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

Percentage Accurate: 96.0% → 98.0%

Time: 9.8s

Alternatives: 13

Speedup: 1.3×

• 41.1% 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 96.1%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-+l+ N/A

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

   7. lift-+.f64 N/A

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

   8. +-commutative N/A

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

   9. associate-+l+ N/A

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

   10. lift-*.f64 N/A

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

   11. lower-fma.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. lower-fma.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. lower-fma.f64 97.6

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

4. Applied egg-rr 97.6%

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

Alternative 2: 64.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, x, c \cdot i\right)\\ \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+226}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{-281}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{-105}:\\ \;\;\;\;\mathsf{fma}\left(i, c, a \cdot b\right)\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+36}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t, a \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma y x (* c i))))
   (if (<= (* z t) -5e+226)
     (* z t)
     (if (<= (* z t) 5e-281)
       t_1
       (if (<= (* z t) 5e-105)
         (fma i c (* a b))
         (if (<= (* z t) 5e+36) t_1 (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 t_1 = fma(y, x, (c * i));
	double tmp;
	if ((z * t) <= -5e+226) {
		tmp = z * t;
	} else if ((z * t) <= 5e-281) {
		tmp = t_1;
	} else if ((z * t) <= 5e-105) {
		tmp = fma(i, c, (a * b));
	} else if ((z * t) <= 5e+36) {
		tmp = t_1;
	} else {
		tmp = fma(z, t, (a * b));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(y, x, Float64(c * i))
	tmp = 0.0
	if (Float64(z * t) <= -5e+226)
		tmp = Float64(z * t);
	elseif (Float64(z * t) <= 5e-281)
		tmp = t_1;
	elseif (Float64(z * t) <= 5e-105)
		tmp = fma(i, c, Float64(a * b));
	elseif (Float64(z * t) <= 5e+36)
		tmp = t_1;
	else
		tmp = fma(z, t, Float64(a * b));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(y * x + N[(c * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(z * t), $MachinePrecision], -5e+226], N[(z * t), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 5e-281], t$95$1, If[LessEqual[N[(z * t), $MachinePrecision], 5e-105], N[(i * c + N[(a * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 5e+36], t$95$1, N[(z * t + N[(a * b), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 z t) < -5.0000000000000005e226

   1. Initial program 85.7%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 96.4

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

   5. Simplified 96.4%

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

   if -5.0000000000000005e226 < (*.f64 z t) < 4.9999999999999998e-281 or 4.99999999999999963e-105 < (*.f64 z t) < 4.99999999999999977e36

   1. Initial program 100.0%

      \[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \]
   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. lower-*.f64 70.4

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

   5. Simplified 70.4%

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

      1. *-commutative N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 70.4

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

   7. Applied egg-rr 70.4%

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

   if 4.9999999999999998e-281 < (*.f64 z t) < 4.99999999999999963e-105

   1. Initial program 96.7%

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

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 81.4

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

   5. Simplified 81.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 84.6

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

   7. Applied egg-rr 84.6%

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

   if 4.99999999999999977e36 < (*.f64 z t)

   1. Initial program 89.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-+l+ N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-fma.f64 95.8

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

   4. Applied egg-rr 95.8%

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

   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. lower-*.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 4 regimes into one program.

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+226}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{-281}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c \cdot i\right)\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{-105}:\\ \;\;\;\;\mathsf{fma}\left(i, c, a \cdot b\right)\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+36}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t, a \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 62.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, x, c \cdot i\right)\\ \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+226}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{-281}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{-105}:\\ \;\;\;\;\mathsf{fma}\left(i, c, a \cdot b\right)\\ \mathbf{elif}\;z \cdot t \leq 4 \cdot 10^{+231}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;z \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma y x (* c i))))
   (if (<= (* z t) -5e+226)
     (* z t)
     (if (<= (* z t) 5e-281)
       t_1
       (if (<= (* z t) 5e-105)
         (fma i c (* a b))
         (if (<= (* z t) 4e+231) t_1 (* z t)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma(y, x, (c * i));
	double tmp;
	if ((z * t) <= -5e+226) {
		tmp = z * t;
	} else if ((z * t) <= 5e-281) {
		tmp = t_1;
	} else if ((z * t) <= 5e-105) {
		tmp = fma(i, c, (a * b));
	} else if ((z * t) <= 4e+231) {
		tmp = t_1;
	} else {
		tmp = z * t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(y, x, Float64(c * i))
	tmp = 0.0
	if (Float64(z * t) <= -5e+226)
		tmp = Float64(z * t);
	elseif (Float64(z * t) <= 5e-281)
		tmp = t_1;
	elseif (Float64(z * t) <= 5e-105)
		tmp = fma(i, c, Float64(a * b));
	elseif (Float64(z * t) <= 4e+231)
		tmp = t_1;
	else
		tmp = Float64(z * t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(y * x + N[(c * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(z * t), $MachinePrecision], -5e+226], N[(z * t), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 5e-281], t$95$1, If[LessEqual[N[(z * t), $MachinePrecision], 5e-105], N[(i * c + N[(a * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 4e+231], t$95$1, N[(z * t), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -5.0000000000000005e226 or 4.0000000000000002e231 < (*.f64 z t)

