Graphics.Rasterific.CubicBezier:cachedBezierAt from Rasterific-0.6.1

Percentage Accurate: 92.4% → 96.3%

Time: 12.7s

Alternatives: 13

Speedup: 0.7×

• 31.7% 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: 92.4% 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: 96.3% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (+ (+ x (* y z)) (* t a)) (* (* z a) b))))
   (if (<= t_1 INFINITY) t_1 (* a (+ t (* z b))))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (+.f64 x (*.f64 y z)) (*.f64 t a)) (*.f64 (*.f64 a z) b)) < +inf.0

   1. Initial program 99.5%

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

   if +inf.0 < (+.f64 (+.f64 (+.f64 x (*.f64 y z)) (*.f64 t a)) (*.f64 (*.f64 a z) b))

   1. Initial program 0.0%

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

      1. associate-+l+ 0.0%

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

      2. associate-*l* 6.7%

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

   3. Simplified 6.7%

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

   5. Taylor expanded in a around inf 93.3%

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

4. Final simplification 99.2%

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

Alternative 2: 38.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.8 \cdot 10^{-13}:\\ \;\;\;\;t \cdot a\\ \mathbf{elif}\;a \leq -3.1 \cdot 10^{-213}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;a \leq -6.6 \cdot 10^{-245}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 7.6 \cdot 10^{-123}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;a \leq 1.25 \cdot 10^{-57}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 8.4 \cdot 10^{+67}:\\ \;\;\;\;t \cdot a\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(z \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= a -3.8e-13)
   (* t a)
   (if (<= a -3.1e-213)
     (* y z)
     (if (<= a -6.6e-245)
       x
       (if (<= a 7.6e-123)
         (* y z)
         (if (<= a 1.25e-57) x (if (<= a 8.4e+67) (* t a) (* a (* z b)))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= -3.8e-13) {
		tmp = t * a;
	} else if (a <= -3.1e-213) {
		tmp = y * z;
	} else if (a <= -6.6e-245) {
		tmp = x;
	} else if (a <= 7.6e-123) {
		tmp = y * z;
	} else if (a <= 1.25e-57) {
		tmp = x;
	} else if (a <= 8.4e+67) {
		tmp = t * a;
	} else {
		tmp = a * (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 <= (-3.8d-13)) then
        tmp = t * a
    else if (a <= (-3.1d-213)) then
        tmp = y * z
    else if (a <= (-6.6d-245)) then
        tmp = x
    else if (a <= 7.6d-123) then
        tmp = y * z
    else if (a <= 1.25d-57) then
        tmp = x
    else if (a <= 8.4d+67) then
        tmp = t * a
    else
        tmp = a * (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 <= -3.8e-13) {
		tmp = t * a;
	} else if (a <= -3.1e-213) {
		tmp = y * z;
	} else if (a <= -6.6e-245) {
		tmp = x;
	} else if (a <= 7.6e-123) {
		tmp = y * z;
	} else if (a <= 1.25e-57) {
		tmp = x;
	} else if (a <= 8.4e+67) {
		tmp = t * a;
	} else {
		tmp = a * (z * b);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if a <= -3.8e-13:
		tmp = t * a
	elif a <= -3.1e-213:
		tmp = y * z
	elif a <= -6.6e-245:
		tmp = x
	elif a <= 7.6e-123:
		tmp = y * z
	elif a <= 1.25e-57:
		tmp = x
	elif a <= 8.4e+67:
		tmp = t * a
	else:
		tmp = a * (z * b)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (a <= -3.8e-13)
		tmp = Float64(t * a);
	elseif (a <= -3.1e-213)
		tmp = Float64(y * z);
	elseif (a <= -6.6e-245)
		tmp = x;
	elseif (a <= 7.6e-123)
		tmp = Float64(y * z);
	elseif (a <= 1.25e-57)
		tmp = x;
	elseif (a <= 8.4e+67)
		tmp = Float64(t * a);
	else
		tmp = Float64(a * Float64(z * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (a <= -3.8e-13)
		tmp = t * a;
	elseif (a <= -3.1e-213)
		tmp = y * z;
	elseif (a <= -6.6e-245)
		tmp = x;
	elseif (a <= 7.6e-123)
		tmp = y * z;
	elseif (a <= 1.25e-57)
		tmp = x;
	elseif (a <= 8.4e+67)
		tmp = t * a;
	else
		tmp = a * (z * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[a, -3.8e-13], N[(t * a), $MachinePrecision], If[LessEqual[a, -3.1e-213], N[(y * z), $MachinePrecision], If[LessEqual[a, -6.6e-245], x, If[LessEqual[a, 7.6e-123], N[(y * z), $MachinePrecision], If[LessEqual[a, 1.25e-57], x, If[LessEqual[a, 8.4e+67], N[(t * a), $MachinePrecision], N[(a * N[(z * b), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if a < -3.8e-13 or 1.25e-57 < a < 8.4000000000000005e67

   1. Initial program 89.9%

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

      1. associate-+l+ 89.9%

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

      2. associate-*l* 91.0%

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

   3. Simplified 91.0%

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

   5. Taylor expanded in t around inf 45.4%

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

   if -3.8e-13 < a < -3.0999999999999998e-213 or -6.6000000000000002e-245 < a < 7.59999999999999991e-123

   1. Initial program 98.9%

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

      1. associate-+l+ 98.9%

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

      2. associate-*l* 90.6%

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

   3. Simplified 90.6%

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

   5. Taylor expanded in y around inf 58.3%

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

      1. *-commutative 58.3%

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

   7. Simplified 58.3%

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

   if -3.0999999999999998e-213 < a < -6.6000000000000002e-245 or 7.59999999999999991e-123 < a < 1.25e-57

