Linear.V4:$cdot from linear-1.19.1.3, C

Percentage Accurate: 96.0% → 97.7%

Time: 7.7s

Alternatives: 12

Speedup: 0.5×

• 40.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 96.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.7% accurate, 0.1× speedup

Math

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

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (+ (+ (* a b) (+ (* x y) (* z t))) (* c i)) INFINITY)
   (fma c i (+ (+ (* a b) (* x y)) (* z t)))
   (fma a b (* z t))))

C

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

Julia

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

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(N[(N[(a * b), $MachinePrecision] + N[(N[(x * y), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(c * i), $MachinePrecision]), $MachinePrecision], Infinity], N[(c * i + N[(N[(N[(a * b), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * b + N[(z * t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-def 100.0%

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

      3. associate-+l+ 100.0%

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

      4. fma-def 100.0%

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

      5. fma-def 100.0%

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

   3. Simplified 100.0%

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

      1. fma-udef 100.0%

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

      2. fma-udef 100.0%

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

      3. associate-+l+ 100.0%

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

      4. +-commutative 100.0%

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

      5. associate-+r+ 100.0%

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

   5. Applied egg-rr 100.0%

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

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

   1. Initial program 0.0%

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

   2. Taylor expanded in x around 0 20.0%

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

      1. *-commutative 20.0%

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

      2. fma-def 40.0%

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

      3. *-commutative 40.0%

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

   4. Applied egg-rr 40.0%

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

   5. Taylor expanded in c around 0 35.2%

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

      1. fma-def 55.2%

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

   7. Simplified 55.2%

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

4. Final simplification 96.5%

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

Alternative 2: 98.0% accurate, 0.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 92.2%

