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

Percentage Accurate: 95.9% → 98.0%

Time: 8.6s

Alternatives: 16

Speedup: 1.3×

• 41.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 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

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

FPCore

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

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, (a * b))));
}

Julia

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

Wolfram

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

Derivation

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. 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 99.2

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

   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 99.2

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

4. Applied rewrites 99.2%

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

5. Final simplification 99.2%

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

Alternative 2: 67.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z, t, a \cdot b\right)\\ \mathbf{if}\;a \cdot b \leq -1 \cdot 10^{+59}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \cdot b \leq 5 \cdot 10^{-111}:\\ \;\;\;\;\mathsf{fma}\left(i, c, x \cdot y\right)\\ \mathbf{elif}\;a \cdot b \leq 5 \cdot 10^{+94}:\\ \;\;\;\;\mathsf{fma}\left(i, c, t \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma z t (* a b))))
   (if (<= (* a b) -1e+59)
     t_1
     (if (<= (* a b) 5e-111)
       (fma i c (* x y))
       (if (<= (* a b) 5e+94) (fma i c (* t z)) 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, (a * b));
	double tmp;
	if ((a * b) <= -1e+59) {
		tmp = t_1;
	} else if ((a * b) <= 5e-111) {
		tmp = fma(i, c, (x * y));
	} else if ((a * b) <= 5e+94) {
		tmp = fma(i, c, (t * z));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(z, t, Float64(a * b))
	tmp = 0.0
	if (Float64(a * b) <= -1e+59)
		tmp = t_1;
	elseif (Float64(a * b) <= 5e-111)
		tmp = fma(i, c, Float64(x * y));
	elseif (Float64(a * b) <= 5e+94)
		tmp = fma(i, c, Float64(t * z));
	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[(a * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(a * b), $MachinePrecision], -1e+59], t$95$1, If[LessEqual[N[(a * b), $MachinePrecision], 5e-111], N[(i * c + N[(x * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], 5e+94], N[(i * c + N[(t * z), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 a b) < -9.99999999999999972e58 or 5.0000000000000001e94 < (*.f64 a b)

   1. Initial program 93.6%

      \[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \]
   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 85.6

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

   5. Applied rewrites 85.6%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 76.8%

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

   10. Applied rewrites 79.4%

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

   if -9.99999999999999972e58 < (*.f64 a b) < 5.0000000000000003e-111

   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 z around inf

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

      1. associate-+r+ N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-inverses N/A

         \[\leadsto \left(\left(t + \frac{a \cdot b}{z}\right) \cdot z + \left(x \cdot y\right) \cdot \color{blue}{1}\right) + c \cdot i \]

      8. *-rgt-identity N/A

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

      9. lower-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 94.8

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

   5. Applied rewrites 94.8%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 71.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 71.3

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

   10. Applied rewrites 71.3%

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

   if 5.0000000000000003e-111 < (*.f64 a b) < 5.0000000000000001e94

   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 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 91.2

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

   5. Applied rewrites 91.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 72.7%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b \leq -1 \cdot 10^{+59}:\\ \;\;\;\;\mathsf{fma}\left(z, t, a \cdot b\right)\\ \mathbf{elif}\;a \cdot b \leq 5 \cdot 10^{-111}:\\ \;\;\;\;\mathsf{fma}\left(i, c, x \cdot y\right)\\ \mathbf{elif}\;a \cdot b \leq 5 \cdot 10^{+94}:\\ \;\;\;\;\mathsf{fma}\left(i, c, t \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t, a \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \cdot z \leq -5 \cdot 10^{+96}:\\ \;\;\;\;\mathsf{fma}\left(z, t, \mathsf{fma}\left(a, b, x \cdot y\right)\right)\\ \mathbf{elif}\;t \cdot z \leq 2 \cdot 10^{+75}:\\ \;\;\;\;\mathsf{fma}\left(x, y, \mathsf{fma}\left(c, i, a \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t, \mathsf{fma}\left(c, i, x \cdot y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* t z) -5e+96)
   (fma z t (fma a b (* x y)))
   (if (<= (* t z) 2e+75)
     (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 ((t * z) <= -5e+96) {
		tmp = fma(z, t, fma(a, b, (x * y)));
	} else if ((t * z) <= 2e+75) {
		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(t * z) <= -5e+96)
		tmp = fma(z, t, fma(a, b, Float64(x * y)));
	elseif (Float64(t * z) <= 2e+75)
		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[LessEqual[N[(t * z), $MachinePrecision], -5e+96], N[(z * t + N[(a * b + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(t * z), $MachinePrecision], 2e+75], 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 3 regimes

