Graphics.Rasterific.CubicBezier:cachedBezierAt from Rasterific-0.6.1

Percentage Accurate: 91.9% → 94.2%

Time: 11.1s

Alternatives: 15

Speedup: 0.9×

• 31.9% 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: 91.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = ((x + (y * z)) + (t * a)) + ((a * z) * b)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 94.2% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 94.3%

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

   1. associate-+l+ 94.3%

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

   2. +-commutative 94.3%

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

   3. *-commutative 94.3%

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

   4. associate-*l* 95.4%

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

   5. *-commutative 95.4%

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

   6. distribute-lft-out 96.6%

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

   7. fma-def 97.0%

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

   8. *-commutative 97.0%

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

   9. +-commutative 97.0%

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

   10. fma-def 97.0%

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

3. Simplified 97.0%

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

4. Final simplification 97.0%

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

Alternative 2: 93.6% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 94.3%

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

   1. associate-+l+ 94.3%

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

   2. +-commutative 94.3%

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

   3. fma-def 94.3%

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

   4. associate-*l* 95.4%

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

   5. *-commutative 95.4%

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

   6. *-commutative 95.4%

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

   7. distribute-rgt-out 96.6%

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

   8. *-commutative 96.6%

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

3. Simplified 96.6%

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

4. Final simplification 96.6%

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

Alternative 3: 95.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(x + Float64(z * y)) + Float64(a * t)) + Float64(b * Float64(a * z)))
	tmp = 0.0
	if (t_1 <= Inf)
		tmp = t_1;
	else
		tmp = Float64(Float64(z * y) + 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 + (z * y)) + (a * t)) + (b * (a * z));
	tmp = 0.0;
	if (t_1 <= Inf)
		tmp = t_1;
	else
		tmp = (z * y) + (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[(z * y), $MachinePrecision]), $MachinePrecision] + N[(a * t), $MachinePrecision]), $MachinePrecision] + N[(b * N[(a * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, Infinity], t$95$1, N[(N[(z * y), $MachinePrecision] + N[(a * N[(t + N[(z * b), $MachinePrecision]), $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 97.3%

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

   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. 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. +-commutative 0.0%

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

      3. fma-def 0.0%

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

      4. associate-*l* 25.0%

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

      5. *-commutative 25.0%

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

      6. *-commutative 25.0%

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

      7. distribute-rgt-out 62.5%

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

      8. *-commutative 62.5%

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

   3. Simplified 62.5%

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

   4. Taylor expanded in x around 0 62.5%

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

4. Final simplification 96.2%

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

Alternative 4: 69.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot y + a \cdot t\\ t_2 := a \cdot \left(t + z \cdot b\right)\\ \mathbf{if}\;a \leq -6 \cdot 10^{-52}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq -9.8 \cdot 10^{-164}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -2.4 \cdot 10^{-233}:\\ \;\;\;\;x + b \cdot \left(a \cdot z\right)\\ \mathbf{elif}\;a \leq 1.2 \cdot 10^{-134}:\\ \;\;\;\;x + z \cdot y\\ \mathbf{elif}\;a \leq 250:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (* z y) (* a t))) (t_2 (* a (+ t (* z b)))))
   (if (<= a -6e-52)
     t_2
     (if (<= a -9.8e-164)
       t_1
       (if (<= a -2.4e-233)
         (+ x (* b (* a z)))
         (if (<= a 1.2e-134) (+ x (* z y)) (if (<= a 250.0) t_1 t_2)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (z * y) + (a * t);
	double t_2 = a * (t + (z * b));
	double tmp;
	if (a <= -6e-52) {
		tmp = t_2;
	} else if (a <= -9.8e-164) {
		tmp = t_1;
	} else if (a <= -2.4e-233) {
		tmp = x + (b * (a * z));
	} else if (a <= 1.2e-134) {
		tmp = x + (z * y);
	} else if (a <= 250.0) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (z * y) + (a * t)
    t_2 = a * (t + (z * b))
    if (a <= (-6d-52)) then
        tmp = t_2
    else if (a <= (-9.8d-164)) then
        tmp = t_1
    else if (a <= (-2.4d-233)) then
        tmp = x + (b * (a * z))
    else if (a <= 1.2d-134) then
        tmp = x + (z * y)
    else if (a <= 250.0d0) then
        tmp = t_1
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (z * y) + (a * t);
	double t_2 = a * (t + (z * b));
	double tmp;
	if (a <= -6e-52) {
		tmp = t_2;
	} else if (a <= -9.8e-164) {
		tmp = t_1;
	} else if (a <= -2.4e-233) {
		tmp = x + (b * (a * z));
	} else if (a <= 1.2e-134) {
		tmp = x + (z * y);
	} else if (a <= 250.0) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (z * y) + (a * t)
	t_2 = a * (t + (z * b))
	tmp = 0
	if a <= -6e-52:
		tmp = t_2
	elif a <= -9.8e-164:
		tmp = t_1
	elif a <= -2.4e-233:
		tmp = x + (b * (a * z))
	elif a <= 1.2e-134:
		tmp = x + (z * y)
	elif a <= 250.0:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(z * y) + Float64(a * t))
	t_2 = Float64(a * Float64(t + Float64(z * b)))
	tmp = 0.0
	if (a <= -6e-52)
		tmp = t_2;
	elseif (a <= -9.8e-164)
		tmp = t_1;
	elseif (a <= -2.4e-233)
		tmp = Float64(x + Float64(b * Float64(a * z)));
	elseif (a <= 1.2e-134)
		tmp = Float64(x + Float64(z * y));
	elseif (a <= 250.0)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (z * y) + (a * t);
	t_2 = a * (t + (z * b));
	tmp = 0.0;
	if (a <= -6e-52)
		tmp = t_2;
	elseif (a <= -9.8e-164)
		tmp = t_1;
	elseif (a <= -2.4e-233)
		tmp = x + (b * (a * z));
	elseif (a <= 1.2e-134)
		tmp = x + (z * y);
	elseif (a <= 250.0)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(z * y), $MachinePrecision] + N[(a * t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(a * N[(t + N[(z * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -6e-52], t$95$2, If[LessEqual[a, -9.8e-164], t$95$1, If[LessEqual[a, -2.4e-233], N[(x + N[(b * N[(a * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1.2e-134], N[(x + N[(z * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 250.0], t$95$1, t$95$2]]]]]]]

