Graphics.Rasterific.CubicBezier:cachedBezierAt from Rasterific-0.6.1

Percentage Accurate: 92.7% → 94.4%

Time: 11.1s

Alternatives: 15

Speedup: 0.8×

• 32.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

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)) + (t * a)) + ((a * z) * b)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 92.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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)) + (t * a)) + ((a * z) * b)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 94.4% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 4.8 \cdot 10^{+37}:\\ \;\;\;\;\mathsf{fma}\left(z, a \cdot b, \mathsf{fma}\left(t, a, \mathsf{fma}\left(y, z, x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + a \cdot \left(t + z \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= a 4.8e+37)
   (fma z (* a b) (fma t a (fma y z x)))
   (+ x (* a (+ t (* z b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= 4.8e+37) {
		tmp = fma(z, (a * b), fma(t, a, fma(y, z, x)));
	} else {
		tmp = x + (a * (t + (z * b)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (a <= 4.8e+37)
		tmp = fma(z, Float64(a * b), fma(t, a, fma(y, z, x)));
	else
		tmp = Float64(x + Float64(a * Float64(t + Float64(z * b))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[a, 4.8e+37], N[(z * N[(a * b), $MachinePrecision] + N[(t * a + N[(y * z + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(a * N[(t + N[(z * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 4.8e37

   1. Initial program 93.6%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   2. Step-by-step derivation

      1. +-commutative 93.6%

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

      2. *-commutative 93.6%

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

      3. associate-*l* 95.9%

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

      4. *-commutative 95.9%

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

      5. fma-define 96.9%

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

      6. *-commutative 96.9%

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

      7. +-commutative 96.9%

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

      8. fma-define 97.4%

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

      9. +-commutative 97.4%

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

      10. fma-define 97.4%

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

   3. Simplified 97.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, a \cdot b, \mathsf{fma}\left(t, a, \mathsf{fma}\left(y, z, x\right)\right)\right)} \]
   4. Add Preprocessing

   if 4.8e37 < a

   1. Initial program 83.5%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0 85.3%

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

   4. Taylor expanded in a around inf 96.8%

      \[\leadsto x + \color{blue}{a \cdot \left(t + b \cdot z\right)} \]
   5. Step-by-step derivation

      1. +-commutative 96.8%

         \[\leadsto x + a \cdot \color{blue}{\left(b \cdot z + t\right)} \]

   6. Simplified 96.8%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 4.8 \cdot 10^{+37}:\\ \;\;\;\;\mathsf{fma}\left(z, a \cdot b, \mathsf{fma}\left(t, a, \mathsf{fma}\left(y, z, x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + a \cdot \left(t + z \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 38.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot \left(z \cdot b\right)\\ \mathbf{if}\;a \leq -1.65 \cdot 10^{+131}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq -5.5 \cdot 10^{+73}:\\ \;\;\;\;z \cdot y\\ \mathbf{elif}\;a \leq -6 \cdot 10^{-7}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq -2 \cdot 10^{-71}:\\ \;\;\;\;z \cdot y\\ \mathbf{elif}\;a \leq -1.45 \cdot 10^{-157}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 5 \cdot 10^{-171}:\\ \;\;\;\;z \cdot y\\ \mathbf{elif}\;a \leq 8.2 \cdot 10^{+96}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 1.6 \cdot 10^{+215}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;a \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* a (* z b))))
   (if (<= a -1.65e+131)
     t_1
     (if (<= a -5.5e+73)
       (* z y)
       (if (<= a -6e-7)
         t_1
         (if (<= a -2e-71)
           (* z y)
           (if (<= a -1.45e-157)
             x
             (if (<= a 5e-171)
               (* z y)
               (if (<= a 8.2e+96) x (if (<= a 1.6e+215) t_1 (* a t)))))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = a * (z * b);
	double tmp;
	if (a <= -1.65e+131) {
		tmp = t_1;
	} else if (a <= -5.5e+73) {
		tmp = z * y;
	} else if (a <= -6e-7) {
		tmp = t_1;
	} else if (a <= -2e-71) {
		tmp = z * y;
	} else if (a <= -1.45e-157) {
		tmp = x;
	} else if (a <= 5e-171) {
		tmp = z * y;
	} else if (a <= 8.2e+96) {
		tmp = x;
	} else if (a <= 1.6e+215) {
		tmp = t_1;
	} else {
		tmp = a * t;
	}
	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 = a * (z * b)
    if (a <= (-1.65d+131)) then
        tmp = t_1
    else if (a <= (-5.5d+73)) then
        tmp = z * y
    else if (a <= (-6d-7)) then
        tmp = t_1
    else if (a <= (-2d-71)) then
        tmp = z * y
    else if (a <= (-1.45d-157)) then
        tmp = x
    else if (a <= 5d-171) then
        tmp = z * y
    else if (a <= 8.2d+96) then
        tmp = x
    else if (a <= 1.6d+215) then
        tmp = t_1
    else
        tmp = a * t
    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 = a * (z * b);
	double tmp;
	if (a <= -1.65e+131) {
		tmp = t_1;
	} else if (a <= -5.5e+73) {
		tmp = z * y;
	} else if (a <= -6e-7) {
		tmp = t_1;
	} else if (a <= -2e-71) {
		tmp = z * y;
	} else if (a <= -1.45e-157) {
		tmp = x;
	} else if (a <= 5e-171) {
		tmp = z * y;
	} else if (a <= 8.2e+96) {
		tmp = x;
	} else if (a <= 1.6e+215) {
		tmp = t_1;
	} else {
		tmp = a * t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = a * (z * b)
	tmp = 0
	if a <= -1.65e+131:
		tmp = t_1
	elif a <= -5.5e+73:
		tmp = z * y
	elif a <= -6e-7:
		tmp = t_1
	elif a <= -2e-71:
		tmp = z * y
	elif a <= -1.45e-157:
		tmp = x
	elif a <= 5e-171:
		tmp = z * y
	elif a <= 8.2e+96:
		tmp = x
	elif a <= 1.6e+215:
		tmp = t_1
	else:
		tmp = a * t
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(a * Float64(z * b))
	tmp = 0.0
	if (a <= -1.65e+131)
		tmp = t_1;
	elseif (a <= -5.5e+73)
		tmp = Float64(z * y);
	elseif (a <= -6e-7)
		tmp = t_1;
	elseif (a <= -2e-71)
		tmp = Float64(z * y);
	elseif (a <= -1.45e-157)
		tmp = x;
	elseif (a <= 5e-171)
		tmp = Float64(z * y);
	elseif (a <= 8.2e+96)
		tmp = x;
	elseif (a <= 1.6e+215)
		tmp = t_1;
	else
		tmp = Float64(a * t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = a * (z * b);
	tmp = 0.0;
	if (a <= -1.65e+131)
		tmp = t_1;
	elseif (a <= -5.5e+73)
		tmp = z * y;
	elseif (a <= -6e-7)
		tmp = t_1;
	elseif (a <= -2e-71)
		tmp = z * y;
	elseif (a <= -1.45e-157)
		tmp = x;
	elseif (a <= 5e-171)
		tmp = z * y;
	elseif (a <= 8.2e+96)
		tmp = x;
	elseif (a <= 1.6e+215)
		tmp = t_1;
	else
		tmp = a * t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(a * N[(z * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -1.65e+131], t$95$1, If[LessEqual[a, -5.5e+73], N[(z * y), $MachinePrecision], If[LessEqual[a, -6e-7], t$95$1, If[LessEqual[a, -2e-71], N[(z * y), $MachinePrecision], If[LessEqual[a, -1.45e-157], x, If[LessEqual[a, 5e-171], N[(z * y), $MachinePrecision], If[LessEqual[a, 8.2e+96], x, If[LessEqual[a, 1.6e+215], t$95$1, N[(a * t), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if a < -1.6499999999999999e131 or -5.5000000000000003e73 < a < -5.9999999999999997e-7 or 8.19999999999999996e96 < a < 1.5999999999999999e215