   1. Initial program 85.7%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 95.2

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

   5. Simplified 95.2%

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

   if -5.0000000000000005e226 < (*.f64 z t) < 4.9999999999999998e-281 or 4.99999999999999963e-105 < (*.f64 z t) < 4.0000000000000002e231

   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 x around inf

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

      1. lower-*.f64 66.7

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

   5. Simplified 66.7%

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

      1. *-commutative N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 66.7

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

   7. Applied egg-rr 66.7%

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

   if 4.9999999999999998e-281 < (*.f64 z t) < 4.99999999999999963e-105

   1. Initial program 96.7%

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

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 81.4

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

   5. Simplified 81.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 84.6

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

   7. Applied egg-rr 84.6%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+226}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{-281}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c \cdot i\right)\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{-105}:\\ \;\;\;\;\mathsf{fma}\left(i, c, a \cdot b\right)\\ \mathbf{elif}\;z \cdot t \leq 4 \cdot 10^{+231}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 60.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+226}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{-72}:\\ \;\;\;\;\mathsf{fma}\left(i, c, a \cdot b\right)\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+36}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;z \cdot t \leq 4 \cdot 10^{+231}:\\ \;\;\;\;\mathsf{fma}\left(b, a, c \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* z t) -5e+226)
   (* z t)
   (if (<= (* z t) 2e-72)
     (fma i c (* a b))
     (if (<= (* z t) 5e+36)
       (* x y)
       (if (<= (* z t) 4e+231) (fma b a (* 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 ((z * t) <= -5e+226) {
		tmp = z * t;
	} else if ((z * t) <= 2e-72) {
		tmp = fma(i, c, (a * b));
	} else if ((z * t) <= 5e+36) {
		tmp = x * y;
	} else if ((z * t) <= 4e+231) {
		tmp = fma(b, a, (c * i));
	} else {
		tmp = z * t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(z * t) <= -5e+226)
		tmp = Float64(z * t);
	elseif (Float64(z * t) <= 2e-72)
		tmp = fma(i, c, Float64(a * b));
	elseif (Float64(z * t) <= 5e+36)
		tmp = Float64(x * y);
	elseif (Float64(z * t) <= 4e+231)
		tmp = fma(b, a, Float64(c * i));
	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], -5e+226], N[(z * t), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 2e-72], N[(i * c + N[(a * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 5e+36], N[(x * y), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 4e+231], N[(b * a + N[(c * i), $MachinePrecision]), $MachinePrecision], N[(z * t), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 z t) < -5.0000000000000005e226 or 4.0000000000000002e231 < (*.f64 z t)