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

   1. +-commutative 92.2%

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

   2. fma-def 93.7%

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

   3. associate-+l+ 93.7%

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

   4. fma-def 94.9%

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

   5. fma-def 96.1%

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

3. Simplified 96.1%

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

4. Final simplification 96.1%

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

Alternative 3: 97.7% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

      1. associate-+l+ 100.0%

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

      2. fma-def 100.0%

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

   3. Simplified 100.0%

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

      1. fma-def 100.0%

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

      2. +-commutative 100.0%

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

   5. Applied egg-rr 100.0%

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

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

   1. Initial program 0.0%

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

   2. Taylor expanded in x around 0 20.0%

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

      1. *-commutative 20.0%

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

      2. fma-def 40.0%

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

      3. *-commutative 40.0%

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

   4. Applied egg-rr 40.0%

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

   5. Taylor expanded in c around 0 35.2%

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

      1. fma-def 55.2%

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

   7. Simplified 55.2%

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

4. Final simplification 96.5%

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

Alternative 4: 97.3% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

      1. associate-+l+ 100.0%

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

      2. fma-def 100.0%

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

   3. Simplified 100.0%

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

      1. fma-def 100.0%

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

      2. +-commutative 100.0%

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

   5. Applied egg-rr 100.0%

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

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

   1. Initial program 0.0%

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

   2. Taylor expanded in a around 0 40.0%

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

   3. Taylor expanded in c around 0 45.6%

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

4. Final simplification 95.7%

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

Alternative 5: 65.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot y + z \cdot t\\ t_2 := a \cdot b + c \cdot i\\ t_3 := a \cdot b + z \cdot t\\ \mathbf{if}\;c \cdot i \leq -3.6 \cdot 10^{+57}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;c \cdot i \leq -6.4 \cdot 10^{-60}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \cdot i \leq -1 \cdot 10^{-275}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;c \cdot i \leq 1.06 \cdot 10^{-194}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \cdot i \leq 2.75 \cdot 10^{+45}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ (* x y) (* z t)))
        (t_2 (+ (* a b) (* c i)))
        (t_3 (+ (* a b) (* z t))))
   (if (<= (* c i) -3.6e+57)
     t_2
     (if (<= (* c i) -6.4e-60)
       t_1
       (if (<= (* c i) -1e-275)
         t_3
         (if (<= (* c i) 1.06e-194)
           t_1
           (if (<= (* c i) 2.75e+45) t_3 t_2)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (x * y) + (z * t);
	double t_2 = (a * b) + (c * i);
	double t_3 = (a * b) + (z * t);
	double tmp;
	if ((c * i) <= -3.6e+57) {
		tmp = t_2;
	} else if ((c * i) <= -6.4e-60) {
		tmp = t_1;
	} else if ((c * i) <= -1e-275) {
		tmp = t_3;
	} else if ((c * i) <= 1.06e-194) {
		tmp = t_1;
	} else if ((c * i) <= 2.75e+45) {
		tmp = t_3;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i)
    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), intent (in) :: c
    real(8), intent (in) :: i
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = (x * y) + (z * t)
    t_2 = (a * b) + (c * i)
    t_3 = (a * b) + (z * t)
    if ((c * i) <= (-3.6d+57)) then
        tmp = t_2
    else if ((c * i) <= (-6.4d-60)) then
        tmp = t_1
    else if ((c * i) <= (-1d-275)) then
        tmp = t_3
    else if ((c * i) <= 1.06d-194) then
        tmp = t_1
    else if ((c * i) <= 2.75d+45) then
        tmp = t_3
    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 c, double i) {
	double t_1 = (x * y) + (z * t);
	double t_2 = (a * b) + (c * i);
	double t_3 = (a * b) + (z * t);
	double tmp;
	if ((c * i) <= -3.6e+57) {
		tmp = t_2;
	} else if ((c * i) <= -6.4e-60) {
		tmp = t_1;
	} else if ((c * i) <= -1e-275) {
		tmp = t_3;
	} else if ((c * i) <= 1.06e-194) {
		tmp = t_1;
	} else if ((c * i) <= 2.75e+45) {
		tmp = t_3;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	t_1 = (x * y) + (z * t)
	t_2 = (a * b) + (c * i)
	t_3 = (a * b) + (z * t)
	tmp = 0
	if (c * i) <= -3.6e+57:
		tmp = t_2
	elif (c * i) <= -6.4e-60:
		tmp = t_1
	elif (c * i) <= -1e-275:
		tmp = t_3
	elif (c * i) <= 1.06e-194:
		tmp = t_1
	elif (c * i) <= 2.75e+45:
		tmp = t_3
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(x * y) + Float64(z * t))
	t_2 = Float64(Float64(a * b) + Float64(c * i))
	t_3 = Float64(Float64(a * b) + Float64(z * t))
	tmp = 0.0
	if (Float64(c * i) <= -3.6e+57)
		tmp = t_2;
	elseif (Float64(c * i) <= -6.4e-60)
		tmp = t_1;
	elseif (Float64(c * i) <= -1e-275)
		tmp = t_3;
	elseif (Float64(c * i) <= 1.06e-194)
		tmp = t_1;
	elseif (Float64(c * i) <= 2.75e+45)
		tmp = t_3;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	t_1 = (x * y) + (z * t);
	t_2 = (a * b) + (c * i);
	t_3 = (a * b) + (z * t);
	tmp = 0.0;
	if ((c * i) <= -3.6e+57)
		tmp = t_2;
	elseif ((c * i) <= -6.4e-60)
		tmp = t_1;
	elseif ((c * i) <= -1e-275)
		tmp = t_3;
	elseif ((c * i) <= 1.06e-194)
		tmp = t_1;
	elseif ((c * i) <= 2.75e+45)
		tmp = t_3;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(x * y), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(a * b), $MachinePrecision] + N[(c * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(a * b), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(c * i), $MachinePrecision], -3.6e+57], t$95$2, If[LessEqual[N[(c * i), $MachinePrecision], -6.4e-60], t$95$1, If[LessEqual[N[(c * i), $MachinePrecision], -1e-275], t$95$3, If[LessEqual[N[(c * i), $MachinePrecision], 1.06e-194], t$95$1, If[LessEqual[N[(c * i), $MachinePrecision], 2.75e+45], t$95$3, t$95$2]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 c i) < -3.6000000000000002e57 or 2.75e45 < (*.f64 c i)

   1. Initial program 90.2%

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

   2. Taylor expanded in a around inf 79.5%

      \[\leadsto \color{blue}{a \cdot b} + c \cdot i \]

   if -3.6000000000000002e57 < (*.f64 c i) < -6.4000000000000003e-60 or -9.99999999999999934e-276 < (*.f64 c i) < 1.06000000000000002e-194