   1. Initial program 100.0%

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

      1. associate-+l+ 100.0%

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

      2. associate-*l* 91.2%

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

   3. Simplified 91.2%

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

   5. Taylor expanded in x around inf 65.2%

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

   if 8.4000000000000005e67 < a

   1. Initial program 88.4%

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

      1. associate-+l+ 88.4%

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

      2. +-commutative 88.4%

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

      3. fma-def 88.4%

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

      4. associate-*l* 90.3%

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

      5. *-commutative 90.3%

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

      6. *-commutative 90.3%

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

      7. distribute-rgt-out 92.2%

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

      8. *-commutative 92.2%

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

   3. Simplified 92.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, z, x\right) + a \cdot \left(t + z \cdot b\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-cube-cbrt 91.8%

         \[\leadsto \mathsf{fma}\left(y, z, x\right) + a \cdot \color{blue}{\left(\left(\sqrt[3]{t + z \cdot b} \cdot \sqrt[3]{t + z \cdot b}\right) \cdot \sqrt[3]{t + z \cdot b}\right)} \]

      2. pow3 91.8%

         \[\leadsto \mathsf{fma}\left(y, z, x\right) + a \cdot \color{blue}{{\left(\sqrt[3]{t + z \cdot b}\right)}^{3}} \]

      3. +-commutative 91.8%

         \[\leadsto \mathsf{fma}\left(y, z, x\right) + a \cdot {\left(\sqrt[3]{\color{blue}{z \cdot b + t}}\right)}^{3} \]

      4. fma-def 91.8%

         \[\leadsto \mathsf{fma}\left(y, z, x\right) + a \cdot {\left(\sqrt[3]{\color{blue}{\mathsf{fma}\left(z, b, t\right)}}\right)}^{3} \]

   6. Applied egg-rr 91.8%

      \[\leadsto \mathsf{fma}\left(y, z, x\right) + a \cdot \color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(z, b, t\right)}\right)}^{3}} \]

   7. Taylor expanded in z around inf 51.9%

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

      1. *-commutative 51.9%

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

   9. Simplified 51.9%

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

   10. Taylor expanded in y around 0 53.2%

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

       1. *-commutative 53.2%

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

   12. Simplified 53.2%

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

4. Final simplification 53.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.8 \cdot 10^{-13}:\\ \;\;\;\;t \cdot a\\ \mathbf{elif}\;a \leq -3.1 \cdot 10^{-213}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;a \leq -6.6 \cdot 10^{-245}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 7.6 \cdot 10^{-123}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;a \leq 1.25 \cdot 10^{-57}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 8.4 \cdot 10^{+67}:\\ \;\;\;\;t \cdot a\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(z \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 38.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.3 \cdot 10^{-12}:\\ \;\;\;\;t \cdot a\\ \mathbf{elif}\;a \leq -6 \cdot 10^{-210}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;a \leq -1.95 \cdot 10^{-244}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 10^{-120}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;a \leq 1.25 \cdot 10^{-57}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= a -2.3e-12)
   (* t a)
   (if (<= a -6e-210)
     (* y z)
     (if (<= a -1.95e-244)
       x
       (if (<= a 1e-120) (* y z) (if (<= a 1.25e-57) x (* t a)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= -2.3e-12) {
		tmp = t * a;
	} else if (a <= -6e-210) {
		tmp = y * z;
	} else if (a <= -1.95e-244) {
		tmp = x;
	} else if (a <= 1e-120) {
		tmp = y * z;
	} else if (a <= 1.25e-57) {
		tmp = x;
	} else {
		tmp = t * a;
	}
	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 <= (-2.3d-12)) then
        tmp = t * a
    else if (a <= (-6d-210)) then
        tmp = y * z
    else if (a <= (-1.95d-244)) then
        tmp = x
    else if (a <= 1d-120) then
        tmp = y * z
    else if (a <= 1.25d-57) then
        tmp = x
    else
        tmp = t * a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= -2.3e-12) {
		tmp = t * a;
	} else if (a <= -6e-210) {
		tmp = y * z;
	} else if (a <= -1.95e-244) {
		tmp = x;
	} else if (a <= 1e-120) {
		tmp = y * z;
	} else if (a <= 1.25e-57) {
		tmp = x;
	} else {
		tmp = t * a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if a <= -2.3e-12:
		tmp = t * a
	elif a <= -6e-210:
		tmp = y * z
	elif a <= -1.95e-244:
		tmp = x
	elif a <= 1e-120:
		tmp = y * z
	elif a <= 1.25e-57:
		tmp = x
	else:
		tmp = t * a
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (a <= -2.3e-12)
		tmp = Float64(t * a);
	elseif (a <= -6e-210)
		tmp = Float64(y * z);
	elseif (a <= -1.95e-244)
		tmp = x;
	elseif (a <= 1e-120)
		tmp = Float64(y * z);
	elseif (a <= 1.25e-57)
		tmp = x;
	else
		tmp = Float64(t * a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (a <= -2.3e-12)
		tmp = t * a;
	elseif (a <= -6e-210)
		tmp = y * z;
	elseif (a <= -1.95e-244)
		tmp = x;
	elseif (a <= 1e-120)
		tmp = y * z;
	elseif (a <= 1.25e-57)
		tmp = x;
	else
		tmp = t * a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[a, -2.3e-12], N[(t * a), $MachinePrecision], If[LessEqual[a, -6e-210], N[(y * z), $MachinePrecision], If[LessEqual[a, -1.95e-244], x, If[LessEqual[a, 1e-120], N[(y * z), $MachinePrecision], If[LessEqual[a, 1.25e-57], x, N[(t * a), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if a < -2.29999999999999989e-12 or 1.25e-57 < a

