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

Percentage Accurate: 95.9% → 98.0%

Time: 8.2s

Alternatives: 14

Speedup: 1.3×

• 41.3% 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: 95.9% 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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

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. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. associate-+l+ N/A

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

   4. lift-+.f64 N/A

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

   5. +-commutative N/A

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

   6. associate-+l+ N/A

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

   7. lift-*.f64 N/A

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

   8. lower-fma.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-fma.f64 N/A

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

   12. +-commutative N/A

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-fma.f64 99.6

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

   16. lift-*.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 99.6

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

4. Applied rewrites 99.6%

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

Alternative 2: 74.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(t, z, x \cdot y\right)\\ t_2 := x \cdot y + z \cdot t\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+87}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+35}:\\ \;\;\;\;\mathsf{fma}\left(b, a, c \cdot i\right)\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+191}:\\ \;\;\;\;\mathsf{fma}\left(y, x, a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma t z (* x y))) (t_2 (+ (* x y) (* z t))))
   (if (<= t_2 -1e+87)
     t_1
     (if (<= t_2 2e+35)
       (fma b a (* c i))
       (if (<= t_2 5e+191) (fma y x (* 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(t, z, (x * y));
	double t_2 = (x * y) + (z * t);
	double tmp;
	if (t_2 <= -1e+87) {
		tmp = t_1;
	} else if (t_2 <= 2e+35) {
		tmp = fma(b, a, (c * i));
	} else if (t_2 <= 5e+191) {
		tmp = fma(y, x, (a * b));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(t, z, Float64(x * y))
	t_2 = Float64(Float64(x * y) + Float64(z * t))
	tmp = 0.0
	if (t_2 <= -1e+87)
		tmp = t_1;
	elseif (t_2 <= 2e+35)
		tmp = fma(b, a, Float64(c * i));
	elseif (t_2 <= 5e+191)
		tmp = fma(y, x, 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[(t * z + N[(x * y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * y), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+87], t$95$1, If[LessEqual[t$95$2, 2e+35], N[(b * a + N[(c * i), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e+191], N[(y * x + N[(a * b), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (*.f64 x y) (*.f64 z t)) < -9.9999999999999996e86 or 5.0000000000000002e191 < (+.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 91.1%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 85.8

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

   5. Applied rewrites 85.8%

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

   7. Applied rewrites 87.5%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 58.5%

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

   11. Taylor expanded in c around 0

       \[\leadsto t \cdot z + \color{blue}{x \cdot y} \]
   12. Step-by-step derivation

   13. Applied rewrites 77.6%

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

   if -9.9999999999999996e86 < (+.f64 (*.f64 x y) (*.f64 z t)) < 1.9999999999999999e35

   1. Initial program 98.9%

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

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 89.0

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

   5. Applied rewrites 89.0%

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

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(b, a, c \cdot i\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 83.1%

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

   if 1.9999999999999999e35 < (+.f64 (*.f64 x y) (*.f64 z t)) < 5.0000000000000002e191

   1. Initial program 99.9%

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

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 82.9

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

   5. Applied rewrites 82.9%

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

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(b, a, c \cdot i\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 48.9%

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

   9. Taylor expanded in c around 0

      \[\leadsto a \cdot b + \color{blue}{x \cdot y} \]
   10. Step-by-step derivation

   11. Applied rewrites 71.1%

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

   13. Applied rewrites 71.2%

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

Alternative 3: 74.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(t, z, x \cdot y\right)\\ t_2 := x \cdot y + z \cdot t\\ \mathbf{if}\;t\_2 \leq -4 \cdot 10^{+87}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+35}:\\ \;\;\;\;\mathsf{fma}\left(i, c, b \cdot a\right)\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+191}:\\ \;\;\;\;\mathsf{fma}\left(y, x, a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma t z (* x y))) (t_2 (+ (* x y) (* z t))))
   (if (<= t_2 -4e+87)
     t_1
     (if (<= t_2 2e+35)
       (fma i c (* b a))
       (if (<= t_2 5e+191) (fma y x (* 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(t, z, (x * y));
	double t_2 = (x * y) + (z * t);
	double tmp;
	if (t_2 <= -4e+87) {
		tmp = t_1;
	} else if (t_2 <= 2e+35) {
		tmp = fma(i, c, (b * a));
	} else if (t_2 <= 5e+191) {
		tmp = fma(y, x, (a * b));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(t, z, Float64(x * y))
	t_2 = Float64(Float64(x * y) + Float64(z * t))
	tmp = 0.0
	if (t_2 <= -4e+87)
		tmp = t_1;
	elseif (t_2 <= 2e+35)
		tmp = fma(i, c, Float64(b * a));
	elseif (t_2 <= 5e+191)
		tmp = fma(y, x, 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[(t * z + N[(x * y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * y), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -4e+87], t$95$1, If[LessEqual[t$95$2, 2e+35], N[(i * c + N[(b * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e+191], N[(y * x + N[(a * b), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (*.f64 x y) (*.f64 z t)) < -3.9999999999999998e87 or 5.0000000000000002e191 < (+.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 91.9%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 86.5

