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

Percentage Accurate: 95.8% → 98.0%

Time: 8.4s

Alternatives: 13

Speedup: 0.5×

• 42.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 95.8% 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, 0.5× speedup

Math

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

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 b a (fma y x (* t z))))))

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(b, a, fma(y, x, (t * z)));
	}
	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(b, a, fma(y, x, Float64(t * z)));
	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[(b * a + N[(y * x + N[(t * z), $MachinePrecision]), $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 c around 0

      \[\leadsto \color{blue}{a \cdot b + \left(t \cdot z + x \cdot y\right)} \]
   4. 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 78.6

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

   5. Applied rewrites 78.6%

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

4. Final simplification 98.8%

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

Alternative 2: 43.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot x \leq -5 \cdot 10^{+97}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \cdot x \leq -5 \cdot 10^{-44}:\\ \;\;\;\;i \cdot c\\ \mathbf{elif}\;y \cdot x \leq -1 \cdot 10^{-199}:\\ \;\;\;\;t \cdot z\\ \mathbf{elif}\;y \cdot x \leq 10^{-265}:\\ \;\;\;\;b \cdot a\\ \mathbf{elif}\;y \cdot x \leq 5 \cdot 10^{-84}:\\ \;\;\;\;t \cdot z\\ \mathbf{elif}\;y \cdot x \leq 5 \cdot 10^{+34}:\\ \;\;\;\;i \cdot c\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* y x) -5e+97)
   (* y x)
   (if (<= (* y x) -5e-44)
     (* i c)
     (if (<= (* y x) -1e-199)
       (* t z)
       (if (<= (* y x) 1e-265)
         (* b a)
         (if (<= (* y x) 5e-84)
           (* t z)
           (if (<= (* y x) 5e+34) (* i c) (* y x))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((y * x) <= -5e+97) {
		tmp = y * x;
	} else if ((y * x) <= -5e-44) {
		tmp = i * c;
	} else if ((y * x) <= -1e-199) {
		tmp = t * z;
	} else if ((y * x) <= 1e-265) {
		tmp = b * a;
	} else if ((y * x) <= 5e-84) {
		tmp = t * z;
	} else if ((y * x) <= 5e+34) {
		tmp = i * c;
	} else {
		tmp = y * x;
	}
	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 ((y * x) <= (-5d+97)) then
        tmp = y * x
    else if ((y * x) <= (-5d-44)) then
        tmp = i * c
    else if ((y * x) <= (-1d-199)) then
        tmp = t * z
    else if ((y * x) <= 1d-265) then
        tmp = b * a
    else if ((y * x) <= 5d-84) then
        tmp = t * z
    else if ((y * x) <= 5d+34) then
        tmp = i * c
    else
        tmp = y * x
    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 ((y * x) <= -5e+97) {
		tmp = y * x;
	} else if ((y * x) <= -5e-44) {
		tmp = i * c;
	} else if ((y * x) <= -1e-199) {
		tmp = t * z;
	} else if ((y * x) <= 1e-265) {
		tmp = b * a;
	} else if ((y * x) <= 5e-84) {
		tmp = t * z;
	} else if ((y * x) <= 5e+34) {
		tmp = i * c;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if (y * x) <= -5e+97:
		tmp = y * x
	elif (y * x) <= -5e-44:
		tmp = i * c
	elif (y * x) <= -1e-199:
		tmp = t * z
	elif (y * x) <= 1e-265:
		tmp = b * a
	elif (y * x) <= 5e-84:
		tmp = t * z
	elif (y * x) <= 5e+34:
		tmp = i * c
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(y * x) <= -5e+97)
		tmp = Float64(y * x);
	elseif (Float64(y * x) <= -5e-44)
		tmp = Float64(i * c);
	elseif (Float64(y * x) <= -1e-199)
		tmp = Float64(t * z);
	elseif (Float64(y * x) <= 1e-265)
		tmp = Float64(b * a);
	elseif (Float64(y * x) <= 5e-84)
		tmp = Float64(t * z);
	elseif (Float64(y * x) <= 5e+34)
		tmp = Float64(i * c);
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	tmp = 0.0;
	if ((y * x) <= -5e+97)
		tmp = y * x;
	elseif ((y * x) <= -5e-44)
		tmp = i * c;
	elseif ((y * x) <= -1e-199)
		tmp = t * z;
	elseif ((y * x) <= 1e-265)
		tmp = b * a;
	elseif ((y * x) <= 5e-84)
		tmp = t * z;
	elseif ((y * x) <= 5e+34)
		tmp = i * c;
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(y * x), $MachinePrecision], -5e+97], N[(y * x), $MachinePrecision], If[LessEqual[N[(y * x), $MachinePrecision], -5e-44], N[(i * c), $MachinePrecision], If[LessEqual[N[(y * x), $MachinePrecision], -1e-199], N[(t * z), $MachinePrecision], If[LessEqual[N[(y * x), $MachinePrecision], 1e-265], N[(b * a), $MachinePrecision], If[LessEqual[N[(y * x), $MachinePrecision], 5e-84], N[(t * z), $MachinePrecision], If[LessEqual[N[(y * x), $MachinePrecision], 5e+34], N[(i * c), $MachinePrecision], N[(y * x), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 x y) < -4.99999999999999999e97 or 4.9999999999999998e34 < (*.f64 x y)

