Graphics.Rasterific.CubicBezier:cachedBezierAt from Rasterific-0.6.1

Percentage Accurate: 92.4% → 96.6%

Time: 11.1s

Alternatives: 11

Speedup: 0.8×

• 31.5% 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: 96.6% accurate, 0.5× 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 99.9%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]

   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. *-commutative 0.0%

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

      2. associate-*l* 15.8%

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

   3. Simplified 15.8%

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

   4. Taylor expanded in a around inf 84.2%

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

4. Final simplification 98.8%

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

Alternative 2: 95.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.5 \cdot 10^{+49} \lor \neg \left(a \leq 1.25 \cdot 10^{+93}\right):\\ \;\;\;\;x + a \cdot \left(t + z \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x + y \cdot z\right) + t \cdot a\right) + z \cdot \left(a \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= a -1.5e+49) (not (<= a 1.25e+93)))
   (+ x (* a (+ t (* z b))))
   (+ (+ (+ x (* y z)) (* t a)) (* z (* a b)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((a <= -1.5e+49) || !(a <= 1.25e+93)) {
		tmp = x + (a * (t + (z * b)));
	} else {
		tmp = ((x + (y * z)) + (t * a)) + (z * (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 ((a <= (-1.5d+49)) .or. (.not. (a <= 1.25d+93))) then
        tmp = x + (a * (t + (z * b)))
    else
        tmp = ((x + (y * z)) + (t * a)) + (z * (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 ((a <= -1.5e+49) || !(a <= 1.25e+93)) {
		tmp = x + (a * (t + (z * b)));
	} else {
		tmp = ((x + (y * z)) + (t * a)) + (z * (a * b));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (a <= -1.5e+49) or not (a <= 1.25e+93):
		tmp = x + (a * (t + (z * b)))
	else:
		tmp = ((x + (y * z)) + (t * a)) + (z * (a * b))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((a <= -1.5e+49) || !(a <= 1.25e+93))
		tmp = Float64(x + Float64(a * Float64(t + Float64(z * b))));
	else
		tmp = Float64(Float64(Float64(x + Float64(y * z)) + Float64(t * a)) + Float64(z * Float64(a * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((a <= -1.5e+49) || ~((a <= 1.25e+93)))
		tmp = x + (a * (t + (z * b)));
	else
		tmp = ((x + (y * z)) + (t * a)) + (z * (a * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[a, -1.5e+49], N[Not[LessEqual[a, 1.25e+93]], $MachinePrecision]], N[(x + N[(a * N[(t + N[(z * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x + N[(y * z), $MachinePrecision]), $MachinePrecision] + N[(t * a), $MachinePrecision]), $MachinePrecision] + N[(z * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.5000000000000001e49 or 1.25e93 < a

   1. Initial program 84.0%

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

      1. associate-+l+ 84.0%

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

      2. +-commutative 84.0%

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

      3. *-commutative 84.0%

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

      4. *-commutative 84.0%

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

      5. associate-*l* 86.8%

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

      6. distribute-rgt-out 91.5%

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

      7. fma-def 93.4%

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

      8. *-commutative 93.4%

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

      9. +-commutative 93.4%

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

      10. fma-def 93.4%

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

   3. Simplified 93.4%

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

   4. Taylor expanded in y around 0 94.4%

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

   if -1.5000000000000001e49 < a < 1.25e93

   1. Initial program 98.6%

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

      1. *-commutative 98.6%

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

      2. associate-*l* 98.6%

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

   3. Simplified 98.6%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.5 \cdot 10^{+49} \lor \neg \left(a \leq 1.25 \cdot 10^{+93}\right):\\ \;\;\;\;x + a \cdot \left(t + z \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x + y \cdot z\right) + t \cdot a\right) + z \cdot \left(a \cdot b\right)\\ \end{array} \]

