11.9400905721), $MachinePrecision] * z + 0.607771387771), $MachinePrecision] / y), $MachinePrecision] / N[(N[(N[(N[(3.13060547623 * z + 11.1667541262), $MachinePrecision] * z + t), $MachinePrecision] * z + a), $MachinePrecision] * z + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(3.13060547623 * y + x), $MachinePrecision] - N[(N[(N[(y / z), $MachinePrecision] / z), $MachinePrecision] * (-t)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 313060547623/100000000000 binary64)) #s(literal 55833770631/5000000000 binary64)) z) t) z) a) z) b)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 z #s(literal 15234687407/1000000000 binary64)) z) #s(literal 314690115749/10000000000 binary64)) z) #s(literal 119400905721/10000000000 binary64)) z) #s(literal 607771387771/1000000000000 binary64))) < +inf.0

   1. Initial program 93.9%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto x + \color{blue}{\frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}}} \]

      2. clear-num N/A

         \[\leadsto x + \color{blue}{\frac{1}{\frac{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}}{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}}} \]

      3. lower-/.f64 N/A

         \[\leadsto x + \color{blue}{\frac{1}{\frac{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}}{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}}} \]

      4. lift-*.f64 N/A

         \[\leadsto x + \frac{1}{\frac{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}}{\color{blue}{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}}} \]

      5. associate-/r* N/A

         \[\leadsto x + \frac{1}{\color{blue}{\frac{\frac{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}}{y}}{\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}}} \]

      6. lower-/.f64 N/A

         \[\leadsto x + \frac{1}{\color{blue}{\frac{\frac{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}}{y}}{\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}}} \]

   4. Applied rewrites 96.0%

      \[\leadsto x + \color{blue}{\frac{1}{\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(15.234687407 + z, z, 31.4690115749\right), z, 11.9400905721\right), z, 0.607771387771\right)}{y}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(3.13060547623, z, 11.1667541262\right), z, t\right), z, a\right), z, b\right)}}} \]

   if +inf.0 < (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 313060547623/100000000000 binary64)) #s(literal 55833770631/5000000000 binary64)) z) t) z) a) z) b)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 z #s(literal 15234687407/1000000000 binary64)) z) #s(literal 314690115749/10000000000 binary64)) z) #s(literal 119400905721/10000000000 binary64)) z) #s(literal 607771387771/1000000000000 binary64)))

   1. Initial program 0.0%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
   2. Add Preprocessing

   3. Taylor expanded in z around -inf

      \[\leadsto \color{blue}{x + \left(-1 \cdot \frac{\left(\frac{-55833770631}{5000000000} \cdot y + -1 \cdot \frac{t \cdot y - \left(\frac{-15234687407}{1000000000} \cdot \left(\frac{-55833770631}{5000000000} \cdot y - \frac{-4769379582500641883561}{100000000000000000000} \cdot y\right) + \frac{98517059967927196814627}{1000000000000000000000} \cdot y\right)}{z}\right) - \frac{-4769379582500641883561}{100000000000000000000} \cdot y}{z} + \frac{313060547623}{100000000000} \cdot y\right)} \]

   5. Applied rewrites 82.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(3.13060547623, y, x\right) - \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(t, y, \mathsf{fma}\left(-98.5170599679272, y, y \cdot 556.47806218377\right)\right)}{z}, -1, 36.52704169880642 \cdot y\right)}{z}} \]

   6. Taylor expanded in t around inf

      \[\leadsto \mathsf{fma}\left(\frac{313060547623}{100000000000}, y, x\right) - -1 \cdot \color{blue}{\frac{t \cdot y}{{z}^{2}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 99.9%

      \[\leadsto \mathsf{fma}\left(3.13060547623, y, x\right) - \left(-t\right) \cdot \color{blue}{\frac{\frac{y}{z}}{z}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(b + \left(a + \left(t + \left(11.1667541262 + 3.13060547623 \cdot z\right) \cdot z\right) \cdot z\right) \cdot z\right) \cdot y}{0.607771387771 + \left(11.9400905721 + \left(31.4690115749 + \left(15.234687407 + z\right) \cdot z\right) \cdot z\right) \cdot z} \leq \infty:\\ \;\;\;\;x - \frac{-1}{\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(15.234687407 + z, z, 31.4690115749\right), z, 11.9400905721\right), z, 0.607771387771\right)}{y}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(3.13060547623, z, 11.1667541262\right), z, t\right), z, a\right), z, b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(3.13060547623, y, x\right) - \frac{\frac{y}{z}}{z} \cdot \left(-t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 92.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -7.5 \cdot 10^{+14}:\\ \;\;\;\;\mathsf{fma}\left(3.13060547623 - \frac{36.52704169880642 - \frac{457.9610022158428 + t}{z}}{z}, y, x\right)\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{+21}:\\ \;\;\;\;\frac{\left(\left(\left(\left(z \cdot z\right) \cdot 3.13060547623\right) \cdot z + a\right) \cdot z + b\right) \cdot y}{0.607771387771} + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(3.13060547623, y, x\right) - \frac{\frac{y}{z}}{z} \cdot \left(-t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -7.5e+14)
   (fma
    (- 3.13060547623 (/ (- 36.52704169880642 (/ (+ 457.9610022158428 t) z)) z))
    y
    x)
   (if (<= z 9.5e+21)
     (+
      (/
       (* (+ (* (+ (* (* (* z z) 3.13060547623) z) a) z) b) y)
       0.607771387771)
      x)
     (- (fma 3.13060547623 y x) (* (/ (/ y z) z) (- t))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -7.5e+14) {
		tmp = fma((3.13060547623 - ((36.52704169880642 - ((457.9610022158428 + t) / z)) / z)), y, x);
	} else if (z <= 9.5e+21) {
		tmp = ((((((((z * z) * 3.13060547623) * z) + a) * z) + b) * y) / 0.607771387771) + x;
	} else {
		tmp = fma(3.13060547623, y, x) - (((y / z) / z) * -t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -7.5e+14)
		tmp = fma(Float64(3.13060547623 - Float64(Float64(36.52704169880642 - Float64(Float64(457.9610022158428 + t) / z)) / z)), y, x);
	elseif (z <= 9.5e+21)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(z * z) * 3.13060547623) * z) + a) * z) + b) * y) / 0.607771387771) + x);
	else
		tmp = Float64(fma(3.13060547623, y, x) - Float64(Float64(Float64(y / z) / z) * Float64(-t)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -7.5e+14], N[(N[(3.13060547623 - N[(N[(36.52704169880642 - N[(N[(457.9610022158428 + t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[z, 9.5e+21], N[(N[(N[(N[(N[(N[(N[(N[(N[(z * z), $MachinePrecision] * 3.13060547623), $MachinePrecision] * z), $MachinePrecision] + a), $MachinePrecision] * z), $MachinePrecision] + b), $MachinePrecision] * y), $MachinePrecision] / 0.607771387771), $MachinePrecision] + x), $MachinePrecision], N[(N[(3.13060547623 * y + x), $MachinePrecision] - N[(N[(N[(y / z), $MachinePrecision] / z), $MachinePrecision] * (-t)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -7.5e14