   1. Initial program 89.3%

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

      1. associate-+l+ 89.3%

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

      2. associate-*l* 90.7%

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

   3. Simplified 90.7%

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

   5. Taylor expanded in t around inf 43.0%

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

   if -2.29999999999999989e-12 < a < -6.0000000000000003e-210 or -1.9499999999999999e-244 < a < 9.99999999999999979e-121

   1. Initial program 98.9%

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

      1. associate-+l+ 98.9%

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

      2. associate-*l* 90.6%

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

   3. Simplified 90.6%

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

   5. Taylor expanded in y around inf 58.3%

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

      1. *-commutative 58.3%

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

   7. Simplified 58.3%

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

   if -6.0000000000000003e-210 < a < -1.9499999999999999e-244 or 9.99999999999999979e-121 < a < 1.25e-57

   1. Initial program 100.0%

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

      1. associate-+l+ 100.0%

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

      2. associate-*l* 91.2%

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

   3. Simplified 91.2%

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

   5. Taylor expanded in x around inf 65.2%

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

4. Final simplification 50.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.3 \cdot 10^{-12}:\\ \;\;\;\;t \cdot a\\ \mathbf{elif}\;a \leq -6 \cdot 10^{-210}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;a \leq -1.95 \cdot 10^{-244}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 10^{-120}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;a \leq 1.25 \cdot 10^{-57}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 63.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + t \cdot a\\ \mathbf{if}\;a \leq -6 \cdot 10^{-13}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 8000000000:\\ \;\;\;\;x + y \cdot z\\ \mathbf{elif}\;a \leq 4.2 \cdot 10^{+195} \lor \neg \left(a \leq 1.5 \cdot 10^{+292}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(z \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ x (* t a))))
   (if (<= a -6e-13)
     t_1
     (if (<= a 8000000000.0)
       (+ x (* y z))
       (if (or (<= a 4.2e+195) (not (<= a 1.5e+292))) t_1 (* a (* z b)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x + (t * a);
	double tmp;
	if (a <= -6e-13) {
		tmp = t_1;
	} else if (a <= 8000000000.0) {
		tmp = x + (y * z);
	} else if ((a <= 4.2e+195) || !(a <= 1.5e+292)) {
		tmp = t_1;
	} else {
		tmp = a * (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) :: t_1
    real(8) :: tmp
    t_1 = x + (t * a)
    if (a <= (-6d-13)) then
        tmp = t_1
    else if (a <= 8000000000.0d0) then
        tmp = x + (y * z)
    else if ((a <= 4.2d+195) .or. (.not. (a <= 1.5d+292))) then
        tmp = t_1
    else
        tmp = a * (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 t_1 = x + (t * a);
	double tmp;
	if (a <= -6e-13) {
		tmp = t_1;
	} else if (a <= 8000000000.0) {
		tmp = x + (y * z);
	} else if ((a <= 4.2e+195) || !(a <= 1.5e+292)) {
		tmp = t_1;
	} else {
		tmp = a * (z * b);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x + (t * a)
	tmp = 0
	if a <= -6e-13:
		tmp = t_1
	elif a <= 8000000000.0:
		tmp = x + (y * z)
	elif (a <= 4.2e+195) or not (a <= 1.5e+292):
		tmp = t_1
	else:
		tmp = a * (z * b)
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x + Float64(t * a))
	tmp = 0.0
	if (a <= -6e-13)
		tmp = t_1;
	elseif (a <= 8000000000.0)
		tmp = Float64(x + Float64(y * z));
	elseif ((a <= 4.2e+195) || !(a <= 1.5e+292))
		tmp = t_1;
	else
		tmp = Float64(a * Float64(z * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x + (t * a);
	tmp = 0.0;
	if (a <= -6e-13)
		tmp = t_1;
	elseif (a <= 8000000000.0)
		tmp = x + (y * z);
	elseif ((a <= 4.2e+195) || ~((a <= 1.5e+292)))
		tmp = t_1;
	else
		tmp = a * (z * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x + N[(t * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -6e-13], t$95$1, If[LessEqual[a, 8000000000.0], N[(x + N[(y * z), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[a, 4.2e+195], N[Not[LessEqual[a, 1.5e+292]], $MachinePrecision]], t$95$1, N[(a * N[(z * b), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if a < -5.99999999999999968e-13 or 8e9 < a < 4.20000000000000019e195 or 1.50000000000000002e292 < a

   1. Initial program 90.5%

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

      1. associate-+l+ 90.5%

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

      2. associate-*l* 91.4%

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

   3. Simplified 91.4%

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

   5. Taylor expanded in z around 0 60.7%

      \[\leadsto \color{blue}{x + a \cdot t} \]
   6. Step-by-step derivation

      1. +-commutative 60.7%

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

   7. Simplified 60.7%

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

   if -5.99999999999999968e-13 < a < 8e9

   1. Initial program 99.2%

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

      1. associate-+l+ 99.2%

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

      2. associate-*l* 91.6%

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

   3. Simplified 91.6%

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

   5. Taylor expanded in a around 0 78.6%

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

   if 4.20000000000000019e195 < a < 1.50000000000000002e292

   1. Initial program 79.1%

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

      1. associate-+l+ 79.1%

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

      2. +-commutative 79.1%

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

      3. fma-def 79.1%

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

      4. associate-*l* 83.3%

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

      5. *-commutative 83.3%

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

      6. *-commutative 83.3%

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

      7. distribute-rgt-out 87.4%

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

      8. *-commutative 87.4%

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

   3. Simplified 87.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, z, x\right) + a \cdot \left(t + z \cdot b\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-cube-cbrt 87.3%

         \[\leadsto \mathsf{fma}\left(y, z, x\right) + a \cdot \color{blue}{\left(\left(\sqrt[3]{t + z \cdot b} \cdot \sqrt[3]{t + z \cdot b}\right) \cdot \sqrt[3]{t + z \cdot b}\right)} \]