Alternative 3: 58.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot \left(z \cdot b\right)\\ \mathbf{if}\;z \leq -4 \cdot 10^{+238}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -6.8 \cdot 10^{+192}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq -4.9 \cdot 10^{-37}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2 \cdot 10^{+79}:\\ \;\;\;\;x + t \cdot a\\ \mathbf{elif}\;z \leq 2 \cdot 10^{+151} \lor \neg \left(z \leq 6.8 \cdot 10^{+224}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* a (* z b))))
   (if (<= z -4e+238)
     t_1
     (if (<= z -6.8e+192)
       (* y z)
       (if (<= z -4.9e-37)
         t_1
         (if (<= z 2e+79)
           (+ x (* t a))
           (if (or (<= z 2e+151) (not (<= z 6.8e+224))) t_1 (* y z))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = a * (z * b);
	double tmp;
	if (z <= -4e+238) {
		tmp = t_1;
	} else if (z <= -6.8e+192) {
		tmp = y * z;
	} else if (z <= -4.9e-37) {
		tmp = t_1;
	} else if (z <= 2e+79) {
		tmp = x + (t * a);
	} else if ((z <= 2e+151) || !(z <= 6.8e+224)) {
		tmp = t_1;
	} else {
		tmp = 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) :: t_1
    real(8) :: tmp
    t_1 = a * (z * b)
    if (z <= (-4d+238)) then
        tmp = t_1
    else if (z <= (-6.8d+192)) then
        tmp = y * z
    else if (z <= (-4.9d-37)) then
        tmp = t_1
    else if (z <= 2d+79) then
        tmp = x + (t * a)
    else if ((z <= 2d+151) .or. (.not. (z <= 6.8d+224))) then
        tmp = t_1
    else
        tmp = 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 t_1 = a * (z * b);
	double tmp;
	if (z <= -4e+238) {
		tmp = t_1;
	} else if (z <= -6.8e+192) {
		tmp = y * z;
	} else if (z <= -4.9e-37) {
		tmp = t_1;
	} else if (z <= 2e+79) {
		tmp = x + (t * a);
	} else if ((z <= 2e+151) || !(z <= 6.8e+224)) {
		tmp = t_1;
	} else {
		tmp = y * z;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = a * (z * b)
	tmp = 0
	if z <= -4e+238:
		tmp = t_1
	elif z <= -6.8e+192:
		tmp = y * z
	elif z <= -4.9e-37:
		tmp = t_1
	elif z <= 2e+79:
		tmp = x + (t * a)
	elif (z <= 2e+151) or not (z <= 6.8e+224):
		tmp = t_1
	else:
		tmp = y * z
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(a * Float64(z * b))
	tmp = 0.0
	if (z <= -4e+238)
		tmp = t_1;
	elseif (z <= -6.8e+192)
		tmp = Float64(y * z);
	elseif (z <= -4.9e-37)
		tmp = t_1;
	elseif (z <= 2e+79)
		tmp = Float64(x + Float64(t * a));
	elseif ((z <= 2e+151) || !(z <= 6.8e+224))
		tmp = t_1;
	else
		tmp = Float64(y * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = a * (z * b);
	tmp = 0.0;
	if (z <= -4e+238)
		tmp = t_1;
	elseif (z <= -6.8e+192)
		tmp = y * z;
	elseif (z <= -4.9e-37)
		tmp = t_1;
	elseif (z <= 2e+79)
		tmp = x + (t * a);
	elseif ((z <= 2e+151) || ~((z <= 6.8e+224)))
		tmp = t_1;
	else
		tmp = y * z;
	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[z, -4e+238], t$95$1, If[LessEqual[z, -6.8e+192], N[(y * z), $MachinePrecision], If[LessEqual[z, -4.9e-37], t$95$1, If[LessEqual[z, 2e+79], N[(x + N[(t * a), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[z, 2e+151], N[Not[LessEqual[z, 6.8e+224]], $MachinePrecision]], t$95$1, N[(y * z), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.0000000000000002e238 or -6.79999999999999992e192 < z < -4.90000000000000018e-37 or 1.99999999999999993e79 < z < 2.00000000000000003e151 or 6.8000000000000004e224 < z

   1. Initial program 85.9%

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

      1. *-commutative 85.9%

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

      2. associate-*l* 87.0%

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

   3. Simplified 87.0%

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

   4. Taylor expanded in a around inf 65.3%

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

   5. Taylor expanded in t around 0 54.5%

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

   if -4.0000000000000002e238 < z < -6.79999999999999992e192 or 2.00000000000000003e151 < z < 6.8000000000000004e224