2. if (*.f64 z t) < -5.0000000000000004e96

   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. 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 95.2

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

      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 95.2

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

   4. Applied rewrites 95.2%

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

   5. Taylor expanded in c around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 97.5

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

   7. Applied rewrites 97.5%

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

   if -5.0000000000000004e96 < (*.f64 z t) < 1.99999999999999985e75

   1. Initial program 98.7%

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

   3. Taylor expanded in z around 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 93.2

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

   5. Applied rewrites 93.2%

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

   7. Applied rewrites 94.5%

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

   if 1.99999999999999985e75 < (*.f64 z t)

   1. Initial program 95.9%

      \[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \]
   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 89.0

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

   5. Applied rewrites 89.0%

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

   7. Applied rewrites 91.0%

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

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot z \leq -5 \cdot 10^{+96}:\\ \;\;\;\;\mathsf{fma}\left(z, t, \mathsf{fma}\left(a, b, x \cdot y\right)\right)\\ \mathbf{elif}\;t \cdot z \leq 2 \cdot 10^{+75}:\\ \;\;\;\;\mathsf{fma}\left(x, y, \mathsf{fma}\left(c, i, a \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t, \mathsf{fma}\left(c, i, x \cdot y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \cdot z \leq -1000000000:\\ \;\;\;\;\mathsf{fma}\left(i, c, \mathsf{fma}\left(t, z, x \cdot y\right)\right)\\ \mathbf{elif}\;t \cdot z \leq 2 \cdot 10^{+75}:\\ \;\;\;\;\mathsf{fma}\left(x, y, \mathsf{fma}\left(c, i, a \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t, \mathsf{fma}\left(c, i, x \cdot y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* t z) -1000000000.0)
   (fma i c (fma t z (* x y)))
   (if (<= (* t z) 2e+75)
     (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 ((t * z) <= -1000000000.0) {
		tmp = fma(i, c, fma(t, z, (x * y)));
	} else if ((t * z) <= 2e+75) {
		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(t * z) <= -1000000000.0)
		tmp = fma(i, c, fma(t, z, Float64(x * y)));
	elseif (Float64(t * z) <= 2e+75)
		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[LessEqual[N[(t * z), $MachinePrecision], -1000000000.0], N[(i * c + N[(t * z + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(t * z), $MachinePrecision], 2e+75], 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 3 regimes

2. if (*.f64 z t) < -1e9

   1. Initial program 92.0%

      \[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \]
   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 90.7

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

   5. Applied rewrites 90.7%

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

   if -1e9 < (*.f64 z t) < 1.99999999999999985e75

   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. *-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 94.7

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

   5. Applied rewrites 94.7%

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

   7. Applied rewrites 96.1%

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

   if 1.99999999999999985e75 < (*.f64 z t)

   1. Initial program 95.9%

      \[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \]
   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 89.0

