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

Percentage Accurate: 96.0% → 98.0%

Time: 10.3s

Alternatives: 11

Speedup: 1.3×

• 41.9% 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: 98.0% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 96.5%

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

   1. associate-+l+ N/A

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

   2. +-commutative N/A

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

   3. associate-+l+ N/A

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

   4. accelerator-lowering-fma.f64 N/A

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

   5. accelerator-lowering-fma.f64 N/A

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

   6. accelerator-lowering-fma.f64 N/A

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

   7. *-lowering-*.f64 98.0

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

4. Applied egg-rr 98.0%

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

Alternative 2: 67.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(x, y, a \cdot b\right)\\ \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+50}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \cdot y \leq -1 \cdot 10^{-64}:\\ \;\;\;\;\mathsf{fma}\left(z, t, c \cdot i\right)\\ \mathbf{elif}\;x \cdot y \leq 10^{+54}:\\ \;\;\;\;\mathsf{fma}\left(z, t, a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma x y (* a b))))
   (if (<= (* x y) -2e+50)
     t_1
     (if (<= (* x y) -1e-64)
       (fma z t (* c i))
       (if (<= (* x y) 1e+54) (fma z t (* a b)) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma(x, y, (a * b));
	double tmp;
	if ((x * y) <= -2e+50) {
		tmp = t_1;
	} else if ((x * y) <= -1e-64) {
		tmp = fma(z, t, (c * i));
	} else if ((x * y) <= 1e+54) {
		tmp = fma(z, t, (a * b));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(x, y, Float64(a * b))
	tmp = 0.0
	if (Float64(x * y) <= -2e+50)
		tmp = t_1;
	elseif (Float64(x * y) <= -1e-64)
		tmp = fma(z, t, Float64(c * i));
	elseif (Float64(x * y) <= 1e+54)
		tmp = fma(z, t, Float64(a * b));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(x * y + N[(a * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * y), $MachinePrecision], -2e+50], t$95$1, If[LessEqual[N[(x * y), $MachinePrecision], -1e-64], N[(z * t + N[(c * i), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 1e+54], N[(z * t + N[(a * b), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -2.0000000000000002e50 or 1.0000000000000001e54 < (*.f64 x y)

   1. Initial program 94.2%

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

   3. Taylor expanded in c around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. *-lowering-*.f64 92.0

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

   5. Simplified 92.0%

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

   6. Taylor expanded in a around inf

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

      1. *-lowering-*.f64 81.9

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

   8. Simplified 81.9%

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

   if -2.0000000000000002e50 < (*.f64 x y) < -9.99999999999999965e-65

   1. Initial program 90.8%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{t \cdot z} + c \cdot i \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 70.0

         \[\leadsto \color{blue}{t \cdot z} + c \cdot i \]

   5. Simplified 70.0%

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{z \cdot t} + c \cdot i \]

      2. accelerator-lowering-fma.f64 N/A

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

      3. *-lowering-*.f64 74.6

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

   7. Applied egg-rr 74.6%

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

   if -9.99999999999999965e-65 < (*.f64 x y) < 1.0000000000000001e54

   1. Initial program 99.2%

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

      1. associate-+l+ N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. *-lowering-*.f64 99.2

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

   4. Applied egg-rr 99.2%

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

   5. Taylor expanded in a around inf

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

      1. *-lowering-*.f64 74.2

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

   7. Simplified 74.2%

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

Alternative 3: 89.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z, t, \mathsf{fma}\left(x, y, c \cdot i\right)\right)\\ \mathbf{if}\;c \cdot i \leq -1.95 \cdot 10^{+63}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;c \cdot i \leq 7.5 \cdot 10^{+59}:\\ \;\;\;\;\mathsf{fma}\left(x, y, \mathsf{fma}\left(a, b, z \cdot t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma z t (fma x y (* c i)))))
   (if (<= (* c i) -1.95e+63)
     t_1
     (if (<= (* c i) 7.5e+59) (fma x y (fma a b (* z t))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma(z, t, fma(x, y, (c * i)));
	double tmp;
	if ((c * i) <= -1.95e+63) {
		tmp = t_1;
	} else if ((c * i) <= 7.5e+59) {
		tmp = fma(x, y, fma(a, b, (z * t)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(z, t, fma(x, y, Float64(c * i)))
	tmp = 0.0
	if (Float64(c * i) <= -1.95e+63)
		tmp = t_1;
	elseif (Float64(c * i) <= 7.5e+59)
		tmp = fma(x, y, fma(a, b, Float64(z * t)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(z * t + N[(x * y + N[(c * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(c * i), $MachinePrecision], -1.95e+63], t$95$1, If[LessEqual[N[(c * i), $MachinePrecision], 7.5e+59], N[(x * y + N[(a * b + N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 c i) < -1.95e63 or 7.4999999999999996e59 < (*.f64 c i)