   1. Initial program 84.5%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf 57.3%

      \[\leadsto \color{blue}{a \cdot \left(b \cdot z\right)} \]

   if -1.6499999999999999e131 < a < -5.5000000000000003e73 or -5.9999999999999997e-7 < a < -1.9999999999999998e-71 or -1.44999999999999994e-157 < a < 4.99999999999999992e-171

   1. Initial program 95.6%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 53.9%

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

      1. *-commutative 53.9%

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

   5. Simplified 53.9%

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

   if -1.9999999999999998e-71 < a < -1.44999999999999994e-157 or 4.99999999999999992e-171 < a < 8.19999999999999996e96

   1. Initial program 99.9%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 58.9%

      \[\leadsto \color{blue}{x} \]

   if 1.5999999999999999e215 < a

   1. Initial program 77.3%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf 69.5%

      \[\leadsto \color{blue}{a \cdot t} \]
3. Recombined 4 regimes into one program.

4. Final simplification 57.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.65 \cdot 10^{+131}:\\ \;\;\;\;a \cdot \left(z \cdot b\right)\\ \mathbf{elif}\;a \leq -5.5 \cdot 10^{+73}:\\ \;\;\;\;z \cdot y\\ \mathbf{elif}\;a \leq -6 \cdot 10^{-7}:\\ \;\;\;\;a \cdot \left(z \cdot b\right)\\ \mathbf{elif}\;a \leq -2 \cdot 10^{-71}:\\ \;\;\;\;z \cdot y\\ \mathbf{elif}\;a \leq -1.45 \cdot 10^{-157}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 5 \cdot 10^{-171}:\\ \;\;\;\;z \cdot y\\ \mathbf{elif}\;a \leq 8.2 \cdot 10^{+96}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 1.6 \cdot 10^{+215}:\\ \;\;\;\;a \cdot \left(z \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 38.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5.2 \cdot 10^{-9}:\\ \;\;\;\;z \cdot \left(a \cdot b\right)\\ \mathbf{elif}\;a \leq -1.25 \cdot 10^{-71}:\\ \;\;\;\;z \cdot y\\ \mathbf{elif}\;a \leq -1.55 \cdot 10^{-157}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 1.95 \cdot 10^{-171}:\\ \;\;\;\;z \cdot y\\ \mathbf{elif}\;a \leq 8.4 \cdot 10^{+96}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 5.8 \cdot 10^{+214}:\\ \;\;\;\;a \cdot \left(z \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= a -5.2e-9)
   (* z (* a b))
   (if (<= a -1.25e-71)
     (* z y)
     (if (<= a -1.55e-157)
       x
       (if (<= a 1.95e-171)
         (* z y)
         (if (<= a 8.4e+96) x (if (<= a 5.8e+214) (* a (* z b)) (* a t))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= -5.2e-9) {
		tmp = z * (a * b);
	} else if (a <= -1.25e-71) {
		tmp = z * y;
	} else if (a <= -1.55e-157) {
		tmp = x;
	} else if (a <= 1.95e-171) {
		tmp = z * y;
	} else if (a <= 8.4e+96) {
		tmp = x;
	} else if (a <= 5.8e+214) {
		tmp = a * (z * b);
	} else {
		tmp = a * t;
	}
	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) :: tmp
    if (a <= (-5.2d-9)) then
        tmp = z * (a * b)
    else if (a <= (-1.25d-71)) then
        tmp = z * y
    else if (a <= (-1.55d-157)) then
        tmp = x
    else if (a <= 1.95d-171) then
        tmp = z * y
    else if (a <= 8.4d+96) then
        tmp = x
    else if (a <= 5.8d+214) then
        tmp = a * (z * b)
    else
        tmp = a * t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= -5.2e-9) {
		tmp = z * (a * b);
	} else if (a <= -1.25e-71) {
		tmp = z * y;
	} else if (a <= -1.55e-157) {
		tmp = x;
	} else if (a <= 1.95e-171) {
		tmp = z * y;
	} else if (a <= 8.4e+96) {
		tmp = x;
	} else if (a <= 5.8e+214) {
		tmp = a * (z * b);
	} else {
		tmp = a * t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if a <= -5.2e-9:
		tmp = z * (a * b)
	elif a <= -1.25e-71:
		tmp = z * y
	elif a <= -1.55e-157:
		tmp = x
	elif a <= 1.95e-171:
		tmp = z * y
	elif a <= 8.4e+96:
		tmp = x
	elif a <= 5.8e+214:
		tmp = a * (z * b)
	else:
		tmp = a * t
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (a <= -5.2e-9)
		tmp = Float64(z * Float64(a * b));
	elseif (a <= -1.25e-71)
		tmp = Float64(z * y);
	elseif (a <= -1.55e-157)
		tmp = x;
	elseif (a <= 1.95e-171)
		tmp = Float64(z * y);
	elseif (a <= 8.4e+96)
		tmp = x;
	elseif (a <= 5.8e+214)
		tmp = Float64(a * Float64(z * b));
	else
		tmp = Float64(a * t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (a <= -5.2e-9)
		tmp = z * (a * b);
	elseif (a <= -1.25e-71)
		tmp = z * y;
	elseif (a <= -1.55e-157)
		tmp = x;
	elseif (a <= 1.95e-171)
		tmp = z * y;
	elseif (a <= 8.4e+96)
		tmp = x;
	elseif (a <= 5.8e+214)
		tmp = a * (z * b);
	else
		tmp = a * t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[a, -5.2e-9], N[(z * N[(a * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, -1.25e-71], N[(z * y), $MachinePrecision], If[LessEqual[a, -1.55e-157], x, If[LessEqual[a, 1.95e-171], N[(z * y), $MachinePrecision], If[LessEqual[a, 8.4e+96], x, If[LessEqual[a, 5.8e+214], N[(a * N[(z * b), $MachinePrecision]), $MachinePrecision], N[(a * t), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if a < -5.2000000000000002e-9