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

   5. Applied rewrites 89.0%

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

   7. Applied rewrites 91.0%

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

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot z \leq -1000000000:\\ \;\;\;\;\mathsf{fma}\left(i, c, \mathsf{fma}\left(t, z, x \cdot y\right)\right)\\ \mathbf{elif}\;t \cdot z \leq 2 \cdot 10^{+75}:\\ \;\;\;\;\mathsf{fma}\left(x, y, \mathsf{fma}\left(c, i, a \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t, \mathsf{fma}\left(c, i, x \cdot y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(i, c, \mathsf{fma}\left(t, z, x \cdot y\right)\right)\\ \mathbf{if}\;t \cdot z \leq -1000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \cdot z \leq 2 \cdot 10^{+75}:\\ \;\;\;\;\mathsf{fma}\left(x, y, \mathsf{fma}\left(c, i, a \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma i c (fma t z (* x y)))))
   (if (<= (* t z) -1000000000.0)
     t_1
     (if (<= (* t z) 2e+75) (fma x y (fma c i (* 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(i, c, fma(t, z, (x * y)));
	double tmp;
	if ((t * z) <= -1000000000.0) {
		tmp = t_1;
	} else if ((t * z) <= 2e+75) {
		tmp = fma(x, y, fma(c, i, (a * b)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(i, c, fma(t, z, Float64(x * y)))
	tmp = 0.0
	if (Float64(t * z) <= -1000000000.0)
		tmp = t_1;
	elseif (Float64(t * z) <= 2e+75)
		tmp = fma(x, y, fma(c, i, 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[(i * c + N[(t * z + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t * z), $MachinePrecision], -1000000000.0], t$95$1, If[LessEqual[N[(t * z), $MachinePrecision], 2e+75], N[(x * y + N[(c * i + N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -1e9 or 1.99999999999999985e75 < (*.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 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 90.0

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

   5. Applied rewrites 90.0%

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

   if -1e9 < (*.f64 z t) < 1.99999999999999985e75

   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. *-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 94.7

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

   5. Applied rewrites 94.7%

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

   7. Applied rewrites 96.1%

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

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot z \leq -1000000000:\\ \;\;\;\;\mathsf{fma}\left(i, c, \mathsf{fma}\left(t, z, x \cdot y\right)\right)\\ \mathbf{elif}\;t \cdot z \leq 2 \cdot 10^{+75}:\\ \;\;\;\;\mathsf{fma}\left(x, y, \mathsf{fma}\left(c, i, a \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, c, \mathsf{fma}\left(t, z, x \cdot y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 88.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{-33}:\\ \;\;\;\;\mathsf{fma}\left(i, c, \mathsf{fma}\left(t, z, x \cdot y\right)\right)\\ \mathbf{elif}\;x \cdot y \leq 10^{-67}:\\ \;\;\;\;\mathsf{fma}\left(b, a, \mathsf{fma}\left(i, c, t \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a, \mathsf{fma}\left(i, c, x \cdot y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* x y) -5e-33)
   (fma i c (fma t z (* x y)))
   (if (<= (* x y) 1e-67)
     (fma b a (fma i c (* t z)))
     (fma b a (fma i c (* x y))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((x * y) <= -5e-33) {
		tmp = fma(i, c, fma(t, z, (x * y)));
	} else if ((x * y) <= 1e-67) {
		tmp = fma(b, a, fma(i, c, (t * z)));
	} else {
		tmp = fma(b, a, fma(i, c, (x * y)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(x * y) <= -5e-33)
		tmp = fma(i, c, fma(t, z, Float64(x * y)));
	elseif (Float64(x * y) <= 1e-67)
		tmp = fma(b, a, fma(i, c, Float64(t * z)));
	else
		tmp = fma(b, a, fma(i, c, Float64(x * y)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(x * y), $MachinePrecision], -5e-33], N[(i * c + N[(t * z + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 1e-67], N[(b * a + N[(i * c + N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(b * a + N[(i * c + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -5.00000000000000028e-33

   1. Initial program 96.8%

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

   3. Taylor expanded in 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 85.9

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

   5. Applied rewrites 85.9%

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

   if -5.00000000000000028e-33 < (*.f64 x y) < 9.99999999999999943e-68

   1. Initial program 99.1%

      \[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \]
   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 97.5