   1. Initial program 85.2%

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

      1. *-commutative 85.2%

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

      2. associate-*l* 92.5%

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

   3. Simplified 92.5%

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

   4. Taylor expanded in y around inf 67.8%

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

      1. *-commutative 67.8%

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

   6. Simplified 67.8%

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

   if -4.90000000000000018e-37 < z < 1.99999999999999993e79

   1. Initial program 98.5%

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

      1. *-commutative 98.5%

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

      2. associate-*l* 91.4%

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

   3. Simplified 91.4%

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

   4. Taylor expanded in z around 0 77.9%

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

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{+238}:\\ \;\;\;\;a \cdot \left(z \cdot b\right)\\ \mathbf{elif}\;z \leq -6.8 \cdot 10^{+192}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq -4.9 \cdot 10^{-37}:\\ \;\;\;\;a \cdot \left(z \cdot b\right)\\ \mathbf{elif}\;z \leq 2 \cdot 10^{+79}:\\ \;\;\;\;x + t \cdot a\\ \mathbf{elif}\;z \leq 2 \cdot 10^{+151} \lor \neg \left(z \leq 6.8 \cdot 10^{+224}\right):\\ \;\;\;\;a \cdot \left(z \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \]

Alternative 4: 38.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.85 \cdot 10^{-11}:\\ \;\;\;\;t \cdot a\\ \mathbf{elif}\;a \leq -4 \cdot 10^{-177}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq -8 \cdot 10^{-211}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;a \leq 2.7 \cdot 10^{-301}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 3.5 \cdot 10^{-94}:\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;t \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= a -1.85e-11)
   (* t a)
   (if (<= a -4e-177)
     x
     (if (<= a -8e-211)
       (* y z)
       (if (<= a 2.7e-301) x (if (<= a 3.5e-94) (* y z) (* t a)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= -1.85e-11) {
		tmp = t * a;
	} else if (a <= -4e-177) {
		tmp = x;
	} else if (a <= -8e-211) {
		tmp = y * z;
	} else if (a <= 2.7e-301) {
		tmp = x;
	} else if (a <= 3.5e-94) {
		tmp = y * z;
	} 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 <= (-1.85d-11)) then
        tmp = t * a
    else if (a <= (-4d-177)) then
        tmp = x
    else if (a <= (-8d-211)) then
        tmp = y * z
    else if (a <= 2.7d-301) then
        tmp = x
    else if (a <= 3.5d-94) then
        tmp = y * z
    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 <= -1.85e-11) {
		tmp = t * a;
	} else if (a <= -4e-177) {
		tmp = x;
	} else if (a <= -8e-211) {
		tmp = y * z;
	} else if (a <= 2.7e-301) {
		tmp = x;
	} else if (a <= 3.5e-94) {
		tmp = y * z;
	} else {
		tmp = t * a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if a <= -1.85e-11:
		tmp = t * a
	elif a <= -4e-177:
		tmp = x
	elif a <= -8e-211:
		tmp = y * z
	elif a <= 2.7e-301:
		tmp = x
	elif a <= 3.5e-94:
		tmp = y * z
	else:
		tmp = t * a
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (a <= -1.85e-11)
		tmp = Float64(t * a);
	elseif (a <= -4e-177)
		tmp = x;
	elseif (a <= -8e-211)
		tmp = Float64(y * z);
	elseif (a <= 2.7e-301)
		tmp = x;
	elseif (a <= 3.5e-94)
		tmp = Float64(y * z);
	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 <= -1.85e-11)
		tmp = t * a;
	elseif (a <= -4e-177)
		tmp = x;
	elseif (a <= -8e-211)
		tmp = y * z;
	elseif (a <= 2.7e-301)
		tmp = x;
	elseif (a <= 3.5e-94)
		tmp = y * z;
	else
		tmp = t * a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[a, -1.85e-11], N[(t * a), $MachinePrecision], If[LessEqual[a, -4e-177], x, If[LessEqual[a, -8e-211], N[(y * z), $MachinePrecision], If[LessEqual[a, 2.7e-301], x, If[LessEqual[a, 3.5e-94], N[(y * z), $MachinePrecision], N[(t * a), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.8500000000000001e-11 or 3.49999999999999998e-94 < a