   1. Initial program 20.3%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto x + \frac{y \cdot \color{blue}{\left(a \cdot z\right)}}{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}} \]
   4. Step-by-step derivation

      1. lower-*.f64 43.1

         \[\leadsto x + \frac{y \cdot \color{blue}{\left(a \cdot z\right)}}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]

   5. Applied rewrites 43.1%

      \[\leadsto x + \frac{y \cdot \color{blue}{\left(a \cdot z\right)}}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{y \cdot \left(a \cdot z\right)}{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{y \cdot \left(a \cdot z\right)}{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}} + x} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y \cdot \left(a \cdot z\right)}{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}}} + x \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(a \cdot z\right)}}{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}} + x \]

      5. associate-/l* N/A

         \[\leadsto \color{blue}{y \cdot \frac{a \cdot z}{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}}} + x \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\frac{a \cdot z}{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}} \cdot y} + x \]

   7. Applied rewrites 43.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z \cdot a}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(15.234687407 + z, z, 31.4690115749\right), z, 11.9400905721\right), z, 0.607771387771\right)}, y, x\right)} \]

   8. Taylor expanded in z around -inf

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{313060547623}{100000000000} + -1 \cdot \frac{\frac{3652704169880641883561}{100000000000000000000} + -1 \cdot \frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z}}{z}}, y, x\right) \]

   10. Applied rewrites 97.0%

       \[\leadsto \mathsf{fma}\left(\color{blue}{3.13060547623 - \frac{36.52704169880642 - \frac{457.9610022158428 + t}{z}}{z}}, y, x\right) \]

   if -7.5e14 < z < 9.500000000000001e21

   1. Initial program 99.7%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto x + \frac{y \cdot \left(\left(\color{blue}{\left(\frac{313060547623}{100000000000} \cdot {z}^{2}\right)} \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\color{blue}{\left({z}^{2} \cdot \frac{313060547623}{100000000000}\right)} \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}} \]

      2. lower-*.f64 N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\color{blue}{\left({z}^{2} \cdot \frac{313060547623}{100000000000}\right)} \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}} \]

      3. unpow2 N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\color{blue}{\left(z \cdot z\right)} \cdot \frac{313060547623}{100000000000}\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}} \]

      4. lower-*.f64 93.3

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\color{blue}{\left(z \cdot z\right)} \cdot 3.13060547623\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]

   5. Applied rewrites 93.3%

      \[\leadsto x + \frac{y \cdot \left(\left(\color{blue}{\left(\left(z \cdot z\right) \cdot 3.13060547623\right)} \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]

   6. Taylor expanded in z around 0

      \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot z\right) \cdot \frac{313060547623}{100000000000}\right) \cdot z + a\right) \cdot z + b\right)}{\color{blue}{\frac{607771387771}{1000000000000}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 90.8%

      \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot z\right) \cdot 3.13060547623\right) \cdot z + a\right) \cdot z + b\right)}{\color{blue}{0.607771387771}} \]

   if 9.500000000000001e21 < z

   1. Initial program 9.5%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
   2. Add Preprocessing

   3. Taylor expanded in z around -inf

      \[\leadsto \color{blue}{x + \left(-1 \cdot \frac{\left(\frac{-55833770631}{5000000000} \cdot y + -1 \cdot \frac{t \cdot y - \left(\frac{-15234687407}{1000000000} \cdot \left(\frac{-55833770631}{5000000000} \cdot y - \frac{-4769379582500641883561}{100000000000000000000} \cdot y\right) + \frac{98517059967927196814627}{1000000000000000000000} \cdot y\right)}{z}\right) - \frac{-4769379582500641883561}{100000000000000000000} \cdot y}{z} + \frac{313060547623}{100000000000} \cdot y\right)} \]

   5. Applied rewrites 81.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(3.13060547623, y, x\right) - \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(t, y, \mathsf{fma}\left(-98.5170599679272, y, y \cdot 556.47806218377\right)\right)}{z}, -1, 36.52704169880642 \cdot y\right)}{z}} \]

   6. Taylor expanded in t around inf

      \[\leadsto \mathsf{fma}\left(\frac{313060547623}{100000000000}, y, x\right) - -1 \cdot \color{blue}{\frac{t \cdot y}{{z}^{2}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 95.6%

      \[\leadsto \mathsf{fma}\left(3.13060547623, y, x\right) - \left(-t\right) \cdot \color{blue}{\frac{\frac{y}{z}}{z}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.5 \cdot 10^{+14}:\\ \;\;\;\;\mathsf{fma}\left(3.13060547623 - \frac{36.52704169880642 - \frac{457.9610022158428 + t}{z}}{z}, y, x\right)\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{+21}:\\ \;\;\;\;\frac{\left(\left(\left(\left(z \cdot z\right) \cdot 3.13060547623\right) \cdot z + a\right) \cdot z + b\right) \cdot y}{0.607771387771} + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(3.13060547623, y, x\right) - \frac{\frac{y}{z}}{z} \cdot \left(-t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 92.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2200000000000:\\ \;\;\;\;\mathsf{fma}\left(3.13060547623 - \frac{36.52704169880642 - \frac{457.9610022158428 + t}{z}}{z}, y, x\right)\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{+21}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(1.6453555072203998, a, -32.324150453290734 \cdot b\right), z, 1.6453555072203998 \cdot b\right), y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(3.13060547623, y, x\right) - \frac{\frac{y}{z}}{z} \cdot \left(-t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -2200000000000.0)
   (fma
    (- 3.13060547623 (/ (- 36.52704169880642 (/ (+ 457.9610022158428 t) z)) z))
    y
    x)
   (if (<= z 9.5e+21)
     (fma
      (fma
       (fma 1.6453555072203998 a (* -32.324150453290734 b))
       z
       (* 1.6453555072203998 b))
      y
      x)
     (- (fma 3.13060547623 y x) (* (/ (/ y z) z) (- t))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -2200000000000.0) {
		tmp = fma((3.13060547623 - ((36.52704169880642 - ((457.9610022158428 + t) / z)) / z)), y, x);
	} else if (z <= 9.5e+21) {
		tmp = fma(fma(fma(1.6453555072203998, a, (-32.324150453290734 * b)), z, (1.6453555072203998 * b)), y, x);
	} else {
		tmp = fma(3.13060547623, y, x) - (((y / z) / z) * -t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -2200000000000.0)
		tmp = fma(Float64(3.13060547623 - Float64(Float64(36.52704169880642 - Float64(Float64(457.9610022158428 + t) / z)) / z)), y, x);
	elseif (z <= 9.5e+21)
		tmp = fma(fma(fma(1.6453555072203998, a, Float64(-32.324150453290734 * b)), z, Float64(1.6453555072203998 * b)), y, x);
	else
		tmp = Float64(fma(3.13060547623, y, x) - Float64(Float64(Float64(y / z) / z) * Float64(-t)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -2200000000000.0], N[(N[(3.13060547623 - N[(N[(36.52704169880642 - N[(N[(457.9610022158428 + t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[z, 9.5e+21], N[(N[(N[(1.6453555072203998 * a + N[(-32.324150453290734 * b), $MachinePrecision]), $MachinePrecision] * z + N[(1.6453555072203998 * b), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision], N[(N[(3.13060547623 * y + x), $MachinePrecision] - N[(N[(N[(y / z), $MachinePrecision] / z), $MachinePrecision] * (-t)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.2e12