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

   5. Applied rewrites 97.5%

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

   if 9.99999999999999943e-68 < (*.f64 x y)

   1. Initial program 92.2%

      \[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \]
   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 91.2

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

   5. Applied rewrites 91.2%

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

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{-33}:\\ \;\;\;\;\mathsf{fma}\left(i, c, \mathsf{fma}\left(t, z, x \cdot y\right)\right)\\ \mathbf{elif}\;x \cdot y \leq 10^{-67}:\\ \;\;\;\;\mathsf{fma}\left(b, a, \mathsf{fma}\left(i, c, t \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a, \mathsf{fma}\left(i, c, x \cdot y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 88.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(b, a, \mathsf{fma}\left(i, c, x \cdot y\right)\right)\\ \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{+65}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \cdot y \leq 10^{-67}:\\ \;\;\;\;\mathsf{fma}\left(b, a, \mathsf{fma}\left(i, c, t \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma b a (fma i c (* x y)))))
   (if (<= (* x y) -5e+65)
     t_1
     (if (<= (* x y) 1e-67) (fma b a (fma i c (* t z))) 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(b, a, fma(i, c, (x * y)));
	double tmp;
	if ((x * y) <= -5e+65) {
		tmp = t_1;
	} else if ((x * y) <= 1e-67) {
		tmp = fma(b, a, fma(i, c, (t * z)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(b, a, fma(i, c, Float64(x * y)))
	tmp = 0.0
	if (Float64(x * y) <= -5e+65)
		tmp = t_1;
	elseif (Float64(x * y) <= 1e-67)
		tmp = fma(b, a, fma(i, c, Float64(t * z)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(b * a + N[(i * c + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * y), $MachinePrecision], -5e+65], t$95$1, If[LessEqual[N[(x * y), $MachinePrecision], 1e-67], N[(b * a + N[(i * c + N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -4.99999999999999973e65 or 9.99999999999999943e-68 < (*.f64 x y)

   1. Initial program 94.1%

      \[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \]
   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 88.0

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

   5. Applied rewrites 88.0%

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

   if -4.99999999999999973e65 < (*.f64 x y) < 9.99999999999999943e-68

   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 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 94.9

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

   5. Applied rewrites 94.9%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{+65}:\\ \;\;\;\;\mathsf{fma}\left(b, a, \mathsf{fma}\left(i, c, x \cdot y\right)\right)\\ \mathbf{elif}\;x \cdot y \leq 10^{-67}:\\ \;\;\;\;\mathsf{fma}\left(b, a, \mathsf{fma}\left(i, c, t \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a, \mathsf{fma}\left(i, c, x \cdot y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 86.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(i, c, x \cdot y\right)\\ \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{+189}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{+132}:\\ \;\;\;\;\mathsf{fma}\left(b, a, \mathsf{fma}\left(i, c, t \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma i c (* x y))))
   (if (<= (* x y) -1e+189)
     t_1
     (if (<= (* x y) 2e+132) (fma b a (fma i c (* t z))) 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(i, c, (x * y));
	double tmp;
	if ((x * y) <= -1e+189) {
		tmp = t_1;
	} else if ((x * y) <= 2e+132) {
		tmp = fma(b, a, fma(i, c, (t * z)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(i, c, Float64(x * y))
	tmp = 0.0
	if (Float64(x * y) <= -1e+189)
		tmp = t_1;
	elseif (Float64(x * y) <= 2e+132)
		tmp = fma(b, a, fma(i, c, Float64(t * z)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(i * c + N[(x * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * y), $MachinePrecision], -1e+189], t$95$1, If[LessEqual[N[(x * y), $MachinePrecision], 2e+132], N[(b * a + N[(i * c + N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -1e189 or 1.99999999999999998e132 < (*.f64 x y)

   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 z around inf

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

      1. associate-+r+ N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-inverses N/A

         \[\leadsto \left(\left(t + \frac{a \cdot b}{z}\right) \cdot z + \left(x \cdot y\right) \cdot \color{blue}{1}\right) + c \cdot i \]