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

   5. Applied rewrites 86.5%

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

   7. Applied rewrites 88.2%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 59.0%

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

   11. Taylor expanded in c around 0

       \[\leadsto t \cdot z + \color{blue}{x \cdot y} \]
   12. Step-by-step derivation

   13. Applied rewrites 78.2%

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

   if -3.9999999999999998e87 < (+.f64 (*.f64 x y) (*.f64 z t)) < 1.9999999999999999e35

   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. 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. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 88.1

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

   5. Applied rewrites 88.1%

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

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(b, a, c \cdot i\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 82.3%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 82.3%

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

   if 1.9999999999999999e35 < (+.f64 (*.f64 x y) (*.f64 z t)) < 5.0000000000000002e191

   1. Initial program 99.9%

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

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 82.9

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

   5. Applied rewrites 82.9%

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

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(b, a, c \cdot i\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 48.9%

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

   9. Taylor expanded in c around 0

      \[\leadsto a \cdot b + \color{blue}{x \cdot y} \]
   10. Step-by-step derivation

   11. Applied rewrites 71.1%

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

   13. Applied rewrites 71.2%

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

Alternative 4: 42.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+127}:\\ \;\;\;\;t \cdot z\\ \mathbf{elif}\;z \cdot t \leq -2 \cdot 10^{-91}:\\ \;\;\;\;i \cdot c\\ \mathbf{elif}\;z \cdot t \leq -5 \cdot 10^{-321}:\\ \;\;\;\;b \cdot a\\ \mathbf{elif}\;z \cdot t \leq 10^{+24}:\\ \;\;\;\;i \cdot c\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+193}:\\ \;\;\;\;b \cdot a\\ \mathbf{else}:\\ \;\;\;\;t \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* z t) -5e+127)
   (* t z)
   (if (<= (* z t) -2e-91)
     (* i c)
     (if (<= (* z t) -5e-321)
       (* b a)
       (if (<= (* z t) 1e+24)
         (* i c)
         (if (<= (* z t) 2e+193) (* b a) (* t z)))))))

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+127) {
		tmp = t * z;
	} else if ((z * t) <= -2e-91) {
		tmp = i * c;
	} else if ((z * t) <= -5e-321) {
		tmp = b * a;
	} else if ((z * t) <= 1e+24) {
		tmp = i * c;
	} else if ((z * t) <= 2e+193) {
		tmp = b * a;
	} else {
		tmp = t * z;
	}
	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) <= (-5d+127)) then
        tmp = t * z
    else if ((z * t) <= (-2d-91)) then
        tmp = i * c
    else if ((z * t) <= (-5d-321)) then
        tmp = b * a
    else if ((z * t) <= 1d+24) then
        tmp = i * c
    else if ((z * t) <= 2d+193) then
        tmp = b * a
    else
        tmp = t * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((z * t) <= -5e+127) {
		tmp = t * z;
	} else if ((z * t) <= -2e-91) {
		tmp = i * c;
	} else if ((z * t) <= -5e-321) {
		tmp = b * a;
	} else if ((z * t) <= 1e+24) {
		tmp = i * c;
	} else if ((z * t) <= 2e+193) {
		tmp = b * a;
	} else {
		tmp = t * z;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if (z * t) <= -5e+127:
		tmp = t * z
	elif (z * t) <= -2e-91:
		tmp = i * c
	elif (z * t) <= -5e-321:
		tmp = b * a
	elif (z * t) <= 1e+24:
		tmp = i * c
	elif (z * t) <= 2e+193:
		tmp = b * a
	else:
		tmp = t * z
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(z * t) <= -5e+127)
		tmp = Float64(t * z);
	elseif (Float64(z * t) <= -2e-91)
		tmp = Float64(i * c);
	elseif (Float64(z * t) <= -5e-321)
		tmp = Float64(b * a);
	elseif (Float64(z * t) <= 1e+24)
		tmp = Float64(i * c);
	elseif (Float64(z * t) <= 2e+193)
		tmp = Float64(b * a);
	else
		tmp = Float64(t * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	tmp = 0.0;
	if ((z * t) <= -5e+127)
		tmp = t * z;
	elseif ((z * t) <= -2e-91)
		tmp = i * c;
	elseif ((z * t) <= -5e-321)
		tmp = b * a;
	elseif ((z * t) <= 1e+24)
		tmp = i * c;
	elseif ((z * t) <= 2e+193)
		tmp = b * a;
	else
		tmp = t * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(z * t), $MachinePrecision], -5e+127], N[(t * z), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], -2e-91], N[(i * c), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], -5e-321], N[(b * a), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 1e+24], N[(i * c), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 2e+193], N[(b * a), $MachinePrecision], N[(t * z), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -5.0000000000000004e127 or 2.00000000000000013e193 < (*.f64 z t)