   1. Initial program 85.7%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 95.2

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

   5. Simplified 95.2%

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

   if -5.0000000000000005e226 < (*.f64 z t) < 1.9999999999999999e-72

   1. Initial program 99.3%

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

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 63.8

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

   5. Simplified 63.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 64.4

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

   7. Applied egg-rr 64.4%

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

   if 1.9999999999999999e-72 < (*.f64 z t) < 4.99999999999999977e36

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 66.2

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

   5. Simplified 66.2%

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

   if 4.99999999999999977e36 < (*.f64 z t) < 4.0000000000000002e231

   1. Initial program 92.5%

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

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 46.1

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

   5. Simplified 46.1%

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

      1. *-commutative N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 49.8

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

   7. Applied egg-rr 49.8%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+226}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{-72}:\\ \;\;\;\;\mathsf{fma}\left(i, c, a \cdot b\right)\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+36}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;z \cdot t \leq 4 \cdot 10^{+231}:\\ \;\;\;\;\mathsf{fma}\left(b, a, c \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 60.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(b, a, c \cdot i\right)\\ \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+226}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{-72}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+36}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;z \cdot t \leq 4 \cdot 10^{+231}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;z \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma b a (* c i))))
   (if (<= (* z t) -5e+226)
     (* z t)
     (if (<= (* z t) 2e-72)
       t_1
       (if (<= (* z t) 5e+36) (* x y) (if (<= (* z t) 4e+231) t_1 (* z t)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma(b, a, (c * i));
	double tmp;
	if ((z * t) <= -5e+226) {
		tmp = z * t;
	} else if ((z * t) <= 2e-72) {
		tmp = t_1;
	} else if ((z * t) <= 5e+36) {
		tmp = x * y;
	} else if ((z * t) <= 4e+231) {
		tmp = t_1;
	} else {
		tmp = z * t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(b, a, Float64(c * i))
	tmp = 0.0
	if (Float64(z * t) <= -5e+226)
		tmp = Float64(z * t);
	elseif (Float64(z * t) <= 2e-72)
		tmp = t_1;
	elseif (Float64(z * t) <= 5e+36)
		tmp = Float64(x * y);
	elseif (Float64(z * t) <= 4e+231)
		tmp = t_1;
	else
		tmp = Float64(z * t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(b * a + N[(c * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(z * t), $MachinePrecision], -5e+226], N[(z * t), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 2e-72], t$95$1, If[LessEqual[N[(z * t), $MachinePrecision], 5e+36], N[(x * y), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 4e+231], t$95$1, N[(z * t), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -5.0000000000000005e226 or 4.0000000000000002e231 < (*.f64 z t)

   1. Initial program 85.7%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 95.2

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

   5. Simplified 95.2%

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

   if -5.0000000000000005e226 < (*.f64 z t) < 1.9999999999999999e-72 or 4.99999999999999977e36 < (*.f64 z t) < 4.0000000000000002e231

   1. Initial program 98.4%

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

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 61.2

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

   5. Simplified 61.2%

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

      1. *-commutative N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 61.8

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

   7. Applied egg-rr 61.8%

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

   if 1.9999999999999999e-72 < (*.f64 z t) < 4.99999999999999977e36

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 66.2

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

   5. Simplified 66.2%

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

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+226}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{-72}:\\ \;\;\;\;\mathsf{fma}\left(b, a, c \cdot i\right)\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+36}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;z \cdot t \leq 4 \cdot 10^{+231}:\\ \;\;\;\;\mathsf{fma}\left(b, a, c \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z, t, x \cdot y\right)\\ t_2 := x \cdot y + z \cdot t\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+120}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{+68}:\\ \;\;\;\;\mathsf{fma}\left(i, c, a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma z t (* x y))) (t_2 (+ (* x y) (* z t))))
   (if (<= t_2 -5e+120) t_1 (if (<= t_2 1e+68) (fma i c (* 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(z, t, (x * y));
	double t_2 = (x * y) + (z * t);
	double tmp;
	if (t_2 <= -5e+120) {
		tmp = t_1;
	} else if (t_2 <= 1e+68) {
		tmp = fma(i, c, (a * b));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(z, t, Float64(x * y))
	t_2 = Float64(Float64(x * y) + Float64(z * t))
	tmp = 0.0
	if (t_2 <= -5e+120)
		tmp = t_1;
	elseif (t_2 <= 1e+68)
		tmp = fma(i, c, 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[(z * t + N[(x * y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * y), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+120], t$95$1, If[LessEqual[t$95$2, 1e+68], N[(i * c + N[(a * b), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 x y) (*.f64 z t)) < -5.00000000000000019e120 or 9.99999999999999953e67 < (+.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 93.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-+l+ N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-fma.f64 95.8

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

   4. Applied egg-rr 95.8%

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

   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. lower-*.f64 82.0

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

   7. Simplified 82.0%

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

   if -5.00000000000000019e120 < (+.f64 (*.f64 x y) (*.f64 z t)) < 9.99999999999999953e67

