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

Percentage Accurate: 96.0% → 97.6%

Time: 8.4s

Alternatives: 12

Speedup: 0.5×

• 41.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 96.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in t 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 50.0

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

   5. Applied rewrites 50.0%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 75.2%

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

4. Final simplification 98.8%

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

Alternative 2: 43.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \cdot z \leq -3.8 \cdot 10^{+93}:\\ \;\;\;\;t \cdot z\\ \mathbf{elif}\;t \cdot z \leq -2.16 \cdot 10^{-113}:\\ \;\;\;\;b \cdot a\\ \mathbf{elif}\;t \cdot z \leq -4 \cdot 10^{-318}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;t \cdot z \leq 2.15 \cdot 10^{-164}:\\ \;\;\;\;i \cdot c\\ \mathbf{elif}\;t \cdot z \leq 1.6 \cdot 10^{+124}:\\ \;\;\;\;b \cdot a\\ \mathbf{else}:\\ \;\;\;\;t \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* t z) -3.8e+93)
   (* t z)
   (if (<= (* t z) -2.16e-113)
     (* b a)
     (if (<= (* t z) -4e-318)
       (* y x)
       (if (<= (* t z) 2.15e-164)
         (* i c)
         (if (<= (* t z) 1.6e+124) (* 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 ((t * z) <= -3.8e+93) {
		tmp = t * z;
	} else if ((t * z) <= -2.16e-113) {
		tmp = b * a;
	} else if ((t * z) <= -4e-318) {
		tmp = y * x;
	} else if ((t * z) <= 2.15e-164) {
		tmp = i * c;
	} else if ((t * z) <= 1.6e+124) {
		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 ((t * z) <= (-3.8d+93)) then
        tmp = t * z
    else if ((t * z) <= (-2.16d-113)) then
        tmp = b * a
    else if ((t * z) <= (-4d-318)) then
        tmp = y * x
    else if ((t * z) <= 2.15d-164) then
        tmp = i * c
    else if ((t * z) <= 1.6d+124) 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 ((t * z) <= -3.8e+93) {
		tmp = t * z;
	} else if ((t * z) <= -2.16e-113) {
		tmp = b * a;
	} else if ((t * z) <= -4e-318) {
		tmp = y * x;
	} else if ((t * z) <= 2.15e-164) {
		tmp = i * c;
	} else if ((t * z) <= 1.6e+124) {
		tmp = b * a;
	} else {
		tmp = t * z;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if (t * z) <= -3.8e+93:
		tmp = t * z
	elif (t * z) <= -2.16e-113:
		tmp = b * a
	elif (t * z) <= -4e-318:
		tmp = y * x
	elif (t * z) <= 2.15e-164:
		tmp = i * c
	elif (t * z) <= 1.6e+124:
		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(t * z) <= -3.8e+93)
		tmp = Float64(t * z);
	elseif (Float64(t * z) <= -2.16e-113)
		tmp = Float64(b * a);
	elseif (Float64(t * z) <= -4e-318)
		tmp = Float64(y * x);
	elseif (Float64(t * z) <= 2.15e-164)
		tmp = Float64(i * c);
	elseif (Float64(t * z) <= 1.6e+124)
		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 ((t * z) <= -3.8e+93)
		tmp = t * z;
	elseif ((t * z) <= -2.16e-113)
		tmp = b * a;
	elseif ((t * z) <= -4e-318)
		tmp = y * x;
	elseif ((t * z) <= 2.15e-164)
		tmp = i * c;
	elseif ((t * z) <= 1.6e+124)
		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[(t * z), $MachinePrecision], -3.8e+93], N[(t * z), $MachinePrecision], If[LessEqual[N[(t * z), $MachinePrecision], -2.16e-113], N[(b * a), $MachinePrecision], If[LessEqual[N[(t * z), $MachinePrecision], -4e-318], N[(y * x), $MachinePrecision], If[LessEqual[N[(t * z), $MachinePrecision], 2.15e-164], N[(i * c), $MachinePrecision], If[LessEqual[N[(t * z), $MachinePrecision], 1.6e+124], N[(b * a), $MachinePrecision], N[(t * z), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 z t) < -3.7999999999999998e93 or 1.59999999999999996e124 < (*.f64 z t)