   1. Initial program 84.9%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf 70.2%

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

   4. Taylor expanded in t around 0 48.1%

      \[\leadsto \color{blue}{a \cdot \left(b \cdot z\right)} \]
   5. Step-by-step derivation

      1. associate-*r* 50.5%

         \[\leadsto \color{blue}{\left(a \cdot b\right) \cdot z} \]

      2. *-commutative 50.5%

         \[\leadsto \color{blue}{z \cdot \left(a \cdot b\right)} \]

   6. Simplified 50.5%

      \[\leadsto \color{blue}{z \cdot \left(a \cdot b\right)} \]

   if -5.2000000000000002e-9 < a < -1.24999999999999999e-71 or -1.5499999999999999e-157 < a < 1.9499999999999999e-171

   1. Initial program 99.0%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 55.3%

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

      1. *-commutative 55.3%

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

   5. Simplified 55.3%

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

   if -1.24999999999999999e-71 < a < -1.5499999999999999e-157 or 1.9499999999999999e-171 < a < 8.4000000000000005e96

   1. Initial program 99.9%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 58.9%

      \[\leadsto \color{blue}{x} \]

   if 8.4000000000000005e96 < a < 5.7999999999999999e214

   1. Initial program 81.9%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf 60.8%

      \[\leadsto \color{blue}{a \cdot \left(b \cdot z\right)} \]

   if 5.7999999999999999e214 < a

   1. Initial program 77.3%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf 69.5%

      \[\leadsto \color{blue}{a \cdot t} \]
3. Recombined 5 regimes into one program.

4. Final simplification 56.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5.2 \cdot 10^{-9}:\\ \;\;\;\;z \cdot \left(a \cdot b\right)\\ \mathbf{elif}\;a \leq -1.25 \cdot 10^{-71}:\\ \;\;\;\;z \cdot y\\ \mathbf{elif}\;a \leq -1.55 \cdot 10^{-157}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 1.95 \cdot 10^{-171}:\\ \;\;\;\;z \cdot y\\ \mathbf{elif}\;a \leq 8.4 \cdot 10^{+96}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 5.8 \cdot 10^{+214}:\\ \;\;\;\;a \cdot \left(z \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.55 \cdot 10^{+140} \lor \neg \left(a \leq -5.2 \cdot 10^{+88} \lor \neg \left(a \leq -1.05 \cdot 10^{-5}\right) \land a \leq 1.15 \cdot 10^{-14}\right):\\ \;\;\;\;x + a \cdot \left(t + z \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(a \cdot t + z \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= a -1.55e+140)
         (not
          (or (<= a -5.2e+88) (and (not (<= a -1.05e-5)) (<= a 1.15e-14)))))
   (+ x (* a (+ t (* z b))))
   (+ x (+ (* a t) (* z y)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((a <= -1.55e+140) || !((a <= -5.2e+88) || (!(a <= -1.05e-5) && (a <= 1.15e-14)))) {
		tmp = x + (a * (t + (z * b)));
	} else {
		tmp = x + ((a * t) + (z * y));
	}
	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) :: tmp
    if ((a <= (-1.55d+140)) .or. (.not. (a <= (-5.2d+88)) .or. (.not. (a <= (-1.05d-5))) .and. (a <= 1.15d-14))) then
        tmp = x + (a * (t + (z * b)))
    else
        tmp = x + ((a * t) + (z * y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((a <= -1.55e+140) || !((a <= -5.2e+88) || (!(a <= -1.05e-5) && (a <= 1.15e-14)))) {
		tmp = x + (a * (t + (z * b)));
	} else {
		tmp = x + ((a * t) + (z * y));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (a <= -1.55e+140) or not ((a <= -5.2e+88) or (not (a <= -1.05e-5) and (a <= 1.15e-14))):
		tmp = x + (a * (t + (z * b)))
	else:
		tmp = x + ((a * t) + (z * y))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((a <= -1.55e+140) || !((a <= -5.2e+88) || (!(a <= -1.05e-5) && (a <= 1.15e-14))))
		tmp = Float64(x + Float64(a * Float64(t + Float64(z * b))));
	else
		tmp = Float64(x + Float64(Float64(a * t) + Float64(z * y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((a <= -1.55e+140) || ~(((a <= -5.2e+88) || (~((a <= -1.05e-5)) && (a <= 1.15e-14)))))
		tmp = x + (a * (t + (z * b)));
	else
		tmp = x + ((a * t) + (z * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[a, -1.55e+140], N[Not[Or[LessEqual[a, -5.2e+88], And[N[Not[LessEqual[a, -1.05e-5]], $MachinePrecision], LessEqual[a, 1.15e-14]]]], $MachinePrecision]], N[(x + N[(a * N[(t + N[(z * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(a * t), $MachinePrecision] + N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.55e140 or -5.2000000000000001e88 < a < -1.04999999999999994e-5 or 1.14999999999999999e-14 < a

   1. Initial program 86.0%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0 89.0%

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

   4. Taylor expanded in a around inf 90.6%

      \[\leadsto x + \color{blue}{a \cdot \left(t + b \cdot z\right)} \]
   5. Step-by-step derivation

      1. +-commutative 90.6%

         \[\leadsto x + a \cdot \color{blue}{\left(b \cdot z + t\right)} \]

   6. Simplified 90.6%

      \[\leadsto x + \color{blue}{a \cdot \left(b \cdot z + t\right)} \]

   if -1.55e140 < a < -5.2000000000000001e88 or -1.04999999999999994e-5 < a < 1.14999999999999999e-14