   1. Initial program 100.0%

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

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 77.6

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

   5. Simplified 77.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 77.7

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

   7. Applied egg-rr 77.7%

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

Alternative 7: 87.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(a, b, z \cdot t\right)\\ \mathbf{if}\;c \cdot i \leq -5 \cdot 10^{-108}:\\ \;\;\;\;\mathsf{fma}\left(a, b, \mathsf{fma}\left(c, i, x \cdot y\right)\right)\\ \mathbf{elif}\;c \cdot i \leq 10000000000:\\ \;\;\;\;\mathsf{fma}\left(x, y, t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(c, i, t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma a b (* z t))))
   (if (<= (* c i) -5e-108)
     (fma a b (fma c i (* x y)))
     (if (<= (* c i) 10000000000.0) (fma x y t_1) (fma c i 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(a, b, (z * t));
	double tmp;
	if ((c * i) <= -5e-108) {
		tmp = fma(a, b, fma(c, i, (x * y)));
	} else if ((c * i) <= 10000000000.0) {
		tmp = fma(x, y, t_1);
	} else {
		tmp = fma(c, i, t_1);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(a, b, Float64(z * t))
	tmp = 0.0
	if (Float64(c * i) <= -5e-108)
		tmp = fma(a, b, fma(c, i, Float64(x * y)));
	elseif (Float64(c * i) <= 10000000000.0)
		tmp = fma(x, y, t_1);
	else
		tmp = fma(c, i, t_1);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(a * b + N[(z * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(c * i), $MachinePrecision], -5e-108], N[(a * b + N[(c * i + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(c * i), $MachinePrecision], 10000000000.0], N[(x * y + t$95$1), $MachinePrecision], N[(c * i + t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 c i) < -5e-108

   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 0

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

      1. lower-fma.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 89.5

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

   5. Simplified 89.5%

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

   if -5e-108 < (*.f64 c i) < 1e10

   1. Initial program 95.7%

      \[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \]
   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. lower-fma.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 97.3

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

   5. Simplified 97.3%

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

   if 1e10 < (*.f64 c i)

   1. Initial program 96.3%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 87.1

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

   5. Simplified 87.1%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \cdot i \leq -5 \cdot 10^{-108}:\\ \;\;\;\;\mathsf{fma}\left(a, b, \mathsf{fma}\left(c, i, x \cdot y\right)\right)\\ \mathbf{elif}\;c \cdot i \leq 10000000000:\\ \;\;\;\;\mathsf{fma}\left(x, y, \mathsf{fma}\left(a, b, z \cdot t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(c, i, \mathsf{fma}\left(a, b, z \cdot t\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 87.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+226}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;z \cdot t \leq 6 \cdot 10^{+68}:\\ \;\;\;\;\mathsf{fma}\left(a, b, \mathsf{fma}\left(c, i, x \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(c, i, \mathsf{fma}\left(a, b, z \cdot t\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* z t) -5e+226)
   (* z t)
   (if (<= (* z t) 6e+68)
     (fma a b (fma c i (* x y)))
     (fma c i (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 ((z * t) <= -5e+226) {
		tmp = z * t;
	} else if ((z * t) <= 6e+68) {
		tmp = fma(a, b, fma(c, i, (x * y)));
	} else {
		tmp = fma(c, i, fma(a, b, (z * t)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(z * t) <= -5e+226)
		tmp = Float64(z * t);
	elseif (Float64(z * t) <= 6e+68)
		tmp = fma(a, b, fma(c, i, Float64(x * y)));
	else
		tmp = fma(c, i, fma(a, b, Float64(z * t)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(z * t), $MachinePrecision], -5e+226], N[(z * t), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 6e+68], N[(a * b + N[(c * i + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c * i + N[(a * b + N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -5.0000000000000005e226

   1. Initial program 85.7%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 96.4

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

   5. Simplified 96.4%

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

   if -5.0000000000000005e226 < (*.f64 z t) < 6.0000000000000004e68

   1. Initial program 99.4%

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

   3. Taylor expanded in z around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 92.5

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

   5. Simplified 92.5%

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

   if 6.0000000000000004e68 < (*.f64 z t)