   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 t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 70.9

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

   5. Applied rewrites 70.9%

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

   if -3.7999999999999998e93 < (*.f64 z t) < -2.16e-113 or 2.1499999999999999e-164 < (*.f64 z t) < 1.59999999999999996e124

   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 b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 42.1

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

   5. Applied rewrites 42.1%

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

   if -2.16e-113 < (*.f64 z t) < -3.9999999e-318

   1. Initial program 97.1%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 54.7

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

   5. Applied rewrites 54.7%

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

   if -3.9999999e-318 < (*.f64 z t) < 2.1499999999999999e-164

   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 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 49.1

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

   5. Applied rewrites 49.1%

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

4. Final simplification 54.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot z \leq -3.8 \cdot 10^{+93}:\\ \;\;\;\;t \cdot z\\ \mathbf{elif}\;t \cdot z \leq -2.16 \cdot 10^{-113}:\\ \;\;\;\;b \cdot a\\ \mathbf{elif}\;t \cdot z \leq -4 \cdot 10^{-318}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;t \cdot z \leq 2.15 \cdot 10^{-164}:\\ \;\;\;\;i \cdot c\\ \mathbf{elif}\;t \cdot z \leq 1.6 \cdot 10^{+124}:\\ \;\;\;\;b \cdot a\\ \mathbf{else}:\\ \;\;\;\;t \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 66.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, x, b \cdot a\right)\\ \mathbf{if}\;t \cdot z \leq -5 \cdot 10^{+95}:\\ \;\;\;\;\mathsf{fma}\left(i, c, t \cdot z\right)\\ \mathbf{elif}\;t \cdot z \leq -1 \cdot 10^{-112}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \cdot z \leq 10^{-164}:\\ \;\;\;\;\mathsf{fma}\left(i, c, y \cdot x\right)\\ \mathbf{elif}\;t \cdot z \leq 2 \cdot 10^{+126}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t, b \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma y x (* b a))))
   (if (<= (* t z) -5e+95)
     (fma i c (* t z))
     (if (<= (* t z) -1e-112)
       t_1
       (if (<= (* t z) 1e-164)
         (fma i c (* y x))
         (if (<= (* t z) 2e+126) t_1 (fma z t (* b a))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma(y, x, (b * a));
	double tmp;
	if ((t * z) <= -5e+95) {
		tmp = fma(i, c, (t * z));
	} else if ((t * z) <= -1e-112) {
		tmp = t_1;
	} else if ((t * z) <= 1e-164) {
		tmp = fma(i, c, (y * x));
	} else if ((t * z) <= 2e+126) {
		tmp = t_1;
	} else {
		tmp = fma(z, t, (b * a));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(y, x, Float64(b * a))
	tmp = 0.0
	if (Float64(t * z) <= -5e+95)
		tmp = fma(i, c, Float64(t * z));
	elseif (Float64(t * z) <= -1e-112)
		tmp = t_1;
	elseif (Float64(t * z) <= 1e-164)
		tmp = fma(i, c, Float64(y * x));
	elseif (Float64(t * z) <= 2e+126)
		tmp = t_1;
	else
		tmp = fma(z, t, Float64(b * a));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(y * x + N[(b * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t * z), $MachinePrecision], -5e+95], N[(i * c + N[(t * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(t * z), $MachinePrecision], -1e-112], t$95$1, If[LessEqual[N[(t * z), $MachinePrecision], 1e-164], N[(i * c + N[(y * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(t * z), $MachinePrecision], 2e+126], t$95$1, N[(z * t + N[(b * a), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 z t) < -5.00000000000000025e95

   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 t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 77.6

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

   5. Applied rewrites 77.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 79.6

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

   7. Applied rewrites 79.6%

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

   if -5.00000000000000025e95 < (*.f64 z t) < -9.9999999999999995e-113 or 9.99999999999999962e-165 < (*.f64 z t) < 1.99999999999999985e126