   1. Initial program 93.0%

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

   2. Taylor expanded in a around 0 78.4%

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

   3. Taylor expanded in c around 0 76.7%

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

   if -6.4000000000000003e-60 < (*.f64 c i) < -9.99999999999999934e-276 or 1.06000000000000002e-194 < (*.f64 c i) < 2.75e45

   1. Initial program 94.1%

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

   2. Taylor expanded in x around 0 81.5%

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

      1. *-commutative 81.5%

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

      2. fma-def 83.8%

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

      3. *-commutative 83.8%

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

   4. Applied egg-rr 83.8%

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

   5. Taylor expanded in c around 0 77.4%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \cdot i \leq -3.6 \cdot 10^{+57}:\\ \;\;\;\;a \cdot b + c \cdot i\\ \mathbf{elif}\;c \cdot i \leq -6.4 \cdot 10^{-60}:\\ \;\;\;\;x \cdot y + z \cdot t\\ \mathbf{elif}\;c \cdot i \leq -1 \cdot 10^{-275}:\\ \;\;\;\;a \cdot b + z \cdot t\\ \mathbf{elif}\;c \cdot i \leq 1.06 \cdot 10^{-194}:\\ \;\;\;\;x \cdot y + z \cdot t\\ \mathbf{elif}\;c \cdot i \leq 2.75 \cdot 10^{+45}:\\ \;\;\;\;a \cdot b + z \cdot t\\ \mathbf{else}:\\ \;\;\;\;a \cdot b + c \cdot i\\ \end{array} \]

Alternative 6: 88.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot b \leq -\infty:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \cdot b \leq -1.8 \cdot 10^{+54} \lor \neg \left(a \cdot b \leq 4.7 \cdot 10^{+57}\right):\\ \;\;\;\;c \cdot i + \left(a \cdot b + z \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot i + \left(x \cdot y + z \cdot t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* a b) (- INFINITY))
   (* a b)
   (if (or (<= (* a b) -1.8e+54) (not (<= (* a b) 4.7e+57)))
     (+ (* c i) (+ (* a b) (* z t)))
     (+ (* c i) (+ (* x y) (* z t))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((a * b) <= -((double) INFINITY)) {
		tmp = a * b;
	} else if (((a * b) <= -1.8e+54) || !((a * b) <= 4.7e+57)) {
		tmp = (c * i) + ((a * b) + (z * t));
	} else {
		tmp = (c * i) + ((x * y) + (z * t));
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((a * b) <= -Double.POSITIVE_INFINITY) {
		tmp = a * b;
	} else if (((a * b) <= -1.8e+54) || !((a * b) <= 4.7e+57)) {
		tmp = (c * i) + ((a * b) + (z * t));
	} else {
		tmp = (c * i) + ((x * y) + (z * t));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if (a * b) <= -math.inf:
		tmp = a * b
	elif ((a * b) <= -1.8e+54) or not ((a * b) <= 4.7e+57):
		tmp = (c * i) + ((a * b) + (z * t))
	else:
		tmp = (c * i) + ((x * y) + (z * t))
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(a * b) <= Float64(-Inf))
		tmp = Float64(a * b);
	elseif ((Float64(a * b) <= -1.8e+54) || !(Float64(a * b) <= 4.7e+57))
		tmp = Float64(Float64(c * i) + Float64(Float64(a * b) + Float64(z * t)));
	else
		tmp = Float64(Float64(c * i) + Float64(Float64(x * y) + Float64(z * t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	tmp = 0.0;
	if ((a * b) <= -Inf)
		tmp = a * b;
	elseif (((a * b) <= -1.8e+54) || ~(((a * b) <= 4.7e+57)))
		tmp = (c * i) + ((a * b) + (z * t));
	else
		tmp = (c * i) + ((x * y) + (z * t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(a * b), $MachinePrecision], (-Infinity)], N[(a * b), $MachinePrecision], If[Or[LessEqual[N[(a * b), $MachinePrecision], -1.8e+54], N[Not[LessEqual[N[(a * b), $MachinePrecision], 4.7e+57]], $MachinePrecision]], N[(N[(c * i), $MachinePrecision] + N[(N[(a * b), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c * i), $MachinePrecision] + N[(N[(x * y), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 a b) < -inf.0