   1. Initial program 88.0%

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

      1. *-commutative 88.0%

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

      2. associate-*l* 83.9%

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

   3. Simplified 83.9%

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

   4. Taylor expanded in t around inf 44.7%

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

   if -1.8500000000000001e-11 < a < -3.99999999999999981e-177 or -8.00000000000000069e-211 < a < 2.7e-301

   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. *-commutative 99.9%

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

      2. associate-*l* 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around inf 53.9%

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

   if -3.99999999999999981e-177 < a < -8.00000000000000069e-211 or 2.7e-301 < a < 3.49999999999999998e-94

   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. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around inf 56.9%

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

      1. *-commutative 56.9%

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

   6. Simplified 56.9%

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

4. Final simplification 48.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.85 \cdot 10^{-11}:\\ \;\;\;\;t \cdot a\\ \mathbf{elif}\;a \leq -4 \cdot 10^{-177}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq -8 \cdot 10^{-211}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;a \leq 2.7 \cdot 10^{-301}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 3.5 \cdot 10^{-94}:\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;t \cdot a\\ \end{array} \]

Alternative 5: 36.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot \left(z \cdot b\right)\\ \mathbf{if}\;b \leq -3.5 \cdot 10^{+105}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -9.5 \cdot 10^{+31}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;b \leq -2.3 \cdot 10^{-30} \lor \neg \left(b \leq 1.55 \cdot 10^{+52}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* a (* z b))))
   (if (<= b -3.5e+105)
     t_1
     (if (<= b -9.5e+31)
       (* y z)
       (if (or (<= b -2.3e-30) (not (<= b 1.55e+52))) t_1 (* t a))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = a * (z * b);
	double tmp;
	if (b <= -3.5e+105) {
		tmp = t_1;
	} else if (b <= -9.5e+31) {
		tmp = y * z;
	} else if ((b <= -2.3e-30) || !(b <= 1.55e+52)) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_1 = a * (z * b)
    if (b <= (-3.5d+105)) then
        tmp = t_1
    else if (b <= (-9.5d+31)) then
        tmp = y * z
    else if ((b <= (-2.3d-30)) .or. (.not. (b <= 1.55d+52))) then
        tmp = t_1
    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 t_1 = a * (z * b);
	double tmp;
	if (b <= -3.5e+105) {
		tmp = t_1;
	} else if (b <= -9.5e+31) {
		tmp = y * z;
	} else if ((b <= -2.3e-30) || !(b <= 1.55e+52)) {
		tmp = t_1;
	} else {
		tmp = t * a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = a * (z * b)
	tmp = 0
	if b <= -3.5e+105:
		tmp = t_1
	elif b <= -9.5e+31:
		tmp = y * z
	elif (b <= -2.3e-30) or not (b <= 1.55e+52):
		tmp = t_1
	else:
		tmp = t * a
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(a * Float64(z * b))
	tmp = 0.0
	if (b <= -3.5e+105)
		tmp = t_1;
	elseif (b <= -9.5e+31)
		tmp = Float64(y * z);
	elseif ((b <= -2.3e-30) || !(b <= 1.55e+52))
		tmp = t_1;
	else
		tmp = Float64(t * a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = a * (z * b);
	tmp = 0.0;
	if (b <= -3.5e+105)
		tmp = t_1;
	elseif (b <= -9.5e+31)
		tmp = y * z;
	elseif ((b <= -2.3e-30) || ~((b <= 1.55e+52)))
		tmp = t_1;
	else
		tmp = t * a;
	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[b, -3.5e+105], t$95$1, If[LessEqual[b, -9.5e+31], N[(y * z), $MachinePrecision], If[Or[LessEqual[b, -2.3e-30], N[Not[LessEqual[b, 1.55e+52]], $MachinePrecision]], t$95$1, N[(t * a), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.49999999999999991e105 or -9.5000000000000008e31 < b < -2.29999999999999984e-30 or 1.55e52 < b