   1. Initial program 20.3%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto x + \frac{y \cdot \color{blue}{\left(a \cdot z\right)}}{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}} \]
   4. Step-by-step derivation

      1. lower-*.f64 43.1

         \[\leadsto x + \frac{y \cdot \color{blue}{\left(a \cdot z\right)}}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]

   5. Applied rewrites 43.1%

      \[\leadsto x + \frac{y \cdot \color{blue}{\left(a \cdot z\right)}}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{y \cdot \left(a \cdot z\right)}{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{y \cdot \left(a \cdot z\right)}{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}} + x} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y \cdot \left(a \cdot z\right)}{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}}} + x \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(a \cdot z\right)}}{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}} + x \]

      5. associate-/l* N/A

         \[\leadsto \color{blue}{y \cdot \frac{a \cdot z}{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}}} + x \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\frac{a \cdot z}{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}} \cdot y} + x \]

   7. Applied rewrites 43.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z \cdot a}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(15.234687407 + z, z, 31.4690115749\right), z, 11.9400905721\right), z, 0.607771387771\right)}, y, x\right)} \]

   8. Taylor expanded in z around -inf

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{313060547623}{100000000000} + -1 \cdot \frac{\frac{3652704169880641883561}{100000000000000000000} + -1 \cdot \frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z}}{z}}, y, x\right) \]

   10. Applied rewrites 97.0%

       \[\leadsto \mathsf{fma}\left(\color{blue}{3.13060547623 - \frac{36.52704169880642 - \frac{457.9610022158428 + t}{z}}{z}}, y, x\right) \]

   if -2.2e12 < z < 9.500000000000001e21

   1. Initial program 99.7%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto x + \frac{y \cdot \color{blue}{\left(a \cdot z\right)}}{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}} \]
   4. Step-by-step derivation

      1. lower-*.f64 59.4

         \[\leadsto x + \frac{y \cdot \color{blue}{\left(a \cdot z\right)}}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]

   5. Applied rewrites 59.4%

      \[\leadsto x + \frac{y \cdot \color{blue}{\left(a \cdot z\right)}}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{y \cdot \left(a \cdot z\right)}{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{y \cdot \left(a \cdot z\right)}{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}} + x} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y \cdot \left(a \cdot z\right)}{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}}} + x \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(a \cdot z\right)}}{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}} + x \]

      5. associate-/l* N/A

         \[\leadsto \color{blue}{y \cdot \frac{a \cdot z}{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}}} + x \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\frac{a \cdot z}{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}} \cdot y} + x \]

   7. Applied rewrites 59.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z \cdot a}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(15.234687407 + z, z, 31.4690115749\right), z, 11.9400905721\right), z, 0.607771387771\right)}, y, x\right)} \]

   8. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1000000000000}{607771387771} \cdot b + z \cdot \left(\frac{1000000000000}{607771387771} \cdot a - \frac{11940090572100000000000000}{369386059793087248348441} \cdot b\right)}, y, x\right) \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{z \cdot \left(\frac{1000000000000}{607771387771} \cdot a - \frac{11940090572100000000000000}{369386059793087248348441} \cdot b\right) + \frac{1000000000000}{607771387771} \cdot b}, y, x\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1000000000000}{607771387771} \cdot a - \frac{11940090572100000000000000}{369386059793087248348441} \cdot b\right) \cdot z} + \frac{1000000000000}{607771387771} \cdot b, y, x\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1000000000000}{607771387771} \cdot a - \frac{11940090572100000000000000}{369386059793087248348441} \cdot b, z, \frac{1000000000000}{607771387771} \cdot b\right)}, y, x\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1000000000000}{607771387771} \cdot a + \left(\mathsf{neg}\left(\frac{11940090572100000000000000}{369386059793087248348441} \cdot b\right)\right)}, z, \frac{1000000000000}{607771387771} \cdot b\right), y, x\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1000000000000}{607771387771}, a, \mathsf{neg}\left(\frac{11940090572100000000000000}{369386059793087248348441} \cdot b\right)\right)}, z, \frac{1000000000000}{607771387771} \cdot b\right), y, x\right) \]

      6. distribute-lft-neg-in N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1000000000000}{607771387771}, a, \color{blue}{\left(\mathsf{neg}\left(\frac{11940090572100000000000000}{369386059793087248348441}\right)\right) \cdot b}\right), z, \frac{1000000000000}{607771387771} \cdot b\right), y, x\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1000000000000}{607771387771}, a, \color{blue}{\frac{-11940090572100000000000000}{369386059793087248348441}} \cdot b\right), z, \frac{1000000000000}{607771387771} \cdot b\right), y, x\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1000000000000}{607771387771}, a, \color{blue}{\frac{-11940090572100000000000000}{369386059793087248348441} \cdot b}\right), z, \frac{1000000000000}{607771387771} \cdot b\right), y, x\right) \]

      9. lower-*.f64 90.3

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(1.6453555072203998, a, -32.324150453290734 \cdot b\right), z, \color{blue}{1.6453555072203998 \cdot b}\right), y, x\right) \]

   10. Applied rewrites 90.3%

       \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(1.6453555072203998, a, -32.324150453290734 \cdot b\right), z, 1.6453555072203998 \cdot b\right)}, y, x\right) \]

   if 9.500000000000001e21 < z

   1. Initial program 9.5%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
   2. Add Preprocessing

   3. Taylor expanded in z around -inf

      \[\leadsto \color{blue}{x + \left(-1 \cdot \frac{\left(\frac{-55833770631}{5000000000} \cdot y + -1 \cdot \frac{t \cdot y - \left(\frac{-15234687407}{1000000000} \cdot \left(\frac{-55833770631}{5000000000} \cdot y - \frac{-4769379582500641883561}{100000000000000000000} \cdot y\right) + \frac{98517059967927196814627}{1000000000000000000000} \cdot y\right)}{z}\right) - \frac{-4769379582500641883561}{100000000000000000000} \cdot y}{z} + \frac{313060547623}{100000000000} \cdot y\right)} \]

   5. Applied rewrites 81.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(3.13060547623, y, x\right) - \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(t, y, \mathsf{fma}\left(-98.5170599679272, y, y \cdot 556.47806218377\right)\right)}{z}, -1, 36.52704169880642 \cdot y\right)}{z}} \]