   1. Initial program 96.9%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0 94.1%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.55 \cdot 10^{+140} \lor \neg \left(a \leq -5.2 \cdot 10^{+88} \lor \neg \left(a \leq -1.05 \cdot 10^{-5}\right) \land a \leq 1.15 \cdot 10^{-14}\right):\\ \;\;\;\;x + a \cdot \left(t + z \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(a \cdot t + z \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \left(a \cdot t + z \cdot y\right)\\ t_2 := x + a \cdot \left(t + z \cdot b\right)\\ \mathbf{if}\;a \leq -1.55 \cdot 10^{+140}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;a \leq -1.35 \cdot 10^{+86}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq -1.08 \cdot 10^{-23}:\\ \;\;\;\;a \cdot t + z \cdot \left(a \cdot b + y\right)\\ \mathbf{elif}\;a \leq 2.4 \cdot 10^{-14}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ x (+ (* a t) (* z y)))) (t_2 (+ x (* a (+ t (* z b))))))
   (if (<= a -1.55e+140)
     t_2
     (if (<= a -1.35e+86)
       t_1
       (if (<= a -1.08e-23)
         (+ (* a t) (* z (+ (* a b) y)))
         (if (<= a 2.4e-14) t_1 t_2))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x + ((a * t) + (z * y));
	double t_2 = x + (a * (t + (z * b)));
	double tmp;
	if (a <= -1.55e+140) {
		tmp = t_2;
	} else if (a <= -1.35e+86) {
		tmp = t_1;
	} else if (a <= -1.08e-23) {
		tmp = (a * t) + (z * ((a * b) + y));
	} else if (a <= 2.4e-14) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	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) :: t_2
    real(8) :: tmp
    t_1 = x + ((a * t) + (z * y))
    t_2 = x + (a * (t + (z * b)))
    if (a <= (-1.55d+140)) then
        tmp = t_2
    else if (a <= (-1.35d+86)) then
        tmp = t_1
    else if (a <= (-1.08d-23)) then
        tmp = (a * t) + (z * ((a * b) + y))
    else if (a <= 2.4d-14) then
        tmp = t_1
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x + ((a * t) + (z * y));
	double t_2 = x + (a * (t + (z * b)));
	double tmp;
	if (a <= -1.55e+140) {
		tmp = t_2;
	} else if (a <= -1.35e+86) {
		tmp = t_1;
	} else if (a <= -1.08e-23) {
		tmp = (a * t) + (z * ((a * b) + y));
	} else if (a <= 2.4e-14) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x + ((a * t) + (z * y))
	t_2 = x + (a * (t + (z * b)))
	tmp = 0
	if a <= -1.55e+140:
		tmp = t_2
	elif a <= -1.35e+86:
		tmp = t_1
	elif a <= -1.08e-23:
		tmp = (a * t) + (z * ((a * b) + y))
	elif a <= 2.4e-14:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x + Float64(Float64(a * t) + Float64(z * y)))
	t_2 = Float64(x + Float64(a * Float64(t + Float64(z * b))))
	tmp = 0.0
	if (a <= -1.55e+140)
		tmp = t_2;
	elseif (a <= -1.35e+86)
		tmp = t_1;
	elseif (a <= -1.08e-23)
		tmp = Float64(Float64(a * t) + Float64(z * Float64(Float64(a * b) + y)));
	elseif (a <= 2.4e-14)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x + ((a * t) + (z * y));
	t_2 = x + (a * (t + (z * b)));
	tmp = 0.0;
	if (a <= -1.55e+140)
		tmp = t_2;
	elseif (a <= -1.35e+86)
		tmp = t_1;
	elseif (a <= -1.08e-23)
		tmp = (a * t) + (z * ((a * b) + y));
	elseif (a <= 2.4e-14)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x + N[(N[(a * t), $MachinePrecision] + N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x + N[(a * N[(t + N[(z * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -1.55e+140], t$95$2, If[LessEqual[a, -1.35e+86], t$95$1, If[LessEqual[a, -1.08e-23], N[(N[(a * t), $MachinePrecision] + N[(z * N[(N[(a * b), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 2.4e-14], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.55e140 or 2.4e-14 < a

   1. Initial program 83.5%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0 87.1%

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

   4. Taylor expanded in a around inf 92.3%

      \[\leadsto x + \color{blue}{a \cdot \left(t + b \cdot z\right)} \]
   5. Step-by-step derivation

      1. +-commutative 92.3%

         \[\leadsto x + a \cdot \color{blue}{\left(b \cdot z + t\right)} \]

   6. Simplified 92.3%

      \[\leadsto x + \color{blue}{a \cdot \left(b \cdot z + t\right)} \]

   if -1.55e140 < a < -1.35000000000000009e86 or -1.08000000000000003e-23 < a < 2.4e-14

   1. Initial program 96.9%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0 94.7%

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

   if -1.35000000000000009e86 < a < -1.08000000000000003e-23

   1. Initial program 99.9%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0 99.9%

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

   4. Taylor expanded in x around 0 86.1%

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

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.55 \cdot 10^{+140}:\\ \;\;\;\;x + a \cdot \left(t + z \cdot b\right)\\ \mathbf{elif}\;a \leq -1.35 \cdot 10^{+86}:\\ \;\;\;\;x + \left(a \cdot t + z \cdot y\right)\\ \mathbf{elif}\;a \leq -1.08 \cdot 10^{-23}:\\ \;\;\;\;a \cdot t + z \cdot \left(a \cdot b + y\right)\\ \mathbf{elif}\;a \leq 2.4 \cdot 10^{-14}:\\ \;\;\;\;x + \left(a \cdot t + z \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x + a \cdot \left(t + z \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \left(a \cdot t + z \cdot y\right)\\ t_2 := x + a \cdot \left(t + z \cdot b\right)\\ \mathbf{if}\;a \leq -1.56 \cdot 10^{+140}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;a \leq -4.1 \cdot 10^{+86}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq -1.6 \cdot 10^{-8}:\\ \;\;\;\;x + \left(a \cdot t + a \cdot \left(z \cdot b\right)\right)\\ \mathbf{elif}\;a \leq 3.3 \cdot 10^{-15}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ x (+ (* a t) (* z y)))) (t_2 (+ x (* a (+ t (* z b))))))
   (if (<= a -1.56e+140)
     t_2
     (if (<= a -4.1e+86)
       t_1
       (if (<= a -1.6e-8)
         (+ x (+ (* a t) (* a (* z b))))
         (if (<= a 3.3e-15) t_1 t_2))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x + ((a * t) + (z * y));
	double t_2 = x + (a * (t + (z * b)));
	double tmp;
	if (a <= -1.56e+140) {
		tmp = t_2;
	} else if (a <= -4.1e+86) {
		tmp = t_1;
	} else if (a <= -1.6e-8) {
		tmp = x + ((a * t) + (a * (z * b)));
	} else if (a <= 3.3e-15) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	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) :: t_2
    real(8) :: tmp
    t_1 = x + ((a * t) + (z * y))
    t_2 = x + (a * (t + (z * b)))
    if (a <= (-1.56d+140)) then
        tmp = t_2
    else if (a <= (-4.1d+86)) then
        tmp = t_1
    else if (a <= (-1.6d-8)) then
        tmp = x + ((a * t) + (a * (z * b)))
    else if (a <= 3.3d-15) then
        tmp = t_1
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x + ((a * t) + (z * y));
	double t_2 = x + (a * (t + (z * b)));
	double tmp;
	if (a <= -1.56e+140) {
		tmp = t_2;
	} else if (a <= -4.1e+86) {
		tmp = t_1;
	} else if (a <= -1.6e-8) {
		tmp = x + ((a * t) + (a * (z * b)));
	} else if (a <= 3.3e-15) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x + ((a * t) + (z * y))
	t_2 = x + (a * (t + (z * b)))
	tmp = 0
	if a <= -1.56e+140:
		tmp = t_2
	elif a <= -4.1e+86:
		tmp = t_1
	elif a <= -1.6e-8:
		tmp = x + ((a * t) + (a * (z * b)))
	elif a <= 3.3e-15:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x + Float64(Float64(a * t) + Float64(z * y)))
	t_2 = Float64(x + Float64(a * Float64(t + Float64(z * b))))
	tmp = 0.0
	if (a <= -1.56e+140)
		tmp = t_2;
	elseif (a <= -4.1e+86)
		tmp = t_1;
	elseif (a <= -1.6e-8)
		tmp = Float64(x + Float64(Float64(a * t) + Float64(a * Float64(z * b))));
	elseif (a <= 3.3e-15)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x + ((a * t) + (z * y));
	t_2 = x + (a * (t + (z * b)));
	tmp = 0.0;
	if (a <= -1.56e+140)
		tmp = t_2;
	elseif (a <= -4.1e+86)
		tmp = t_1;
	elseif (a <= -1.6e-8)
		tmp = x + ((a * t) + (a * (z * b)));
	elseif (a <= 3.3e-15)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x + N[(N[(a * t), $MachinePrecision] + N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x + N[(a * N[(t + N[(z * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -1.56e+140], t$95$2, If[LessEqual[a, -4.1e+86], t$95$1, If[LessEqual[a, -1.6e-8], N[(x + N[(N[(a * t), $MachinePrecision] + N[(a * N[(z * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 3.3e-15], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.56000000000000002e140 or 3.3e-15 < a