   1. Initial program 92.3%

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

      1. associate-+l+ N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. *-lowering-*.f64 95.6

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

   4. Applied egg-rr 95.6%

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

   5. Taylor expanded in a around 0

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

      1. *-lowering-*.f64 83.4

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

   7. Simplified 83.4%

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

   if -1.95e63 < (*.f64 c i) < 7.4999999999999996e59

   1. Initial program 98.8%

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

   3. Taylor expanded in c around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. *-lowering-*.f64 96.0

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

   5. Simplified 96.0%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \cdot i \leq -1.95 \cdot 10^{+63}:\\ \;\;\;\;\mathsf{fma}\left(z, t, \mathsf{fma}\left(x, y, c \cdot i\right)\right)\\ \mathbf{elif}\;c \cdot i \leq 7.5 \cdot 10^{+59}:\\ \;\;\;\;\mathsf{fma}\left(x, y, \mathsf{fma}\left(a, b, z \cdot t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t, \mathsf{fma}\left(x, y, c \cdot i\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \cdot i \leq -9.2 \cdot 10^{+120}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c \cdot i\right)\\ \mathbf{elif}\;c \cdot i \leq 3.4 \cdot 10^{+183}:\\ \;\;\;\;\mathsf{fma}\left(x, y, \mathsf{fma}\left(a, b, z \cdot t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t, c \cdot i\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* c i) -9.2e+120)
   (fma y x (* c i))
   (if (<= (* c i) 3.4e+183) (fma x y (fma a b (* z t))) (fma z t (* c i)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((c * i) <= -9.2e+120) {
		tmp = fma(y, x, (c * i));
	} else if ((c * i) <= 3.4e+183) {
		tmp = fma(x, y, fma(a, b, (z * t)));
	} else {
		tmp = fma(z, t, (c * i));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(c * i) <= -9.2e+120)
		tmp = fma(y, x, Float64(c * i));
	elseif (Float64(c * i) <= 3.4e+183)
		tmp = fma(x, y, fma(a, b, Float64(z * t)));
	else
		tmp = fma(z, t, Float64(c * i));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(c * i), $MachinePrecision], -9.2e+120], N[(y * x + N[(c * i), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(c * i), $MachinePrecision], 3.4e+183], N[(x * y + N[(a * b + N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z * t + N[(c * i), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 c i) < -9.1999999999999997e120

   1. Initial program 87.1%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot y} + c \cdot i \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 78.0

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

   5. Simplified 78.0%

      \[\leadsto \color{blue}{x \cdot y} + c \cdot i \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. *-lowering-*.f64 81.2

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

   7. Applied egg-rr 81.2%

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

   if -9.1999999999999997e120 < (*.f64 c i) < 3.4e183

   1. Initial program 98.5%

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

   3. Taylor expanded in c around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. *-lowering-*.f64 91.5

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

   5. Simplified 91.5%

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

   if 3.4e183 < (*.f64 c i)

   1. Initial program 91.7%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{t \cdot z} + c \cdot i \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 75.4

         \[\leadsto \color{blue}{t \cdot z} + c \cdot i \]

   5. Simplified 75.4%

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{z \cdot t} + c \cdot i \]