      2. pow3 87.3%

         \[\leadsto \mathsf{fma}\left(y, z, x\right) + a \cdot \color{blue}{{\left(\sqrt[3]{t + z \cdot b}\right)}^{3}} \]

      3. +-commutative 87.3%

         \[\leadsto \mathsf{fma}\left(y, z, x\right) + a \cdot {\left(\sqrt[3]{\color{blue}{z \cdot b + t}}\right)}^{3} \]

      4. fma-def 87.3%

         \[\leadsto \mathsf{fma}\left(y, z, x\right) + a \cdot {\left(\sqrt[3]{\color{blue}{\mathsf{fma}\left(z, b, t\right)}}\right)}^{3} \]

   6. Applied egg-rr 87.3%

      \[\leadsto \mathsf{fma}\left(y, z, x\right) + a \cdot \color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(z, b, t\right)}\right)}^{3}} \]

   7. Taylor expanded in z around inf 59.0%

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

      1. *-commutative 59.0%

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

   9. Simplified 59.0%

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

   10. Taylor expanded in y around 0 66.8%

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

       1. *-commutative 66.8%

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

   12. Simplified 66.8%

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

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -6 \cdot 10^{-13}:\\ \;\;\;\;x + t \cdot a\\ \mathbf{elif}\;a \leq 8000000000:\\ \;\;\;\;x + y \cdot z\\ \mathbf{elif}\;a \leq 4.2 \cdot 10^{+195} \lor \neg \left(a \leq 1.5 \cdot 10^{+292}\right):\\ \;\;\;\;x + t \cdot a\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(z \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + a \cdot \left(t + z \cdot b\right)\\ \mathbf{if}\;a \leq -9.5 \cdot 10^{-14}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -4.5 \cdot 10^{-160}:\\ \;\;\;\;z \cdot \left(y + a \cdot b\right)\\ \mathbf{elif}\;a \leq 3.05 \cdot 10^{-120}:\\ \;\;\;\;x + y \cdot z\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ x (* a (+ t (* z b))))))
   (if (<= a -9.5e-14)
     t_1
     (if (<= a -4.5e-160)
       (* z (+ y (* a b)))
       (if (<= a 3.05e-120) (+ x (* y z)) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x + (a * (t + (z * b)));
	double tmp;
	if (a <= -9.5e-14) {
		tmp = t_1;
	} else if (a <= -4.5e-160) {
		tmp = z * (y + (a * b));
	} else if (a <= 3.05e-120) {
		tmp = x + (y * z);
	} 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 + (a * (t + (z * b)))
    if (a <= (-9.5d-14)) then
        tmp = t_1
    else if (a <= (-4.5d-160)) then
        tmp = z * (y + (a * b))
    else if (a <= 3.05d-120) then
        tmp = x + (y * z)
    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 + (a * (t + (z * b)));
	double tmp;
	if (a <= -9.5e-14) {
		tmp = t_1;
	} else if (a <= -4.5e-160) {
		tmp = z * (y + (a * b));
	} else if (a <= 3.05e-120) {
		tmp = x + (y * z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x + (a * (t + (z * b)))
	tmp = 0
	if a <= -9.5e-14:
		tmp = t_1
	elif a <= -4.5e-160:
		tmp = z * (y + (a * b))
	elif a <= 3.05e-120:
		tmp = x + (y * z)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x + Float64(a * Float64(t + Float64(z * b))))
	tmp = 0.0
	if (a <= -9.5e-14)
		tmp = t_1;
	elseif (a <= -4.5e-160)
		tmp = Float64(z * Float64(y + Float64(a * b)));
	elseif (a <= 3.05e-120)
		tmp = Float64(x + Float64(y * z));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x + (a * (t + (z * b)));
	tmp = 0.0;
	if (a <= -9.5e-14)
		tmp = t_1;
	elseif (a <= -4.5e-160)
		tmp = z * (y + (a * b));
	elseif (a <= 3.05e-120)
		tmp = x + (y * z);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x + N[(a * N[(t + N[(z * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -9.5e-14], t$95$1, If[LessEqual[a, -4.5e-160], N[(z * N[(y + N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 3.05e-120], N[(x + N[(y * z), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if a < -9.4999999999999999e-14 or 3.05e-120 < a