   1. Initial program 91.5%

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

      1. *-commutative 91.5%

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

      2. associate-*l* 84.3%

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

   3. Simplified 84.3%

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

   4. Taylor expanded in a around inf 65.6%

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

   5. Taylor expanded in t around 0 57.5%

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

   if -3.49999999999999991e105 < b < -9.5000000000000008e31

   1. Initial program 94.1%

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

      1. *-commutative 94.1%

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

      2. associate-*l* 88.2%

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

   3. Simplified 88.2%

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

   4. Taylor expanded in y around inf 48.4%

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

      1. *-commutative 48.4%

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

   6. Simplified 48.4%

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

   if -2.29999999999999984e-30 < b < 1.55e52

   1. Initial program 93.3%

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

      1. *-commutative 93.3%

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

      2. associate-*l* 95.8%

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

   3. Simplified 95.8%

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

   4. Taylor expanded in t around inf 45.2%

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

4. Final simplification 51.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.5 \cdot 10^{+105}:\\ \;\;\;\;a \cdot \left(z \cdot b\right)\\ \mathbf{elif}\;b \leq -9.5 \cdot 10^{+31}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;b \leq -2.3 \cdot 10^{-30} \lor \neg \left(b \leq 1.55 \cdot 10^{+52}\right):\\ \;\;\;\;a \cdot \left(z \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot a\\ \end{array} \]

Alternative 6: 83.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.38 \cdot 10^{-145} \lor \neg \left(a \leq 4.6 \cdot 10^{-95}\right):\\ \;\;\;\;x + a \cdot \left(t + z \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= a -1.38e-145) (not (<= a 4.6e-95)))
   (+ x (* 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 <= -1.38e-145) || !(a <= 4.6e-95)) {
		tmp = x + (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 <= (-1.38d-145)) .or. (.not. (a <= 4.6d-95))) then
        tmp = x + (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 <= -1.38e-145) || !(a <= 4.6e-95)) {
		tmp = x + (a * (t + (z * b)));
	} else {
		tmp = x + (y * z);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (a <= -1.38e-145) or not (a <= 4.6e-95):
		tmp = x + (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 <= -1.38e-145) || !(a <= 4.6e-95))
		tmp = Float64(x + 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 <= -1.38e-145) || ~((a <= 4.6e-95)))
		tmp = x + (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, -1.38e-145], N[Not[LessEqual[a, 4.6e-95]], $MachinePrecision]], N[(x + N[(a * N[(t + N[(z * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(y * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.38e-145 or 4.59999999999999998e-95 < a

   1. Initial program 89.4%

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

      1. associate-+l+ 89.4%

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

      2. +-commutative 89.4%

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

      3. *-commutative 89.4%

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

      4. *-commutative 89.4%

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

      5. associate-*l* 90.6%

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

      6. distribute-rgt-out 93.4%

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

      7. fma-def 94.5%

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

      8. *-commutative 94.5%

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

      9. +-commutative 94.5%

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

      10. fma-def 94.5%

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

   3. Simplified 94.5%

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

   4. Taylor expanded in y around 0 89.2%

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

   if -1.38e-145 < a < 4.59999999999999998e-95

   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. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in a around 0 83.5%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.38 \cdot 10^{-145} \lor \neg \left(a \leq 4.6 \cdot 10^{-95}\right):\\ \;\;\;\;x + a \cdot \left(t + z \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot z\\ \end{array} \]

Alternative 7: 87.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.28 \cdot 10^{-45} \lor \neg \left(a \leq 3.8 \cdot 10^{-72}\right):\\ \;\;\;\;x + a \cdot \left(t + z \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(t \cdot a + y \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= a -1.28e-45) (not (<= a 3.8e-72)))
   (+ 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 <= -1.28e-45) || !(a <= 3.8e-72)) {
		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 <= (-1.28d-45)) .or. (.not. (a <= 3.8d-72))) 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 <= -1.28e-45) || !(a <= 3.8e-72)) {
		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 <= -1.28e-45) or not (a <= 3.8e-72):
		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 <= -1.28e-45) || !(a <= 3.8e-72))
		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 <= -1.28e-45) || ~((a <= 3.8e-72)))
		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, -1.28e-45], N[Not[LessEqual[a, 3.8e-72]], $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 < -1.28e-45 or 3.80000000000000002e-72 < a