   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 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 62.0

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

   5. Applied rewrites 62.0%

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

   if -4.99999999999999999e97 < (*.f64 x y) < -5.00000000000000039e-44 or 5.0000000000000002e-84 < (*.f64 x y) < 4.9999999999999998e34

   1. Initial program 93.5%

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

   3. Taylor expanded in c around 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 59.0

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

   5. Applied rewrites 59.0%

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

   if -5.00000000000000039e-44 < (*.f64 x y) < -9.99999999999999982e-200 or 9.99999999999999985e-266 < (*.f64 x y) < 5.0000000000000002e-84

   1. Initial program 97.8%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 60.9

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

   5. Applied rewrites 60.9%

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

   if -9.99999999999999982e-200 < (*.f64 x y) < 9.99999999999999985e-266

   1. Initial program 98.1%

      \[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \]
   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 45.3

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

   5. Applied rewrites 45.3%

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

4. Final simplification 57.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot x \leq -5 \cdot 10^{+97}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \cdot x \leq -5 \cdot 10^{-44}:\\ \;\;\;\;i \cdot c\\ \mathbf{elif}\;y \cdot x \leq -1 \cdot 10^{-199}:\\ \;\;\;\;t \cdot z\\ \mathbf{elif}\;y \cdot x \leq 10^{-265}:\\ \;\;\;\;b \cdot a\\ \mathbf{elif}\;y \cdot x \leq 5 \cdot 10^{-84}:\\ \;\;\;\;t \cdot z\\ \mathbf{elif}\;y \cdot x \leq 5 \cdot 10^{+34}:\\ \;\;\;\;i \cdot c\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, x, t \cdot z\right)\\ t_2 := t \cdot z + y \cdot x\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+97}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{+41}:\\ \;\;\;\;\mathsf{fma}\left(i, c, b \cdot a\right)\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+112}:\\ \;\;\;\;\mathsf{fma}\left(y, x, b \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma y x (* t z))) (t_2 (+ (* t z) (* y x))))
   (if (<= t_2 -1e+97)
     t_1
     (if (<= t_2 4e+41)
       (fma i c (* b a))
       (if (<= t_2 2e+112) (fma y x (* b a)) 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, (t * z));
	double t_2 = (t * z) + (y * x);
	double tmp;
	if (t_2 <= -1e+97) {
		tmp = t_1;
	} else if (t_2 <= 4e+41) {
		tmp = fma(i, c, (b * a));
	} else if (t_2 <= 2e+112) {
		tmp = fma(y, x, (b * a));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(y, x, Float64(t * z))
	t_2 = Float64(Float64(t * z) + Float64(y * x))
	tmp = 0.0
	if (t_2 <= -1e+97)
		tmp = t_1;
	elseif (t_2 <= 4e+41)
		tmp = fma(i, c, Float64(b * a));
	elseif (t_2 <= 2e+112)
		tmp = fma(y, x, Float64(b * a));
	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[(t * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t * z), $MachinePrecision] + N[(y * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+97], t$95$1, If[LessEqual[t$95$2, 4e+41], N[(i * c + N[(b * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e+112], N[(y * x + N[(b * a), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (*.f64 x y) (*.f64 z t)) < -1.0000000000000001e97 or 1.9999999999999999e112 < (+.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 92.6%