      2. accelerator-lowering-fma.f64 N/A

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

      3. *-lowering-*.f64 75.4

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

   7. Applied egg-rr 75.4%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \cdot i \leq -9.2 \cdot 10^{+120}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c \cdot i\right)\\ \mathbf{elif}\;c \cdot i \leq 3.4 \cdot 10^{+183}:\\ \;\;\;\;\mathsf{fma}\left(x, y, \mathsf{fma}\left(a, b, z \cdot t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t, c \cdot i\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 43.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+114}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{-285}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;z \cdot t \leq 10^{+62}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* z t) -1e+114)
   (* z t)
   (if (<= (* z t) 5e-285) (* a b) (if (<= (* z t) 1e+62) (* 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 ((z * t) <= -1e+114) {
		tmp = z * t;
	} else if ((z * t) <= 5e-285) {
		tmp = a * b;
	} else if ((z * t) <= 1e+62) {
		tmp = x * y;
	} 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 ((z * t) <= (-1d+114)) then
        tmp = z * t
    else if ((z * t) <= 5d-285) then
        tmp = a * b
    else if ((z * t) <= 1d+62) then
        tmp = x * y
    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 ((z * t) <= -1e+114) {
		tmp = z * t;
	} else if ((z * t) <= 5e-285) {
		tmp = a * b;
	} else if ((z * t) <= 1e+62) {
		tmp = x * y;
	} else {
		tmp = z * t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if (z * t) <= -1e+114:
		tmp = z * t
	elif (z * t) <= 5e-285:
		tmp = a * b
	elif (z * t) <= 1e+62:
		tmp = x * y
	else:
		tmp = z * t
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(z * t) <= -1e+114)
		tmp = Float64(z * t);
	elseif (Float64(z * t) <= 5e-285)
		tmp = Float64(a * b);
	elseif (Float64(z * t) <= 1e+62)
		tmp = Float64(x * y);
	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 ((z * t) <= -1e+114)
		tmp = z * t;
	elseif ((z * t) <= 5e-285)
		tmp = a * b;
	elseif ((z * t) <= 1e+62)
		tmp = x * y;
	else
		tmp = z * t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(z * t), $MachinePrecision], -1e+114], N[(z * t), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 5e-285], N[(a * b), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 1e+62], N[(x * y), $MachinePrecision], N[(z * t), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -1e114 or 1.00000000000000004e62 < (*.f64 z t)

   1. Initial program 96.6%

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

   3. Taylor expanded in z around inf

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

      1. *-lowering-*.f64 67.6

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

   5. Simplified 67.6%

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

   if -1e114 < (*.f64 z t) < 5.00000000000000018e-285

   1. Initial program 96.0%

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

   3. Taylor expanded in a around inf

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

      1. *-lowering-*.f64 42.4

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

   5. Simplified 42.4%

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

   if 5.00000000000000018e-285 < (*.f64 z t) < 1.00000000000000004e62

   1. Initial program 96.9%

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

   3. Taylor expanded in x around inf

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

      1. *-lowering-*.f64 44.0

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

   5. Simplified 44.0%

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

4. Final simplification 51.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+114}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{-285}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;z \cdot t \leq 10^{+62}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 66.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+116}:\\ \;\;\;\;\mathsf{fma}\left(z, t, x \cdot y\right)\\ \mathbf{elif}\;z \cdot t \leq 10^{+62}:\\ \;\;\;\;\mathsf{fma}\left(x, y, a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t, a \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* z t) -2e+116)
   (fma z t (* x y))
   (if (<= (* z t) 1e+62) (fma x y (* a b)) (fma 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 ((z * t) <= -2e+116) {
		tmp = fma(z, t, (x * y));
	} else if ((z * t) <= 1e+62) {
		tmp = fma(x, y, (a * b));
	} else {
		tmp = fma(z, t, (a * b));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(z * t) <= -2e+116)
		tmp = fma(z, t, Float64(x * y));
	elseif (Float64(z * t) <= 1e+62)
		tmp = fma(x, y, Float64(a * b));
	else
		tmp = fma(z, t, Float64(a * b));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(z * t), $MachinePrecision], -2e+116], N[(z * t + N[(x * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 1e+62], N[(x * y + N[(a * b), $MachinePrecision]), $MachinePrecision], N[(z * t + N[(a * b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -2.00000000000000003e116

   1. Initial program 95.5%

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

      1. associate-+l+ N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. *-lowering-*.f64 97.7

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

   4. Applied egg-rr 97.7%

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

   5. Taylor expanded in x around inf

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

      1. *-lowering-*.f64 88.8

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

   7. Simplified 88.8%

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

   if -2.00000000000000003e116 < (*.f64 z t) < 1.00000000000000004e62

   1. Initial program 96.4%

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

   3. Taylor expanded in c around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. *-lowering-*.f64 78.4

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

   5. Simplified 78.4%

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

   6. Taylor expanded in a around inf

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

      1. *-lowering-*.f64 73.1

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

   8. Simplified 73.1%

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

   if 1.00000000000000004e62 < (*.f64 z t)