   6. Taylor expanded in t around inf

      \[\leadsto \mathsf{fma}\left(\frac{313060547623}{100000000000}, y, x\right) - -1 \cdot \color{blue}{\frac{t \cdot y}{{z}^{2}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 95.6%

      \[\leadsto \mathsf{fma}\left(3.13060547623, y, x\right) - \left(-t\right) \cdot \color{blue}{\frac{\frac{y}{z}}{z}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2200000000000:\\ \;\;\;\;\mathsf{fma}\left(3.13060547623 - \frac{36.52704169880642 - \frac{457.9610022158428 + t}{z}}{z}, y, x\right)\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{+21}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(1.6453555072203998, a, -32.324150453290734 \cdot b\right), z, 1.6453555072203998 \cdot b\right), y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(3.13060547623, y, x\right) - \frac{\frac{y}{z}}{z} \cdot \left(-t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 92.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(3.13060547623 - \frac{36.52704169880642 - \frac{457.9610022158428 + t}{z}}{z}, y, x\right)\\ \mathbf{if}\;z \leq -2200000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{+21}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(1.6453555072203998, a, -32.324150453290734 \cdot b\right), z, 1.6453555072203998 \cdot b\right), y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1
         (fma
          (-
           3.13060547623
           (/ (- 36.52704169880642 (/ (+ 457.9610022158428 t) z)) z))
          y
          x)))
   (if (<= z -2200000000000.0)
     t_1
     (if (<= z 9.5e+21)
       (fma
        (fma
         (fma 1.6453555072203998 a (* -32.324150453290734 b))
         z
         (* 1.6453555072203998 b))
        y
        x)
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma((3.13060547623 - ((36.52704169880642 - ((457.9610022158428 + t) / z)) / z)), y, x);
	double tmp;
	if (z <= -2200000000000.0) {
		tmp = t_1;
	} else if (z <= 9.5e+21) {
		tmp = fma(fma(fma(1.6453555072203998, a, (-32.324150453290734 * b)), z, (1.6453555072203998 * b)), y, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(Float64(3.13060547623 - Float64(Float64(36.52704169880642 - Float64(Float64(457.9610022158428 + t) / z)) / z)), y, x)
	tmp = 0.0
	if (z <= -2200000000000.0)
		tmp = t_1;
	elseif (z <= 9.5e+21)
		tmp = fma(fma(fma(1.6453555072203998, a, Float64(-32.324150453290734 * b)), z, Float64(1.6453555072203998 * b)), y, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(3.13060547623 - N[(N[(36.52704169880642 - N[(N[(457.9610022158428 + t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision]}, If[LessEqual[z, -2200000000000.0], t$95$1, If[LessEqual[z, 9.5e+21], N[(N[(N[(1.6453555072203998 * a + N[(-32.324150453290734 * b), $MachinePrecision]), $MachinePrecision] * z + N[(1.6453555072203998 * b), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.2e12 or 9.500000000000001e21 < z

   1. Initial program 14.4%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto x + \frac{y \cdot \color{blue}{\left(a \cdot z\right)}}{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}} \]
   4. Step-by-step derivation

      1. lower-*.f64 41.2

         \[\leadsto x + \frac{y \cdot \color{blue}{\left(a \cdot z\right)}}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]

   5. Applied rewrites 41.2%

      \[\leadsto x + \frac{y \cdot \color{blue}{\left(a \cdot z\right)}}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{y \cdot \left(a \cdot z\right)}{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{y \cdot \left(a \cdot z\right)}{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}} + x} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y \cdot \left(a \cdot z\right)}{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}}} + x \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(a \cdot z\right)}}{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}} + x \]

      5. associate-/l* N/A

         \[\leadsto \color{blue}{y \cdot \frac{a \cdot z}{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}}} + x \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\frac{a \cdot z}{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}} \cdot y} + x \]

   7. Applied rewrites 42.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z \cdot a}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(15.234687407 + z, z, 31.4690115749\right), z, 11.9400905721\right), z, 0.607771387771\right)}, y, x\right)} \]

   8. Taylor expanded in z around -inf

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{313060547623}{100000000000} + -1 \cdot \frac{\frac{3652704169880641883561}{100000000000000000000} + -1 \cdot \frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z}}{z}}, y, x\right) \]

   10. Applied rewrites 96.2%

       \[\leadsto \mathsf{fma}\left(\color{blue}{3.13060547623 - \frac{36.52704169880642 - \frac{457.9610022158428 + t}{z}}{z}}, y, x\right) \]

   if -2.2e12 < z < 9.500000000000001e21

   1. Initial program 99.7%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto x + \frac{y \cdot \color{blue}{\left(a \cdot z\right)}}{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}} \]
   4. Step-by-step derivation

      1. lower-*.f64 59.4

         \[\leadsto x + \frac{y \cdot \color{blue}{\left(a \cdot z\right)}}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]

   5. Applied rewrites 59.4%

      \[\leadsto x + \frac{y \cdot \color{blue}{\left(a \cdot z\right)}}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{y \cdot \left(a \cdot z\right)}{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{y \cdot \left(a \cdot z\right)}{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}} + x} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y \cdot \left(a \cdot z\right)}{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}}} + x \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(a \cdot z\right)}}{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}} + x \]

      5. associate-/l* N/A

         \[\leadsto \color{blue}{y \cdot \frac{a \cdot z}{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}}} + x \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\frac{a \cdot z}{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}} \cdot y} + x \]

   7. Applied rewrites 59.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z \cdot a}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(15.234687407 + z, z, 31.4690115749\right), z, 11.9400905721\right), z, 0.607771387771\right)}, y, x\right)} \]

   8. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1000000000000}{607771387771} \cdot b + z \cdot \left(\frac{1000000000000}{607771387771} \cdot a - \frac{11940090572100000000000000}{369386059793087248348441} \cdot b\right)}, y, x\right) \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{z \cdot \left(\frac{1000000000000}{607771387771} \cdot a - \frac{11940090572100000000000000}{369386059793087248348441} \cdot b\right) + \frac{1000000000000}{607771387771} \cdot b}, y, x\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1000000000000}{607771387771} \cdot a - \frac{11940090572100000000000000}{369386059793087248348441} \cdot b\right) \cdot z} + \frac{1000000000000}{607771387771} \cdot b, y, x\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1000000000000}{607771387771} \cdot a - \frac{11940090572100000000000000}{369386059793087248348441} \cdot b, z, \frac{1000000000000}{607771387771} \cdot b\right)}, y, x\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1000000000000}{607771387771} \cdot a + \left(\mathsf{neg}\left(\frac{11940090572100000000000000}{369386059793087248348441} \cdot b\right)\right)}, z, \frac{1000000000000}{607771387771} \cdot b\right), y, x\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1000000000000}{607771387771}, a, \mathsf{neg}\left(\frac{11940090572100000000000000}{369386059793087248348441} \cdot b\right)\right)}, z, \frac{1000000000000}{607771387771} \cdot b\right), y, x\right) \]

      6. distribute-lft-neg-in N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1000000000000}{607771387771}, a, \color{blue}{\left(\mathsf{neg}\left(\frac{11940090572100000000000000}{369386059793087248348441}\right)\right) \cdot b}\right), z, \frac{1000000000000}{607771387771} \cdot b\right), y, x\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1000000000000}{607771387771}, a, \color{blue}{\frac{-11940090572100000000000000}{369386059793087248348441}} \cdot b\right), z, \frac{1000000000000}{607771387771} \cdot b\right), y, x\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1000000000000}{607771387771}, a, \color{blue}{\frac{-11940090572100000000000000}{369386059793087248348441} \cdot b}\right), z, \frac{1000000000000}{607771387771} \cdot b\right), y, x\right) \]