      8. *-rgt-identity N/A

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

      9. lower-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 90.5

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

   5. Applied rewrites 90.5%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 86.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 86.8

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

   10. Applied rewrites 86.8%

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

   if -1e189 < (*.f64 x y) < 1.99999999999999998e132

   1. Initial program 97.9%

      \[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \]
   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 87.6

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

   5. Applied rewrites 87.6%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{+189}:\\ \;\;\;\;\mathsf{fma}\left(i, c, x \cdot y\right)\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{+132}:\\ \;\;\;\;\mathsf{fma}\left(b, a, \mathsf{fma}\left(i, c, t \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, c, x \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 66.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \cdot z \leq -5 \cdot 10^{+96}:\\ \;\;\;\;\mathsf{fma}\left(z, t, a \cdot b\right)\\ \mathbf{elif}\;t \cdot z \leq 2 \cdot 10^{+75}:\\ \;\;\;\;\mathsf{fma}\left(i, c, a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t, c \cdot i\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* t z) -5e+96)
   (fma z t (* a b))
   (if (<= (* t z) 2e+75) (fma i c (* a b)) (fma z t (* c i)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((t * z) <= -5e+96) {
		tmp = fma(z, t, (a * b));
	} else if ((t * z) <= 2e+75) {
		tmp = fma(i, c, (a * b));
	} else {
		tmp = fma(z, t, (c * i));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(t * z) <= -5e+96)
		tmp = fma(z, t, Float64(a * b));
	elseif (Float64(t * z) <= 2e+75)
		tmp = fma(i, c, Float64(a * b));
	else
		tmp = fma(z, t, Float64(c * i));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(t * z), $MachinePrecision], -5e+96], N[(z * t + N[(a * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(t * z), $MachinePrecision], 2e+75], N[(i * c + N[(a * b), $MachinePrecision]), $MachinePrecision], N[(z * t + N[(c * i), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -5.0000000000000004e96

   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

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

      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 75.7

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

   5. Applied rewrites 75.7%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 73.2%

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

   10. Applied rewrites 75.6%

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

   if -5.0000000000000004e96 < (*.f64 z t) < 1.99999999999999985e75

   1. Initial program 98.7%

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

   3. Taylor expanded in z around 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 93.2

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

   5. Applied rewrites 93.2%

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

   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 61.7%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 62.3%

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

   if 1.99999999999999985e75 < (*.f64 z t)

   1. Initial program 95.9%

      \[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \]
   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 89.0

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

   5. Applied rewrites 89.0%

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

   7. Applied rewrites 91.0%

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

   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 81.3%

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

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot z \leq -5 \cdot 10^{+96}:\\ \;\;\;\;\mathsf{fma}\left(z, t, a \cdot b\right)\\ \mathbf{elif}\;t \cdot z \leq 2 \cdot 10^{+75}:\\ \;\;\;\;\mathsf{fma}\left(i, c, a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t, c \cdot i\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 66.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \cdot z \leq -5 \cdot 10^{+96}:\\ \;\;\;\;\mathsf{fma}\left(z, t, a \cdot b\right)\\ \mathbf{elif}\;t \cdot z \leq 2 \cdot 10^{+75}:\\ \;\;\;\;\mathsf{fma}\left(i, c, a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, c, t \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* t z) -5e+96)
   (fma z t (* a b))
   (if (<= (* t z) 2e+75) (fma i c (* a b)) (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 ((t * z) <= -5e+96) {
		tmp = fma(z, t, (a * b));
	} else if ((t * z) <= 2e+75) {
		tmp = fma(i, c, (a * b));
	} else {
		tmp = fma(i, c, (t * z));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(t * z) <= -5e+96)
		tmp = fma(z, t, Float64(a * b));
	elseif (Float64(t * z) <= 2e+75)
		tmp = fma(i, c, Float64(a * b));
	else
		tmp = fma(i, c, Float64(t * z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(t * z), $MachinePrecision], -5e+96], N[(z * t + N[(a * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(t * z), $MachinePrecision], 2e+75], N[(i * c + N[(a * b), $MachinePrecision]), $MachinePrecision], N[(i * c + N[(t * z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -5.0000000000000004e96