   1. Initial program 97.7%

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

      1. associate-+l+ N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. *-lowering-*.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in a around inf

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

      1. *-lowering-*.f64 81.7

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

   7. Simplified 81.7%

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

Alternative 7: 66.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(x, y, a \cdot b\right)\\ \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+71}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \cdot y \leq 10^{+54}:\\ \;\;\;\;\mathsf{fma}\left(z, t, a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma x y (* a b))))
   (if (<= (* x y) -2e+71) t_1 (if (<= (* x y) 1e+54) (fma z t (* a b)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma(x, y, (a * b));
	double tmp;
	if ((x * y) <= -2e+71) {
		tmp = t_1;
	} else if ((x * y) <= 1e+54) {
		tmp = fma(z, t, (a * b));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(x, y, Float64(a * b))
	tmp = 0.0
	if (Float64(x * y) <= -2e+71)
		tmp = t_1;
	elseif (Float64(x * y) <= 1e+54)
		tmp = fma(z, t, Float64(a * b));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(x * y + N[(a * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * y), $MachinePrecision], -2e+71], t$95$1, If[LessEqual[N[(x * y), $MachinePrecision], 1e+54], N[(z * t + N[(a * b), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -2.0000000000000001e71 or 1.0000000000000001e54 < (*.f64 x y)

   1. Initial program 94.0%

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

   3. Taylor expanded in c around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. *-lowering-*.f64 91.7

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

   5. Simplified 91.7%

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

   6. Taylor expanded in a around inf

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

      1. *-lowering-*.f64 82.3

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

   8. Simplified 82.3%

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

   if -2.0000000000000001e71 < (*.f64 x y) < 1.0000000000000001e54

   1. Initial program 98.0%

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

      1. associate-+l+ N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. *-lowering-*.f64 98.7

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

   4. Applied egg-rr 98.7%

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

   5. Taylor expanded in a around inf

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

      1. *-lowering-*.f64 70.8

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

   7. Simplified 70.8%

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

Alternative 8: 62.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+255}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+123}:\\ \;\;\;\;\mathsf{fma}\left(x, y, a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* z t) -1e+255)
   (* z t)
   (if (<= (* z t) 5e+123) (fma x y (* 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 ((z * t) <= -1e+255) {
		tmp = z * t;
	} else if ((z * t) <= 5e+123) {
		tmp = fma(x, y, (a * b));
	} else {
		tmp = z * t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(z * t) <= -1e+255)
		tmp = Float64(z * t);
	elseif (Float64(z * t) <= 5e+123)
		tmp = fma(x, y, Float64(a * b));
	else
		tmp = Float64(z * t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(z * t), $MachinePrecision], -1e+255], N[(z * t), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 5e+123], N[(x * y + N[(a * b), $MachinePrecision]), $MachinePrecision], N[(z * t), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -9.99999999999999988e254 or 4.99999999999999974e123 < (*.f64 z t)