   1. Initial program 96.7%

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

   3. Taylor expanded in t 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 88.3

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

   5. Applied rewrites 88.3%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 68.5%

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

   if -9.9999999999999995e-113 < (*.f64 z t) < 9.99999999999999962e-165

   1. Initial program 97.4%

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

   3. Taylor expanded in t 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 98.7

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

   5. Applied rewrites 98.7%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 77.0%

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

   if 1.99999999999999985e126 < (*.f64 z t)

   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 t 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 34.0

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

   5. Applied rewrites 34.0%

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

   6. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 89.1

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

   8. Applied rewrites 89.1%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 89.2%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot z \leq -5 \cdot 10^{+95}:\\ \;\;\;\;\mathsf{fma}\left(i, c, t \cdot z\right)\\ \mathbf{elif}\;t \cdot z \leq -1 \cdot 10^{-112}:\\ \;\;\;\;\mathsf{fma}\left(y, x, b \cdot a\right)\\ \mathbf{elif}\;t \cdot z \leq 10^{-164}:\\ \;\;\;\;\mathsf{fma}\left(i, c, y \cdot x\right)\\ \mathbf{elif}\;t \cdot z \leq 2 \cdot 10^{+126}:\\ \;\;\;\;\mathsf{fma}\left(y, x, b \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t, b \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 87.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \cdot z \leq -2 \cdot 10^{+92}:\\ \;\;\;\;\mathsf{fma}\left(b, a, \mathsf{fma}\left(y, x, t \cdot z\right)\right)\\ \mathbf{elif}\;t \cdot z \leq 2 \cdot 10^{+126}:\\ \;\;\;\;\mathsf{fma}\left(b, a, \mathsf{fma}\left(i, c, y \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t, b \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* t z) -2e+92)
   (fma b a (fma y x (* t z)))
   (if (<= (* t z) 2e+126) (fma b a (fma i c (* y x))) (fma z t (* b a)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((t * z) <= -2e+92) {
		tmp = fma(b, a, fma(y, x, (t * z)));
	} else if ((t * z) <= 2e+126) {
		tmp = fma(b, a, fma(i, c, (y * x)));
	} else {
		tmp = fma(z, t, (b * a));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(t * z) <= -2e+92)
		tmp = fma(b, a, fma(y, x, Float64(t * z)));
	elseif (Float64(t * z) <= 2e+126)
		tmp = fma(b, a, fma(i, c, Float64(y * x)));
	else
		tmp = fma(z, t, Float64(b * a));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(t * z), $MachinePrecision], -2e+92], N[(b * a + N[(y * x + N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(t * z), $MachinePrecision], 2e+126], N[(b * a + N[(i * c + N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z * t + N[(b * a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -2.0000000000000001e92

   1. Initial program 92.1%

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

   3. Taylor expanded in t 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 39.8

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

   5. Applied rewrites 39.8%

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

   6. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 89.2

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

   8. Applied rewrites 89.2%

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

   if -2.0000000000000001e92 < (*.f64 z t) < 1.99999999999999985e126

   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 t 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 93.7

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

   5. Applied rewrites 93.7%

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

   if 1.99999999999999985e126 < (*.f64 z t)

   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 t 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 34.0

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

   5. Applied rewrites 34.0%

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

   6. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 89.1

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

   8. Applied rewrites 89.1%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 89.2%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot z \leq -2 \cdot 10^{+92}:\\ \;\;\;\;\mathsf{fma}\left(b, a, \mathsf{fma}\left(y, x, t \cdot z\right)\right)\\ \mathbf{elif}\;t \cdot z \leq 2 \cdot 10^{+126}:\\ \;\;\;\;\mathsf{fma}\left(b, a, \mathsf{fma}\left(i, c, y \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t, b \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 87.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \cdot z \leq -5 \cdot 10^{+95}:\\ \;\;\;\;\mathsf{fma}\left(b, a, \mathsf{fma}\left(i, c, t \cdot z\right)\right)\\ \mathbf{elif}\;t \cdot z \leq 2 \cdot 10^{+126}:\\ \;\;\;\;\mathsf{fma}\left(b, a, \mathsf{fma}\left(i, c, y \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t, b \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* t z) -5e+95)
   (fma b a (fma i c (* t z)))
   (if (<= (* t z) 2e+126) (fma b a (fma i c (* y x))) (fma z t (* b a)))))