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

   3. Taylor expanded in b around 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. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 90.6

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

   5. Applied rewrites 90.6%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 79.9%

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

   if -1.0000000000000001e97 < (+.f64 (*.f64 x y) (*.f64 z t)) < 4.00000000000000002e41

   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

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

      1. *-commutative N/A

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

      2. lower-*.f64 87.2

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

   5. Applied rewrites 87.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 89.5

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

   7. Applied rewrites 89.5%

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

   if 4.00000000000000002e41 < (+.f64 (*.f64 x y) (*.f64 z t)) < 1.9999999999999999e112

   1. Initial program 99.9%

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

   3. Taylor expanded in c around 0

      \[\leadsto \color{blue}{a \cdot b + \left(t \cdot z + x \cdot y\right)} \]
   4. 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 96.0

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

   5. Applied rewrites 96.0%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 82.1%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot z + y \cdot x \leq -1 \cdot 10^{+97}:\\ \;\;\;\;\mathsf{fma}\left(y, x, t \cdot z\right)\\ \mathbf{elif}\;t \cdot z + y \cdot x \leq 4 \cdot 10^{+41}:\\ \;\;\;\;\mathsf{fma}\left(i, c, b \cdot a\right)\\ \mathbf{elif}\;t \cdot z + y \cdot x \leq 2 \cdot 10^{+112}:\\ \;\;\;\;\mathsf{fma}\left(y, x, b \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, t \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 65.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(i, c, y \cdot x\right)\\ t_2 := \mathsf{fma}\left(b, a, t \cdot z\right)\\ \mathbf{if}\;b \cdot a \leq -2 \cdot 10^{+68}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;b \cdot a \leq -1 \cdot 10^{-275}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \cdot a \leq 2 \cdot 10^{-148}:\\ \;\;\;\;\mathsf{fma}\left(y, x, t \cdot z\right)\\ \mathbf{elif}\;b \cdot a \leq 5 \cdot 10^{+90}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma i c (* y x))) (t_2 (fma b a (* t z))))
   (if (<= (* b a) -2e+68)
     t_2
     (if (<= (* b a) -1e-275)
       t_1
       (if (<= (* b a) 2e-148)
         (fma y x (* t z))
         (if (<= (* b a) 5e+90) t_1 t_2))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma(i, c, (y * x));
	double t_2 = fma(b, a, (t * z));
	double tmp;
	if ((b * a) <= -2e+68) {
		tmp = t_2;
	} else if ((b * a) <= -1e-275) {
		tmp = t_1;
	} else if ((b * a) <= 2e-148) {
		tmp = fma(y, x, (t * z));
	} else if ((b * a) <= 5e+90) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(i, c, Float64(y * x))
	t_2 = fma(b, a, Float64(t * z))
	tmp = 0.0
	if (Float64(b * a) <= -2e+68)
		tmp = t_2;
	elseif (Float64(b * a) <= -1e-275)
		tmp = t_1;
	elseif (Float64(b * a) <= 2e-148)
		tmp = fma(y, x, Float64(t * z));
	elseif (Float64(b * a) <= 5e+90)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(i * c + N[(y * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(b * a + N[(t * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(b * a), $MachinePrecision], -2e+68], t$95$2, If[LessEqual[N[(b * a), $MachinePrecision], -1e-275], t$95$1, If[LessEqual[N[(b * a), $MachinePrecision], 2e-148], N[(y * x + N[(t * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(b * a), $MachinePrecision], 5e+90], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 a b) < -1.99999999999999991e68 or 5.0000000000000004e90 < (*.f64 a b)

   1. Initial program 88.5%

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

   3. Taylor expanded in c around 0

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

      1. *-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 87.2

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

   5. Applied rewrites 87.2%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 78.0%

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

   if -1.99999999999999991e68 < (*.f64 a b) < -9.99999999999999934e-276 or 1.99999999999999987e-148 < (*.f64 a b) < 5.0000000000000004e90

   1. Initial program 98.1%

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

   3. Taylor expanded in b around 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. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 89.8