   1. Initial program 93.8%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 35.1

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

   5. Applied rewrites 35.1%

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

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(b, a, c \cdot i\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 24.2%

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

   9. Taylor expanded in i around inf

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

   11. Applied rewrites 36.6%

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

   12. Taylor expanded in z around inf

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

       1. lower-*.f64 69.0

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

   14. Applied rewrites 69.0%

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

   if -5.0000000000000004e127 < (*.f64 z t) < -2.00000000000000004e-91 or -4.99994e-321 < (*.f64 z t) < 9.9999999999999998e23

   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 c around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 43.2

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

   5. Applied rewrites 43.2%

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

   if -2.00000000000000004e-91 < (*.f64 z t) < -4.99994e-321 or 9.9999999999999998e23 < (*.f64 z t) < 2.00000000000000013e193

   1. Initial program 94.6%

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

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 68.9

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

   5. Applied rewrites 68.9%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 60.9%

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

   9. Taylor expanded in z around 0

      \[\leadsto a \cdot b \]
   10. Step-by-step derivation

   11. Applied rewrites 50.3%

       \[\leadsto b \cdot a \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 66.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(a, b, t \cdot z\right)\\ t_2 := \mathsf{fma}\left(i, c, b \cdot a\right)\\ \mathbf{if}\;c \cdot i \leq -2 \cdot 10^{+162}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;c \cdot i \leq -1 \cdot 10^{-308}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;c \cdot i \leq 2 \cdot 10^{-172}:\\ \;\;\;\;\mathsf{fma}\left(b, a, y \cdot x\right)\\ \mathbf{elif}\;c \cdot i \leq 2 \cdot 10^{-7}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma a b (* t z))) (t_2 (fma i c (* b a))))
   (if (<= (* c i) -2e+162)
     t_2
     (if (<= (* c i) -1e-308)
       t_1
       (if (<= (* c i) 2e-172)
         (fma b a (* y x))
         (if (<= (* c i) 2e-7) t_1 t_2))))))

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, (t * z));
	double t_2 = fma(i, c, (b * a));
	double tmp;
	if ((c * i) <= -2e+162) {
		tmp = t_2;
	} else if ((c * i) <= -1e-308) {
		tmp = t_1;
	} else if ((c * i) <= 2e-172) {
		tmp = fma(b, a, (y * x));
	} else if ((c * i) <= 2e-7) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(a, b, Float64(t * z))
	t_2 = fma(i, c, Float64(b * a))
	tmp = 0.0
	if (Float64(c * i) <= -2e+162)
		tmp = t_2;
	elseif (Float64(c * i) <= -1e-308)
		tmp = t_1;
	elseif (Float64(c * i) <= 2e-172)
		tmp = fma(b, a, Float64(y * x));
	elseif (Float64(c * i) <= 2e-7)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(a * b + N[(t * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(i * c + N[(b * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(c * i), $MachinePrecision], -2e+162], t$95$2, If[LessEqual[N[(c * i), $MachinePrecision], -1e-308], t$95$1, If[LessEqual[N[(c * i), $MachinePrecision], 2e-172], N[(b * a + N[(y * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(c * i), $MachinePrecision], 2e-7], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 c i) < -1.9999999999999999e162 or 1.9999999999999999e-7 < (*.f64 c i)

