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

Percentage Accurate: 95.6% → 97.7%

Time: 7.7s

Alternatives: 11

Speedup: 1.3×

• 40.7% 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: 95.6% 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: 97.7% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 97.6%

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

   1. lift-+.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. associate-+l+ N/A

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

   4. lift-+.f64 N/A

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

   5. +-commutative N/A

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

   6. associate-+l+ N/A

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

   7. lift-*.f64 N/A

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

   8. lower-fma.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-fma.f64 N/A

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

   12. +-commutative N/A

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-fma.f64 98.4

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

   16. lift-*.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 98.4

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

4. Applied rewrites 98.4%

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

5. Final simplification 98.4%

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

Alternative 2: 75.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(t, z, x \cdot y\right)\\ t_2 := x \cdot y + t \cdot z\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+138}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{+205}:\\ \;\;\;\;\mathsf{fma}\left(b, a, c \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma t z (* x y))) (t_2 (+ (* x y) (* t z))))
   (if (<= t_2 -1e+138) t_1 (if (<= t_2 1e+205) (fma b a (* c i)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma(t, z, (x * y));
	double t_2 = (x * y) + (t * z);
	double tmp;
	if (t_2 <= -1e+138) {
		tmp = t_1;
	} else if (t_2 <= 1e+205) {
		tmp = fma(b, a, (c * i));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(t, z, Float64(x * y))
	t_2 = Float64(Float64(x * y) + Float64(t * z))
	tmp = 0.0
	if (t_2 <= -1e+138)
		tmp = t_1;
	elseif (t_2 <= 1e+205)
		tmp = fma(b, a, Float64(c * i));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(t * z + N[(x * y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * y), $MachinePrecision] + N[(t * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+138], t$95$1, If[LessEqual[t$95$2, 1e+205], N[(b * a + N[(c * i), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 x y) (*.f64 z t)) < -1e138 or 1.00000000000000002e205 < (+.f64 (*.f64 x y) (*.f64 z t))

   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 a around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 91.3

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

   5. Applied rewrites 91.3%

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

   7. Applied rewrites 91.3%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 50.7%

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

   11. Taylor expanded in c around 0

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

   13. Applied rewrites 83.2%

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

   if -1e138 < (+.f64 (*.f64 x y) (*.f64 z t)) < 1.00000000000000002e205

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 90.1

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

   5. Applied rewrites 90.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 77.1%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y + t \cdot z \leq -1 \cdot 10^{+138}:\\ \;\;\;\;\mathsf{fma}\left(t, z, x \cdot y\right)\\ \mathbf{elif}\;x \cdot y + t \cdot z \leq 10^{+205}:\\ \;\;\;\;\mathsf{fma}\left(b, a, c \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, z, x \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 62.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \cdot i \leq -5 \cdot 10^{+26}:\\ \;\;\;\;c \cdot i\\ \mathbf{elif}\;c \cdot i \leq -5 \cdot 10^{-230}:\\ \;\;\;\;\mathsf{fma}\left(t, z, x \cdot y\right)\\ \mathbf{elif}\;c \cdot i \leq 2 \cdot 10^{+140}:\\ \;\;\;\;\mathsf{fma}\left(b, a, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot i\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* c i) -5e+26)
   (* c i)
   (if (<= (* c i) -5e-230)
     (fma t z (* x y))
     (if (<= (* c i) 2e+140) (fma b a (* x y)) (* c i)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((c * i) <= -5e+26) {
		tmp = c * i;
	} else if ((c * i) <= -5e-230) {
		tmp = fma(t, z, (x * y));
	} else if ((c * i) <= 2e+140) {
		tmp = fma(b, a, (x * y));
	} else {
		tmp = c * i;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(c * i) <= -5e+26)
		tmp = Float64(c * i);
	elseif (Float64(c * i) <= -5e-230)
		tmp = fma(t, z, Float64(x * y));
	elseif (Float64(c * i) <= 2e+140)
		tmp = fma(b, a, Float64(x * y));
	else
		tmp = Float64(c * i);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(c * i), $MachinePrecision], -5e+26], N[(c * i), $MachinePrecision], If[LessEqual[N[(c * i), $MachinePrecision], -5e-230], N[(t * z + N[(x * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(c * i), $MachinePrecision], 2e+140], N[(b * a + N[(x * y), $MachinePrecision]), $MachinePrecision], N[(c * i), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 c i) < -5.0000000000000001e26 or 2.00000000000000012e140 < (*.f64 c i)