   1. Initial program 90.3%

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

      1. associate-+l+ 90.3%

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

      2. +-commutative 90.3%

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

      3. fma-def 90.3%

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

      4. associate-*l* 91.5%

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

      5. *-commutative 91.5%

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

      6. *-commutative 91.5%

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

      7. distribute-rgt-out 96.1%

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

      8. *-commutative 96.1%

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

   3. Simplified 96.1%

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

   5. Taylor expanded in y around 0 91.7%

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

   if -9.4999999999999999e-14 < a < -4.50000000000000026e-160

   1. Initial program 96.3%

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

      1. associate-+l+ 96.3%

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

      2. associate-*l* 86.5%

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

   3. Simplified 86.5%

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

   5. Taylor expanded in z around inf 78.7%

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

   if -4.50000000000000026e-160 < a < 3.05e-120

   1. Initial program 100.0%

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

      1. associate-+l+ 100.0%

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

      2. associate-*l* 90.5%

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

   3. Simplified 90.5%

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

   5. Taylor expanded in a around 0 91.9%

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

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -9.5 \cdot 10^{-14}:\\ \;\;\;\;x + a \cdot \left(t + z \cdot b\right)\\ \mathbf{elif}\;a \leq -4.5 \cdot 10^{-160}:\\ \;\;\;\;z \cdot \left(y + a \cdot b\right)\\ \mathbf{elif}\;a \leq 3.05 \cdot 10^{-120}:\\ \;\;\;\;x + y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x + a \cdot \left(t + z \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t \cdot a + y \cdot z\\ t_2 := x + a \cdot \left(t + z \cdot b\right)\\ \mathbf{if}\;a \leq -3.1 \cdot 10^{-40}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq -4.1 \cdot 10^{-160}:\\ \;\;\;\;\left(z \cdot a\right) \cdot b + t_1\\ \mathbf{elif}\;a \leq 1.18 \cdot 10^{-32}:\\ \;\;\;\;x + t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (* t a) (* y z))) (t_2 (+ x (* a (+ t (* z b))))))
   (if (<= a -3.1e-40)
     t_2
     (if (<= a -4.1e-160)
       (+ (* (* z a) b) t_1)
       (if (<= a 1.18e-32) (+ x t_1) t_2)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t * a) + (y * z);
	double t_2 = x + (a * (t + (z * b)));
	double tmp;
	if (a <= -3.1e-40) {
		tmp = t_2;
	} else if (a <= -4.1e-160) {
		tmp = ((z * a) * b) + t_1;
	} else if (a <= 1.18e-32) {
		tmp = x + 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 = (t * a) + (y * z)
    t_2 = x + (a * (t + (z * b)))
    if (a <= (-3.1d-40)) then
        tmp = t_2
    else if (a <= (-4.1d-160)) then
        tmp = ((z * a) * b) + t_1
    else if (a <= 1.18d-32) then
        tmp = x + 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 = (t * a) + (y * z);
	double t_2 = x + (a * (t + (z * b)));
	double tmp;
	if (a <= -3.1e-40) {
		tmp = t_2;
	} else if (a <= -4.1e-160) {
		tmp = ((z * a) * b) + t_1;
	} else if (a <= 1.18e-32) {
		tmp = x + t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (t * a) + (y * z)
	t_2 = x + (a * (t + (z * b)))
	tmp = 0
	if a <= -3.1e-40:
		tmp = t_2
	elif a <= -4.1e-160:
		tmp = ((z * a) * b) + t_1
	elif a <= 1.18e-32:
		tmp = x + t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t * a) + Float64(y * z))
	t_2 = Float64(x + Float64(a * Float64(t + Float64(z * b))))
	tmp = 0.0
	if (a <= -3.1e-40)
		tmp = t_2;
	elseif (a <= -4.1e-160)
		tmp = Float64(Float64(Float64(z * a) * b) + t_1);
	elseif (a <= 1.18e-32)
		tmp = Float64(x + t_1);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (t * a) + (y * z);
	t_2 = x + (a * (t + (z * b)));
	tmp = 0.0;
	if (a <= -3.1e-40)
		tmp = t_2;
	elseif (a <= -4.1e-160)
		tmp = ((z * a) * b) + t_1;
	elseif (a <= 1.18e-32)
		tmp = x + t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t * a), $MachinePrecision] + N[(y * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x + N[(a * N[(t + N[(z * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -3.1e-40], t$95$2, If[LessEqual[a, -4.1e-160], N[(N[(N[(z * a), $MachinePrecision] * b), $MachinePrecision] + t$95$1), $MachinePrecision], If[LessEqual[a, 1.18e-32], N[(x + t$95$1), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if a < -3.10000000000000011e-40 or 1.17999999999999997e-32 < a