Derivation

1. Split input into 4 regimes

2. if a < -6e-52 or 250 < 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. +-commutative 89.3%

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

      3. fma-def 89.3%

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

      4. associate-*l* 95.6%

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

      5. *-commutative 95.6%

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

      6. *-commutative 95.6%

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

      7. distribute-rgt-out 97.8%

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

      8. *-commutative 97.8%

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

   3. Simplified 97.8%

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

   4. Taylor expanded in y around 0 91.5%

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

   5. Taylor expanded in x around 0 76.2%

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

   if -6e-52 < a < -9.7999999999999993e-164 or 1.20000000000000005e-134 < a < 250

   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. *-commutative 100.0%

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

      3. associate-*l* 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in t around inf 87.5%

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

   5. Taylor expanded in x around 0 76.4%

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

   if -9.7999999999999993e-164 < a < -2.3999999999999999e-233

   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. +-commutative 99.9%

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

      3. fma-def 99.9%

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

      4. associate-*l* 83.4%

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

      5. *-commutative 83.4%

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

      6. *-commutative 83.4%

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

      7. distribute-rgt-out 83.4%

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

      8. *-commutative 83.4%

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

   3. Simplified 83.4%

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

   4. Taylor expanded in y around 0 77.8%

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

   5. Taylor expanded in t around 0 66.3%

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

      1. associate-*r* 82.8%

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

      2. *-commutative 82.8%

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

      3. associate-*l* 82.8%

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

   7. Simplified 82.8%

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

   if -2.3999999999999999e-233 < a < 1.20000000000000005e-134

   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. *-commutative 100.0%

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

      3. associate-*l* 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in a around 0 95.0%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -6 \cdot 10^{-52}:\\ \;\;\;\;a \cdot \left(t + z \cdot b\right)\\ \mathbf{elif}\;a \leq -9.8 \cdot 10^{-164}:\\ \;\;\;\;z \cdot y + a \cdot t\\ \mathbf{elif}\;a \leq -2.4 \cdot 10^{-233}:\\ \;\;\;\;x + b \cdot \left(a \cdot z\right)\\ \mathbf{elif}\;a \leq 1.2 \cdot 10^{-134}:\\ \;\;\;\;x + z \cdot y\\ \mathbf{elif}\;a \leq 250:\\ \;\;\;\;z \cdot y + a \cdot t\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(t + z \cdot b\right)\\ \end{array} \]

Alternative 5: 80.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + a \cdot \left(t + z \cdot b\right)\\ \mathbf{if}\;a \leq -3 \cdot 10^{-65}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -4.6 \cdot 10^{-163}:\\ \;\;\;\;z \cdot y + a \cdot t\\ \mathbf{elif}\;a \leq -3.1 \cdot 10^{-201} \lor \neg \left(a \leq 1.75 \cdot 10^{-126}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ x (* a (+ t (* z b))))))
   (if (<= a -3e-65)
     t_1
     (if (<= a -4.6e-163)
       (+ (* z y) (* a t))
       (if (or (<= a -3.1e-201) (not (<= a 1.75e-126))) t_1 (+ x (* z y)))))))