   1. Initial program 87.8%

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

      1. associate-+l+ 87.8%

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

      2. +-commutative 87.8%

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

      3. *-commutative 87.8%

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

      4. *-commutative 87.8%

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

      5. associate-*l* 89.7%

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

      6. distribute-rgt-out 92.9%

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

      7. fma-def 94.2%

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

      8. *-commutative 94.2%

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

      9. +-commutative 94.2%

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

      10. fma-def 94.2%

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

   3. Simplified 94.2%

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

   4. Taylor expanded in y around 0 91.3%

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

   if -1.28e-45 < a < 3.80000000000000002e-72

   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. *-commutative 99.9%

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

      2. associate-*l* 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in b around 0 89.6%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.28 \cdot 10^{-45} \lor \neg \left(a \leq 3.8 \cdot 10^{-72}\right):\\ \;\;\;\;x + a \cdot \left(t + z \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(t \cdot a + y \cdot z\right)\\ \end{array} \]

Alternative 8: 62.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + t \cdot a\\ \mathbf{if}\;a \leq -1.15 \cdot 10^{+188}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -62000000000000:\\ \;\;\;\;a \cdot \left(z \cdot b\right)\\ \mathbf{elif}\;a \leq 5 \cdot 10^{-72}:\\ \;\;\;\;x + y \cdot z\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ x (* t a))))
   (if (<= a -1.15e+188)
     t_1
     (if (<= a -62000000000000.0)
       (* a (* z b))
       (if (<= a 5e-72) (+ x (* y z)) t_1)))))

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 <= -1.15e+188) {
		tmp = t_1;
	} else if (a <= -62000000000000.0) {
		tmp = a * (z * b);
	} else if (a <= 5e-72) {
		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 + (t * a)
    if (a <= (-1.15d+188)) then
        tmp = t_1
    else if (a <= (-62000000000000.0d0)) then
        tmp = a * (z * b)
    else if (a <= 5d-72) 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 + (t * a);
	double tmp;
	if (a <= -1.15e+188) {
		tmp = t_1;
	} else if (a <= -62000000000000.0) {
		tmp = a * (z * b);
	} else if (a <= 5e-72) {
		tmp = x + (y * z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x + (t * a)
	tmp = 0
	if a <= -1.15e+188:
		tmp = t_1
	elif a <= -62000000000000.0:
		tmp = a * (z * b)
	elif a <= 5e-72:
		tmp = x + (y * z)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x + Float64(t * a))
	tmp = 0.0
	if (a <= -1.15e+188)
		tmp = t_1;
	elseif (a <= -62000000000000.0)
		tmp = Float64(a * Float64(z * b));
	elseif (a <= 5e-72)
		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 + (t * a);
	tmp = 0.0;
	if (a <= -1.15e+188)
		tmp = t_1;
	elseif (a <= -62000000000000.0)
		tmp = a * (z * b);
	elseif (a <= 5e-72)
		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[(t * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -1.15e+188], t$95$1, If[LessEqual[a, -62000000000000.0], N[(a * N[(z * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 5e-72], N[(x + N[(y * z), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.15000000000000006e188 or 4.9999999999999996e-72 < a

   1. Initial program 86.1%

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

      1. *-commutative 86.1%

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

      2. associate-*l* 82.1%

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

   3. Simplified 82.1%

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

   4. Taylor expanded in z around 0 62.9%

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

   if -1.15000000000000006e188 < a < -6.2e13

   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. *-commutative 90.5%

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

      2. associate-*l* 84.7%

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

   3. Simplified 84.7%

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

   4. Taylor expanded in a around inf 93.4%

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

   5. Taylor expanded in t around 0 63.1%

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

   if -6.2e13 < a < 4.9999999999999996e-72

   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. *-commutative 99.9%

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

      2. associate-*l* 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in a around 0 77.1%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.15 \cdot 10^{+188}:\\ \;\;\;\;x + t \cdot a\\ \mathbf{elif}\;a \leq -62000000000000:\\ \;\;\;\;a \cdot \left(z \cdot b\right)\\ \mathbf{elif}\;a \leq 5 \cdot 10^{-72}:\\ \;\;\;\;x + y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x + t \cdot a\\ \end{array} \]