   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

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

      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 75.7

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

   5. Applied rewrites 75.7%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 73.2%

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

   10. Applied rewrites 75.6%

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

   if -5.0000000000000004e96 < (*.f64 z t) < 1.99999999999999985e75

   1. Initial program 98.7%

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

   3. Taylor expanded in z around 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 93.2

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

   5. Applied rewrites 93.2%

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

   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 61.7%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 62.3%

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

   if 1.99999999999999985e75 < (*.f64 z t)

   1. Initial program 95.9%

      \[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \]
   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 89.0

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

   5. Applied rewrites 89.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 79.3%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot z \leq -5 \cdot 10^{+96}:\\ \;\;\;\;\mathsf{fma}\left(z, t, a \cdot b\right)\\ \mathbf{elif}\;t \cdot z \leq 2 \cdot 10^{+75}:\\ \;\;\;\;\mathsf{fma}\left(i, c, a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, c, t \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 65.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z, t, a \cdot b\right)\\ \mathbf{if}\;t \cdot z \leq -5 \cdot 10^{+96}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \cdot z \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(i, c, a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma z t (* a b))))
   (if (<= (* t z) -5e+96) t_1 (if (<= (* t z) 5e-7) (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, (a * b));
	double tmp;
	if ((t * z) <= -5e+96) {
		tmp = t_1;
	} else if ((t * z) <= 5e-7) {
		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(a * b))
	tmp = 0.0
	if (Float64(t * z) <= -5e+96)
		tmp = t_1;
	elseif (Float64(t * z) <= 5e-7)
		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[(a * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t * z), $MachinePrecision], -5e+96], t$95$1, If[LessEqual[N[(t * z), $MachinePrecision], 5e-7], N[(i * c + N[(a * b), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -5.0000000000000004e96 or 4.99999999999999977e-7 < (*.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. 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 79.1

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

   5. Applied rewrites 79.1%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 71.2%

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

   10. Applied rewrites 73.2%

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

   if -5.0000000000000004e96 < (*.f64 z t) < 4.99999999999999977e-7

   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 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 94.1

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

   5. Applied rewrites 94.1%

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

   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 63.2%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 63.8%

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

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot z \leq -5 \cdot 10^{+96}:\\ \;\;\;\;\mathsf{fma}\left(z, t, a \cdot b\right)\\ \mathbf{elif}\;t \cdot z \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(i, c, a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t, a \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 62.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \cdot i \leq -5 \cdot 10^{+101}:\\ \;\;\;\;c \cdot i\\ \mathbf{elif}\;c \cdot i \leq 4 \cdot 10^{+117}:\\ \;\;\;\;\mathsf{fma}\left(z, t, a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot i\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* c i) -5e+101)
   (* c i)
   (if (<= (* c i) 4e+117) (fma 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) {
	double tmp;
	if ((c * i) <= -5e+101) {
		tmp = c * i;
	} else if ((c * i) <= 4e+117) {
		tmp = fma(z, t, (a * b));
	} else {
		tmp = c * i;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(c * i) <= -5e+101)
		tmp = Float64(c * i);
	elseif (Float64(c * i) <= 4e+117)
		tmp = fma(z, t, Float64(a * b));
	else
		tmp = Float64(c * i);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(c * i), $MachinePrecision], -5e+101], N[(c * i), $MachinePrecision], If[LessEqual[N[(c * i), $MachinePrecision], 4e+117], N[(z * t + N[(a * b), $MachinePrecision]), $MachinePrecision], N[(c * i), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 c i) < -4.99999999999999989e101 or 4.0000000000000002e117 < (*.f64 c i)