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 <= -3e-65) {
		tmp = t_1;
	} else if (a <= -4.6e-163) {
		tmp = (z * y) + (a * t);
	} else if ((a <= -3.1e-201) || !(a <= 1.75e-126)) {
		tmp = t_1;
	} else {
		tmp = x + (z * y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x + (a * (t + (z * b)))
    if (a <= (-3d-65)) then
        tmp = t_1
    else if (a <= (-4.6d-163)) then
        tmp = (z * y) + (a * t)
    else if ((a <= (-3.1d-201)) .or. (.not. (a <= 1.75d-126))) then
        tmp = t_1
    else
        tmp = x + (z * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x + (a * (t + (z * b)));
	double tmp;
	if (a <= -3e-65) {
		tmp = t_1;
	} else if (a <= -4.6e-163) {
		tmp = (z * y) + (a * t);
	} else if ((a <= -3.1e-201) || !(a <= 1.75e-126)) {
		tmp = t_1;
	} else {
		tmp = x + (z * y);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x + (a * (t + (z * b)))
	tmp = 0
	if a <= -3e-65:
		tmp = t_1
	elif a <= -4.6e-163:
		tmp = (z * y) + (a * t)
	elif (a <= -3.1e-201) or not (a <= 1.75e-126):
		tmp = t_1
	else:
		tmp = x + (z * y)
	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 <= -3e-65)
		tmp = t_1;
	elseif (a <= -4.6e-163)
		tmp = Float64(Float64(z * y) + Float64(a * t));
	elseif ((a <= -3.1e-201) || !(a <= 1.75e-126))
		tmp = t_1;
	else
		tmp = Float64(x + Float64(z * y));
	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 <= -3e-65)
		tmp = t_1;
	elseif (a <= -4.6e-163)
		tmp = (z * y) + (a * t);
	elseif ((a <= -3.1e-201) || ~((a <= 1.75e-126)))
		tmp = t_1;
	else
		tmp = x + (z * y);
	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, -3e-65], t$95$1, If[LessEqual[a, -4.6e-163], N[(N[(z * y), $MachinePrecision] + N[(a * t), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[a, -3.1e-201], N[Not[LessEqual[a, 1.75e-126]], $MachinePrecision]], t$95$1, N[(x + N[(z * y), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if a < -2.99999999999999998e-65 or -4.5999999999999999e-163 < a < -3.0999999999999999e-201 or 1.75e-126 < a

   1. Initial program 91.3%

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

      1. associate-+l+ 91.3%

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

      2. +-commutative 91.3%

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

      3. fma-def 91.3%

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

      4. associate-*l* 94.7%

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

      5. *-commutative 94.7%

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

      6. *-commutative 94.7%

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

      7. distribute-rgt-out 96.5%

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

      8. *-commutative 96.5%

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

   3. Simplified 96.5%

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

   4. Taylor expanded in y around 0 87.7%

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

   if -2.99999999999999998e-65 < a < -4.5999999999999999e-163

   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. *-commutative 99.9%

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

      3. associate-*l* 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in t around inf 88.6%

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

   5. Taylor expanded in x around 0 79.5%

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

   if -3.0999999999999999e-201 < a < 1.75e-126

   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. *-commutative 100.0%

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

      3. associate-*l* 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in a around 0 94.1%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3 \cdot 10^{-65}:\\ \;\;\;\;x + a \cdot \left(t + z \cdot b\right)\\ \mathbf{elif}\;a \leq -4.6 \cdot 10^{-163}:\\ \;\;\;\;z \cdot y + a \cdot t\\ \mathbf{elif}\;a \leq -3.1 \cdot 10^{-201} \lor \neg \left(a \leq 1.75 \cdot 10^{-126}\right):\\ \;\;\;\;x + a \cdot \left(t + z \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot y\\ \end{array} \]

Alternative 6: 92.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -6 \cdot 10^{+220}:\\ \;\;\;\;x + a \cdot \left(t + z \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + z \cdot y\right) + \left(z \cdot \left(a \cdot b\right) + a \cdot t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= a -6e+220)
   (+ x (* a (+ t (* z b))))
   (+ (+ x (* z y)) (+ (* z (* a b)) (* a t)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= -6e+220) {
		tmp = x + (a * (t + (z * b)));
	} else {
		tmp = (x + (z * y)) + ((z * (a * b)) + (a * t));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-6d+220)) then
        tmp = x + (a * (t + (z * b)))
    else
        tmp = (x + (z * y)) + ((z * (a * b)) + (a * t))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= -6e+220) {
		tmp = x + (a * (t + (z * b)));
	} else {
		tmp = (x + (z * y)) + ((z * (a * b)) + (a * t));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if a <= -6e+220:
		tmp = x + (a * (t + (z * b)))
	else:
		tmp = (x + (z * y)) + ((z * (a * b)) + (a * t))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (a <= -6e+220)
		tmp = Float64(x + Float64(a * Float64(t + Float64(z * b))));
	else
		tmp = Float64(Float64(x + Float64(z * y)) + Float64(Float64(z * Float64(a * b)) + Float64(a * t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (a <= -6e+220)
		tmp = x + (a * (t + (z * b)));
	else
		tmp = (x + (z * y)) + ((z * (a * b)) + (a * t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[a, -6e+220], N[(x + N[(a * N[(t + N[(z * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x + N[(z * y), $MachinePrecision]), $MachinePrecision] + N[(N[(z * N[(a * b), $MachinePrecision]), $MachinePrecision] + N[(a * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -6.00000000000000048e220