      9. lower-*.f64 90.3

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(1.6453555072203998, a, -32.324150453290734 \cdot b\right), z, \color{blue}{1.6453555072203998 \cdot b}\right), y, x\right) \]

   10. Applied rewrites 90.3%

       \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(1.6453555072203998, a, -32.324150453290734 \cdot b\right), z, 1.6453555072203998 \cdot b\right)}, y, x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 90.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.5 \cdot 10^{+155}:\\ \;\;\;\;\mathsf{fma}\left(3.13060547623, y, x\right)\\ \mathbf{elif}\;z \leq -2200000000000:\\ \;\;\;\;\mathsf{fma}\left(3.13060547623, y, x\right) - \frac{\mathsf{fma}\left(-1, t, -457.9610022158428\right) \cdot y}{z \cdot z}\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{+24}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(1.6453555072203998, a, -32.324150453290734 \cdot b\right), z, 1.6453555072203998 \cdot b\right), y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(3.13060547623, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -3.5e+155)
   (fma 3.13060547623 y x)
   (if (<= z -2200000000000.0)
     (-
      (fma 3.13060547623 y x)
      (/ (* (fma -1.0 t -457.9610022158428) y) (* z z)))
     (if (<= z 2.7e+24)
       (fma
        (fma
         (fma 1.6453555072203998 a (* -32.324150453290734 b))
         z
         (* 1.6453555072203998 b))
        y
        x)
       (fma 3.13060547623 y x)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -3.5e+155) {
		tmp = fma(3.13060547623, y, x);
	} else if (z <= -2200000000000.0) {
		tmp = fma(3.13060547623, y, x) - ((fma(-1.0, t, -457.9610022158428) * y) / (z * z));
	} else if (z <= 2.7e+24) {
		tmp = fma(fma(fma(1.6453555072203998, a, (-32.324150453290734 * b)), z, (1.6453555072203998 * b)), y, x);
	} else {
		tmp = fma(3.13060547623, y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -3.5e+155)
		tmp = fma(3.13060547623, y, x);
	elseif (z <= -2200000000000.0)
		tmp = Float64(fma(3.13060547623, y, x) - Float64(Float64(fma(-1.0, t, -457.9610022158428) * y) / Float64(z * z)));
	elseif (z <= 2.7e+24)
		tmp = fma(fma(fma(1.6453555072203998, a, Float64(-32.324150453290734 * b)), z, Float64(1.6453555072203998 * b)), y, x);
	else
		tmp = fma(3.13060547623, y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -3.5e+155], N[(3.13060547623 * y + x), $MachinePrecision], If[LessEqual[z, -2200000000000.0], N[(N[(3.13060547623 * y + x), $MachinePrecision] - N[(N[(N[(-1.0 * t + -457.9610022158428), $MachinePrecision] * y), $MachinePrecision] / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.7e+24], N[(N[(N[(1.6453555072203998 * a + N[(-32.324150453290734 * b), $MachinePrecision]), $MachinePrecision] * z + N[(1.6453555072203998 * b), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision], N[(3.13060547623 * y + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.49999999999999985e155 or 2.7e24 < z

   1. Initial program 6.4%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{x + \frac{313060547623}{100000000000} \cdot y} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{313060547623}{100000000000} \cdot y + x} \]

      2. lower-fma.f64 93.3

         \[\leadsto \color{blue}{\mathsf{fma}\left(3.13060547623, y, x\right)} \]

   5. Applied rewrites 93.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(3.13060547623, y, x\right)} \]

   if -3.49999999999999985e155 < z < -2.2e12

   1. Initial program 50.7%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
   2. Add Preprocessing

   3. Taylor expanded in z around -inf

      \[\leadsto \color{blue}{x + \left(-1 \cdot \frac{\left(\frac{-55833770631}{5000000000} \cdot y + -1 \cdot \frac{t \cdot y - \left(\frac{-15234687407}{1000000000} \cdot \left(\frac{-55833770631}{5000000000} \cdot y - \frac{-4769379582500641883561}{100000000000000000000} \cdot y\right) + \frac{98517059967927196814627}{1000000000000000000000} \cdot y\right)}{z}\right) - \frac{-4769379582500641883561}{100000000000000000000} \cdot y}{z} + \frac{313060547623}{100000000000} \cdot y\right)} \]

   5. Applied rewrites 92.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(3.13060547623, y, x\right) - \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(t, y, \mathsf{fma}\left(-98.5170599679272, y, y \cdot 556.47806218377\right)\right)}{z}, -1, 36.52704169880642 \cdot y\right)}{z}} \]

   6. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma}\left(\frac{313060547623}{100000000000}, y, x\right) - -1 \cdot \color{blue}{\frac{\frac{-98517059967927196814627}{1000000000000000000000} \cdot y + \left(\frac{55647806218377003596563527016327}{100000000000000000000000000000} \cdot y + t \cdot y\right)}{{z}^{2}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 92.5%

      \[\leadsto \mathsf{fma}\left(3.13060547623, y, x\right) - \frac{\mathsf{fma}\left(-1, t, -457.9610022158428\right) \cdot y}{\color{blue}{z \cdot z}} \]

   if -2.2e12 < z < 2.7e24

   1. Initial program 99.0%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto x + \frac{y \cdot \color{blue}{\left(a \cdot z\right)}}{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}} \]
   4. Step-by-step derivation

      1. lower-*.f64 59.0

         \[\leadsto x + \frac{y \cdot \color{blue}{\left(a \cdot z\right)}}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]

   5. Applied rewrites 59.0%

      \[\leadsto x + \frac{y \cdot \color{blue}{\left(a \cdot z\right)}}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{y \cdot \left(a \cdot z\right)}{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{y \cdot \left(a \cdot z\right)}{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}} + x} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y \cdot \left(a \cdot z\right)}{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}}} + x \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(a \cdot z\right)}}{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}} + x \]

      5. associate-/l* N/A

         \[\leadsto \color{blue}{y \cdot \frac{a \cdot z}{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}}} + x \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\frac{a \cdot z}{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}} \cdot y} + x \]

   7. Applied rewrites 59.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z \cdot a}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(15.234687407 + z, z, 31.4690115749\right), z, 11.9400905721\right), z, 0.607771387771\right)}, y, x\right)} \]

   8. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1000000000000}{607771387771} \cdot b + z \cdot \left(\frac{1000000000000}{607771387771} \cdot a - \frac{11940090572100000000000000}{369386059793087248348441} \cdot b\right)}, y, x\right) \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{z \cdot \left(\frac{1000000000000}{607771387771} \cdot a - \frac{11940090572100000000000000}{369386059793087248348441} \cdot b\right) + \frac{1000000000000}{607771387771} \cdot b}, y, x\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1000000000000}{607771387771} \cdot a - \frac{11940090572100000000000000}{369386059793087248348441} \cdot b\right) \cdot z} + \frac{1000000000000}{607771387771} \cdot b, y, x\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1000000000000}{607771387771} \cdot a - \frac{11940090572100000000000000}{369386059793087248348441} \cdot b, z, \frac{1000000000000}{607771387771} \cdot b\right)}, y, x\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1000000000000}{607771387771} \cdot a + \left(\mathsf{neg}\left(\frac{11940090572100000000000000}{369386059793087248348441} \cdot b\right)\right)}, z, \frac{1000000000000}{607771387771} \cdot b\right), y, x\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1000000000000}{607771387771}, a, \mathsf{neg}\left(\frac{11940090572100000000000000}{369386059793087248348441} \cdot b\right)\right)}, z, \frac{1000000000000}{607771387771} \cdot b\right), y, x\right) \]