   1. Initial program 93.6%

      \[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \]
   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 68.6

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

   5. Applied rewrites 68.6%

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

   if -4.99999999999999989e101 < (*.f64 c i) < 4.0000000000000002e117

   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 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 66.0

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

   5. Applied rewrites 66.0%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 58.3%

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

   10. Applied rewrites 59.5%

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

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \cdot i \leq -5 \cdot 10^{+101}:\\ \;\;\;\;c \cdot i\\ \mathbf{elif}\;c \cdot i \leq 4 \cdot 10^{+117}:\\ \;\;\;\;\mathsf{fma}\left(z, t, a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot i\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 62.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \cdot i \leq -5 \cdot 10^{+101}:\\ \;\;\;\;c \cdot i\\ \mathbf{elif}\;c \cdot i \leq 4 \cdot 10^{+117}:\\ \;\;\;\;\mathsf{fma}\left(a, b, t \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot i\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* c i) -5e+101)
   (* c i)
   (if (<= (* c i) 4e+117) (fma a b (* t z)) (* 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) <= -5e+101) {
		tmp = c * i;
	} else if ((c * i) <= 4e+117) {
		tmp = fma(a, b, (t * z));
	} else {
		tmp = c * i;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(c * i) <= -5e+101)
		tmp = Float64(c * i);
	elseif (Float64(c * i) <= 4e+117)
		tmp = fma(a, b, Float64(t * z));
	else
		tmp = Float64(c * i);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(c * i), $MachinePrecision], -5e+101], N[(c * i), $MachinePrecision], If[LessEqual[N[(c * i), $MachinePrecision], 4e+117], N[(a * b + N[(t * z), $MachinePrecision]), $MachinePrecision], N[(c * i), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 c i) < -4.99999999999999989e101 or 4.0000000000000002e117 < (*.f64 c i)

   1. Initial program 93.6%

      \[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \]
   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 68.6

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

   5. Applied rewrites 68.6%

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

   if -4.99999999999999989e101 < (*.f64 c i) < 4.0000000000000002e117

   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 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 66.0

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

   5. Applied rewrites 66.0%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 58.3%

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

4. Final simplification 61.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \cdot i \leq -5 \cdot 10^{+101}:\\ \;\;\;\;c \cdot i\\ \mathbf{elif}\;c \cdot i \leq 4 \cdot 10^{+117}:\\ \;\;\;\;\mathsf{fma}\left(a, b, t \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot i\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 42.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \cdot z \leq -5 \cdot 10^{+96}:\\ \;\;\;\;t \cdot z\\ \mathbf{elif}\;t \cdot z \leq 5 \cdot 10^{-7}:\\ \;\;\;\;c \cdot i\\ \mathbf{else}:\\ \;\;\;\;t \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* t z) -5e+96) (* t z) (if (<= (* t z) 5e-7) (* c i) (* t z))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((t * z) <= -5e+96) {
		tmp = t * z;
	} else if ((t * z) <= 5e-7) {
		tmp = c * i;
	} 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 ((t * z) <= (-5d+96)) then
        tmp = t * z
    else if ((t * z) <= 5d-7) then
        tmp = c * i
    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 ((t * z) <= -5e+96) {
		tmp = t * z;
	} else if ((t * z) <= 5e-7) {
		tmp = c * i;
	} else {
		tmp = t * z;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if (t * z) <= -5e+96:
		tmp = t * z
	elif (t * z) <= 5e-7:
		tmp = c * i
	else:
		tmp = t * z
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(t * z) <= -5e+96)
		tmp = Float64(t * z);
	elseif (Float64(t * z) <= 5e-7)
		tmp = Float64(c * i);
	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 ((t * z) <= -5e+96)
		tmp = t * z;
	elseif ((t * z) <= 5e-7)
		tmp = c * i;
	else
		tmp = t * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(t * z), $MachinePrecision], -5e+96], N[(t * z), $MachinePrecision], If[LessEqual[N[(t * z), $MachinePrecision], 5e-7], N[(c * i), $MachinePrecision], N[(t * z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -5.0000000000000004e96 or 4.99999999999999977e-7 < (*.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. 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 87.7