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

   5. Applied rewrites 89.8%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 73.3%

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

   if -9.99999999999999934e-276 < (*.f64 a b) < 1.99999999999999987e-148

   1. Initial program 98.1%

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

   3. Taylor expanded in b around 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. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 86.6%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot a \leq -2 \cdot 10^{+68}:\\ \;\;\;\;\mathsf{fma}\left(b, a, t \cdot z\right)\\ \mathbf{elif}\;b \cdot a \leq -1 \cdot 10^{-275}:\\ \;\;\;\;\mathsf{fma}\left(i, c, y \cdot x\right)\\ \mathbf{elif}\;b \cdot a \leq 2 \cdot 10^{-148}:\\ \;\;\;\;\mathsf{fma}\left(y, x, t \cdot z\right)\\ \mathbf{elif}\;b \cdot a \leq 5 \cdot 10^{+90}:\\ \;\;\;\;\mathsf{fma}\left(i, c, y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a, t \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 82.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(b, a, \mathsf{fma}\left(y, x, t \cdot z\right)\right)\\ t_2 := t \cdot z + y \cdot x\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+97}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{+41}:\\ \;\;\;\;\mathsf{fma}\left(i, c, b \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma b a (fma y x (* t z)))) (t_2 (+ (* t z) (* y x))))
   (if (<= t_2 -1e+97) t_1 (if (<= t_2 4e+41) (fma i c (* b a)) 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(y, x, (t * z)));
	double t_2 = (t * z) + (y * x);
	double tmp;
	if (t_2 <= -1e+97) {
		tmp = t_1;
	} else if (t_2 <= 4e+41) {
		tmp = fma(i, c, (b * a));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(b, a, fma(y, x, Float64(t * z)))
	t_2 = Float64(Float64(t * z) + Float64(y * x))
	tmp = 0.0
	if (t_2 <= -1e+97)
		tmp = t_1;
	elseif (t_2 <= 4e+41)
		tmp = fma(i, c, Float64(b * a));
	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[(y * x + N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t * z), $MachinePrecision] + N[(y * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+97], t$95$1, If[LessEqual[t$95$2, 4e+41], N[(i * c + N[(b * a), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 x y) (*.f64 z t)) < -1.0000000000000001e97 or 4.00000000000000002e41 < (+.f64 (*.f64 x y) (*.f64 z t))

   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 c around 0

      \[\leadsto \color{blue}{a \cdot b + \left(t \cdot z + x \cdot y\right)} \]
   4. 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 88.8

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

   5. Applied rewrites 88.8%

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

   if -1.0000000000000001e97 < (+.f64 (*.f64 x y) (*.f64 z t)) < 4.00000000000000002e41