      6. distribute-lft-neg-in N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1000000000000}{607771387771}, a, \color{blue}{\left(\mathsf{neg}\left(\frac{11940090572100000000000000}{369386059793087248348441}\right)\right) \cdot b}\right), z, \frac{1000000000000}{607771387771} \cdot b\right), y, x\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1000000000000}{607771387771}, a, \color{blue}{\frac{-11940090572100000000000000}{369386059793087248348441}} \cdot b\right), z, \frac{1000000000000}{607771387771} \cdot b\right), y, x\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1000000000000}{607771387771}, a, \color{blue}{\frac{-11940090572100000000000000}{369386059793087248348441} \cdot b}\right), z, \frac{1000000000000}{607771387771} \cdot b\right), y, x\right) \]

      9. lower-*.f64 89.6

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(1.6453555072203998, a, -32.324150453290734 \cdot b\right), z, \color{blue}{1.6453555072203998 \cdot b}\right), y, x\right) \]

   10. Applied rewrites 89.6%

       \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(1.6453555072203998, a, -32.324150453290734 \cdot b\right), z, 1.6453555072203998 \cdot b\right)}, y, x\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 90.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.05 \cdot 10^{+16}:\\ \;\;\;\;\mathsf{fma}\left(3.13060547623, y, x\right)\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{+24}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(1.6453555072203998, a, -32.324150453290734 \cdot b\right), z, 1.6453555072203998 \cdot b\right), y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(3.13060547623, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -3.05e+16)
   (fma 3.13060547623 y x)
   (if (<= z 2.7e+24)
     (fma
      (fma
       (fma 1.6453555072203998 a (* -32.324150453290734 b))
       z
       (* 1.6453555072203998 b))
      y
      x)
     (fma 3.13060547623 y x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -3.05e+16) {
		tmp = fma(3.13060547623, y, x);
	} else if (z <= 2.7e+24) {
		tmp = fma(fma(fma(1.6453555072203998, a, (-32.324150453290734 * b)), z, (1.6453555072203998 * b)), y, x);
	} else {
		tmp = fma(3.13060547623, y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -3.05e+16)
		tmp = fma(3.13060547623, y, x);
	elseif (z <= 2.7e+24)
		tmp = fma(fma(fma(1.6453555072203998, a, Float64(-32.324150453290734 * b)), z, Float64(1.6453555072203998 * b)), y, x);
	else
		tmp = fma(3.13060547623, y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -3.05e+16], N[(3.13060547623 * y + x), $MachinePrecision], If[LessEqual[z, 2.7e+24], N[(N[(N[(1.6453555072203998 * a + N[(-32.324150453290734 * b), $MachinePrecision]), $MachinePrecision] * z + N[(1.6453555072203998 * b), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision], N[(3.13060547623 * y + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.05e16 or 2.7e24 < z

   1. Initial program 14.5%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{x + \frac{313060547623}{100000000000} \cdot y} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{313060547623}{100000000000} \cdot y + x} \]

      2. lower-fma.f64 88.7

         \[\leadsto \color{blue}{\mathsf{fma}\left(3.13060547623, y, x\right)} \]

   5. Applied rewrites 88.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(3.13060547623, y, x\right)} \]

   if -3.05e16 < z < 2.7e24

   1. Initial program 99.0%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto x + \frac{y \cdot \color{blue}{\left(a \cdot z\right)}}{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}} \]
   4. Step-by-step derivation

      1. lower-*.f64 59.0

         \[\leadsto x + \frac{y \cdot \color{blue}{\left(a \cdot z\right)}}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]

   5. Applied rewrites 59.0%

      \[\leadsto x + \frac{y \cdot \color{blue}{\left(a \cdot z\right)}}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{y \cdot \left(a \cdot z\right)}{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{y \cdot \left(a \cdot z\right)}{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}} + x} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y \cdot \left(a \cdot z\right)}{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}}} + x \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(a \cdot z\right)}}{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}} + x \]

      5. associate-/l* N/A

         \[\leadsto \color{blue}{y \cdot \frac{a \cdot z}{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}}} + x \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\frac{a \cdot z}{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}} \cdot y} + x \]

   7. Applied rewrites 59.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z \cdot a}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(15.234687407 + z, z, 31.4690115749\right), z, 11.9400905721\right), z, 0.607771387771\right)}, y, x\right)} \]

   8. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1000000000000}{607771387771} \cdot b + z \cdot \left(\frac{1000000000000}{607771387771} \cdot a - \frac{11940090572100000000000000}{369386059793087248348441} \cdot b\right)}, y, x\right) \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{z \cdot \left(\frac{1000000000000}{607771387771} \cdot a - \frac{11940090572100000000000000}{369386059793087248348441} \cdot b\right) + \frac{1000000000000}{607771387771} \cdot b}, y, x\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1000000000000}{607771387771} \cdot a - \frac{11940090572100000000000000}{369386059793087248348441} \cdot b\right) \cdot z} + \frac{1000000000000}{607771387771} \cdot b, y, x\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1000000000000}{607771387771} \cdot a - \frac{11940090572100000000000000}{369386059793087248348441} \cdot b, z, \frac{1000000000000}{607771387771} \cdot b\right)}, y, x\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1000000000000}{607771387771} \cdot a + \left(\mathsf{neg}\left(\frac{11940090572100000000000000}{369386059793087248348441} \cdot b\right)\right)}, z, \frac{1000000000000}{607771387771} \cdot b\right), y, x\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1000000000000}{607771387771}, a, \mathsf{neg}\left(\frac{11940090572100000000000000}{369386059793087248348441} \cdot b\right)\right)}, z, \frac{1000000000000}{607771387771} \cdot b\right), y, x\right) \]

      6. distribute-lft-neg-in N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1000000000000}{607771387771}, a, \color{blue}{\left(\mathsf{neg}\left(\frac{11940090572100000000000000}{369386059793087248348441}\right)\right) \cdot b}\right), z, \frac{1000000000000}{607771387771} \cdot b\right), y, x\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1000000000000}{607771387771}, a, \color{blue}{\frac{-11940090572100000000000000}{369386059793087248348441}} \cdot b\right), z, \frac{1000000000000}{607771387771} \cdot b\right), y, x\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1000000000000}{607771387771}, a, \color{blue}{\frac{-11940090572100000000000000}{369386059793087248348441} \cdot b}\right), z, \frac{1000000000000}{607771387771} \cdot b\right), y, x\right) \]

      9. lower-*.f64 89.6

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(1.6453555072203998, a, -32.324150453290734 \cdot b\right), z, \color{blue}{1.6453555072203998 \cdot b}\right), y, x\right) \]