   1. Initial program 83.5%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0 87.1%

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

   4. Taylor expanded in a around inf 92.3%

      \[\leadsto x + \color{blue}{a \cdot \left(t + b \cdot z\right)} \]
   5. Step-by-step derivation

      1. +-commutative 92.3%

         \[\leadsto x + a \cdot \color{blue}{\left(b \cdot z + t\right)} \]

   6. Simplified 92.3%

      \[\leadsto x + \color{blue}{a \cdot \left(b \cdot z + t\right)} \]

   if -1.56000000000000002e140 < a < -4.0999999999999999e86 or -1.6000000000000001e-8 < a < 3.3e-15

   1. Initial program 96.9%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0 94.1%

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

   if -4.0999999999999999e86 < a < -1.6000000000000001e-8

   1. Initial program 100.0%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 80.5%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.56 \cdot 10^{+140}:\\ \;\;\;\;x + a \cdot \left(t + z \cdot b\right)\\ \mathbf{elif}\;a \leq -4.1 \cdot 10^{+86}:\\ \;\;\;\;x + \left(a \cdot t + z \cdot y\right)\\ \mathbf{elif}\;a \leq -1.6 \cdot 10^{-8}:\\ \;\;\;\;x + \left(a \cdot t + a \cdot \left(z \cdot b\right)\right)\\ \mathbf{elif}\;a \leq 3.3 \cdot 10^{-15}:\\ \;\;\;\;x + \left(a \cdot t + z \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x + a \cdot \left(t + z \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 71.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.65 \cdot 10^{+131} \lor \neg \left(a \leq -7.6 \cdot 10^{+88} \lor \neg \left(a \leq -1.12 \cdot 10^{-23}\right) \land a \leq 8.4 \cdot 10^{+96}\right):\\ \;\;\;\;a \cdot \left(t + z \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= a -1.65e+131)
         (not
          (or (<= a -7.6e+88) (and (not (<= a -1.12e-23)) (<= a 8.4e+96)))))
   (* a (+ t (* z b)))
   (+ x (* z y))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((a <= -1.65e+131) || !((a <= -7.6e+88) || (!(a <= -1.12e-23) && (a <= 8.4e+96)))) {
		tmp = a * (t + (z * b));
	} else {
		tmp = x + (z * y);
	}
	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) :: tmp
    if ((a <= (-1.65d+131)) .or. (.not. (a <= (-7.6d+88)) .or. (.not. (a <= (-1.12d-23))) .and. (a <= 8.4d+96))) then
        tmp = a * (t + (z * b))
    else
        tmp = x + (z * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((a <= -1.65e+131) || !((a <= -7.6e+88) || (!(a <= -1.12e-23) && (a <= 8.4e+96)))) {
		tmp = a * (t + (z * b));
	} else {
		tmp = x + (z * y);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (a <= -1.65e+131) or not ((a <= -7.6e+88) or (not (a <= -1.12e-23) and (a <= 8.4e+96))):
		tmp = a * (t + (z * b))
	else:
		tmp = x + (z * y)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((a <= -1.65e+131) || !((a <= -7.6e+88) || (!(a <= -1.12e-23) && (a <= 8.4e+96))))
		tmp = Float64(a * Float64(t + Float64(z * b)));
	else
		tmp = Float64(x + Float64(z * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((a <= -1.65e+131) || ~(((a <= -7.6e+88) || (~((a <= -1.12e-23)) && (a <= 8.4e+96)))))
		tmp = a * (t + (z * b));
	else
		tmp = x + (z * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[a, -1.65e+131], N[Not[Or[LessEqual[a, -7.6e+88], And[N[Not[LessEqual[a, -1.12e-23]], $MachinePrecision], LessEqual[a, 8.4e+96]]]], $MachinePrecision]], N[(a * N[(t + N[(z * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(z * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.6499999999999999e131 or -7.5999999999999993e88 < a < -1.1200000000000001e-23 or 8.4000000000000005e96 < a

   1. Initial program 84.2%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf 82.2%

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

   if -1.6499999999999999e131 < a < -7.5999999999999993e88 or -1.1200000000000001e-23 < a < 8.4000000000000005e96