   1. Initial program 88.5%

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

      1. associate-+l+ 88.5%

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

      2. +-commutative 88.5%

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

      3. fma-def 88.5%

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

      4. associate-*l* 89.8%

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

      5. *-commutative 89.8%

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

      6. *-commutative 89.8%

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

      7. distribute-rgt-out 94.9%

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

      8. *-commutative 94.9%

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

   3. Simplified 94.9%

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

   5. Taylor expanded in y around 0 92.9%

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

   if -3.10000000000000011e-40 < a < -4.10000000000000002e-160

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 95.3%

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

   if -4.10000000000000002e-160 < a < 1.17999999999999997e-32

   1. Initial program 100.0%

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

      1. associate-+l+ 100.0%

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

      2. associate-*l* 92.6%

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

   3. Simplified 92.6%

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

   5. Taylor expanded in b around 0 95.7%

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

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.1 \cdot 10^{-40}:\\ \;\;\;\;x + a \cdot \left(t + z \cdot b\right)\\ \mathbf{elif}\;a \leq -4.1 \cdot 10^{-160}:\\ \;\;\;\;\left(z \cdot a\right) \cdot b + \left(t \cdot a + y \cdot z\right)\\ \mathbf{elif}\;a \leq 1.18 \cdot 10^{-32}:\\ \;\;\;\;x + \left(t \cdot a + y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x + a \cdot \left(t + z \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 71.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot \left(t + z \cdot b\right)\\ \mathbf{if}\;a \leq -1.62 \cdot 10^{-12}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -4 \cdot 10^{-160}:\\ \;\;\;\;z \cdot \left(y + a \cdot b\right)\\ \mathbf{elif}\;a \leq 4.6 \cdot 10^{-54}:\\ \;\;\;\;x + y \cdot z\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* a (+ t (* z b)))))
   (if (<= a -1.62e-12)
     t_1
     (if (<= a -4e-160)
       (* z (+ y (* a b)))
       (if (<= a 4.6e-54) (+ x (* y z)) 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 <= -1.62e-12) {
		tmp = t_1;
	} else if (a <= -4e-160) {
		tmp = z * (y + (a * b));
	} else if (a <= 4.6e-54) {
		tmp = x + (y * z);
	} 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 <= (-1.62d-12)) then
        tmp = t_1
    else if (a <= (-4d-160)) then
        tmp = z * (y + (a * b))
    else if (a <= 4.6d-54) then
        tmp = x + (y * z)
    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 <= -1.62e-12) {
		tmp = t_1;
	} else if (a <= -4e-160) {
		tmp = z * (y + (a * b));
	} else if (a <= 4.6e-54) {
		tmp = x + (y * z);
	} 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 <= -1.62e-12:
		tmp = t_1
	elif a <= -4e-160:
		tmp = z * (y + (a * b))
	elif a <= 4.6e-54:
		tmp = x + (y * z)
	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 <= -1.62e-12)
		tmp = t_1;
	elseif (a <= -4e-160)
		tmp = Float64(z * Float64(y + Float64(a * b)));
	elseif (a <= 4.6e-54)
		tmp = Float64(x + Float64(y * z));
	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 <= -1.62e-12)
		tmp = t_1;
	elseif (a <= -4e-160)
		tmp = z * (y + (a * b));
	elseif (a <= 4.6e-54)
		tmp = x + (y * z);
	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, -1.62e-12], t$95$1, If[LessEqual[a, -4e-160], N[(z * N[(y + N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 4.6e-54], N[(x + N[(y * z), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.62e-12 or 4.5999999999999998e-54 < a

   1. Initial program 89.3%

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

      1. associate-+l+ 89.3%

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

      2. associate-*l* 90.6%

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

   3. Simplified 90.6%

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

   5. Taylor expanded in a around inf 79.6%

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

   if -1.62e-12 < a < -4e-160

   1. Initial program 96.3%

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

      1. associate-+l+ 96.3%

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

      2. associate-*l* 86.5%

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

   3. Simplified 86.5%

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

   5. Taylor expanded in z around inf 78.7%

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

   if -4e-160 < a < 4.5999999999999998e-54

   1. Initial program 100.0%

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

      1. associate-+l+ 100.0%

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

      2. associate-*l* 92.2%

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

   3. Simplified 92.2%

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

   5. Taylor expanded in a around 0 88.9%

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

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.62 \cdot 10^{-12}:\\ \;\;\;\;a \cdot \left(t + z \cdot b\right)\\ \mathbf{elif}\;a \leq -4 \cdot 10^{-160}:\\ \;\;\;\;z \cdot \left(y + a \cdot b\right)\\ \mathbf{elif}\;a \leq 4.6 \cdot 10^{-54}:\\ \;\;\;\;x + y \cdot z\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(t + z \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 93.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 6.2 \cdot 10^{+173}:\\ \;\;\;\;\left(x + y \cdot z\right) + \left(a \cdot \left(z \cdot b\right) + t \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y + a \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z 6.2e+173)
   (+ (+ x (* y z)) (+ (* a (* z b)) (* t a)))
   (* z (+ y (* a b)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= 6.2e+173) {
		tmp = (x + (y * z)) + ((a * (z * b)) + (t * a));
	} else {
		tmp = z * (y + (a * 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 (z <= 6.2d+173) then
        tmp = (x + (y * z)) + ((a * (z * b)) + (t * a))
    else
        tmp = z * (y + (a * 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 (z <= 6.2e+173) {
		tmp = (x + (y * z)) + ((a * (z * b)) + (t * a));
	} else {
		tmp = z * (y + (a * b));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if z <= 6.2e+173:
		tmp = (x + (y * z)) + ((a * (z * b)) + (t * a))
	else:
		tmp = z * (y + (a * b))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= 6.2e+173)
		tmp = Float64(Float64(x + Float64(y * z)) + Float64(Float64(a * Float64(z * b)) + Float64(t * a)));
	else
		tmp = Float64(z * Float64(y + Float64(a * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (z <= 6.2e+173)
		tmp = (x + (y * z)) + ((a * (z * b)) + (t * a));
	else
		tmp = z * (y + (a * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, 6.2e+173], N[(N[(x + N[(y * z), $MachinePrecision]), $MachinePrecision] + N[(N[(a * N[(z * b), $MachinePrecision]), $MachinePrecision] + N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z * N[(y + N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 6.2e173