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+95) {
		tmp = fma(b, a, fma(i, c, (t * z)));
	} else if ((t * z) <= 2e+126) {
		tmp = fma(b, a, fma(i, c, (y * x)));
	} else {
		tmp = fma(z, t, (b * a));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(t * z) <= -5e+95)
		tmp = fma(b, a, fma(i, c, Float64(t * z)));
	elseif (Float64(t * z) <= 2e+126)
		tmp = fma(b, a, fma(i, c, Float64(y * x)));
	else
		tmp = fma(z, t, Float64(b * a));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(t * z), $MachinePrecision], -5e+95], N[(b * a + N[(i * c + N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(t * z), $MachinePrecision], 2e+126], N[(b * a + N[(i * c + N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z * t + N[(b * a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -5.00000000000000025e95

   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 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. *-commutative N/A

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

      6. lower-*.f64 84.0

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

   5. Applied rewrites 84.0%

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

   if -5.00000000000000025e95 < (*.f64 z t) < 1.99999999999999985e126

   1. Initial program 97.0%

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

   3. Taylor expanded in t 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%

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

   if 1.99999999999999985e126 < (*.f64 z t)

   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 t 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 34.0

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

   5. Applied rewrites 34.0%

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

   6. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 89.1

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

   8. Applied rewrites 89.1%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 89.2%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot z \leq -5 \cdot 10^{+95}:\\ \;\;\;\;\mathsf{fma}\left(b, a, \mathsf{fma}\left(i, c, t \cdot z\right)\right)\\ \mathbf{elif}\;t \cdot z \leq 2 \cdot 10^{+126}:\\ \;\;\;\;\mathsf{fma}\left(b, a, \mathsf{fma}\left(i, c, y \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t, b \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \cdot z \leq -5 \cdot 10^{+95}:\\ \;\;\;\;\mathsf{fma}\left(i, c, t \cdot z\right)\\ \mathbf{elif}\;t \cdot z \leq 2 \cdot 10^{+126}:\\ \;\;\;\;\mathsf{fma}\left(b, a, \mathsf{fma}\left(i, c, y \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t, b \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* t z) -5e+95)
   (fma i c (* t z))
   (if (<= (* t z) 2e+126) (fma b a (fma i c (* y x))) (fma z t (* b a)))))

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+95) {
		tmp = fma(i, c, (t * z));
	} else if ((t * z) <= 2e+126) {
		tmp = fma(b, a, fma(i, c, (y * x)));
	} else {
		tmp = fma(z, t, (b * a));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(t * z) <= -5e+95)
		tmp = fma(i, c, Float64(t * z));
	elseif (Float64(t * z) <= 2e+126)
		tmp = fma(b, a, fma(i, c, Float64(y * x)));
	else
		tmp = fma(z, t, Float64(b * a));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(t * z), $MachinePrecision], -5e+95], N[(i * c + N[(t * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(t * z), $MachinePrecision], 2e+126], N[(b * a + N[(i * c + N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z * t + N[(b * a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -5.00000000000000025e95

   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 t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 77.6

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

   5. Applied rewrites 77.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 79.6

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

   7. Applied rewrites 79.6%

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

   if -5.00000000000000025e95 < (*.f64 z t) < 1.99999999999999985e126

   1. Initial program 97.0%

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

   3. Taylor expanded in t 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%

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

   if 1.99999999999999985e126 < (*.f64 z t)

   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 t 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 34.0