   1. Initial program 72.7%

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

   2. Taylor expanded in x around 0 72.7%

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

      1. *-commutative 72.7%

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

      2. fma-def 77.3%

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

      3. *-commutative 77.3%

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

   4. Applied egg-rr 77.3%

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

   5. Taylor expanded in b around inf 90.9%

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

   if -inf.0 < (*.f64 a b) < -1.8000000000000001e54 or 4.7000000000000003e57 < (*.f64 a b)

   1. Initial program 90.2%

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

   2. Taylor expanded in x around 0 83.1%

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

   if -1.8000000000000001e54 < (*.f64 a b) < 4.7000000000000003e57

   1. Initial program 96.9%

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

   2. Taylor expanded in a around 0 93.0%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b \leq -\infty:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \cdot b \leq -1.8 \cdot 10^{+54} \lor \neg \left(a \cdot b \leq 4.7 \cdot 10^{+57}\right):\\ \;\;\;\;c \cdot i + \left(a \cdot b + z \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot i + \left(x \cdot y + z \cdot t\right)\\ \end{array} \]

Alternative 7: 42.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot b \leq -3.5 \cdot 10^{+224}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \cdot b \leq 1.8 \cdot 10^{-46}:\\ \;\;\;\;c \cdot i\\ \mathbf{elif}\;a \cdot b \leq 1.22 \cdot 10^{+183}:\\ \;\;\;\;z \cdot t\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* a b) -3.5e+224)
   (* a b)
   (if (<= (* a b) 1.8e-46)
     (* c i)
     (if (<= (* a b) 1.22e+183) (* z t) (* a b)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((a * b) <= -3.5e+224) {
		tmp = a * b;
	} else if ((a * b) <= 1.8e-46) {
		tmp = c * i;
	} else if ((a * b) <= 1.22e+183) {
		tmp = z * t;
	} else {
		tmp = a * b;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i)
    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), intent (in) :: c
    real(8), intent (in) :: i
    real(8) :: tmp
    if ((a * b) <= (-3.5d+224)) then
        tmp = a * b
    else if ((a * b) <= 1.8d-46) then
        tmp = c * i
    else if ((a * b) <= 1.22d+183) then
        tmp = z * t
    else
        tmp = 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 c, double i) {
	double tmp;
	if ((a * b) <= -3.5e+224) {
		tmp = a * b;
	} else if ((a * b) <= 1.8e-46) {
		tmp = c * i;
	} else if ((a * b) <= 1.22e+183) {
		tmp = z * t;
	} else {
		tmp = a * b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if (a * b) <= -3.5e+224:
		tmp = a * b
	elif (a * b) <= 1.8e-46:
		tmp = c * i
	elif (a * b) <= 1.22e+183:
		tmp = z * t
	else:
		tmp = a * b
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(a * b) <= -3.5e+224)
		tmp = Float64(a * b);
	elseif (Float64(a * b) <= 1.8e-46)
		tmp = Float64(c * i);
	elseif (Float64(a * b) <= 1.22e+183)
		tmp = Float64(z * t);
	else
		tmp = Float64(a * b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	tmp = 0.0;
	if ((a * b) <= -3.5e+224)
		tmp = a * b;
	elseif ((a * b) <= 1.8e-46)
		tmp = c * i;
	elseif ((a * b) <= 1.22e+183)
		tmp = z * t;
	else
		tmp = a * b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(a * b), $MachinePrecision], -3.5e+224], N[(a * b), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], 1.8e-46], N[(c * i), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], 1.22e+183], N[(z * t), $MachinePrecision], N[(a * b), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 a b) < -3.5e224 or 1.21999999999999991e183 < (*.f64 a b)