   1. Initial program 88.3%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 84.5

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

   5. Applied rewrites 84.5%

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

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(b, a, c \cdot i\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 78.0%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 82.3%

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

   if -1.9999999999999999e162 < (*.f64 c i) < -9.9999999999999991e-309 or 2.0000000000000001e-172 < (*.f64 c i) < 1.9999999999999999e-7

   1. Initial program 99.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 78.0

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

   5. Applied rewrites 78.0%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 71.3%

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

   if -9.9999999999999991e-309 < (*.f64 c i) < 2.0000000000000001e-172

   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 z around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 72.0

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

   5. Applied rewrites 72.0%

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

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(b, a, c \cdot i\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 24.3%

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

   9. Taylor expanded in c around 0

      \[\leadsto a \cdot b + \color{blue}{x \cdot y} \]
   10. Step-by-step derivation

   11. Applied rewrites 72.0%

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

Alternative 6: 63.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(a, b, t \cdot z\right)\\ \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+21}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \cdot t \leq -1 \cdot 10^{-86}:\\ \;\;\;\;i \cdot c\\ \mathbf{elif}\;z \cdot t \leq 10^{+118}:\\ \;\;\;\;\mathsf{fma}\left(b, a, y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma a b (* t z))))
   (if (<= (* z t) -5e+21)
     t_1
     (if (<= (* z t) -1e-86)
       (* i c)
       (if (<= (* z t) 1e+118) (fma b a (* y x)) 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, (t * z));
	double tmp;
	if ((z * t) <= -5e+21) {
		tmp = t_1;
	} else if ((z * t) <= -1e-86) {
		tmp = i * c;
	} else if ((z * t) <= 1e+118) {
		tmp = fma(b, a, (y * x));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(a, b, Float64(t * z))
	tmp = 0.0
	if (Float64(z * t) <= -5e+21)
		tmp = t_1;
	elseif (Float64(z * t) <= -1e-86)
		tmp = Float64(i * c);
	elseif (Float64(z * t) <= 1e+118)
		tmp = fma(b, a, Float64(y * x));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(a * b + N[(t * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(z * t), $MachinePrecision], -5e+21], t$95$1, If[LessEqual[N[(z * t), $MachinePrecision], -1e-86], N[(i * c), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 1e+118], N[(b * a + N[(y * x), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -5e21 or 9.99999999999999967e117 < (*.f64 z t)

   1. Initial program 93.7%

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

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 82.6

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

   5. Applied rewrites 82.6%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 77.2%

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

   if -5e21 < (*.f64 z t) < -1.00000000000000008e-86

   1. Initial program 86.7%

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

   3. Taylor expanded in c around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 74.5

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

   5. Applied rewrites 74.5%

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

   if -1.00000000000000008e-86 < (*.f64 z t) < 9.99999999999999967e117

   1. Initial program 97.6%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 91.8

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

   5. Applied rewrites 91.8%

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

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(b, a, c \cdot i\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 63.3%

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

   9. Taylor expanded in c around 0

      \[\leadsto a \cdot b + \color{blue}{x \cdot y} \]
   10. Step-by-step derivation

   11. Applied rewrites 62.7%

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

Alternative 7: 89.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot b \leq -2 \cdot 10^{+23} \lor \neg \left(a \cdot b \leq 4 \cdot 10^{-13}\right):\\ \;\;\;\;\mathsf{fma}\left(x, y, \mathsf{fma}\left(c, i, a \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t, \mathsf{fma}\left(c, i, x \cdot y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (or (<= (* a b) -2e+23) (not (<= (* a b) 4e-13)))
   (fma x y (fma c i (* a b)))
   (fma z t (fma c i (* x y)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (((a * b) <= -2e+23) || !((a * b) <= 4e-13)) {
		tmp = fma(x, y, fma(c, i, (a * b)));
	} else {
		tmp = fma(z, t, fma(c, i, (x * y)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if ((Float64(a * b) <= -2e+23) || !(Float64(a * b) <= 4e-13))
		tmp = fma(x, y, fma(c, i, Float64(a * b)));
	else
		tmp = fma(z, t, fma(c, i, Float64(x * y)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[Or[LessEqual[N[(a * b), $MachinePrecision], -2e+23], N[Not[LessEqual[N[(a * b), $MachinePrecision], 4e-13]], $MachinePrecision]], N[(x * y + N[(c * i + N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z * t + N[(c * i + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a b) < -1.9999999999999998e23 or 4.0000000000000001e-13 < (*.f64 a b)