   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

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

      1. *-commutative N/A

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

      2. lower-*.f64 87.2

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

   5. Applied rewrites 87.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 89.5

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

   7. Applied rewrites 89.5%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot z + y \cdot x \leq -1 \cdot 10^{+97}:\\ \;\;\;\;\mathsf{fma}\left(b, a, \mathsf{fma}\left(y, x, t \cdot z\right)\right)\\ \mathbf{elif}\;t \cdot z + y \cdot x \leq 4 \cdot 10^{+41}:\\ \;\;\;\;\mathsf{fma}\left(i, c, b \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a, \mathsf{fma}\left(y, x, t \cdot z\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 42.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot x \leq -5 \cdot 10^{+97}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \cdot x \leq -2 \cdot 10^{-9}:\\ \;\;\;\;i \cdot c\\ \mathbf{elif}\;y \cdot x \leq 5 \cdot 10^{-247}:\\ \;\;\;\;b \cdot a\\ \mathbf{elif}\;y \cdot x \leq 5 \cdot 10^{+34}:\\ \;\;\;\;i \cdot c\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* y x) -5e+97)
   (* y x)
   (if (<= (* y x) -2e-9)
     (* i c)
     (if (<= (* y x) 5e-247)
       (* b a)
       (if (<= (* y x) 5e+34) (* i c) (* y x))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((y * x) <= -5e+97) {
		tmp = y * x;
	} else if ((y * x) <= -2e-9) {
		tmp = i * c;
	} else if ((y * x) <= 5e-247) {
		tmp = b * a;
	} else if ((y * x) <= 5e+34) {
		tmp = i * c;
	} else {
		tmp = y * x;
	}
	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 ((y * x) <= (-5d+97)) then
        tmp = y * x
    else if ((y * x) <= (-2d-9)) then
        tmp = i * c
    else if ((y * x) <= 5d-247) then
        tmp = b * a
    else if ((y * x) <= 5d+34) then
        tmp = i * c
    else
        tmp = y * x
    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 ((y * x) <= -5e+97) {
		tmp = y * x;
	} else if ((y * x) <= -2e-9) {
		tmp = i * c;
	} else if ((y * x) <= 5e-247) {
		tmp = b * a;
	} else if ((y * x) <= 5e+34) {
		tmp = i * c;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if (y * x) <= -5e+97:
		tmp = y * x
	elif (y * x) <= -2e-9:
		tmp = i * c
	elif (y * x) <= 5e-247:
		tmp = b * a
	elif (y * x) <= 5e+34:
		tmp = i * c
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(y * x) <= -5e+97)
		tmp = Float64(y * x);
	elseif (Float64(y * x) <= -2e-9)
		tmp = Float64(i * c);
	elseif (Float64(y * x) <= 5e-247)
		tmp = Float64(b * a);
	elseif (Float64(y * x) <= 5e+34)
		tmp = Float64(i * c);
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	tmp = 0.0;
	if ((y * x) <= -5e+97)
		tmp = y * x;
	elseif ((y * x) <= -2e-9)
		tmp = i * c;
	elseif ((y * x) <= 5e-247)
		tmp = b * a;
	elseif ((y * x) <= 5e+34)
		tmp = i * c;
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(y * x), $MachinePrecision], -5e+97], N[(y * x), $MachinePrecision], If[LessEqual[N[(y * x), $MachinePrecision], -2e-9], N[(i * c), $MachinePrecision], If[LessEqual[N[(y * x), $MachinePrecision], 5e-247], N[(b * a), $MachinePrecision], If[LessEqual[N[(y * x), $MachinePrecision], 5e+34], N[(i * c), $MachinePrecision], N[(y * x), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -4.99999999999999999e97 or 4.9999999999999998e34 < (*.f64 x y)

   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 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 62.0

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

   5. Applied rewrites 62.0%

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

   if -4.99999999999999999e97 < (*.f64 x y) < -2.00000000000000012e-9 or 4.99999999999999978e-247 < (*.f64 x y) < 4.9999999999999998e34

   1. Initial program 94.5%

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

   3. Taylor expanded in c around 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 49.9

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

   5. Applied rewrites 49.9%

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

   if -2.00000000000000012e-9 < (*.f64 x y) < 4.99999999999999978e-247

   1. Initial program 97.8%

      \[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \]
   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 40.7

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

   5. Applied rewrites 40.7%

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

4. Final simplification 51.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot x \leq -5 \cdot 10^{+97}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \cdot x \leq -2 \cdot 10^{-9}:\\ \;\;\;\;i \cdot c\\ \mathbf{elif}\;y \cdot x \leq 5 \cdot 10^{-247}:\\ \;\;\;\;b \cdot a\\ \mathbf{elif}\;y \cdot x \leq 5 \cdot 10^{+34}:\\ \;\;\;\;i \cdot c\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 89.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, x, t \cdot z\right)\\ \mathbf{if}\;t \cdot z \leq -1.4 \cdot 10^{+97}:\\ \;\;\;\;\mathsf{fma}\left(b, a, t\_1\right)\\ \mathbf{elif}\;t \cdot z \leq 5 \cdot 10^{+43}:\\ \;\;\;\;\mathsf{fma}\left(y, x, b \cdot a\right) + i \cdot c\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, c, t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma y x (* t z))))
   (if (<= (* t z) -1.4e+97)
     (fma b a t_1)
     (if (<= (* t z) 5e+43) (+ (fma y x (* b a)) (* i c)) (fma i c 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, (t * z));
	double tmp;
	if ((t * z) <= -1.4e+97) {
		tmp = fma(b, a, t_1);
	} else if ((t * z) <= 5e+43) {
		tmp = fma(y, x, (b * a)) + (i * c);
	} else {
		tmp = fma(i, c, t_1);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(y, x, Float64(t * z))
	tmp = 0.0
	if (Float64(t * z) <= -1.4e+97)
		tmp = fma(b, a, t_1);
	elseif (Float64(t * z) <= 5e+43)
		tmp = Float64(fma(y, x, Float64(b * a)) + Float64(i * c));
	else
		tmp = fma(i, c, t_1);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(y * x + N[(t * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t * z), $MachinePrecision], -1.4e+97], N[(b * a + t$95$1), $MachinePrecision], If[LessEqual[N[(t * z), $MachinePrecision], 5e+43], N[(N[(y * x + N[(b * a), $MachinePrecision]), $MachinePrecision] + N[(i * c), $MachinePrecision]), $MachinePrecision], N[(i * c + t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -1.4e97