   10. Applied rewrites 89.6%

       \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(1.6453555072203998, a, -32.324150453290734 \cdot b\right), z, 1.6453555072203998 \cdot b\right)}, y, x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 83.0% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -24000000000:\\ \;\;\;\;\mathsf{fma}\left(3.13060547623, y, x\right)\\ \mathbf{elif}\;z \leq 2 \cdot 10^{+50}:\\ \;\;\;\;\mathsf{fma}\left(1.6453555072203998 \cdot y, b, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(3.13060547623, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -24000000000.0)
   (fma 3.13060547623 y x)
   (if (<= z 2e+50)
     (fma (* 1.6453555072203998 y) b x)
     (fma 3.13060547623 y x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -24000000000.0) {
		tmp = fma(3.13060547623, y, x);
	} else if (z <= 2e+50) {
		tmp = fma((1.6453555072203998 * y), b, x);
	} else {
		tmp = fma(3.13060547623, y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -24000000000.0)
		tmp = fma(3.13060547623, y, x);
	elseif (z <= 2e+50)
		tmp = fma(Float64(1.6453555072203998 * y), b, x);
	else
		tmp = fma(3.13060547623, y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -24000000000.0], N[(3.13060547623 * y + x), $MachinePrecision], If[LessEqual[z, 2e+50], N[(N[(1.6453555072203998 * y), $MachinePrecision] * b + x), $MachinePrecision], N[(3.13060547623 * y + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.4e10 or 2.0000000000000002e50 < z

   1. Initial program 13.8%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{x + \frac{313060547623}{100000000000} \cdot y} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{313060547623}{100000000000} \cdot y + x} \]

      2. lower-fma.f64 89.3

         \[\leadsto \color{blue}{\mathsf{fma}\left(3.13060547623, y, x\right)} \]

   5. Applied rewrites 89.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(3.13060547623, y, x\right)} \]

   if -2.4e10 < z < 2.0000000000000002e50

   1. Initial program 98.3%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{x + \frac{313060547623}{100000000000} \cdot y} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{313060547623}{100000000000} \cdot y + x} \]

      2. lower-fma.f64 35.0

         \[\leadsto \color{blue}{\mathsf{fma}\left(3.13060547623, y, x\right)} \]

   5. Applied rewrites 35.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(3.13060547623, y, x\right)} \]

   6. Taylor expanded in y around inf

      \[\leadsto \frac{313060547623}{100000000000} \cdot \color{blue}{y} \]
   7. Step-by-step derivation

   8. Applied rewrites 3.7%

      \[\leadsto 3.13060547623 \cdot \color{blue}{y} \]

   9. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \frac{1000000000000}{607771387771} \cdot \left(b \cdot y\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \color{blue}{\frac{1000000000000}{607771387771} \cdot \left(b \cdot y\right) + x} \]

       2. *-commutative N/A

          \[\leadsto \frac{1000000000000}{607771387771} \cdot \color{blue}{\left(y \cdot b\right)} + x \]

       3. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{1000000000000}{607771387771} \cdot y\right) \cdot b} + x \]

       4. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1000000000000}{607771387771} \cdot y, b, x\right)} \]

       5. lower-*.f64 76.2

          \[\leadsto \mathsf{fma}\left(\color{blue}{1.6453555072203998 \cdot y}, b, x\right) \]

   11. Applied rewrites 76.2%

       \[\leadsto \color{blue}{\mathsf{fma}\left(1.6453555072203998 \cdot y, b, x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 83.0% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -24000000000:\\ \;\;\;\;\mathsf{fma}\left(3.13060547623, y, x\right)\\ \mathbf{elif}\;z \leq 2 \cdot 10^{+50}:\\ \;\;\;\;\mathsf{fma}\left(1.6453555072203998, b \cdot y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(3.13060547623, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -24000000000.0)
   (fma 3.13060547623 y x)
   (if (<= z 2e+50)
     (fma 1.6453555072203998 (* b y) x)
     (fma 3.13060547623 y x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -24000000000.0) {
		tmp = fma(3.13060547623, y, x);
	} else if (z <= 2e+50) {
		tmp = fma(1.6453555072203998, (b * y), x);
	} else {
		tmp = fma(3.13060547623, y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -24000000000.0)
		tmp = fma(3.13060547623, y, x);
	elseif (z <= 2e+50)
		tmp = fma(1.6453555072203998, Float64(b * y), x);
	else
		tmp = fma(3.13060547623, y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -24000000000.0], N[(3.13060547623 * y + x), $MachinePrecision], If[LessEqual[z, 2e+50], N[(1.6453555072203998 * N[(b * y), $MachinePrecision] + x), $MachinePrecision], N[(3.13060547623 * y + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.4e10 or 2.0000000000000002e50 < z

   1. Initial program 13.8%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{x + \frac{313060547623}{100000000000} \cdot y} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{313060547623}{100000000000} \cdot y + x} \]

      2. lower-fma.f64 89.3

         \[\leadsto \color{blue}{\mathsf{fma}\left(3.13060547623, y, x\right)} \]

   5. Applied rewrites 89.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(3.13060547623, y, x\right)} \]

   if -2.4e10 < z < 2.0000000000000002e50

   1. Initial program 98.3%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \frac{1000000000000}{607771387771} \cdot \left(b \cdot y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{1000000000000}{607771387771} \cdot \left(b \cdot y\right) + x} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1000000000000}{607771387771}, b \cdot y, x\right)} \]

      3. lower-*.f64 76.1

         \[\leadsto \mathsf{fma}\left(1.6453555072203998, \color{blue}{b \cdot y}, x\right) \]

   5. Applied rewrites 76.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1.6453555072203998, b \cdot y, x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 61.6% accurate, 11.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(3.13060547623, y, x\right) \end{array} \]

FPCore

(FPCore (x y z t a b) :precision binary64 (fma 3.13060547623 y x))

C

double code(double x, double y, double z, double t, double a, double b) {
	return fma(3.13060547623, y, x);
}

Julia

function code(x, y, z, t, a, b)
	return fma(3.13060547623, y, x)
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(3.13060547623 * y + x), $MachinePrecision]

Derivation

1. Initial program 59.0%

   \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
2. Add Preprocessing

3. Taylor expanded in z around inf

   \[\leadsto \color{blue}{x + \frac{313060547623}{100000000000} \cdot y} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{\frac{313060547623}{100000000000} \cdot y + x} \]

   2. lower-fma.f64 60.2

      \[\leadsto \color{blue}{\mathsf{fma}\left(3.13060547623, y, x\right)} \]

5. Applied rewrites 60.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(3.13060547623, y, x\right)} \]
6. Add Preprocessing

Alternative 14: 22.0% accurate, 13.2× speedup

Math

\[\begin{array}{l} \\ 3.13060547623 \cdot y \end{array} \]

FPCore

(FPCore (x y z t a b) :precision binary64 (* 3.13060547623 y))

C

double code(double x, double y, double z, double t, double a, double b) {
	return 3.13060547623 * y;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    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
    code = 3.13060547623d0 * y
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return 3.13060547623 * y;
}