   1. Initial program 82.7%

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

      1. associate-+l+ 82.7%

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

      2. +-commutative 82.7%

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

      3. fma-def 82.7%

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

      4. associate-*l* 88.1%

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

      5. *-commutative 88.1%

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

      6. *-commutative 88.1%

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

      7. distribute-rgt-out 99.9%

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

      8. *-commutative 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around 0 99.9%

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

   if -6.00000000000000048e220 < a

   1. Initial program 95.1%

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

      1. associate-+l+ 95.1%

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

      2. *-commutative 95.1%

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

      3. associate-*l* 96.3%

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

   3. Simplified 96.3%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -6 \cdot 10^{+220}:\\ \;\;\;\;x + a \cdot \left(t + z \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + z \cdot y\right) + \left(z \cdot \left(a \cdot b\right) + a \cdot t\right)\\ \end{array} \]

Alternative 7: 82.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{+78}:\\ \;\;\;\;x + z \cdot \left(y + a \cdot b\right)\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{+89}:\\ \;\;\;\;z \cdot y + a \cdot \left(t + z \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + z \cdot y\right) + a \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= x -1.45e+78)
   (+ x (* z (+ y (* a b))))
   (if (<= x 2.3e+89)
     (+ (* z y) (* a (+ t (* z b))))
     (+ (+ x (* z y)) (* a t)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (x <= -1.45e+78) {
		tmp = x + (z * (y + (a * b)));
	} else if (x <= 2.3e+89) {
		tmp = (z * y) + (a * (t + (z * b)));
	} else {
		tmp = (x + (z * y)) + (a * t);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (x <= (-1.45d+78)) then
        tmp = x + (z * (y + (a * b)))
    else if (x <= 2.3d+89) then
        tmp = (z * y) + (a * (t + (z * b)))
    else
        tmp = (x + (z * y)) + (a * t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (x <= -1.45e+78) {
		tmp = x + (z * (y + (a * b)));
	} else if (x <= 2.3e+89) {
		tmp = (z * y) + (a * (t + (z * b)));
	} else {
		tmp = (x + (z * y)) + (a * t);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if x <= -1.45e+78:
		tmp = x + (z * (y + (a * b)))
	elif x <= 2.3e+89:
		tmp = (z * y) + (a * (t + (z * b)))
	else:
		tmp = (x + (z * y)) + (a * t)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (x <= -1.45e+78)
		tmp = Float64(x + Float64(z * Float64(y + Float64(a * b))));
	elseif (x <= 2.3e+89)
		tmp = Float64(Float64(z * y) + Float64(a * Float64(t + Float64(z * b))));
	else
		tmp = Float64(Float64(x + Float64(z * y)) + Float64(a * t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (x <= -1.45e+78)
		tmp = x + (z * (y + (a * b)));
	elseif (x <= 2.3e+89)
		tmp = (z * y) + (a * (t + (z * b)));
	else
		tmp = (x + (z * y)) + (a * t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[x, -1.45e+78], N[(x + N[(z * N[(y + N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.3e+89], N[(N[(z * y), $MachinePrecision] + N[(a * N[(t + N[(z * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x + N[(z * y), $MachinePrecision]), $MachinePrecision] + N[(a * t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.45000000000000008e78

   1. Initial program 93.8%

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

      1. associate-+l+ 93.8%

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

      2. *-commutative 93.8%

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

      3. associate-*l* 97.8%

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

   3. Simplified 97.8%

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

   4. Taylor expanded in t around 0 87.6%

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

      1. +-commutative 87.6%

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

      2. associate-*r* 91.5%

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

      3. distribute-rgt-in 93.6%

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

      4. +-commutative 93.6%

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

   6. Simplified 93.6%

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

   if -1.45000000000000008e78 < x < 2.2999999999999999e89

   1. Initial program 95.7%

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

      1. associate-+l+ 95.7%

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

      2. +-commutative 95.7%

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

      3. fma-def 95.7%

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

      4. associate-*l* 97.5%

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

      5. *-commutative 97.5%

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

      6. *-commutative 97.5%

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

      7. distribute-rgt-out 98.8%

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

      8. *-commutative 98.8%

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

   3. Simplified 98.8%

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

   4. Taylor expanded in x around 0 92.2%

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

   if 2.2999999999999999e89 < x

   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. *-commutative 90.5%

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

      3. associate-*l* 87.1%

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

   3. Simplified 87.1%

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

   4. Taylor expanded in t around inf 87.2%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{+78}:\\ \;\;\;\;x + z \cdot \left(y + a \cdot b\right)\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{+89}:\\ \;\;\;\;z \cdot y + a \cdot \left(t + z \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + z \cdot y\right) + a \cdot t\\ \end{array} \]