   1. Initial program 97.3%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0 81.6%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.65 \cdot 10^{+131} \lor \neg \left(a \leq -7.6 \cdot 10^{+88} \lor \neg \left(a \leq -1.12 \cdot 10^{-23}\right) \land a \leq 8.4 \cdot 10^{+96}\right):\\ \;\;\;\;a \cdot \left(t + z \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 71.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot \left(t + z \cdot b\right)\\ \mathbf{if}\;a \leq -1.75 \cdot 10^{+131}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq -6.1 \cdot 10^{-24}:\\ \;\;\;\;z \cdot \left(a \cdot b + y\right)\\ \mathbf{elif}\;a \leq 2.05 \cdot 10^{+97}:\\ \;\;\;\;x + z \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* a (+ t (* z b)))))
   (if (<= a -1.75e+131)
     t_1
     (if (<= a -6.1e-24)
       (* z (+ (* a b) y))
       (if (<= a 2.05e+97) (+ x (* z y)) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = a * (t + (z * b));
	double tmp;
	if (a <= -1.75e+131) {
		tmp = t_1;
	} else if (a <= -6.1e-24) {
		tmp = z * ((a * b) + y);
	} else if (a <= 2.05e+97) {
		tmp = x + (z * y);
	} 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 = a * (t + (z * b))
    if (a <= (-1.75d+131)) then
        tmp = t_1
    else if (a <= (-6.1d-24)) then
        tmp = z * ((a * b) + y)
    else if (a <= 2.05d+97) then
        tmp = x + (z * y)
    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 = a * (t + (z * b));
	double tmp;
	if (a <= -1.75e+131) {
		tmp = t_1;
	} else if (a <= -6.1e-24) {
		tmp = z * ((a * b) + y);
	} else if (a <= 2.05e+97) {
		tmp = x + (z * y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = a * (t + (z * b))
	tmp = 0
	if a <= -1.75e+131:
		tmp = t_1
	elif a <= -6.1e-24:
		tmp = z * ((a * b) + y)
	elif a <= 2.05e+97:
		tmp = x + (z * y)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(a * Float64(t + Float64(z * b)))
	tmp = 0.0
	if (a <= -1.75e+131)
		tmp = t_1;
	elseif (a <= -6.1e-24)
		tmp = Float64(z * Float64(Float64(a * b) + y));
	elseif (a <= 2.05e+97)
		tmp = Float64(x + Float64(z * y));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = a * (t + (z * b));
	tmp = 0.0;
	if (a <= -1.75e+131)
		tmp = t_1;
	elseif (a <= -6.1e-24)
		tmp = z * ((a * b) + y);
	elseif (a <= 2.05e+97)
		tmp = x + (z * y);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(a * N[(t + N[(z * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -1.75e+131], t$95$1, If[LessEqual[a, -6.1e-24], N[(z * N[(N[(a * b), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 2.05e+97], N[(x + N[(z * y), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.7499999999999999e131 or 2.04999999999999994e97 < a

   1. Initial program 80.5%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf 86.4%

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

   if -1.7499999999999999e131 < a < -6.10000000000000036e-24

   1. Initial program 91.3%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf 65.7%

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

   if -6.10000000000000036e-24 < a < 2.04999999999999994e97

   1. Initial program 99.4%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0 82.5%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.75 \cdot 10^{+131}:\\ \;\;\;\;a \cdot \left(t + z \cdot b\right)\\ \mathbf{elif}\;a \leq -6.1 \cdot 10^{-24}:\\ \;\;\;\;z \cdot \left(a \cdot b + y\right)\\ \mathbf{elif}\;a \leq 2.05 \cdot 10^{+97}:\\ \;\;\;\;x + z \cdot y\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(t + z \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 59.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.75 \cdot 10^{+131}:\\ \;\;\;\;z \cdot \left(a \cdot b\right)\\ \mathbf{elif}\;a \leq 3 \cdot 10^{+98}:\\ \;\;\;\;x + z \cdot y\\ \mathbf{elif}\;a \leq 1.92 \cdot 10^{+214}:\\ \;\;\;\;a \cdot \left(z \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;x + a \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= a -1.75e+131)
   (* z (* a b))
   (if (<= a 3e+98)
     (+ x (* z y))
     (if (<= a 1.92e+214) (* a (* z b)) (+ x (* a t))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= -1.75e+131) {
		tmp = z * (a * b);
	} else if (a <= 3e+98) {
		tmp = x + (z * y);
	} else if (a <= 1.92e+214) {
		tmp = a * (z * b);
	} else {
		tmp = x + (a * t);
	}
	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) :: tmp
    if (a <= (-1.75d+131)) then
        tmp = z * (a * b)
    else if (a <= 3d+98) then
        tmp = x + (z * y)
    else if (a <= 1.92d+214) then
        tmp = a * (z * b)
    else
        tmp = x + (a * t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= -1.75e+131) {
		tmp = z * (a * b);
	} else if (a <= 3e+98) {
		tmp = x + (z * y);
	} else if (a <= 1.92e+214) {
		tmp = a * (z * b);
	} else {
		tmp = x + (a * t);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if a <= -1.75e+131:
		tmp = z * (a * b)
	elif a <= 3e+98:
		tmp = x + (z * y)
	elif a <= 1.92e+214:
		tmp = a * (z * b)
	else:
		tmp = x + (a * t)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (a <= -1.75e+131)
		tmp = Float64(z * Float64(a * b));
	elseif (a <= 3e+98)
		tmp = Float64(x + Float64(z * y));
	elseif (a <= 1.92e+214)
		tmp = Float64(a * Float64(z * b));
	else
		tmp = Float64(x + Float64(a * t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (a <= -1.75e+131)
		tmp = z * (a * b);
	elseif (a <= 3e+98)
		tmp = x + (z * y);
	elseif (a <= 1.92e+214)
		tmp = a * (z * b);
	else
		tmp = x + (a * t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[a, -1.75e+131], N[(z * N[(a * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 3e+98], N[(x + N[(z * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1.92e+214], N[(a * N[(z * b), $MachinePrecision]), $MachinePrecision], N[(x + N[(a * t), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if a < -1.7499999999999999e131

   1. Initial program 81.1%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf 84.5%

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

   4. Taylor expanded in t around 0 58.5%

      \[\leadsto \color{blue}{a \cdot \left(b \cdot z\right)} \]
   5. Step-by-step derivation

      1. associate-*r* 60.5%

         \[\leadsto \color{blue}{\left(a \cdot b\right) \cdot z} \]

      2. *-commutative 60.5%

         \[\leadsto \color{blue}{z \cdot \left(a \cdot b\right)} \]

   6. Simplified 60.5%

      \[\leadsto \color{blue}{z \cdot \left(a \cdot b\right)} \]

   if -1.7499999999999999e131 < a < 3.0000000000000001e98

   1. Initial program 97.7%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0 75.3%

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

   if 3.0000000000000001e98 < a < 1.92e214

   1. Initial program 81.9%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf 60.8%

      \[\leadsto \color{blue}{a \cdot \left(b \cdot z\right)} \]

   if 1.92e214 < a

   1. Initial program 77.3%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0 78.1%

      \[\leadsto \color{blue}{x + a \cdot t} \]
3. Recombined 4 regimes into one program.