   1. Initial program 87.7%

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

   3. Taylor expanded in c around 0

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

      1. *-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 94.0

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

   5. Applied rewrites 94.0%

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

   if -1.4e97 < (*.f64 z t) < 5.0000000000000004e43

   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 t around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 92.7

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

   5. Applied rewrites 92.7%

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

   if 5.0000000000000004e43 < (*.f64 z t)

   1. Initial program 93.8%

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

   3. Taylor expanded in b 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. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 95.1

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

   5. Applied rewrites 95.1%

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

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot z \leq -1.4 \cdot 10^{+97}:\\ \;\;\;\;\mathsf{fma}\left(b, a, \mathsf{fma}\left(y, x, t \cdot z\right)\right)\\ \mathbf{elif}\;t \cdot z \leq 5 \cdot 10^{+43}:\\ \;\;\;\;\mathsf{fma}\left(y, x, b \cdot a\right) + i \cdot c\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, c, \mathsf{fma}\left(y, x, t \cdot z\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 89.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, x, t \cdot z\right)\\ t_2 := \mathsf{fma}\left(i, c, t\_1\right)\\ \mathbf{if}\;i \cdot c \leq -1 \cdot 10^{+100}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;i \cdot c \leq 2 \cdot 10^{+19}:\\ \;\;\;\;\mathsf{fma}\left(b, a, t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma y x (* t z))) (t_2 (fma i c t_1)))
   (if (<= (* i c) -1e+100) t_2 (if (<= (* i c) 2e+19) (fma b a t_1) t_2))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma(y, x, (t * z));
	double t_2 = fma(i, c, t_1);
	double tmp;
	if ((i * c) <= -1e+100) {
		tmp = t_2;
	} else if ((i * c) <= 2e+19) {
		tmp = fma(b, a, t_1);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(y, x, Float64(t * z))
	t_2 = fma(i, c, t_1)
	tmp = 0.0
	if (Float64(i * c) <= -1e+100)
		tmp = t_2;
	elseif (Float64(i * c) <= 2e+19)
		tmp = fma(b, a, t_1);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(y * x + N[(t * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(i * c + t$95$1), $MachinePrecision]}, If[LessEqual[N[(i * c), $MachinePrecision], -1e+100], t$95$2, If[LessEqual[N[(i * c), $MachinePrecision], 2e+19], N[(b * a + t$95$1), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 c i) < -1.00000000000000002e100 or 2e19 < (*.f64 c i)

   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 b 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. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 89.9

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

   5. Applied rewrites 89.9%

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

   if -1.00000000000000002e100 < (*.f64 c i) < 2e19

   1. Initial program 96.5%

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

   3. Taylor expanded in c around 0

      \[\leadsto \color{blue}{a \cdot b + \left(t \cdot z + x \cdot y\right)} \]
   4. 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 94.7