   1. Initial program 96.0%

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

      1. associate-+l+ 96.0%

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

      2. associate-*l* 93.1%

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

   3. Simplified 93.1%

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

   if 6.2e173 < z

   1. Initial program 73.4%

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

      1. associate-+l+ 73.4%

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

      2. associate-*l* 69.7%

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

   3. Simplified 69.7%

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

   5. Taylor expanded in z around inf 88.6%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 6.2 \cdot 10^{+173}:\\ \;\;\;\;\left(x + y \cdot z\right) + \left(a \cdot \left(z \cdot b\right) + t \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y + a \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 87.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -7 \cdot 10^{-30} \lor \neg \left(a \leq 2.8 \cdot 10^{-32}\right):\\ \;\;\;\;x + a \cdot \left(t + z \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(t \cdot a + y \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= a -7e-30) (not (<= a 2.8e-32)))
   (+ x (* a (+ t (* z b))))
   (+ x (+ (* t a) (* y z)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((a <= -7e-30) || !(a <= 2.8e-32)) {
		tmp = x + (a * (t + (z * b)));
	} else {
		tmp = x + ((t * a) + (y * z));
	}
	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 <= (-7d-30)) .or. (.not. (a <= 2.8d-32))) then
        tmp = x + (a * (t + (z * b)))
    else
        tmp = x + ((t * a) + (y * z))
    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 <= -7e-30) || !(a <= 2.8e-32)) {
		tmp = x + (a * (t + (z * b)));
	} else {
		tmp = x + ((t * a) + (y * z));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (a <= -7e-30) or not (a <= 2.8e-32):
		tmp = x + (a * (t + (z * b)))
	else:
		tmp = x + ((t * a) + (y * z))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((a <= -7e-30) || !(a <= 2.8e-32))
		tmp = Float64(x + Float64(a * Float64(t + Float64(z * b))));
	else
		tmp = Float64(x + Float64(Float64(t * a) + Float64(y * z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((a <= -7e-30) || ~((a <= 2.8e-32)))
		tmp = x + (a * (t + (z * b)));
	else
		tmp = x + ((t * a) + (y * z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[a, -7e-30], N[Not[LessEqual[a, 2.8e-32]], $MachinePrecision]], N[(x + N[(a * N[(t + N[(z * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(t * a), $MachinePrecision] + N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -7.0000000000000006e-30 or 2.7999999999999999e-32 < a

   1. Initial program 88.3%

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

      1. associate-+l+ 88.3%

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

      2. +-commutative 88.3%

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

      3. fma-def 88.3%

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

      4. associate-*l* 89.7%

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

      5. *-commutative 89.7%

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

      6. *-commutative 89.7%

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

      7. distribute-rgt-out 94.8%

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

      8. *-commutative 94.8%

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

   3. Simplified 94.8%

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

   5. Taylor expanded in y around 0 93.5%

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

   if -7.0000000000000006e-30 < a < 2.7999999999999999e-32

   1. Initial program 99.9%

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

      1. associate-+l+ 99.9%

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

      2. associate-*l* 91.9%

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

   3. Simplified 91.9%

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

   5. Taylor expanded in b around 0 91.2%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -7 \cdot 10^{-30} \lor \neg \left(a \leq 2.8 \cdot 10^{-32}\right):\\ \;\;\;\;x + a \cdot \left(t + z \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(t \cdot a + y \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 73.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5.2 \cdot 10^{-30} \lor \neg \left(a \leq 7 \cdot 10^{-54}\right):\\ \;\;\;\;a \cdot \left(t + z \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= a -5.2e-30) (not (<= a 7e-54)))
   (* a (+ t (* z b)))
   (+ x (* y z))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((a <= -5.2e-30) || !(a <= 7e-54)) {
		tmp = a * (t + (z * b));
	} else {
		tmp = x + (y * z);
	}
	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-30)) .or. (.not. (a <= 7d-54))) then
        tmp = a * (t + (z * b))
    else
        tmp = x + (y * z)
    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-30) || !(a <= 7e-54)) {
		tmp = a * (t + (z * b));
	} else {
		tmp = x + (y * z);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (a <= -5.2e-30) or not (a <= 7e-54):
		tmp = a * (t + (z * b))
	else:
		tmp = x + (y * z)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((a <= -5.2e-30) || !(a <= 7e-54))
		tmp = Float64(a * Float64(t + Float64(z * b)));
	else
		tmp = Float64(x + Float64(y * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((a <= -5.2e-30) || ~((a <= 7e-54)))
		tmp = a * (t + (z * b));
	else
		tmp = x + (y * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[a, -5.2e-30], N[Not[LessEqual[a, 7e-54]], $MachinePrecision]], N[(a * N[(t + N[(z * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(y * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -5.19999999999999973e-30 or 6.99999999999999964e-54 < a

   1. Initial program 88.7%

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

      1. associate-+l+ 88.7%

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

      2. associate-*l* 90.1%

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

   3. Simplified 90.1%

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

   5. Taylor expanded in a around inf 79.2%

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

   if -5.19999999999999973e-30 < a < 6.99999999999999964e-54

   1. Initial program 99.9%

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

      1. associate-+l+ 99.9%

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

      2. associate-*l* 91.5%

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

   3. Simplified 91.5%

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

   5. Taylor expanded in a around 0 82.3%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5.2 \cdot 10^{-30} \lor \neg \left(a \leq 7 \cdot 10^{-54}\right):\\ \;\;\;\;a \cdot \left(t + z \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 57.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.35 \cdot 10^{-12}:\\ \;\;\;\;t \cdot a\\ \mathbf{elif}\;a \leq 1.1 \cdot 10^{+63}:\\ \;\;\;\;x + y \cdot z\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(z \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= a -2.35e-12) (* t a) (if (<= a 1.1e+63) (+ x (* y z)) (* a (* z b)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= -2.35e-12) {
		tmp = t * a;
	} else if (a <= 1.1e+63) {
		tmp = x + (y * z);
	} else {
		tmp = a * (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 <= (-2.35d-12)) then
        tmp = t * a
    else if (a <= 1.1d+63) then
        tmp = x + (y * z)
    else
        tmp = a * (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 <= -2.35e-12) {
		tmp = t * a;
	} else if (a <= 1.1e+63) {
		tmp = x + (y * z);
	} else {
		tmp = a * (z * b);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if a <= -2.35e-12:
		tmp = t * a
	elif a <= 1.1e+63:
		tmp = x + (y * z)
	else:
		tmp = a * (z * b)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (a <= -2.35e-12)
		tmp = Float64(t * a);
	elseif (a <= 1.1e+63)
		tmp = Float64(x + Float64(y * z));
	else
		tmp = Float64(a * Float64(z * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (a <= -2.35e-12)
		tmp = t * a;
	elseif (a <= 1.1e+63)
		tmp = x + (y * z);
	else
		tmp = a * (z * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[a, -2.35e-12], N[(t * a), $MachinePrecision], If[LessEqual[a, 1.1e+63], N[(x + N[(y * z), $MachinePrecision]), $MachinePrecision], N[(a * N[(z * b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -2.34999999999999988e-12