   1. Initial program 80.7%

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

   2. Taylor expanded in x around 0 79.7%

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

      1. *-commutative 79.7%

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

      2. fma-def 84.9%

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

      3. *-commutative 84.9%

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

   4. Applied egg-rr 84.9%

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

   5. Taylor expanded in b around inf 81.3%

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

   if -3.5e224 < (*.f64 a b) < 1.8e-46

   1. Initial program 96.9%

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

   2. Taylor expanded in c around inf 42.7%

      \[\leadsto \color{blue}{c \cdot i} \]

   if 1.8e-46 < (*.f64 a b) < 1.21999999999999991e183

   1. Initial program 97.7%

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

   2. Taylor expanded in x around 0 80.5%

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

      1. *-commutative 80.5%

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

      2. fma-def 80.5%

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

      3. *-commutative 80.5%

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

   4. Applied egg-rr 80.5%

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

   5. Taylor expanded in z around inf 46.7%

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

4. Final simplification 55.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b \leq -3.5 \cdot 10^{+224}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \cdot b \leq 1.8 \cdot 10^{-46}:\\ \;\;\;\;c \cdot i\\ \mathbf{elif}\;a \cdot b \leq 1.22 \cdot 10^{+183}:\\ \;\;\;\;z \cdot t\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \]

Alternative 8: 66.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \cdot i \leq -2.6 \cdot 10^{+84} \lor \neg \left(c \cdot i \leq 1.2 \cdot 10^{+45}\right):\\ \;\;\;\;a \cdot b + c \cdot i\\ \mathbf{else}:\\ \;\;\;\;a \cdot b + z \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (or (<= (* c i) -2.6e+84) (not (<= (* c i) 1.2e+45)))
   (+ (* a b) (* c i))
   (+ (* a b) (* z t))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (((c * i) <= -2.6e+84) || !((c * i) <= 1.2e+45)) {
		tmp = (a * b) + (c * i);
	} else {
		tmp = (a * b) + (z * t);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i)
    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), intent (in) :: c
    real(8), intent (in) :: i
    real(8) :: tmp
    if (((c * i) <= (-2.6d+84)) .or. (.not. ((c * i) <= 1.2d+45))) then
        tmp = (a * b) + (c * i)
    else
        tmp = (a * b) + (z * 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 c, double i) {
	double tmp;
	if (((c * i) <= -2.6e+84) || !((c * i) <= 1.2e+45)) {
		tmp = (a * b) + (c * i);
	} else {
		tmp = (a * b) + (z * t);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if ((c * i) <= -2.6e+84) or not ((c * i) <= 1.2e+45):
		tmp = (a * b) + (c * i)
	else:
		tmp = (a * b) + (z * t)
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if ((Float64(c * i) <= -2.6e+84) || !(Float64(c * i) <= 1.2e+45))
		tmp = Float64(Float64(a * b) + Float64(c * i));
	else
		tmp = Float64(Float64(a * b) + Float64(z * t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	tmp = 0.0;
	if (((c * i) <= -2.6e+84) || ~(((c * i) <= 1.2e+45)))
		tmp = (a * b) + (c * i);
	else
		tmp = (a * b) + (z * t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[Or[LessEqual[N[(c * i), $MachinePrecision], -2.6e+84], N[Not[LessEqual[N[(c * i), $MachinePrecision], 1.2e+45]], $MachinePrecision]], N[(N[(a * b), $MachinePrecision] + N[(c * i), $MachinePrecision]), $MachinePrecision], N[(N[(a * b), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 c i) < -2.6000000000000001e84 or 1.19999999999999995e45 < (*.f64 c i)

   1. Initial program 89.7%

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

   2. Taylor expanded in a around inf 80.2%

      \[\leadsto \color{blue}{a \cdot b} + c \cdot i \]

   if -2.6000000000000001e84 < (*.f64 c i) < 1.19999999999999995e45

   1. Initial program 93.9%

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

   2. Taylor expanded in x around 0 74.3%

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

      1. *-commutative 74.3%

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

      2. fma-def 77.0%

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

      3. *-commutative 77.0%

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

   4. Applied egg-rr 77.0%

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

   5. Taylor expanded in c around 0 70.7%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \cdot i \leq -2.6 \cdot 10^{+84} \lor \neg \left(c \cdot i \leq 1.2 \cdot 10^{+45}\right):\\ \;\;\;\;a \cdot b + c \cdot i\\ \mathbf{else}:\\ \;\;\;\;a \cdot b + z \cdot t\\ \end{array} \]