Alternative 8: 39.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.1 \cdot 10^{+109}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -6.5 \cdot 10^{+37}:\\ \;\;\;\;z \cdot y\\ \mathbf{elif}\;x \leq -6 \cdot 10^{-98}:\\ \;\;\;\;a \cdot t\\ \mathbf{elif}\;x \leq -2.4 \cdot 10^{-293}:\\ \;\;\;\;z \cdot y\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{+91}:\\ \;\;\;\;a \cdot t\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= x -5.1e+109)
   x
   (if (<= x -6.5e+37)
     (* z y)
     (if (<= x -6e-98)
       (* a t)
       (if (<= x -2.4e-293) (* z y) (if (<= x 4.6e+91) (* a t) x))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (x <= -5.1e+109) {
		tmp = x;
	} else if (x <= -6.5e+37) {
		tmp = z * y;
	} else if (x <= -6e-98) {
		tmp = a * t;
	} else if (x <= -2.4e-293) {
		tmp = z * y;
	} else if (x <= 4.6e+91) {
		tmp = a * t;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (x <= (-5.1d+109)) then
        tmp = x
    else if (x <= (-6.5d+37)) then
        tmp = z * y
    else if (x <= (-6d-98)) then
        tmp = a * t
    else if (x <= (-2.4d-293)) then
        tmp = z * y
    else if (x <= 4.6d+91) then
        tmp = a * t
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (x <= -5.1e+109) {
		tmp = x;
	} else if (x <= -6.5e+37) {
		tmp = z * y;
	} else if (x <= -6e-98) {
		tmp = a * t;
	} else if (x <= -2.4e-293) {
		tmp = z * y;
	} else if (x <= 4.6e+91) {
		tmp = a * t;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if x <= -5.1e+109:
		tmp = x
	elif x <= -6.5e+37:
		tmp = z * y
	elif x <= -6e-98:
		tmp = a * t
	elif x <= -2.4e-293:
		tmp = z * y
	elif x <= 4.6e+91:
		tmp = a * t
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (x <= -5.1e+109)
		tmp = x;
	elseif (x <= -6.5e+37)
		tmp = Float64(z * y);
	elseif (x <= -6e-98)
		tmp = Float64(a * t);
	elseif (x <= -2.4e-293)
		tmp = Float64(z * y);
	elseif (x <= 4.6e+91)
		tmp = Float64(a * t);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (x <= -5.1e+109)
		tmp = x;
	elseif (x <= -6.5e+37)
		tmp = z * y;
	elseif (x <= -6e-98)
		tmp = a * t;
	elseif (x <= -2.4e-293)
		tmp = z * y;
	elseif (x <= 4.6e+91)
		tmp = a * t;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[x, -5.1e+109], x, If[LessEqual[x, -6.5e+37], N[(z * y), $MachinePrecision], If[LessEqual[x, -6e-98], N[(a * t), $MachinePrecision], If[LessEqual[x, -2.4e-293], N[(z * y), $MachinePrecision], If[LessEqual[x, 4.6e+91], N[(a * t), $MachinePrecision], x]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.0999999999999999e109 or 4.59999999999999982e91 < x

   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. associate-+l+ 92.5%

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

      2. *-commutative 92.5%

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

      3. associate-*l* 91.5%

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

   3. Simplified 91.5%

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

   4. Taylor expanded in x around inf 55.6%

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

   if -5.0999999999999999e109 < x < -6.4999999999999998e37 or -6e-98 < x < -2.3999999999999999e-293

   1. Initial program 94.7%

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

      1. associate-+l+ 94.7%

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

      2. *-commutative 94.7%

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

      3. associate-*l* 94.5%

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

   3. Simplified 94.5%

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

   4. Taylor expanded in y around inf 45.1%

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

      1. *-commutative 45.1%

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

   6. Simplified 45.1%

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

   if -6.4999999999999998e37 < x < -6e-98 or -2.3999999999999999e-293 < x < 4.59999999999999982e91