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.75 \cdot 10^{+131}:\\ \;\;\;\;z \cdot \left(a \cdot b\right)\\ \mathbf{elif}\;a \leq 3 \cdot 10^{+98}:\\ \;\;\;\;x + z \cdot y\\ \mathbf{elif}\;a \leq 1.92 \cdot 10^{+214}:\\ \;\;\;\;a \cdot \left(z \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;x + a \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 81.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -9.5 \cdot 10^{-24} \lor \neg \left(a \leq 1.15 \cdot 10^{-15}\right):\\ \;\;\;\;x + a \cdot \left(t + z \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= a -9.5e-24) (not (<= a 1.15e-15)))
   (+ x (* a (+ t (* z b))))
   (+ x (* z y))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((a <= -9.5e-24) || !(a <= 1.15e-15)) {
		tmp = x + (a * (t + (z * b)));
	} else {
		tmp = x + (z * y);
	}
	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) :: tmp
    if ((a <= (-9.5d-24)) .or. (.not. (a <= 1.15d-15))) then
        tmp = x + (a * (t + (z * b)))
    else
        tmp = x + (z * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((a <= -9.5e-24) || !(a <= 1.15e-15)) {
		tmp = x + (a * (t + (z * b)));
	} else {
		tmp = x + (z * y);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (a <= -9.5e-24) or not (a <= 1.15e-15):
		tmp = x + (a * (t + (z * b)))
	else:
		tmp = x + (z * y)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((a <= -9.5e-24) || !(a <= 1.15e-15))
		tmp = Float64(x + Float64(a * Float64(t + Float64(z * b))));
	else
		tmp = Float64(x + Float64(z * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((a <= -9.5e-24) || ~((a <= 1.15e-15)))
		tmp = x + (a * (t + (z * b)));
	else
		tmp = x + (z * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[a, -9.5e-24], N[Not[LessEqual[a, 1.15e-15]], $MachinePrecision]], N[(x + N[(a * N[(t + N[(z * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(z * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -9.50000000000000029e-24 or 1.14999999999999995e-15 < a

   1. Initial program 85.6%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0 90.2%

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

   4. Taylor expanded in a around inf 87.1%

      \[\leadsto x + \color{blue}{a \cdot \left(t + b \cdot z\right)} \]
   5. Step-by-step derivation

      1. +-commutative 87.1%

         \[\leadsto x + a \cdot \color{blue}{\left(b \cdot z + t\right)} \]

   6. Simplified 87.1%

      \[\leadsto x + \color{blue}{a \cdot \left(b \cdot z + t\right)} \]

   if -9.50000000000000029e-24 < a < 1.14999999999999995e-15

   1. Initial program 99.3%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0 85.6%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -9.5 \cdot 10^{-24} \lor \neg \left(a \leq 1.15 \cdot 10^{-15}\right):\\ \;\;\;\;x + a \cdot \left(t + z \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 94.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 4.8 \cdot 10^{+37}:\\ \;\;\;\;x + \left(a \cdot t + z \cdot \left(a \cdot b + y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + a \cdot \left(t + z \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= a 4.8e+37)
   (+ x (+ (* a t) (* z (+ (* a b) y))))
   (+ x (* a (+ t (* z b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= 4.8e+37) {
		tmp = x + ((a * t) + (z * ((a * b) + y)));
	} else {
		tmp = x + (a * (t + (z * b)));
	}
	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) :: tmp
    if (a <= 4.8d+37) then
        tmp = x + ((a * t) + (z * ((a * b) + y)))
    else
        tmp = x + (a * (t + (z * b)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= 4.8e+37) {
		tmp = x + ((a * t) + (z * ((a * b) + y)));
	} else {
		tmp = x + (a * (t + (z * b)));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if a <= 4.8e+37:
		tmp = x + ((a * t) + (z * ((a * b) + y)))
	else:
		tmp = x + (a * (t + (z * b)))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (a <= 4.8e+37)
		tmp = Float64(x + Float64(Float64(a * t) + Float64(z * Float64(Float64(a * b) + y))));
	else
		tmp = Float64(x + Float64(a * Float64(t + Float64(z * b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (a <= 4.8e+37)
		tmp = x + ((a * t) + (z * ((a * b) + y)));
	else
		tmp = x + (a * (t + (z * b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[a, 4.8e+37], N[(x + N[(N[(a * t), $MachinePrecision] + N[(z * N[(N[(a * b), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(a * N[(t + N[(z * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 4.8e37

   1. Initial program 93.6%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0 96.9%

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

   if 4.8e37 < a

   1. Initial program 83.5%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0 85.3%

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

   4. Taylor expanded in a around inf 96.8%

      \[\leadsto x + \color{blue}{a \cdot \left(t + b \cdot z\right)} \]
   5. Step-by-step derivation

      1. +-commutative 96.8%

         \[\leadsto x + a \cdot \color{blue}{\left(b \cdot z + t\right)} \]

   6. Simplified 96.8%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 4.8 \cdot 10^{+37}:\\ \;\;\;\;x + \left(a \cdot t + z \cdot \left(a \cdot b + y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + a \cdot \left(t + z \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 59.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.1 \cdot 10^{+103}:\\ \;\;\;\;a \cdot \left(z \cdot b\right)\\ \mathbf{elif}\;z \leq 1.16 \cdot 10^{+70}:\\ \;\;\;\;x + a \cdot t\\ \mathbf{else}:\\ \;\;\;\;z \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -4.1e+103)
   (* a (* z b))
   (if (<= z 1.16e+70) (+ x (* a t)) (* z y))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -4.1e+103) {
		tmp = a * (z * b);
	} else if (z <= 1.16e+70) {
		tmp = x + (a * t);
	} else {
		tmp = z * y;
	}
	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) :: tmp
    if (z <= (-4.1d+103)) then
        tmp = a * (z * b)
    else if (z <= 1.16d+70) then
        tmp = x + (a * t)
    else
        tmp = z * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -4.1e+103) {
		tmp = a * (z * b);
	} else if (z <= 1.16e+70) {
		tmp = x + (a * t);
	} else {
		tmp = z * y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if z <= -4.1e+103:
		tmp = a * (z * b)
	elif z <= 1.16e+70:
		tmp = x + (a * t)
	else:
		tmp = z * y
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -4.1e+103)
		tmp = Float64(a * Float64(z * b));
	elseif (z <= 1.16e+70)
		tmp = Float64(x + Float64(a * t));
	else
		tmp = Float64(z * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (z <= -4.1e+103)
		tmp = a * (z * b);
	elseif (z <= 1.16e+70)
		tmp = x + (a * t);
	else
		tmp = z * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -4.1e+103], N[(a * N[(z * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.16e+70], N[(x + N[(a * t), $MachinePrecision]), $MachinePrecision], N[(z * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.1000000000000002e103