   1. Initial program 88.1%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 81.5

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

   5. Simplified 81.5%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+226}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;z \cdot t \leq 6 \cdot 10^{+68}:\\ \;\;\;\;\mathsf{fma}\left(a, b, \mathsf{fma}\left(c, i, x \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(c, i, \mathsf{fma}\left(a, b, z \cdot t\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 85.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+226}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;z \cdot t \leq 4 \cdot 10^{+231}:\\ \;\;\;\;\mathsf{fma}\left(a, b, \mathsf{fma}\left(c, i, x \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t, x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* z t) -5e+226)
   (* z t)
   (if (<= (* z t) 4e+231) (fma a b (fma c i (* x y))) (fma z t (* x y)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((z * t) <= -5e+226) {
		tmp = z * t;
	} else if ((z * t) <= 4e+231) {
		tmp = fma(a, b, fma(c, i, (x * y)));
	} else {
		tmp = fma(z, t, (x * y));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(z * t) <= -5e+226)
		tmp = Float64(z * t);
	elseif (Float64(z * t) <= 4e+231)
		tmp = fma(a, b, fma(c, i, Float64(x * y)));
	else
		tmp = fma(z, t, Float64(x * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(z * t), $MachinePrecision], -5e+226], N[(z * t), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 4e+231], N[(a * b + N[(c * i + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z * t + N[(x * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -5.0000000000000005e226

   1. Initial program 85.7%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 96.4

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

   5. Simplified 96.4%

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

   if -5.0000000000000005e226 < (*.f64 z t) < 4.0000000000000002e231

   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 z around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 89.6

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

   5. Simplified 89.6%

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

   if 4.0000000000000002e231 < (*.f64 z t)

   1. Initial program 85.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-+l+ N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-fma.f64 90.5

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

   4. Applied egg-rr 90.5%

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

   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. lower-*.f64 98.2

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

   7. Simplified 98.2%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+226}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;z \cdot t \leq 4 \cdot 10^{+231}:\\ \;\;\;\;\mathsf{fma}\left(a, b, \mathsf{fma}\left(c, i, x \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t, x \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 43.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \cdot i \leq -5.7 \cdot 10^{+106}:\\ \;\;\;\;c \cdot i\\ \mathbf{elif}\;c \cdot i \leq -3 \cdot 10^{-131}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;c \cdot i \leq 2.9 \cdot 10^{+36}:\\ \;\;\;\;z \cdot t\\ \mathbf{else}:\\ \;\;\;\;c \cdot i\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* c i) -5.7e+106)
   (* c i)
   (if (<= (* c i) -3e-131)
     (* a b)
     (if (<= (* c i) 2.9e+36) (* 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) <= -5.7e+106) {
		tmp = c * i;
	} else if ((c * i) <= -3e-131) {
		tmp = a * b;
	} else if ((c * i) <= 2.9e+36) {
		tmp = z * t;
	} 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) <= (-5.7d+106)) then
        tmp = c * i
    else if ((c * i) <= (-3d-131)) then
        tmp = a * b
    else if ((c * i) <= 2.9d+36) then
        tmp = z * t
    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) <= -5.7e+106) {
		tmp = c * i;
	} else if ((c * i) <= -3e-131) {
		tmp = a * b;
	} else if ((c * i) <= 2.9e+36) {
		tmp = z * t;
	} else {
		tmp = c * i;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if (c * i) <= -5.7e+106:
		tmp = c * i
	elif (c * i) <= -3e-131:
		tmp = a * b
	elif (c * i) <= 2.9e+36:
		tmp = z * t
	else:
		tmp = c * i
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(c * i) <= -5.7e+106)
		tmp = Float64(c * i);
	elseif (Float64(c * i) <= -3e-131)
		tmp = Float64(a * b);
	elseif (Float64(c * i) <= 2.9e+36)
		tmp = Float64(z * t);
	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) <= -5.7e+106)
		tmp = c * i;
	elseif ((c * i) <= -3e-131)
		tmp = a * b;
	elseif ((c * i) <= 2.9e+36)
		tmp = z * t;
	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], -5.7e+106], N[(c * i), $MachinePrecision], If[LessEqual[N[(c * i), $MachinePrecision], -3e-131], N[(a * b), $MachinePrecision], If[LessEqual[N[(c * i), $MachinePrecision], 2.9e+36], N[(z * t), $MachinePrecision], N[(c * i), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 c i) < -5.6999999999999997e106 or 2.9e36 < (*.f64 c i)