   1. Initial program 95.5%

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

      1. associate-+l+ 95.5%

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

      2. *-commutative 95.5%

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

      3. associate-*l* 97.3%

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

   3. Simplified 97.3%

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

   4. Taylor expanded in t around inf 79.1%

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

   5. Taylor expanded in a around inf 48.4%

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

4. Final simplification 50.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.1 \cdot 10^{+109}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -6.5 \cdot 10^{+37}:\\ \;\;\;\;z \cdot y\\ \mathbf{elif}\;x \leq -6 \cdot 10^{-98}:\\ \;\;\;\;a \cdot t\\ \mathbf{elif}\;x \leq -2.4 \cdot 10^{-293}:\\ \;\;\;\;z \cdot y\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{+91}:\\ \;\;\;\;a \cdot t\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 9: 86.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5.6 \cdot 10^{-37} \lor \neg \left(a \leq 4.8 \cdot 10^{-27}\right):\\ \;\;\;\;x + a \cdot \left(t + z \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(y + a \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= a -5.6e-37) (not (<= a 4.8e-27)))
   (+ x (* a (+ t (* z b))))
   (+ x (* z (+ y (* a b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((a <= -5.6e-37) || !(a <= 4.8e-27)) {
		tmp = x + (a * (t + (z * b)));
	} else {
		tmp = x + (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 ((a <= (-5.6d-37)) .or. (.not. (a <= 4.8d-27))) then
        tmp = x + (a * (t + (z * b)))
    else
        tmp = x + (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 ((a <= -5.6e-37) || !(a <= 4.8e-27)) {
		tmp = x + (a * (t + (z * b)));
	} else {
		tmp = x + (z * (y + (a * b)));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (a <= -5.6e-37) or not (a <= 4.8e-27):
		tmp = x + (a * (t + (z * b)))
	else:
		tmp = x + (z * (y + (a * b)))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((a <= -5.6e-37) || !(a <= 4.8e-27))
		tmp = Float64(x + Float64(a * Float64(t + Float64(z * b))));
	else
		tmp = Float64(x + 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 ((a <= -5.6e-37) || ~((a <= 4.8e-27)))
		tmp = x + (a * (t + (z * b)));
	else
		tmp = x + (z * (y + (a * b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[a, -5.6e-37], N[Not[LessEqual[a, 4.8e-27]], $MachinePrecision]], N[(x + N[(a * N[(t + N[(z * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(z * N[(y + N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -5.6000000000000002e-37 or 4.80000000000000004e-27 < a

   1. Initial program 89.8%

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

      1. associate-+l+ 89.8%

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

      2. +-commutative 89.8%

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

      3. fma-def 89.8%

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

      4. associate-*l* 95.8%

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

      5. *-commutative 95.8%

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

      6. *-commutative 95.8%

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

      7. distribute-rgt-out 97.9%

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

      8. *-commutative 97.9%

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

   3. Simplified 97.9%

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

   4. Taylor expanded in y around 0 89.8%

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

   if -5.6000000000000002e-37 < a < 4.80000000000000004e-27

   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. *-commutative 100.0%

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

      3. associate-*l* 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in t around 0 85.6%

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

      1. +-commutative 85.6%

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

      2. associate-*r* 90.5%

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

      3. distribute-rgt-in 90.5%

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

      4. +-commutative 90.5%

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

   6. Simplified 90.5%

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

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5.6 \cdot 10^{-37} \lor \neg \left(a \leq 4.8 \cdot 10^{-27}\right):\\ \;\;\;\;x + a \cdot \left(t + z \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(y + a \cdot b\right)\\ \end{array} \]

Alternative 10: 86.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.5 \cdot 10^{-53} \lor \neg \left(a \leq 400000\right):\\ \;\;\;\;x + a \cdot \left(t + z \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + z \cdot y\right) + a \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= a -3.5e-53) (not (<= a 400000.0)))
   (+ x (* a (+ t (* z b))))
   (+ (+ x (* z y)) (* a t))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((a <= -3.5e-53) || !(a <= 400000.0)) {
		tmp = x + (a * (t + (z * b)));
	} else {
		tmp = (x + (z * y)) + (a * t);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((a <= (-3.5d-53)) .or. (.not. (a <= 400000.0d0))) then
        tmp = x + (a * (t + (z * b)))
    else
        tmp = (x + (z * y)) + (a * t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((a <= -3.5e-53) || !(a <= 400000.0)) {
		tmp = x + (a * (t + (z * b)));
	} else {
		tmp = (x + (z * y)) + (a * t);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (a <= -3.5e-53) or not (a <= 400000.0):
		tmp = x + (a * (t + (z * b)))
	else:
		tmp = (x + (z * y)) + (a * t)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((a <= -3.5e-53) || !(a <= 400000.0))
		tmp = Float64(x + Float64(a * Float64(t + Float64(z * b))));
	else
		tmp = Float64(Float64(x + Float64(z * y)) + Float64(a * t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((a <= -3.5e-53) || ~((a <= 400000.0)))
		tmp = x + (a * (t + (z * b)));
	else
		tmp = (x + (z * y)) + (a * t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[a, -3.5e-53], N[Not[LessEqual[a, 400000.0]], $MachinePrecision]], N[(x + N[(a * N[(t + N[(z * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x + N[(z * y), $MachinePrecision]), $MachinePrecision] + N[(a * t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -3.49999999999999993e-53 or 4e5 < 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. +-commutative 89.3%

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

      3. fma-def 89.3%

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

      4. associate-*l* 95.6%

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

      5. *-commutative 95.6%

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

      6. *-commutative 95.6%

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

      7. distribute-rgt-out 97.8%

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

      8. *-commutative 97.8%

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

   3. Simplified 97.8%

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

   4. Taylor expanded in y around 0 91.5%

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

   if -3.49999999999999993e-53 < a < 4e5

   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. *-commutative 100.0%

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

      3. associate-*l* 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in t around inf 90.5%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.5 \cdot 10^{-53} \lor \neg \left(a \leq 400000\right):\\ \;\;\;\;x + a \cdot \left(t + z \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + z \cdot y\right) + a \cdot t\\ \end{array} \]