Alternative 9: 73.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.7 \cdot 10^{+18} \lor \neg \left(a \leq 5 \cdot 10^{-72}\right):\\ \;\;\;\;a \cdot \left(t + z \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= a -1.7e+18) (not (<= a 5e-72)))
   (* 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 <= -1.7e+18) || !(a <= 5e-72)) {
		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 <= (-1.7d+18)) .or. (.not. (a <= 5d-72))) 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 <= -1.7e+18) || !(a <= 5e-72)) {
		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 <= -1.7e+18) or not (a <= 5e-72):
		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 <= -1.7e+18) || !(a <= 5e-72))
		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 <= -1.7e+18) || ~((a <= 5e-72)))
		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, -1.7e+18], N[Not[LessEqual[a, 5e-72]], $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 < -1.7e18 or 4.9999999999999996e-72 < a

   1. Initial program 87.1%

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

      1. *-commutative 87.1%

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

      2. associate-*l* 82.7%

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

   3. Simplified 82.7%

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

   4. Taylor expanded in a around inf 79.4%

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

   if -1.7e18 < a < 4.9999999999999996e-72

   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. *-commutative 99.9%

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

      2. associate-*l* 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in a around 0 77.1%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.7 \cdot 10^{+18} \lor \neg \left(a \leq 5 \cdot 10^{-72}\right):\\ \;\;\;\;a \cdot \left(t + z \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot z\\ \end{array} \]

Alternative 10: 39.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.9 \cdot 10^{-10}:\\ \;\;\;\;t \cdot a\\ \mathbf{elif}\;a \leq 1.95 \cdot 10^{-17}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= a -1.9e-10) (* t a) (if (<= a 1.95e-17) x (* t a))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= -1.9e-10) {
		tmp = t * a;
	} else if (a <= 1.95e-17) {
		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 <= (-1.9d-10)) then
        tmp = t * a
    else if (a <= 1.95d-17) 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 <= -1.9e-10) {
		tmp = t * a;
	} else if (a <= 1.95e-17) {
		tmp = x;
	} else {
		tmp = t * a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if a <= -1.9e-10:
		tmp = t * a
	elif a <= 1.95e-17:
		tmp = x
	else:
		tmp = t * a
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (a <= -1.9e-10)
		tmp = Float64(t * a);
	elseif (a <= 1.95e-17)
		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 <= -1.9e-10)
		tmp = t * a;
	elseif (a <= 1.95e-17)
		tmp = x;
	else
		tmp = t * a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[a, -1.9e-10], N[(t * a), $MachinePrecision], If[LessEqual[a, 1.95e-17], x, N[(t * a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.8999999999999999e-10 or 1.94999999999999995e-17 < a

   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. *-commutative 86.4%

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

      2. associate-*l* 81.7%

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

   3. Simplified 81.7%

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

   4. Taylor expanded in t around inf 46.9%

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

   if -1.8999999999999999e-10 < a < 1.94999999999999995e-17

   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. *-commutative 99.9%

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

      2. associate-*l* 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around inf 39.1%

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

4. Final simplification 43.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.9 \cdot 10^{-10}:\\ \;\;\;\;t \cdot a\\ \mathbf{elif}\;a \leq 1.95 \cdot 10^{-17}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t \cdot a\\ \end{array} \]

Alternative 11: 27.4% 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 92.5%

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

   1. *-commutative 92.5%

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

   2. associate-*l* 90.0%

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

3. Simplified 90.0%

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

4. Taylor expanded in x around inf 24.1%

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

5. Final simplification 24.1%

   \[\leadsto x \]

Developer target: 97.4% accurate, 0.9× 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 2023280 
(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)))