   1. Initial program 95.3%

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

   3. Taylor expanded in z around inf

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

      1. *-lowering-*.f64 80.3

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

   5. Simplified 80.3%

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

   if -9.99999999999999988e254 < (*.f64 z t) < 4.99999999999999974e123

   1. Initial program 96.8%

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

   3. Taylor expanded in c around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. *-lowering-*.f64 78.7

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

   5. Simplified 78.7%

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

   6. Taylor expanded in a around inf

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

      1. *-lowering-*.f64 69.8

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

   8. Simplified 69.8%

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

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+255}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+123}:\\ \;\;\;\;\mathsf{fma}\left(x, y, a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 43.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+114}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;z \cdot t \leq 4 \cdot 10^{+101}:\\ \;\;\;\;a \cdot b\\ \mathbf{else}:\\ \;\;\;\;z \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* z t) -1e+114) (* z t) (if (<= (* z t) 4e+101) (* 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 ((z * t) <= -1e+114) {
		tmp = z * t;
	} else if ((z * t) <= 4e+101) {
		tmp = a * b;
	} 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 ((z * t) <= (-1d+114)) then
        tmp = z * t
    else if ((z * t) <= 4d+101) then
        tmp = a * b
    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 ((z * t) <= -1e+114) {
		tmp = z * t;
	} else if ((z * t) <= 4e+101) {
		tmp = a * b;
	} else {
		tmp = z * t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if (z * t) <= -1e+114:
		tmp = z * t
	elif (z * t) <= 4e+101:
		tmp = a * b
	else:
		tmp = z * t
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(z * t) <= -1e+114)
		tmp = Float64(z * t);
	elseif (Float64(z * t) <= 4e+101)
		tmp = Float64(a * b);
	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 ((z * t) <= -1e+114)
		tmp = z * t;
	elseif ((z * t) <= 4e+101)
		tmp = a * b;
	else
		tmp = z * t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(z * t), $MachinePrecision], -1e+114], N[(z * t), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 4e+101], N[(a * b), $MachinePrecision], N[(z * t), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -1e114 or 3.9999999999999999e101 < (*.f64 z t)

   1. Initial program 96.4%

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

   3. Taylor expanded in z around inf

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

      1. *-lowering-*.f64 69.6

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

   5. Simplified 69.6%

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

   if -1e114 < (*.f64 z t) < 3.9999999999999999e101

   1. Initial program 96.5%

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

   3. Taylor expanded in a around inf

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

      1. *-lowering-*.f64 39.4

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

   5. Simplified 39.4%

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

4. Final simplification 49.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+114}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;z \cdot t \leq 4 \cdot 10^{+101}:\\ \;\;\;\;a \cdot b\\ \mathbf{else}:\\ \;\;\;\;z \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 43.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \cdot i \leq -5.8 \cdot 10^{+110}:\\ \;\;\;\;c \cdot i\\ \mathbf{elif}\;c \cdot i \leq 9 \cdot 10^{+59}:\\ \;\;\;\;a \cdot b\\ \mathbf{else}:\\ \;\;\;\;c \cdot i\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* c i) -5.8e+110) (* c i) (if (<= (* c i) 9e+59) (* a b) (* c i))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((c * i) <= -5.8e+110) {
		tmp = c * i;
	} else if ((c * i) <= 9e+59) {
		tmp = a * b;
	} else {
		tmp = c * i;
	}
	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) <= (-5.8d+110)) then
        tmp = c * i
    else if ((c * i) <= 9d+59) then
        tmp = a * b
    else
        tmp = c * i
    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) <= -5.8e+110) {
		tmp = c * i;
	} else if ((c * i) <= 9e+59) {
		tmp = a * b;
	} else {
		tmp = c * i;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if (c * i) <= -5.8e+110:
		tmp = c * i
	elif (c * i) <= 9e+59:
		tmp = a * b
	else:
		tmp = c * i
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(c * i) <= -5.8e+110)
		tmp = Float64(c * i);
	elseif (Float64(c * i) <= 9e+59)
		tmp = Float64(a * b);
	else
		tmp = Float64(c * i);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	tmp = 0.0;
	if ((c * i) <= -5.8e+110)
		tmp = c * i;
	elseif ((c * i) <= 9e+59)
		tmp = a * b;
	else
		tmp = c * i;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(c * i), $MachinePrecision], -5.8e+110], N[(c * i), $MachinePrecision], If[LessEqual[N[(c * i), $MachinePrecision], 9e+59], N[(a * b), $MachinePrecision], N[(c * i), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 c i) < -5.7999999999999999e110 or 8.99999999999999919e59 < (*.f64 c i)

   1. Initial program 92.4%

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

   3. Taylor expanded in c around inf

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

      1. *-lowering-*.f64 54.5

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

   5. Simplified 54.5%

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

   if -5.7999999999999999e110 < (*.f64 c i) < 8.99999999999999919e59

   1. Initial program 98.3%

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

   3. Taylor expanded in a around inf

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

      1. *-lowering-*.f64 35.5

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

   5. Simplified 35.5%

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

Alternative 11: 27.7% 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 96.5%

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

3. Taylor expanded in a around inf

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

   1. *-lowering-*.f64 30.0

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

5. Simplified 30.0%

   \[\leadsto \color{blue}{a \cdot b} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024196 
(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)))