   1. Initial program 91.6%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 82.7

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

   5. Applied rewrites 82.7%

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

   7. Applied rewrites 85.8%

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

   if -1.9999999999999998e23 < (*.f64 a b) < 4.0000000000000001e-13

   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. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 98.2

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

   5. Applied rewrites 98.2%

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

   7. Applied rewrites 98.2%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b \leq -2 \cdot 10^{+23} \lor \neg \left(a \cdot b \leq 4 \cdot 10^{-13}\right):\\ \;\;\;\;\mathsf{fma}\left(x, y, \mathsf{fma}\left(c, i, a \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t, \mathsf{fma}\left(c, i, x \cdot y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 89.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot b \leq -2 \cdot 10^{+23} \lor \neg \left(a \cdot b \leq 4 \cdot 10^{-13}\right):\\ \;\;\;\;\mathsf{fma}\left(x, y, \mathsf{fma}\left(c, i, a \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, c, \mathsf{fma}\left(t, z, y \cdot x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (or (<= (* a b) -2e+23) (not (<= (* a b) 4e-13)))
   (fma x y (fma c i (* a b)))
   (fma i c (fma t z (* y x)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (((a * b) <= -2e+23) || !((a * b) <= 4e-13)) {
		tmp = fma(x, y, fma(c, i, (a * b)));
	} else {
		tmp = fma(i, c, fma(t, z, (y * x)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if ((Float64(a * b) <= -2e+23) || !(Float64(a * b) <= 4e-13))
		tmp = fma(x, y, fma(c, i, Float64(a * b)));
	else
		tmp = fma(i, c, fma(t, z, Float64(y * x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[Or[LessEqual[N[(a * b), $MachinePrecision], -2e+23], N[Not[LessEqual[N[(a * b), $MachinePrecision], 4e-13]], $MachinePrecision]], N[(x * y + N[(c * i + N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(i * c + N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a b) < -1.9999999999999998e23 or 4.0000000000000001e-13 < (*.f64 a b)

   1. Initial program 91.6%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 82.7

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

   5. Applied rewrites 82.7%

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

   7. Applied rewrites 85.8%

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

   if -1.9999999999999998e23 < (*.f64 a b) < 4.0000000000000001e-13

   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. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 98.2

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

   5. Applied rewrites 98.2%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b \leq -2 \cdot 10^{+23} \lor \neg \left(a \cdot b \leq 4 \cdot 10^{-13}\right):\\ \;\;\;\;\mathsf{fma}\left(x, y, \mathsf{fma}\left(c, i, a \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, c, \mathsf{fma}\left(t, z, y \cdot x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 88.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{+186} \lor \neg \left(x \cdot y \leq 10^{+45}\right):\\ \;\;\;\;\mathsf{fma}\left(b, a, \mathsf{fma}\left(i, c, y \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a, \mathsf{fma}\left(i, c, t \cdot z\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (or (<= (* x y) -1e+186) (not (<= (* x y) 1e+45)))
   (fma b a (fma i c (* y x)))
   (fma b a (fma i c (* t z)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (((x * y) <= -1e+186) || !((x * y) <= 1e+45)) {
		tmp = fma(b, a, fma(i, c, (y * x)));
	} else {
		tmp = fma(b, a, fma(i, c, (t * z)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if ((Float64(x * y) <= -1e+186) || !(Float64(x * y) <= 1e+45))
		tmp = fma(b, a, fma(i, c, Float64(y * x)));
	else
		tmp = fma(b, a, fma(i, c, Float64(t * z)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[Or[LessEqual[N[(x * y), $MachinePrecision], -1e+186], N[Not[LessEqual[N[(x * y), $MachinePrecision], 1e+45]], $MachinePrecision]], N[(b * a + N[(i * c + N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(b * a + N[(i * c + N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -9.9999999999999998e185 or 9.9999999999999993e44 < (*.f64 x y)