   1. Initial program 86.4%

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

      1. associate-+l+ 86.4%

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

      2. associate-*l* 87.9%

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

   3. Simplified 87.9%

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

   5. Taylor expanded in t around inf 47.0%

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

   if -2.34999999999999988e-12 < a < 1.0999999999999999e63

   1. Initial program 99.2%

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

      1. associate-+l+ 99.2%

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

      2. associate-*l* 92.2%

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

   3. Simplified 92.2%

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

   5. Taylor expanded in a around 0 75.9%

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

   if 1.0999999999999999e63 < a

   1. Initial program 88.6%

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

      1. associate-+l+ 88.6%

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

      2. +-commutative 88.6%

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

      3. fma-def 88.6%

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

      4. associate-*l* 90.4%

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

      5. *-commutative 90.4%

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

      6. *-commutative 90.4%

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

      7. distribute-rgt-out 92.3%

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

      8. *-commutative 92.3%

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

   3. Simplified 92.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, z, x\right) + a \cdot \left(t + z \cdot b\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-cube-cbrt 91.9%

         \[\leadsto \mathsf{fma}\left(y, z, x\right) + a \cdot \color{blue}{\left(\left(\sqrt[3]{t + z \cdot b} \cdot \sqrt[3]{t + z \cdot b}\right) \cdot \sqrt[3]{t + z \cdot b}\right)} \]

      2. pow3 91.9%

         \[\leadsto \mathsf{fma}\left(y, z, x\right) + a \cdot \color{blue}{{\left(\sqrt[3]{t + z \cdot b}\right)}^{3}} \]

      3. +-commutative 91.9%

         \[\leadsto \mathsf{fma}\left(y, z, x\right) + a \cdot {\left(\sqrt[3]{\color{blue}{z \cdot b + t}}\right)}^{3} \]

      4. fma-def 91.9%

         \[\leadsto \mathsf{fma}\left(y, z, x\right) + a \cdot {\left(\sqrt[3]{\color{blue}{\mathsf{fma}\left(z, b, t\right)}}\right)}^{3} \]

   6. Applied egg-rr 91.9%

      \[\leadsto \mathsf{fma}\left(y, z, x\right) + a \cdot \color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(z, b, t\right)}\right)}^{3}} \]

   7. Taylor expanded in z around inf 51.0%

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

      1. *-commutative 51.0%

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

   9. Simplified 51.0%

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

   10. Taylor expanded in y around 0 52.3%

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

       1. *-commutative 52.3%

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

   12. Simplified 52.3%

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

4. Final simplification 63.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.35 \cdot 10^{-12}:\\ \;\;\;\;t \cdot a\\ \mathbf{elif}\;a \leq 1.1 \cdot 10^{+63}:\\ \;\;\;\;x + y \cdot z\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(z \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 39.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.8 \cdot 10^{+15} \lor \neg \left(t \leq 1.35 \cdot 10^{-47}\right):\\ \;\;\;\;t \cdot a\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= t -2.8e+15) (not (<= t 1.35e-47))) (* t a) x))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((t <= -2.8e+15) || !(t <= 1.35e-47)) {
		tmp = t * a;
	} 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 ((t <= (-2.8d+15)) .or. (.not. (t <= 1.35d-47))) then
        tmp = t * a
    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 ((t <= -2.8e+15) || !(t <= 1.35e-47)) {
		tmp = t * a;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (t <= -2.8e+15) or not (t <= 1.35e-47):
		tmp = t * a
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((t <= -2.8e+15) || !(t <= 1.35e-47))
		tmp = Float64(t * a);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((t <= -2.8e+15) || ~((t <= 1.35e-47)))
		tmp = t * a;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[t, -2.8e+15], N[Not[LessEqual[t, 1.35e-47]], $MachinePrecision]], N[(t * a), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if t < -2.8e15 or 1.3499999999999999e-47 < t

   1. Initial program 90.7%

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

      1. associate-+l+ 90.7%

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

      2. associate-*l* 88.3%

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

   3. Simplified 88.3%

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

   5. Taylor expanded in t around inf 46.8%

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

   if -2.8e15 < t < 1.3499999999999999e-47

   1. Initial program 96.8%

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

      1. associate-+l+ 96.8%

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

      2. associate-*l* 93.1%

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

   3. Simplified 93.1%

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

   5. Taylor expanded in x around inf 32.6%

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

4. Final simplification 39.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.8 \cdot 10^{+15} \lor \neg \left(t \leq 1.35 \cdot 10^{-47}\right):\\ \;\;\;\;t \cdot a\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 26.7% 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 93.7%

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

   1. associate-+l+ 93.7%

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

   2. associate-*l* 90.7%

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

3. Simplified 90.7%

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

5. Taylor expanded in x around inf 22.9%

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

6. Final simplification 22.9%

   \[\leadsto x \]
7. 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 2024024 
(FPCore (x y z t a b)
  :name "Graphics.Rasterific.CubicBezier:cachedBezierAt from Rasterific-0.6.1"
  :precision binary64

  :herbie-target
  (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)))