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

   5. Applied rewrites 34.0%

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

   6. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 89.1

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

   8. Applied rewrites 89.1%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 89.2%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot z \leq -5 \cdot 10^{+95}:\\ \;\;\;\;\mathsf{fma}\left(i, c, t \cdot z\right)\\ \mathbf{elif}\;t \cdot z \leq 2 \cdot 10^{+126}:\\ \;\;\;\;\mathsf{fma}\left(b, a, \mathsf{fma}\left(i, c, y \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t, b \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 41.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot a \leq -1 \cdot 10^{+144}:\\ \;\;\;\;b \cdot a\\ \mathbf{elif}\;b \cdot a \leq -1 \cdot 10^{-249}:\\ \;\;\;\;i \cdot c\\ \mathbf{elif}\;b \cdot a \leq 2 \cdot 10^{+67}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;b \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* b a) -1e+144)
   (* b a)
   (if (<= (* b a) -1e-249) (* i c) (if (<= (* b a) 2e+67) (* y x) (* b a)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((b * a) <= -1e+144) {
		tmp = b * a;
	} else if ((b * a) <= -1e-249) {
		tmp = i * c;
	} else if ((b * a) <= 2e+67) {
		tmp = y * x;
	} else {
		tmp = b * a;
	}
	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 ((b * a) <= (-1d+144)) then
        tmp = b * a
    else if ((b * a) <= (-1d-249)) then
        tmp = i * c
    else if ((b * a) <= 2d+67) then
        tmp = y * x
    else
        tmp = b * a
    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 ((b * a) <= -1e+144) {
		tmp = b * a;
	} else if ((b * a) <= -1e-249) {
		tmp = i * c;
	} else if ((b * a) <= 2e+67) {
		tmp = y * x;
	} else {
		tmp = b * a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if (b * a) <= -1e+144:
		tmp = b * a
	elif (b * a) <= -1e-249:
		tmp = i * c
	elif (b * a) <= 2e+67:
		tmp = y * x
	else:
		tmp = b * a
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(b * a) <= -1e+144)
		tmp = Float64(b * a);
	elseif (Float64(b * a) <= -1e-249)
		tmp = Float64(i * c);
	elseif (Float64(b * a) <= 2e+67)
		tmp = Float64(y * x);
	else
		tmp = Float64(b * a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	tmp = 0.0;
	if ((b * a) <= -1e+144)
		tmp = b * a;
	elseif ((b * a) <= -1e-249)
		tmp = i * c;
	elseif ((b * a) <= 2e+67)
		tmp = y * x;
	else
		tmp = b * a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(b * a), $MachinePrecision], -1e+144], N[(b * a), $MachinePrecision], If[LessEqual[N[(b * a), $MachinePrecision], -1e-249], N[(i * c), $MachinePrecision], If[LessEqual[N[(b * a), $MachinePrecision], 2e+67], N[(y * x), $MachinePrecision], N[(b * a), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 a b) < -1.00000000000000002e144 or 1.99999999999999997e67 < (*.f64 a b)

   1. Initial program 92.7%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 60.0

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

   5. Applied rewrites 60.0%

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

   if -1.00000000000000002e144 < (*.f64 a b) < -1.00000000000000005e-249

   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 42.6

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

   5. Applied rewrites 42.6%

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

   if -1.00000000000000005e-249 < (*.f64 a b) < 1.99999999999999997e67

   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 inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 42.0