   1. Initial program 93.2%

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

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 86.7

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

   5. Applied rewrites 86.7%

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

   if -9.9999999999999998e185 < (*.f64 x y) < 9.9999999999999993e44

   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 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

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 89.2

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

   5. Applied rewrites 89.2%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{+186} \lor \neg \left(x \cdot y \leq 10^{+45}\right):\\ \;\;\;\;\mathsf{fma}\left(b, a, \mathsf{fma}\left(i, c, y \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a, \mathsf{fma}\left(i, c, t \cdot z\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 84.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{+87} \lor \neg \left(x \cdot y \leq 2 \cdot 10^{+64}\right):\\ \;\;\;\;\mathsf{fma}\left(t, z, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a, \mathsf{fma}\left(i, c, t \cdot z\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (or (<= (* x y) -1e+87) (not (<= (* x y) 2e+64)))
   (fma t z (* x y))
   (fma b a (fma i c (* t z)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (((x * y) <= -1e+87) || !((x * y) <= 2e+64)) {
		tmp = fma(t, z, (x * y));
	} else {
		tmp = fma(b, a, fma(i, c, (t * z)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if ((Float64(x * y) <= -1e+87) || !(Float64(x * y) <= 2e+64))
		tmp = fma(t, z, Float64(x * y));
	else
		tmp = fma(b, a, fma(i, c, Float64(t * z)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[Or[LessEqual[N[(x * y), $MachinePrecision], -1e+87], N[Not[LessEqual[N[(x * y), $MachinePrecision], 2e+64]], $MachinePrecision]], N[(t * z + N[(x * y), $MachinePrecision]), $MachinePrecision], N[(b * a + N[(i * c + N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -9.9999999999999996e86 or 2.00000000000000004e64 < (*.f64 x y)

   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. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 86.8

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

   5. Applied rewrites 86.8%

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

   7. Applied rewrites 86.8%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 35.7%

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

   11. Taylor expanded in c around 0

       \[\leadsto t \cdot z + \color{blue}{x \cdot y} \]
   12. Step-by-step derivation

   13. Applied rewrites 76.1%

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

   if -9.9999999999999996e86 < (*.f64 x y) < 2.00000000000000004e64

   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 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

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 91.3

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

   5. Applied rewrites 91.3%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{+87} \lor \neg \left(x \cdot y \leq 2 \cdot 10^{+64}\right):\\ \;\;\;\;\mathsf{fma}\left(t, z, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a, \mathsf{fma}\left(i, c, t \cdot z\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 87.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot b \leq -5 \cdot 10^{+191}:\\ \;\;\;\;\mathsf{fma}\left(i, c, b \cdot a\right)\\ \mathbf{elif}\;a \cdot b \leq 4 \cdot 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(i, c, \mathsf{fma}\left(t, z, y \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a, \mathsf{fma}\left(i, c, y \cdot x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* a b) -5e+191)
   (fma i c (* b a))
   (if (<= (* a b) 4e-13)
     (fma i c (fma t z (* y x)))
     (fma b a (fma i c (* y x))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((a * b) <= -5e+191) {
		tmp = fma(i, c, (b * a));
	} else if ((a * b) <= 4e-13) {
		tmp = fma(i, c, fma(t, z, (y * x)));
	} else {
		tmp = fma(b, a, fma(i, c, (y * x)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(a * b) <= -5e+191)
		tmp = fma(i, c, Float64(b * a));
	elseif (Float64(a * b) <= 4e-13)
		tmp = fma(i, c, fma(t, z, Float64(y * x)));
	else
		tmp = fma(b, a, fma(i, c, Float64(y * x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(a * b), $MachinePrecision], -5e+191], N[(i * c + N[(b * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], 4e-13], N[(i * c + N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(b * a + N[(i * c + N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 a b) < -5.0000000000000002e191

   1. Initial program 89.1%

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

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 84.0

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

   5. Applied rewrites 84.0%

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

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(b, a, c \cdot i\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 84.6%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 90.0%

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

   if -5.0000000000000002e191 < (*.f64 a b) < 4.0000000000000001e-13

   1. Initial program 98.6%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 94.6

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

   5. Applied rewrites 94.6%

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

   if 4.0000000000000001e-13 < (*.f64 a b)