Python

def code(x, y, z, t, a, b):
	return 3.13060547623 * y

Julia

function code(x, y, z, t, a, b)
	return Float64(3.13060547623 * y)
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = 3.13060547623 * y;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(3.13060547623 * y), $MachinePrecision]

Derivation

1. Initial program 59.0%

   \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
2. Add Preprocessing

3. Taylor expanded in z around inf

   \[\leadsto \color{blue}{x + \frac{313060547623}{100000000000} \cdot y} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{\frac{313060547623}{100000000000} \cdot y + x} \]

   2. lower-fma.f64 60.2

      \[\leadsto \color{blue}{\mathsf{fma}\left(3.13060547623, y, x\right)} \]

5. Applied rewrites 60.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(3.13060547623, y, x\right)} \]

6. Taylor expanded in y around inf

   \[\leadsto \frac{313060547623}{100000000000} \cdot \color{blue}{y} \]
7. Step-by-step derivation

8. Applied rewrites 19.8%

   \[\leadsto 3.13060547623 \cdot \color{blue}{y} \]
9. Add Preprocessing

Developer Target 1: 98.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \left(\left(3.13060547623 - \frac{36.527041698806414}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \mathbf{if}\;z < -6.499344996252632 \cdot 10^{+53}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z < 7.066965436914287 \cdot 10^{+59}:\\ \;\;\;\;x + \frac{y}{\frac{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771}{\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1
         (+
          x
          (*
           (+ (- 3.13060547623 (/ 36.527041698806414 z)) (/ t (* z z)))
           (/ y 1.0)))))
   (if (< z -6.499344996252632e+53)
     t_1
     (if (< z 7.066965436914287e+59)
       (+
        x
        (/
         y
         (/
          (+
           (*
            (+ (* (+ (* (+ z 15.234687407) z) 31.4690115749) z) 11.9400905721)
            z)
           0.607771387771)
          (+
           (* (+ (* (+ (* (+ (* z 3.13060547623) 11.1667541262) z) t) z) a) z)
           b))))
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x + (((3.13060547623 - (36.527041698806414 / z)) + (t / (z * z))) * (y / 1.0));
	double tmp;
	if (z < -6.499344996252632e+53) {
		tmp = t_1;
	} else if (z < 7.066965436914287e+59) {
		tmp = x + (y / ((((((((z + 15.234687407) * z) + 31.4690115749) * z) + 11.9400905721) * z) + 0.607771387771) / ((((((((z * 3.13060547623) + 11.1667541262) * z) + t) * z) + a) * z) + b)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    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) :: t_1
    real(8) :: tmp
    t_1 = x + (((3.13060547623d0 - (36.527041698806414d0 / z)) + (t / (z * z))) * (y / 1.0d0))
    if (z < (-6.499344996252632d+53)) then
        tmp = t_1
    else if (z < 7.066965436914287d+59) then
        tmp = x + (y / ((((((((z + 15.234687407d0) * z) + 31.4690115749d0) * z) + 11.9400905721d0) * z) + 0.607771387771d0) / ((((((((z * 3.13060547623d0) + 11.1667541262d0) * z) + t) * z) + a) * z) + b)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x + (((3.13060547623 - (36.527041698806414 / z)) + (t / (z * z))) * (y / 1.0));
	double tmp;
	if (z < -6.499344996252632e+53) {
		tmp = t_1;
	} else if (z < 7.066965436914287e+59) {
		tmp = x + (y / ((((((((z + 15.234687407) * z) + 31.4690115749) * z) + 11.9400905721) * z) + 0.607771387771) / ((((((((z * 3.13060547623) + 11.1667541262) * z) + t) * z) + a) * z) + b)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x + (((3.13060547623 - (36.527041698806414 / z)) + (t / (z * z))) * (y / 1.0))
	tmp = 0
	if z < -6.499344996252632e+53:
		tmp = t_1
	elif z < 7.066965436914287e+59:
		tmp = x + (y / ((((((((z + 15.234687407) * z) + 31.4690115749) * z) + 11.9400905721) * z) + 0.607771387771) / ((((((((z * 3.13060547623) + 11.1667541262) * z) + t) * z) + a) * z) + b)))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x + Float64(Float64(Float64(3.13060547623 - Float64(36.527041698806414 / z)) + Float64(t / Float64(z * z))) * Float64(y / 1.0)))
	tmp = 0.0
	if (z < -6.499344996252632e+53)
		tmp = t_1;
	elseif (z < 7.066965436914287e+59)
		tmp = Float64(x + Float64(y / Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(z + 15.234687407) * z) + 31.4690115749) * z) + 11.9400905721) * z) + 0.607771387771) / Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(z * 3.13060547623) + 11.1667541262) * z) + t) * z) + a) * z) + b))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x + (((3.13060547623 - (36.527041698806414 / z)) + (t / (z * z))) * (y / 1.0));
	tmp = 0.0;
	if (z < -6.499344996252632e+53)
		tmp = t_1;
	elseif (z < 7.066965436914287e+59)
		tmp = x + (y / ((((((((z + 15.234687407) * z) + 31.4690115749) * z) + 11.9400905721) * z) + 0.607771387771) / ((((((((z * 3.13060547623) + 11.1667541262) * z) + t) * z) + a) * z) + b)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x + N[(N[(N[(3.13060547623 - N[(36.527041698806414 / z), $MachinePrecision]), $MachinePrecision] + N[(t / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y / 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Less[z, -6.499344996252632e+53], t$95$1, If[Less[z, 7.066965436914287e+59], N[(x + N[(y / N[(N[(N[(N[(N[(N[(N[(N[(z + 15.234687407), $MachinePrecision] * z), $MachinePrecision] + 31.4690115749), $MachinePrecision] * z), $MachinePrecision] + 11.9400905721), $MachinePrecision] * z), $MachinePrecision] + 0.607771387771), $MachinePrecision] / N[(N[(N[(N[(N[(N[(N[(N[(z * 3.13060547623), $MachinePrecision] + 11.1667541262), $MachinePrecision] * z), $MachinePrecision] + t), $MachinePrecision] * z), $MachinePrecision] + a), $MachinePrecision] * z), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Reproduce

herbie shell --seed 2024249 
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, D"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< z -649934499625263200000000000000000000000000000000000000) (+ x (* (+ (- 313060547623/100000000000 (/ 18263520849403207/500000000000000 z)) (/ t (* z z))) (/ y 1))) (if (< z 706696543691428700000000000000000000000000000000000000000000) (+ x (/ y (/ (+ (* (+ (* (+ (* (+ z 15234687407/1000000000) z) 314690115749/10000000000) z) 119400905721/10000000000) z) 607771387771/1000000000000) (+ (* (+ (* (+ (* (+ (* z 313060547623/100000000000) 55833770631/5000000000) z) t) z) a) z) b)))) (+ x (* (+ (- 313060547623/100000000000 (/ 18263520849403207/500000000000000 z)) (/ t (* z z))) (/ y 1))))))

  (+ x (/ (* y (+ (* (+ (* (+ (* (+ (* z 3.13060547623) 11.1667541262) z) t) z) a) z) b)) (+ (* (+ (* (+ (* (+ z 15.234687407) z) 31.4690115749) z) 11.9400905721) z) 0.607771387771))))