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

   5. Applied rewrites 87.7%

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

   7. Applied rewrites 87.7%

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

   8. Taylor expanded in z around inf

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

      1. lower-*.f64 60.9

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

   10. Applied rewrites 60.9%

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

   if -5.0000000000000004e96 < (*.f64 z t) < 4.99999999999999977e-7

   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 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 38.3

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

   5. Applied rewrites 38.3%

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

4. Final simplification 47.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot z \leq -5 \cdot 10^{+96}:\\ \;\;\;\;t \cdot z\\ \mathbf{elif}\;t \cdot z \leq 5 \cdot 10^{-7}:\\ \;\;\;\;c \cdot i\\ \mathbf{else}:\\ \;\;\;\;t \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 43.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \cdot i \leq -5 \cdot 10^{+88}:\\ \;\;\;\;c \cdot i\\ \mathbf{elif}\;c \cdot i \leq 4 \cdot 10^{+37}:\\ \;\;\;\;a \cdot b\\ \mathbf{else}:\\ \;\;\;\;c \cdot i\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* c i) -5e+88) (* c i) (if (<= (* c i) 4e+37) (* a b) (* c i))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((c * i) <= -5e+88) {
		tmp = c * i;
	} else if ((c * i) <= 4e+37) {
		tmp = a * b;
	} else {
		tmp = c * i;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8) :: tmp
    if ((c * i) <= (-5d+88)) then
        tmp = c * i
    else if ((c * i) <= 4d+37) then
        tmp = a * b
    else
        tmp = c * i
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((c * i) <= -5e+88) {
		tmp = c * i;
	} else if ((c * i) <= 4e+37) {
		tmp = a * b;
	} else {
		tmp = c * i;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if (c * i) <= -5e+88:
		tmp = c * i
	elif (c * i) <= 4e+37:
		tmp = a * b
	else:
		tmp = c * i
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(c * i) <= -5e+88)
		tmp = Float64(c * i);
	elseif (Float64(c * i) <= 4e+37)
		tmp = Float64(a * b);
	else
		tmp = Float64(c * i);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	tmp = 0.0;
	if ((c * i) <= -5e+88)
		tmp = c * i;
	elseif ((c * i) <= 4e+37)
		tmp = a * b;
	else
		tmp = c * i;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(c * i), $MachinePrecision], -5e+88], N[(c * i), $MachinePrecision], If[LessEqual[N[(c * i), $MachinePrecision], 4e+37], N[(a * b), $MachinePrecision], N[(c * i), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 c i) < -4.99999999999999997e88 or 3.99999999999999982e37 < (*.f64 c i)

   1. Initial program 94.2%

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

   3. Taylor expanded in c around 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 61.1

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

   5. Applied rewrites 61.1%

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

   if -4.99999999999999997e88 < (*.f64 c i) < 3.99999999999999982e37

   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 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 63.6

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

   5. Applied rewrites 63.6%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 60.5%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 31.3%

       \[\leadsto b \cdot a \]
3. Recombined 2 regimes into one program.

4. Final simplification 43.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \cdot i \leq -5 \cdot 10^{+88}:\\ \;\;\;\;c \cdot i\\ \mathbf{elif}\;c \cdot i \leq 4 \cdot 10^{+37}:\\ \;\;\;\;a \cdot b\\ \mathbf{else}:\\ \;\;\;\;c \cdot i\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 26.9% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 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 71.8

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

5. Applied rewrites 71.8%

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

6. Taylor expanded in c around 0

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

8. Applied rewrites 47.3%

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

9. Taylor expanded in z around 0

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

11. Applied rewrites 23.8%

    \[\leadsto b \cdot a \]

12. Final simplification 23.8%

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

Reproduce

herbie shell --seed 2024284 
(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)))