   1. Initial program 91.3%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 85.2

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

   5. Applied rewrites 85.2%

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

Alternative 12: 63.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \cdot i \leq -1 \cdot 10^{+286} \lor \neg \left(c \cdot i \leq 10^{+130}\right):\\ \;\;\;\;i \cdot c\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, b, t \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (or (<= (* c i) -1e+286) (not (<= (* c i) 1e+130)))
   (* i c)
   (fma a b (* t z))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (((c * i) <= -1e+286) || !((c * i) <= 1e+130)) {
		tmp = i * c;
	} else {
		tmp = fma(a, b, (t * z));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if ((Float64(c * i) <= -1e+286) || !(Float64(c * i) <= 1e+130))
		tmp = Float64(i * c);
	else
		tmp = fma(a, b, Float64(t * z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[Or[LessEqual[N[(c * i), $MachinePrecision], -1e+286], N[Not[LessEqual[N[(c * i), $MachinePrecision], 1e+130]], $MachinePrecision]], N[(i * c), $MachinePrecision], N[(a * b + N[(t * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 c i) < -1.00000000000000003e286 or 1.0000000000000001e130 < (*.f64 c i)

   1. Initial program 82.0%

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

   3. Taylor expanded in c around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 78.7

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

   5. Applied rewrites 78.7%

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

   if -1.00000000000000003e286 < (*.f64 c i) < 1.0000000000000001e130

   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 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

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 72.9

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

   5. Applied rewrites 72.9%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 63.4%

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

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \cdot i \leq -1 \cdot 10^{+286} \lor \neg \left(c \cdot i \leq 10^{+130}\right):\\ \;\;\;\;i \cdot c\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, b, t \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 43.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot b \leq -5 \cdot 10^{+211} \lor \neg \left(a \cdot b \leq 10^{+65}\right):\\ \;\;\;\;b \cdot a\\ \mathbf{else}:\\ \;\;\;\;i \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (or (<= (* a b) -5e+211) (not (<= (* a b) 1e+65))) (* b a) (* i c)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (((a * b) <= -5e+211) || !((a * b) <= 1e+65)) {
		tmp = b * a;
	} else {
		tmp = i * c;
	}
	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) <= (-5d+211)) .or. (.not. ((a * b) <= 1d+65))) then
        tmp = b * a
    else
        tmp = i * c
    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) <= -5e+211) || !((a * b) <= 1e+65)) {
		tmp = b * a;
	} else {
		tmp = i * c;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if ((a * b) <= -5e+211) or not ((a * b) <= 1e+65):
		tmp = b * a
	else:
		tmp = i * c
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if ((Float64(a * b) <= -5e+211) || !(Float64(a * b) <= 1e+65))
		tmp = Float64(b * a);
	else
		tmp = Float64(i * c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	tmp = 0.0;
	if (((a * b) <= -5e+211) || ~(((a * b) <= 1e+65)))
		tmp = b * a;
	else
		tmp = i * c;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[Or[LessEqual[N[(a * b), $MachinePrecision], -5e+211], N[Not[LessEqual[N[(a * b), $MachinePrecision], 1e+65]], $MachinePrecision]], N[(b * a), $MachinePrecision], N[(i * c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a b) < -4.9999999999999995e211 or 9.9999999999999999e64 < (*.f64 a b)

   1. Initial program 89.9%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 85.4

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

   5. Applied rewrites 85.4%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 80.7%

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

   9. Taylor expanded in z around 0

      \[\leadsto a \cdot b \]
   10. Step-by-step derivation

   11. Applied rewrites 70.5%

       \[\leadsto b \cdot a \]

   if -4.9999999999999995e211 < (*.f64 a b) < 9.9999999999999999e64

   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. *-commutative N/A

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

      2. lower-*.f64 32.6

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

   5. Applied rewrites 32.6%

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

4. Final simplification 45.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b \leq -5 \cdot 10^{+211} \lor \neg \left(a \cdot b \leq 10^{+65}\right):\\ \;\;\;\;b \cdot a\\ \mathbf{else}:\\ \;\;\;\;i \cdot c\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 27.7% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

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 = b * a
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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 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

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

   2. lower-fma.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-*.f64 75.4

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

5. Applied rewrites 75.4%

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

6. Taylor expanded in c around 0

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

8. Applied rewrites 54.1%

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

9. Taylor expanded in z around 0

   \[\leadsto a \cdot b \]
10. Step-by-step derivation

11. Applied rewrites 29.3%

    \[\leadsto b \cdot a \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024320 
(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)))