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

   5. Applied rewrites 94.7%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \cdot c \leq -1 \cdot 10^{+100}:\\ \;\;\;\;\mathsf{fma}\left(i, c, \mathsf{fma}\left(y, x, t \cdot z\right)\right)\\ \mathbf{elif}\;i \cdot c \leq 2 \cdot 10^{+19}:\\ \;\;\;\;\mathsf{fma}\left(b, a, \mathsf{fma}\left(y, x, t \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, c, \mathsf{fma}\left(y, x, t \cdot z\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 67.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \cdot z \leq -7.5 \cdot 10^{+82}:\\ \;\;\;\;\mathsf{fma}\left(b, a, t \cdot z\right)\\ \mathbf{elif}\;t \cdot z \leq 6.3 \cdot 10^{+62}:\\ \;\;\;\;\mathsf{fma}\left(y, x, b \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, t \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* t z) -7.5e+82)
   (fma b a (* t z))
   (if (<= (* t z) 6.3e+62) (fma y x (* b a)) (fma 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) <= -7.5e+82) {
		tmp = fma(b, a, (t * z));
	} else if ((t * z) <= 6.3e+62) {
		tmp = fma(y, x, (b * a));
	} else {
		tmp = fma(y, x, (t * z));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(t * z) <= -7.5e+82)
		tmp = fma(b, a, Float64(t * z));
	elseif (Float64(t * z) <= 6.3e+62)
		tmp = fma(y, x, Float64(b * a));
	else
		tmp = fma(y, x, Float64(t * z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(t * z), $MachinePrecision], -7.5e+82], N[(b * a + N[(t * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(t * z), $MachinePrecision], 6.3e+62], N[(y * x + N[(b * a), $MachinePrecision]), $MachinePrecision], N[(y * x + N[(t * z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -7.4999999999999999e82

   1. Initial program 88.6%

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

   3. Taylor expanded in c around 0

      \[\leadsto \color{blue}{a \cdot b + \left(t \cdot z + x \cdot y\right)} \]
   4. 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 90.8

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

   5. Applied rewrites 90.8%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 83.5%

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

   if -7.4999999999999999e82 < (*.f64 z t) < 6.29999999999999998e62

   1. Initial program 96.8%

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

   3. Taylor expanded in c around 0

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

      1. *-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 67.5

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

   5. Applied rewrites 67.5%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 64.1%

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

   if 6.29999999999999998e62 < (*.f64 z t)

   1. Initial program 93.2%

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

   3. Taylor expanded in b 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. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 96.8

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

   5. Applied rewrites 96.8%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 83.7%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot z \leq -7.5 \cdot 10^{+82}:\\ \;\;\;\;\mathsf{fma}\left(b, a, t \cdot z\right)\\ \mathbf{elif}\;t \cdot z \leq 6.3 \cdot 10^{+62}:\\ \;\;\;\;\mathsf{fma}\left(y, x, b \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, t \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 67.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, x, t \cdot z\right)\\ \mathbf{if}\;t \cdot z \leq -1.45 \cdot 10^{+97}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \cdot z \leq 6.3 \cdot 10^{+62}:\\ \;\;\;\;\mathsf{fma}\left(y, x, b \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma y x (* t z))))
   (if (<= (* t z) -1.45e+97)
     t_1
     (if (<= (* t z) 6.3e+62) (fma y x (* b a)) 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, (t * z));
	double tmp;
	if ((t * z) <= -1.45e+97) {
		tmp = t_1;
	} else if ((t * z) <= 6.3e+62) {
		tmp = fma(y, x, (b * a));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(y, x, Float64(t * z))
	tmp = 0.0
	if (Float64(t * z) <= -1.45e+97)
		tmp = t_1;
	elseif (Float64(t * z) <= 6.3e+62)
		tmp = fma(y, x, Float64(b * a));
	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[(t * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t * z), $MachinePrecision], -1.45e+97], t$95$1, If[LessEqual[N[(t * z), $MachinePrecision], 6.3e+62], N[(y * x + N[(b * a), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -1.44999999999999994e97 or 6.29999999999999998e62 < (*.f64 z t)

   1. Initial program 90.3%

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

   3. Taylor expanded in b around 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. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 91.1

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

   5. Applied rewrites 91.1%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 83.9%

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

   if -1.44999999999999994e97 < (*.f64 z t) < 6.29999999999999998e62

   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 c around 0

      \[\leadsto \color{blue}{a \cdot b + \left(t \cdot z + x \cdot y\right)} \]
   4. 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 67.1