   1. Initial program 95.8%

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

   3. Taylor expanded in c around inf

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

      1. lower-*.f64 55.1

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

   5. Simplified 55.1%

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

   if -5.6999999999999997e106 < (*.f64 c i) < -2.99999999999999996e-131

   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. Taylor expanded in a around inf

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

      1. lower-*.f64 37.8

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

   5. Simplified 37.8%

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

   if -2.99999999999999996e-131 < (*.f64 c i) < 2.9e36

   1. Initial program 95.6%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 44.4

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

   5. Simplified 44.4%

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

4. Final simplification 47.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \cdot i \leq -5.7 \cdot 10^{+106}:\\ \;\;\;\;c \cdot i\\ \mathbf{elif}\;c \cdot i \leq -3 \cdot 10^{-131}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;c \cdot i \leq 2.9 \cdot 10^{+36}:\\ \;\;\;\;z \cdot t\\ \mathbf{else}:\\ \;\;\;\;c \cdot i\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 41.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+207}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;z \cdot t \leq 4 \cdot 10^{+231}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* z t) -2e+207) (* z t) (if (<= (* z t) 4e+231) (* 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) <= -2e+207) {
		tmp = z * t;
	} else if ((z * t) <= 4e+231) {
		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) <= (-2d+207)) then
        tmp = z * t
    else if ((z * t) <= 4d+231) 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) <= -2e+207) {
		tmp = z * t;
	} else if ((z * t) <= 4e+231) {
		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) <= -2e+207:
		tmp = z * t
	elif (z * t) <= 4e+231:
		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) <= -2e+207)
		tmp = Float64(z * t);
	elseif (Float64(z * t) <= 4e+231)
		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) <= -2e+207)
		tmp = z * t;
	elseif ((z * t) <= 4e+231)
		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], -2e+207], N[(z * t), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 4e+231], N[(x * y), $MachinePrecision], N[(z * t), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -2.0000000000000001e207 or 4.0000000000000002e231 < (*.f64 z t)

   1. Initial program 86.0%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 93.4

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

   5. Simplified 93.4%

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

   if -2.0000000000000001e207 < (*.f64 z t) < 4.0000000000000002e231

   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 x around inf

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

      1. lower-*.f64 38.5

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

   5. Simplified 38.5%

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

4. Final simplification 49.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+207}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;z \cdot t \leq 4 \cdot 10^{+231}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 40.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot b \leq -1.5 \cdot 10^{+48}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \cdot b \leq 3.8 \cdot 10^{+280}:\\ \;\;\;\;c \cdot i\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* a b) -1.5e+48) (* a b) (if (<= (* a b) 3.8e+280) (* 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) <= -1.5e+48) {
		tmp = a * b;
	} else if ((a * b) <= 3.8e+280) {
		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) <= (-1.5d+48)) then
        tmp = a * b
    else if ((a * b) <= 3.8d+280) 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) <= -1.5e+48) {
		tmp = a * b;
	} else if ((a * b) <= 3.8e+280) {
		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) <= -1.5e+48:
		tmp = a * b
	elif (a * b) <= 3.8e+280:
		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) <= -1.5e+48)
		tmp = Float64(a * b);
	elseif (Float64(a * b) <= 3.8e+280)
		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) <= -1.5e+48)
		tmp = a * b;
	elseif ((a * b) <= 3.8e+280)
		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], -1.5e+48], N[(a * b), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], 3.8e+280], N[(c * i), $MachinePrecision], N[(a * b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a b) < -1.5e48 or 3.79999999999999964e280 < (*.f64 a b)

   1. Initial program 91.4%

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

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 61.1

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

   5. Simplified 61.1%

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

   if -1.5e48 < (*.f64 a b) < 3.79999999999999964e280

   1. Initial program 98.2%

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

   3. Taylor expanded in c around inf

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

      1. lower-*.f64 32.1

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

   5. Simplified 32.1%

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

Alternative 13: 27.4% 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 96.1%

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

3. Taylor expanded in a around inf

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

   1. lower-*.f64 24.4

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

5. Simplified 24.4%

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

Reproduce

herbie shell --seed 2024208 
(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)))