   1. Initial program 95.9%

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

   3. Taylor expanded in 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 64.7

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

   5. Applied rewrites 64.7%

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

   if -5.0000000000000001e26 < (*.f64 c i) < -5.00000000000000035e-230

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 86.8

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

   5. Applied rewrites 86.8%

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

   7. Applied rewrites 86.8%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 42.3%

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

   11. Taylor expanded in c around 0

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

   13. Applied rewrites 79.2%

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

   if -5.00000000000000035e-230 < (*.f64 c i) < 2.00000000000000012e140

   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 z around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 80.7

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

   5. Applied rewrites 80.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 48.3%

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

   9. Taylor expanded in c around 0

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

   11. Applied rewrites 77.7%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \cdot i \leq -5 \cdot 10^{+26}:\\ \;\;\;\;c \cdot i\\ \mathbf{elif}\;c \cdot i \leq -5 \cdot 10^{-230}:\\ \;\;\;\;\mathsf{fma}\left(t, z, x \cdot y\right)\\ \mathbf{elif}\;c \cdot i \leq 2 \cdot 10^{+140}:\\ \;\;\;\;\mathsf{fma}\left(b, a, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot i\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 88.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot b \leq -2 \cdot 10^{+113}:\\ \;\;\;\;\mathsf{fma}\left(b, a, \mathsf{fma}\left(i, c, t \cdot z\right)\right)\\ \mathbf{elif}\;a \cdot b \leq 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(i, c, \mathsf{fma}\left(t, z, x \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a, \mathsf{fma}\left(i, c, x \cdot y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* a b) -2e+113)
   (fma b a (fma i c (* t z)))
   (if (<= (* a b) 1e-9)
     (fma i c (fma t z (* x y)))
     (fma b a (fma i c (* x y))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((a * b) <= -2e+113) {
		tmp = fma(b, a, fma(i, c, (t * z)));
	} else if ((a * b) <= 1e-9) {
		tmp = fma(i, c, fma(t, z, (x * y)));
	} else {
		tmp = fma(b, a, fma(i, c, (x * y)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(a * b) <= -2e+113)
		tmp = fma(b, a, fma(i, c, Float64(t * z)));
	elseif (Float64(a * b) <= 1e-9)
		tmp = fma(i, c, fma(t, z, Float64(x * y)));
	else
		tmp = fma(b, a, fma(i, c, Float64(x * y)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(a * b), $MachinePrecision], -2e+113], N[(b * a + N[(i * c + N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], 1e-9], N[(i * c + N[(t * z + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(b * a + N[(i * c + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 a b) < -2e113

   1. Initial program 97.5%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 97.5

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

   5. Applied rewrites 97.5%

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

   if -2e113 < (*.f64 a b) < 1.00000000000000006e-9

   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 a around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 93.3

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

   5. Applied rewrites 93.3%

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

   if 1.00000000000000006e-9 < (*.f64 a b)