Alternative 11: 57.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.8 \cdot 10^{+243}:\\ \;\;\;\;a \cdot \left(z \cdot b\right)\\ \mathbf{elif}\;a \leq -2.55 \cdot 10^{+38} \lor \neg \left(a \leq 9.5 \cdot 10^{-27}\right):\\ \;\;\;\;a \cdot t\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= a -2.8e+243)
   (* a (* z b))
   (if (or (<= a -2.55e+38) (not (<= a 9.5e-27))) (* a t) (+ x (* z y)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= -2.8e+243) {
		tmp = a * (z * b);
	} else if ((a <= -2.55e+38) || !(a <= 9.5e-27)) {
		tmp = a * t;
	} else {
		tmp = x + (z * y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-2.8d+243)) then
        tmp = a * (z * b)
    else if ((a <= (-2.55d+38)) .or. (.not. (a <= 9.5d-27))) then
        tmp = a * t
    else
        tmp = x + (z * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= -2.8e+243) {
		tmp = a * (z * b);
	} else if ((a <= -2.55e+38) || !(a <= 9.5e-27)) {
		tmp = a * t;
	} else {
		tmp = x + (z * y);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if a <= -2.8e+243:
		tmp = a * (z * b)
	elif (a <= -2.55e+38) or not (a <= 9.5e-27):
		tmp = a * t
	else:
		tmp = x + (z * y)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (a <= -2.8e+243)
		tmp = Float64(a * Float64(z * b));
	elseif ((a <= -2.55e+38) || !(a <= 9.5e-27))
		tmp = Float64(a * t);
	else
		tmp = Float64(x + Float64(z * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (a <= -2.8e+243)
		tmp = a * (z * b);
	elseif ((a <= -2.55e+38) || ~((a <= 9.5e-27)))
		tmp = a * t;
	else
		tmp = x + (z * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[a, -2.8e+243], N[(a * N[(z * b), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[a, -2.55e+38], N[Not[LessEqual[a, 9.5e-27]], $MachinePrecision]], N[(a * t), $MachinePrecision], N[(x + N[(z * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -2.7999999999999999e243

   1. Initial program 77.4%

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

      1. associate-+l+ 77.4%

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

      2. +-commutative 77.4%

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

      3. fma-def 77.4%

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

      4. associate-*l* 84.5%

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

      5. *-commutative 84.5%

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

      6. *-commutative 84.5%

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

      7. distribute-rgt-out 99.9%

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

      8. *-commutative 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around 0 99.9%

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

   5. Taylor expanded in x around 0 92.2%

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

   6. Taylor expanded in t around 0 69.5%

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

      1. *-commutative 69.5%

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

   8. Simplified 69.5%

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

   if -2.7999999999999999e243 < a < -2.5500000000000001e38 or 9.50000000000000037e-27 < 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%

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

      3. associate-*l* 91.9%

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

   3. Simplified 91.9%

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

   4. Taylor expanded in t around inf 76.0%

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

   5. Taylor expanded in a around inf 50.7%

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

   if -2.5500000000000001e38 < a < 9.50000000000000037e-27

   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. *-commutative 99.9%

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

      3. associate-*l* 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in a around 0 75.4%

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

4. Final simplification 63.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.8 \cdot 10^{+243}:\\ \;\;\;\;a \cdot \left(z \cdot b\right)\\ \mathbf{elif}\;a \leq -2.55 \cdot 10^{+38} \lor \neg \left(a \leq 9.5 \cdot 10^{-27}\right):\\ \;\;\;\;a \cdot t\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot y\\ \end{array} \]

Alternative 12: 71.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.75 \cdot 10^{-51} \lor \neg \left(a \leq 1.85 \cdot 10^{-126}\right):\\ \;\;\;\;a \cdot \left(t + z \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= a -1.75e-51) (not (<= a 1.85e-126)))
   (* a (+ t (* z b)))
   (+ x (* z y))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((a <= -1.75e-51) || !(a <= 1.85e-126)) {
		tmp = a * (t + (z * b));
	} else {
		tmp = x + (z * y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((a <= (-1.75d-51)) .or. (.not. (a <= 1.85d-126))) then
        tmp = a * (t + (z * b))
    else
        tmp = x + (z * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((a <= -1.75e-51) || !(a <= 1.85e-126)) {
		tmp = a * (t + (z * b));
	} else {
		tmp = x + (z * y);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (a <= -1.75e-51) or not (a <= 1.85e-126):
		tmp = a * (t + (z * b))
	else:
		tmp = x + (z * y)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((a <= -1.75e-51) || !(a <= 1.85e-126))
		tmp = Float64(a * Float64(t + Float64(z * b)));
	else
		tmp = Float64(x + Float64(z * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((a <= -1.75e-51) || ~((a <= 1.85e-126)))
		tmp = a * (t + (z * b));
	else
		tmp = x + (z * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[a, -1.75e-51], N[Not[LessEqual[a, 1.85e-126]], $MachinePrecision]], N[(a * N[(t + N[(z * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(z * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.7499999999999999e-51 or 1.85e-126 < a