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

   5. Applied rewrites 42.0%

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

4. Final simplification 48.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot a \leq -1 \cdot 10^{+144}:\\ \;\;\;\;b \cdot a\\ \mathbf{elif}\;b \cdot a \leq -1 \cdot 10^{-249}:\\ \;\;\;\;i \cdot c\\ \mathbf{elif}\;b \cdot a \leq 2 \cdot 10^{+67}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;b \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 64.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot a \leq -5 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(z, t, b \cdot a\right)\\ \mathbf{elif}\;b \cdot a \leq 4 \cdot 10^{+20}:\\ \;\;\;\;\mathsf{fma}\left(i, c, y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, b \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* b a) -5e-8)
   (fma z t (* b a))
   (if (<= (* b a) 4e+20) (fma i c (* y x)) (fma y x (* b a)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((b * a) <= -5e-8) {
		tmp = fma(z, t, (b * a));
	} else if ((b * a) <= 4e+20) {
		tmp = fma(i, c, (y * x));
	} else {
		tmp = fma(y, x, (b * a));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(b * a) <= -5e-8)
		tmp = fma(z, t, Float64(b * a));
	elseif (Float64(b * a) <= 4e+20)
		tmp = fma(i, c, Float64(y * x));
	else
		tmp = fma(y, x, Float64(b * a));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(b * a), $MachinePrecision], -5e-8], N[(z * t + N[(b * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(b * a), $MachinePrecision], 4e+20], N[(i * c + N[(y * x), $MachinePrecision]), $MachinePrecision], N[(y * x + N[(b * a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 a b) < -4.9999999999999998e-8

   1. Initial program 92.7%

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

   3. Taylor expanded in t 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 77.1

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

   5. Applied rewrites 77.1%

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

   6. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 85.6

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

   8. Applied rewrites 85.6%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 75.1%

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

   if -4.9999999999999998e-8 < (*.f64 a b) < 4e20

   1. Initial program 96.9%

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

   3. Taylor expanded in t 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 71.0

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

   5. Applied rewrites 71.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 69.9%

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

   if 4e20 < (*.f64 a b)

   1. Initial program 94.4%

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

   3. Taylor expanded in t 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 78.1

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

   5. Applied rewrites 78.1%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 72.3%

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

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot a \leq -5 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(z, t, b \cdot a\right)\\ \mathbf{elif}\;b \cdot a \leq 4 \cdot 10^{+20}:\\ \;\;\;\;\mathsf{fma}\left(i, c, y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, b \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 65.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, x, b \cdot a\right)\\ \mathbf{if}\;b \cdot a \leq -5 \cdot 10^{+126}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \cdot a \leq 4 \cdot 10^{+20}:\\ \;\;\;\;\mathsf{fma}\left(i, c, y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma y x (* b a))))
   (if (<= (* b a) -5e+126)
     t_1
     (if (<= (* b a) 4e+20) (fma i c (* 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(y, x, (b * a));
	double tmp;
	if ((b * a) <= -5e+126) {
		tmp = t_1;
	} else if ((b * a) <= 4e+20) {
		tmp = fma(i, c, (y * x));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(y, x, Float64(b * a))
	tmp = 0.0
	if (Float64(b * a) <= -5e+126)
		tmp = t_1;
	elseif (Float64(b * a) <= 4e+20)
		tmp = fma(i, c, 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[(y * x + N[(b * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(b * a), $MachinePrecision], -5e+126], t$95$1, If[LessEqual[N[(b * a), $MachinePrecision], 4e+20], N[(i * c + N[(y * x), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a b) < -4.99999999999999977e126 or 4e20 < (*.f64 a b)