Alternative 9: 75.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.3 \cdot 10^{-50}:\\ \;\;\;\;x \cdot y + c \cdot i\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{+163}:\\ \;\;\;\;c \cdot i + \left(a \cdot b + z \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y + z \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= y -1.3e-50)
   (+ (* x y) (* c i))
   (if (<= y 8.5e+163) (+ (* c i) (+ (* a b) (* z t))) (+ (* x y) (* z t)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (y <= -1.3e-50) {
		tmp = (x * y) + (c * i);
	} else if (y <= 8.5e+163) {
		tmp = (c * i) + ((a * b) + (z * t));
	} else {
		tmp = (x * y) + (z * t);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i)
    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), intent (in) :: c
    real(8), intent (in) :: i
    real(8) :: tmp
    if (y <= (-1.3d-50)) then
        tmp = (x * y) + (c * i)
    else if (y <= 8.5d+163) then
        tmp = (c * i) + ((a * b) + (z * t))
    else
        tmp = (x * y) + (z * 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 c, double i) {
	double tmp;
	if (y <= -1.3e-50) {
		tmp = (x * y) + (c * i);
	} else if (y <= 8.5e+163) {
		tmp = (c * i) + ((a * b) + (z * t));
	} else {
		tmp = (x * y) + (z * t);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if y <= -1.3e-50:
		tmp = (x * y) + (c * i)
	elif y <= 8.5e+163:
		tmp = (c * i) + ((a * b) + (z * t))
	else:
		tmp = (x * y) + (z * t)
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (y <= -1.3e-50)
		tmp = Float64(Float64(x * y) + Float64(c * i));
	elseif (y <= 8.5e+163)
		tmp = Float64(Float64(c * i) + Float64(Float64(a * b) + Float64(z * t)));
	else
		tmp = Float64(Float64(x * y) + Float64(z * t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	tmp = 0.0;
	if (y <= -1.3e-50)
		tmp = (x * y) + (c * i);
	elseif (y <= 8.5e+163)
		tmp = (c * i) + ((a * b) + (z * t));
	else
		tmp = (x * y) + (z * t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[y, -1.3e-50], N[(N[(x * y), $MachinePrecision] + N[(c * i), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 8.5e+163], N[(N[(c * i), $MachinePrecision] + N[(N[(a * b), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * y), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.3000000000000001e-50

   1. Initial program 93.3%

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

   2. Taylor expanded in x around inf 65.1%

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

   if -1.3000000000000001e-50 < y < 8.5000000000000003e163

   1. Initial program 93.1%

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

   2. Taylor expanded in x around 0 86.0%

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

   if 8.5000000000000003e163 < y

   1. Initial program 79.9%

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

   2. Taylor expanded in a around 0 82.4%

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

   3. Taylor expanded in c around 0 65.6%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.3 \cdot 10^{-50}:\\ \;\;\;\;x \cdot y + c \cdot i\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{+163}:\\ \;\;\;\;c \cdot i + \left(a \cdot b + z \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y + z \cdot t\\ \end{array} \]

Alternative 10: 42.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot b \leq -4 \cdot 10^{+237}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \cdot b \leq 3.1 \cdot 10^{+89}:\\ \;\;\;\;c \cdot i\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* a b) -4e+237) (* a b) (if (<= (* a b) 3.1e+89) (* c i) (* a b))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((a * b) <= -4e+237) {
		tmp = a * b;
	} else if ((a * b) <= 3.1e+89) {
		tmp = c * i;
	} else {
		tmp = a * b;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i)
    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), intent (in) :: c
    real(8), intent (in) :: i
    real(8) :: tmp
    if ((a * b) <= (-4d+237)) then
        tmp = a * b
    else if ((a * b) <= 3.1d+89) then
        tmp = c * i
    else
        tmp = 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 c, double i) {
	double tmp;
	if ((a * b) <= -4e+237) {
		tmp = a * b;
	} else if ((a * b) <= 3.1e+89) {
		tmp = c * i;
	} else {
		tmp = a * b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if (a * b) <= -4e+237:
		tmp = a * b
	elif (a * b) <= 3.1e+89:
		tmp = c * i
	else:
		tmp = a * b
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(a * b) <= -4e+237)
		tmp = Float64(a * b);
	elseif (Float64(a * b) <= 3.1e+89)
		tmp = Float64(c * i);
	else
		tmp = Float64(a * b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	tmp = 0.0;
	if ((a * b) <= -4e+237)
		tmp = a * b;
	elseif ((a * b) <= 3.1e+89)
		tmp = c * i;
	else
		tmp = a * b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(a * b), $MachinePrecision], -4e+237], N[(a * b), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], 3.1e+89], N[(c * i), $MachinePrecision], N[(a * b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a b) < -3.99999999999999976e237 or 3.1e89 < (*.f64 a b)