   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. +-commutative 90.7%

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

      3. fma-def 90.7%

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

      4. associate-*l* 95.6%

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

      5. *-commutative 95.6%

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

      6. *-commutative 95.6%

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

      7. distribute-rgt-out 97.5%

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

      8. *-commutative 97.5%

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

   3. Simplified 97.5%

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

   4. Taylor expanded in y around 0 88.0%

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

   5. Taylor expanded in x around 0 72.8%

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

   if -1.7499999999999999e-51 < a < 1.85e-126

   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. *-commutative 100.0%

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

      3. associate-*l* 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in a around 0 84.8%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.75 \cdot 10^{-51} \lor \neg \left(a \leq 1.85 \cdot 10^{-126}\right):\\ \;\;\;\;a \cdot \left(t + z \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot y\\ \end{array} \]

Alternative 13: 62.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.5 \cdot 10^{-47} \lor \neg \left(a \leq 1.85 \cdot 10^{-126}\right):\\ \;\;\;\;x + a \cdot t\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= a -1.5e-47) (not (<= a 1.85e-126))) (+ x (* a t)) (+ x (* z y))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((a <= -1.5e-47) || !(a <= 1.85e-126)) {
		tmp = x + (a * t);
	} else {
		tmp = x + (z * y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((a <= (-1.5d-47)) .or. (.not. (a <= 1.85d-126))) then
        tmp = x + (a * t)
    else
        tmp = x + (z * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((a <= -1.5e-47) || !(a <= 1.85e-126)) {
		tmp = x + (a * t);
	} else {
		tmp = x + (z * y);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (a <= -1.5e-47) or not (a <= 1.85e-126):
		tmp = x + (a * t)
	else:
		tmp = x + (z * y)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((a <= -1.5e-47) || !(a <= 1.85e-126))
		tmp = Float64(x + Float64(a * t));
	else
		tmp = Float64(x + Float64(z * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((a <= -1.5e-47) || ~((a <= 1.85e-126)))
		tmp = x + (a * t);
	else
		tmp = x + (z * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[a, -1.5e-47], N[Not[LessEqual[a, 1.85e-126]], $MachinePrecision]], N[(x + N[(a * t), $MachinePrecision]), $MachinePrecision], N[(x + N[(z * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.50000000000000008e-47 or 1.85e-126 < a

   1. Initial program 90.6%

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

      1. associate-+l+ 90.6%

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

      2. *-commutative 90.6%

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

      3. associate-*l* 91.2%

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

   3. Simplified 91.2%

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

   4. Taylor expanded in z around 0 60.5%

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

      1. +-commutative 60.5%

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

   6. Simplified 60.5%

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

   if -1.50000000000000008e-47 < a < 1.85e-126

   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. *-commutative 100.0%

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

      3. associate-*l* 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in a around 0 84.0%

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

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.5 \cdot 10^{-47} \lor \neg \left(a \leq 1.85 \cdot 10^{-126}\right):\\ \;\;\;\;x + a \cdot t\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot y\\ \end{array} \]

Alternative 14: 38.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.5 \cdot 10^{+77}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{+91}:\\ \;\;\;\;a \cdot t\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= x -7.5e+77) x (if (<= x 4.2e+91) (* a t) x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (x <= -7.5e+77) {
		tmp = x;
	} else if (x <= 4.2e+91) {
		tmp = a * t;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (x <= (-7.5d+77)) then
        tmp = x
    else if (x <= 4.2d+91) then
        tmp = a * t
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (x <= -7.5e+77) {
		tmp = x;
	} else if (x <= 4.2e+91) {
		tmp = a * t;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if x <= -7.5e+77:
		tmp = x
	elif x <= 4.2e+91:
		tmp = a * t
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (x <= -7.5e+77)
		tmp = x;
	elseif (x <= 4.2e+91)
		tmp = Float64(a * t);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (x <= -7.5e+77)
		tmp = x;
	elseif (x <= 4.2e+91)
		tmp = a * t;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[x, -7.5e+77], x, If[LessEqual[x, 4.2e+91], N[(a * t), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if x < -7.49999999999999955e77 or 4.20000000000000015e91 < x

   1. Initial program 91.0%

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

      1. associate-+l+ 91.0%

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

      2. *-commutative 91.0%

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

      3. associate-*l* 92.0%

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

   3. Simplified 92.0%

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

   4. Taylor expanded in x around inf 53.4%

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

   if -7.49999999999999955e77 < x < 4.20000000000000015e91

   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. *-commutative 96.3%

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

      3. associate-*l* 96.2%

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

   3. Simplified 96.2%

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

   4. Taylor expanded in t around inf 77.9%

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

   5. Taylor expanded in a around inf 41.7%

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

4. Final simplification 46.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.5 \cdot 10^{+77}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{+91}:\\ \;\;\;\;a \cdot t\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 15: 25.4% 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 94.3%

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

   1. associate-+l+ 94.3%

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

   2. *-commutative 94.3%

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

   3. associate-*l* 94.6%

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

3. Simplified 94.6%

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

4. Taylor expanded in x around inf 25.7%

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

5. Final simplification 25.7%

   \[\leadsto x \]

Developer target: 97.3% 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 2023310 
(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)))