   1. Initial program 76.1%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf 54.6%

      \[\leadsto \color{blue}{a \cdot \left(b \cdot z\right)} \]

   if -4.1000000000000002e103 < z < 1.1599999999999999e70

   1. Initial program 96.4%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0 66.5%

      \[\leadsto \color{blue}{x + a \cdot t} \]

   if 1.1599999999999999e70 < z

   1. Initial program 87.9%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 47.5%

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

      1. *-commutative 47.5%

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

   5. Simplified 47.5%

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

4. Final simplification 60.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.1 \cdot 10^{+103}:\\ \;\;\;\;a \cdot \left(z \cdot b\right)\\ \mathbf{elif}\;z \leq 1.16 \cdot 10^{+70}:\\ \;\;\;\;x + a \cdot t\\ \mathbf{else}:\\ \;\;\;\;z \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 37.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{+94}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 6 \cdot 10^{-44}:\\ \;\;\;\;z \cdot y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= x -2.4e+94) x (if (<= x 6e-44) (* z y) x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (x <= -2.4e+94) {
		tmp = x;
	} else if (x <= 6e-44) {
		tmp = z * y;
	} else {
		tmp = x;
	}
	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) :: tmp
    if (x <= (-2.4d+94)) then
        tmp = x
    else if (x <= 6d-44) then
        tmp = z * y
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (x <= -2.4e+94) {
		tmp = x;
	} else if (x <= 6e-44) {
		tmp = z * y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if x <= -2.4e+94:
		tmp = x
	elif x <= 6e-44:
		tmp = z * y
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (x <= -2.4e+94)
		tmp = x;
	elseif (x <= 6e-44)
		tmp = Float64(z * y);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (x <= -2.4e+94)
		tmp = x;
	elseif (x <= 6e-44)
		tmp = z * y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[x, -2.4e+94], x, If[LessEqual[x, 6e-44], N[(z * y), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if x < -2.39999999999999983e94 or 6.0000000000000005e-44 < x

   1. Initial program 90.9%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 49.0%

      \[\leadsto \color{blue}{x} \]

   if -2.39999999999999983e94 < x < 6.0000000000000005e-44

   1. Initial program 91.5%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 41.8%

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

      1. *-commutative 41.8%

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

   5. Simplified 41.8%

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

Alternative 14: 37.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.4 \cdot 10^{-24}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{-42}:\\ \;\;\;\;a \cdot t\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= x -9.4e-24) x (if (<= x 3.5e-42) (* a t) x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (x <= -9.4e-24) {
		tmp = x;
	} else if (x <= 3.5e-42) {
		tmp = a * t;
	} else {
		tmp = x;
	}
	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) :: tmp
    if (x <= (-9.4d-24)) then
        tmp = x
    else if (x <= 3.5d-42) then
        tmp = a * t
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (x <= -9.4e-24) {
		tmp = x;
	} else if (x <= 3.5e-42) {
		tmp = a * t;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if x <= -9.4e-24:
		tmp = x
	elif x <= 3.5e-42:
		tmp = a * t
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (x <= -9.4e-24)
		tmp = x;
	elseif (x <= 3.5e-42)
		tmp = Float64(a * t);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (x <= -9.4e-24)
		tmp = x;
	elseif (x <= 3.5e-42)
		tmp = a * t;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[x, -9.4e-24], x, If[LessEqual[x, 3.5e-42], N[(a * t), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if x < -9.39999999999999983e-24 or 3.5000000000000002e-42 < x

   1. Initial program 90.2%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 46.6%

      \[\leadsto \color{blue}{x} \]

   if -9.39999999999999983e-24 < x < 3.5000000000000002e-42

   1. Initial program 92.4%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf 31.4%

      \[\leadsto \color{blue}{a \cdot t} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 26.0% accurate, 15.0× speedup

Math

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

FPCore

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

C

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

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
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.2%

   \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
2. Add Preprocessing

3. Taylor expanded in x around inf 27.6%

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

Developer target: 97.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot \left(b \cdot a + y\right) + \left(x + t \cdot a\right)\\ \mathbf{if}\;z < -11820553527347888000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z < 4.7589743188364287 \cdot 10^{-122}:\\ \;\;\;\;\left(b \cdot z + t\right) \cdot a + \left(z \cdot y + x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (* z (+ (* b a) y)) (+ x (* t a)))))
   (if (< z -11820553527347888000.0)
     t_1
     (if (< z 4.7589743188364287e-122)
       (+ (* (+ (* b z) t) a) (+ (* z y) x))
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (z * ((b * a) + y)) + (x + (t * a));
	double tmp;
	if (z < -11820553527347888000.0) {
		tmp = t_1;
	} else if (z < 4.7589743188364287e-122) {
		tmp = (((b * z) + t) * a) + ((z * y) + x);
	} 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 = (z * ((b * a) + y)) + (x + (t * a))
    if (z < (-11820553527347888000.0d0)) then
        tmp = t_1
    else if (z < 4.7589743188364287d-122) then
        tmp = (((b * z) + t) * a) + ((z * y) + x)
    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 = (z * ((b * a) + y)) + (x + (t * a));
	double tmp;
	if (z < -11820553527347888000.0) {
		tmp = t_1;
	} else if (z < 4.7589743188364287e-122) {
		tmp = (((b * z) + t) * a) + ((z * y) + x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (z * ((b * a) + y)) + (x + (t * a))
	tmp = 0
	if z < -11820553527347888000.0:
		tmp = t_1
	elif z < 4.7589743188364287e-122:
		tmp = (((b * z) + t) * a) + ((z * y) + x)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(z * Float64(Float64(b * a) + y)) + Float64(x + Float64(t * a)))
	tmp = 0.0
	if (z < -11820553527347888000.0)
		tmp = t_1;
	elseif (z < 4.7589743188364287e-122)
		tmp = Float64(Float64(Float64(Float64(b * z) + t) * a) + Float64(Float64(z * y) + x));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (z * ((b * a) + y)) + (x + (t * a));
	tmp = 0.0;
	if (z < -11820553527347888000.0)
		tmp = t_1;
	elseif (z < 4.7589743188364287e-122)
		tmp = (((b * z) + t) * a) + ((z * y) + x);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(z * N[(N[(b * a), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision] + N[(x + N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Less[z, -11820553527347888000.0], t$95$1, If[Less[z, 4.7589743188364287e-122], N[(N[(N[(N[(b * z), $MachinePrecision] + t), $MachinePrecision] * a), $MachinePrecision] + N[(N[(z * y), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Reproduce

herbie shell --seed 2024097 
(FPCore (x y z t a b)
  :name "Graphics.Rasterific.CubicBezier:cachedBezierAt from Rasterific-0.6.1"
  :precision binary64

  :alt
  (if (< z -11820553527347888000.0) (+ (* z (+ (* b a) y)) (+ x (* t a))) (if (< z 4.7589743188364287e-122) (+ (* (+ (* b z) t) a) (+ (* z y) x)) (+ (* z (+ (* b a) y)) (+ x (* t a)))))

  (+ (+ (+ x (* y z)) (* t a)) (* (* a z) b)))