   1. Initial program 82.4%

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

   2. Taylor expanded in x around 0 80.4%

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

      1. *-commutative 80.4%

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

      2. fma-def 84.9%

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

      3. *-commutative 84.9%

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

   4. Applied egg-rr 84.9%

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

   5. Taylor expanded in b around inf 73.5%

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

   if -3.99999999999999976e237 < (*.f64 a b) < 3.1e89

   1. Initial program 97.5%

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

   2. Taylor expanded in c around inf 39.7%

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

4. Final simplification 51.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b \leq -4 \cdot 10^{+237}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \cdot b \leq 3.1 \cdot 10^{+89}:\\ \;\;\;\;c \cdot i\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \]

Alternative 11: 53.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 3.3 \cdot 10^{+202}:\\ \;\;\;\;a \cdot b + c \cdot i\\ \mathbf{else}:\\ \;\;\;\;z \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= t 3.3e+202) (+ (* a b) (* c i)) (* z t)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (t <= 3.3e+202) {
		tmp = (a * b) + (c * i);
	} else {
		tmp = z * t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i)
    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), intent (in) :: c
    real(8), intent (in) :: i
    real(8) :: tmp
    if (t <= 3.3d+202) then
        tmp = (a * b) + (c * i)
    else
        tmp = z * 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 c, double i) {
	double tmp;
	if (t <= 3.3e+202) {
		tmp = (a * b) + (c * i);
	} else {
		tmp = z * t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if t <= 3.3e+202:
		tmp = (a * b) + (c * i)
	else:
		tmp = z * t
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (t <= 3.3e+202)
		tmp = Float64(Float64(a * b) + Float64(c * i));
	else
		tmp = Float64(z * t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	tmp = 0.0;
	if (t <= 3.3e+202)
		tmp = (a * b) + (c * i);
	else
		tmp = z * t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[t, 3.3e+202], N[(N[(a * b), $MachinePrecision] + N[(c * i), $MachinePrecision]), $MachinePrecision], N[(z * t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 3.2999999999999999e202

   1. Initial program 93.1%

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

   2. Taylor expanded in a around inf 60.3%

      \[\leadsto \color{blue}{a \cdot b} + c \cdot i \]

   if 3.2999999999999999e202 < t

   1. Initial program 82.6%

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

   2. Taylor expanded in x around 0 68.0%

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

      1. *-commutative 68.0%

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

      2. fma-def 72.4%

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

      3. *-commutative 72.4%

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

   4. Applied egg-rr 72.4%

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

   5. Taylor expanded in z around inf 54.0%

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

4. Final simplification 59.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.3 \cdot 10^{+202}:\\ \;\;\;\;a \cdot b + c \cdot i\\ \mathbf{else}:\\ \;\;\;\;z \cdot t\\ \end{array} \]

Alternative 12: 27.9% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ a \cdot b \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := N[(a * b), $MachinePrecision]

Derivation

1. Initial program 92.2%

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

2. Taylor expanded in x around 0 77.2%

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

   1. *-commutative 77.2%

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

   2. fma-def 78.8%

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

   3. *-commutative 78.8%

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

4. Applied egg-rr 78.8%

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

5. Taylor expanded in b around inf 32.9%

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

6. Final simplification 32.9%

   \[\leadsto a \cdot b \]

Reproduce

herbie shell --seed 2023230 
(FPCore (x y z t a b c i)
  :name "Linear.V4:$cdot from linear-1.19.1.3, C"
  :precision binary64
  (+ (+ (+ (* x y) (* z t)) (* a b)) (* c i)))