   1. Initial program 92.8%

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

   3. Taylor expanded in t 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 81.5

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

   5. Applied rewrites 81.5%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 75.9%

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

   if -4.99999999999999977e126 < (*.f64 a b) < 4e20

   1. Initial program 97.2%

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

   3. Taylor expanded in t 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 68.9

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

   5. Applied rewrites 68.9%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 66.9%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot a \leq -5 \cdot 10^{+126}:\\ \;\;\;\;\mathsf{fma}\left(y, x, b \cdot a\right)\\ \mathbf{elif}\;b \cdot a \leq 4 \cdot 10^{+20}:\\ \;\;\;\;\mathsf{fma}\left(i, c, y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, b \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 64.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \cdot z \leq -1.9 \cdot 10^{+160}:\\ \;\;\;\;t \cdot z\\ \mathbf{elif}\;t \cdot z \leq 1.68 \cdot 10^{+134}:\\ \;\;\;\;\mathsf{fma}\left(i, c, y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* t z) -1.9e+160)
   (* t z)
   (if (<= (* t z) 1.68e+134) (fma i c (* y x)) (* 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) <= -1.9e+160) {
		tmp = t * z;
	} else if ((t * z) <= 1.68e+134) {
		tmp = fma(i, c, (y * x));
	} else {
		tmp = t * z;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(t * z) <= -1.9e+160)
		tmp = Float64(t * z);
	elseif (Float64(t * z) <= 1.68e+134)
		tmp = fma(i, c, Float64(y * x));
	else
		tmp = Float64(t * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(t * z), $MachinePrecision], -1.9e+160], N[(t * z), $MachinePrecision], If[LessEqual[N[(t * z), $MachinePrecision], 1.68e+134], N[(i * c + N[(y * x), $MachinePrecision]), $MachinePrecision], N[(t * z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -1.90000000000000006e160 or 1.68e134 < (*.f64 z t)

   1. Initial program 91.7%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 77.0

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

   5. Applied rewrites 77.0%

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

   if -1.90000000000000006e160 < (*.f64 z t) < 1.68e134

   1. Initial program 96.7%

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

   3. Taylor expanded in t 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 90.5

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

   5. Applied rewrites 90.5%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 63.1%

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

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot z \leq -1.9 \cdot 10^{+160}:\\ \;\;\;\;t \cdot z\\ \mathbf{elif}\;t \cdot z \leq 1.68 \cdot 10^{+134}:\\ \;\;\;\;\mathsf{fma}\left(i, c, y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 41.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot a \leq -1 \cdot 10^{+144}:\\ \;\;\;\;b \cdot a\\ \mathbf{elif}\;b \cdot a \leq 4 \cdot 10^{+20}:\\ \;\;\;\;i \cdot c\\ \mathbf{else}:\\ \;\;\;\;b \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* b a) -1e+144) (* b a) (if (<= (* b a) 4e+20) (* i c) (* b a))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((b * a) <= -1e+144) {
		tmp = b * a;
	} else if ((b * a) <= 4e+20) {
		tmp = i * c;
	} else {
		tmp = b * a;
	}
	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 ((b * a) <= (-1d+144)) then
        tmp = b * a
    else if ((b * a) <= 4d+20) then
        tmp = i * c
    else
        tmp = b * a
    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 ((b * a) <= -1e+144) {
		tmp = b * a;
	} else if ((b * a) <= 4e+20) {
		tmp = i * c;
	} else {
		tmp = b * a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if (b * a) <= -1e+144:
		tmp = b * a
	elif (b * a) <= 4e+20:
		tmp = i * c
	else:
		tmp = b * a
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(b * a) <= -1e+144)
		tmp = Float64(b * a);
	elseif (Float64(b * a) <= 4e+20)
		tmp = Float64(i * c);
	else
		tmp = Float64(b * a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	tmp = 0.0;
	if ((b * a) <= -1e+144)
		tmp = b * a;
	elseif ((b * a) <= 4e+20)
		tmp = i * c;
	else
		tmp = b * a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(b * a), $MachinePrecision], -1e+144], N[(b * a), $MachinePrecision], If[LessEqual[N[(b * a), $MachinePrecision], 4e+20], N[(i * c), $MachinePrecision], N[(b * a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a b) < -1.00000000000000002e144 or 4e20 < (*.f64 a b)

   1. Initial program 93.4%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 56.5

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

   5. Applied rewrites 56.5%

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

   if -1.00000000000000002e144 < (*.f64 a b) < 4e20

   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 34.8

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

   5. Applied rewrites 34.8%

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

4. Final simplification 43.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot a \leq -1 \cdot 10^{+144}:\\ \;\;\;\;b \cdot a\\ \mathbf{elif}\;b \cdot a \leq 4 \cdot 10^{+20}:\\ \;\;\;\;i \cdot c\\ \mathbf{else}:\\ \;\;\;\;b \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 25.9% 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 b around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 26.5

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

5. Applied rewrites 26.5%

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

Reproduce

herbie shell --seed 2024255 
(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)))