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

   5. Applied rewrites 67.1%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 63.6%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot z \leq -1.45 \cdot 10^{+97}:\\ \;\;\;\;\mathsf{fma}\left(y, x, t \cdot z\right)\\ \mathbf{elif}\;t \cdot z \leq 6.3 \cdot 10^{+62}:\\ \;\;\;\;\mathsf{fma}\left(y, x, b \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, t \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 63.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \cdot z \leq -1.35 \cdot 10^{+158}:\\ \;\;\;\;t \cdot z\\ \mathbf{elif}\;t \cdot z \leq 2 \cdot 10^{+115}:\\ \;\;\;\;\mathsf{fma}\left(y, x, b \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* t z) -1.35e+158)
   (* t z)
   (if (<= (* t z) 2e+115) (fma y x (* 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) <= -1.35e+158) {
		tmp = t * z;
	} else if ((t * z) <= 2e+115) {
		tmp = fma(y, x, (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) <= -1.35e+158)
		tmp = Float64(t * z);
	elseif (Float64(t * z) <= 2e+115)
		tmp = fma(y, x, Float64(b * a));
	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], -1.35e+158], N[(t * z), $MachinePrecision], If[LessEqual[N[(t * z), $MachinePrecision], 2e+115], N[(y * x + N[(b * a), $MachinePrecision]), $MachinePrecision], N[(t * z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -1.34999999999999989e158 or 2e115 < (*.f64 z t)

   1. Initial program 88.5%

      \[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \]
   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 77.2

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

   5. Applied rewrites 77.2%

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

   if -1.34999999999999989e158 < (*.f64 z t) < 2e115

   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 c around 0

      \[\leadsto \color{blue}{a \cdot b + \left(t \cdot z + x \cdot y\right)} \]
   4. 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 68.8

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

   5. Applied rewrites 68.8%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 63.8%

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

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot z \leq -1.35 \cdot 10^{+158}:\\ \;\;\;\;t \cdot z\\ \mathbf{elif}\;t \cdot z \leq 2 \cdot 10^{+115}:\\ \;\;\;\;\mathsf{fma}\left(y, x, b \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 42.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \cdot c \leq -1 \cdot 10^{+100}:\\ \;\;\;\;i \cdot c\\ \mathbf{elif}\;i \cdot c \leq 5 \cdot 10^{+22}:\\ \;\;\;\;b \cdot a\\ \mathbf{else}:\\ \;\;\;\;i \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* i c) -1e+100) (* i c) (if (<= (* i c) 5e+22) (* b a) (* i c))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((i * c) <= -1e+100) {
		tmp = i * c;
	} else if ((i * c) <= 5e+22) {
		tmp = b * a;
	} else {
		tmp = i * c;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8) :: tmp
    if ((i * c) <= (-1d+100)) then
        tmp = i * c
    else if ((i * c) <= 5d+22) then
        tmp = b * a
    else
        tmp = i * c
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((i * c) <= -1e+100) {
		tmp = i * c;
	} else if ((i * c) <= 5e+22) {
		tmp = b * a;
	} else {
		tmp = i * c;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if (i * c) <= -1e+100:
		tmp = i * c
	elif (i * c) <= 5e+22:
		tmp = b * a
	else:
		tmp = i * c
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(i * c) <= -1e+100)
		tmp = Float64(i * c);
	elseif (Float64(i * c) <= 5e+22)
		tmp = Float64(b * a);
	else
		tmp = Float64(i * c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	tmp = 0.0;
	if ((i * c) <= -1e+100)
		tmp = i * c;
	elseif ((i * c) <= 5e+22)
		tmp = b * a;
	else
		tmp = i * c;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(i * c), $MachinePrecision], -1e+100], N[(i * c), $MachinePrecision], If[LessEqual[N[(i * c), $MachinePrecision], 5e+22], N[(b * a), $MachinePrecision], N[(i * c), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 c i) < -1.00000000000000002e100 or 4.9999999999999996e22 < (*.f64 c i)

   1. Initial program 91.8%

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

   3. Taylor expanded in c around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 55.9

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

   5. Applied rewrites 55.9%

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

   if -1.00000000000000002e100 < (*.f64 c i) < 4.9999999999999996e22

   1. Initial program 96.5%

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

   3. Taylor expanded in 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 35.4

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

   5. Applied rewrites 35.4%

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

4. Final simplification 44.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \cdot c \leq -1 \cdot 10^{+100}:\\ \;\;\;\;i \cdot c\\ \mathbf{elif}\;i \cdot c \leq 5 \cdot 10^{+22}:\\ \;\;\;\;b \cdot a\\ \mathbf{else}:\\ \;\;\;\;i \cdot c\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 26.5% 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 94.5%

   \[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \]
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 26.2

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

5. Applied rewrites 26.2%

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

Reproduce

herbie shell --seed 2024249 
(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)))