   1. Initial program 98.3%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 92.1

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

   5. Applied rewrites 92.1%

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

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b \leq -2 \cdot 10^{+113}:\\ \;\;\;\;\mathsf{fma}\left(b, a, \mathsf{fma}\left(i, c, t \cdot z\right)\right)\\ \mathbf{elif}\;a \cdot b \leq 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(i, c, \mathsf{fma}\left(t, z, x \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a, \mathsf{fma}\left(i, c, x \cdot y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 87.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \cdot z \leq -1 \cdot 10^{+178}:\\ \;\;\;\;\mathsf{fma}\left(z, t, c \cdot i\right)\\ \mathbf{elif}\;t \cdot z \leq 5 \cdot 10^{+25}:\\ \;\;\;\;\mathsf{fma}\left(b, a, \mathsf{fma}\left(i, c, x \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a, \mathsf{fma}\left(i, c, t \cdot z\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* t z) -1e+178)
   (fma z t (* c i))
   (if (<= (* t z) 5e+25)
     (fma b a (fma i c (* x y)))
     (fma b a (fma i c (* t z))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((t * z) <= -1e+178) {
		tmp = fma(z, t, (c * i));
	} else if ((t * z) <= 5e+25) {
		tmp = fma(b, a, fma(i, c, (x * y)));
	} else {
		tmp = fma(b, a, fma(i, c, (t * z)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(t * z) <= -1e+178)
		tmp = fma(z, t, Float64(c * i));
	elseif (Float64(t * z) <= 5e+25)
		tmp = fma(b, a, fma(i, c, Float64(x * y)));
	else
		tmp = fma(b, a, fma(i, c, Float64(t * z)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(t * z), $MachinePrecision], -1e+178], N[(z * t + N[(c * i), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(t * z), $MachinePrecision], 5e+25], N[(b * a + N[(i * c + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(b * a + N[(i * c + N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -1.0000000000000001e178

   1. Initial program 89.9%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 93.3

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

   5. Applied rewrites 93.3%

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

   7. Applied rewrites 93.3%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 86.0%

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

   if -1.0000000000000001e178 < (*.f64 z t) < 5.00000000000000024e25

   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 z around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 95.1

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

   5. Applied rewrites 95.1%

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

   if 5.00000000000000024e25 < (*.f64 z t)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 87.4

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

   5. Applied rewrites 87.4%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot z \leq -1 \cdot 10^{+178}:\\ \;\;\;\;\mathsf{fma}\left(z, t, c \cdot i\right)\\ \mathbf{elif}\;t \cdot z \leq 5 \cdot 10^{+25}:\\ \;\;\;\;\mathsf{fma}\left(b, a, \mathsf{fma}\left(i, c, x \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a, \mathsf{fma}\left(i, c, t \cdot z\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{+138}:\\ \;\;\;\;\mathsf{fma}\left(t, z, x \cdot y\right)\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+209}:\\ \;\;\;\;\mathsf{fma}\left(b, a, \mathsf{fma}\left(i, c, t \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a, x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* x y) -1e+138)
   (fma t z (* x y))
   (if (<= (* x y) 5e+209) (fma b a (fma i c (* t z))) (fma b a (* x y)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((x * y) <= -1e+138) {
		tmp = fma(t, z, (x * y));
	} else if ((x * y) <= 5e+209) {
		tmp = fma(b, a, fma(i, c, (t * z)));
	} else {
		tmp = fma(b, a, (x * y));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(x * y) <= -1e+138)
		tmp = fma(t, z, Float64(x * y));
	elseif (Float64(x * y) <= 5e+209)
		tmp = fma(b, a, fma(i, c, Float64(t * z)));
	else
		tmp = fma(b, a, Float64(x * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(x * y), $MachinePrecision], -1e+138], N[(t * z + N[(x * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 5e+209], N[(b * a + N[(i * c + N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(b * a + N[(x * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -1e138

   1. Initial program 94.7%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 93.2

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

   5. Applied rewrites 93.2%

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

   7. Applied rewrites 93.2%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 23.6%

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

   11. Taylor expanded in c around 0

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

   13. Applied rewrites 81.3%

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

   if -1e138 < (*.f64 x y) < 4.99999999999999964e209

   1. Initial program 98.9%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 88.8

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

   5. Applied rewrites 88.8%

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

   if 4.99999999999999964e209 < (*.f64 x y)

   1. Initial program 91.2%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 95.7

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

   5. Applied rewrites 95.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 15.4%

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

   9. Taylor expanded in c around 0

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

   11. Applied rewrites 95.7%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{+138}:\\ \;\;\;\;\mathsf{fma}\left(t, z, x \cdot y\right)\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+209}:\\ \;\;\;\;\mathsf{fma}\left(b, a, \mathsf{fma}\left(i, c, t \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a, x \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 42.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \cdot i \leq -10:\\ \;\;\;\;c \cdot i\\ \mathbf{elif}\;c \cdot i \leq -5 \cdot 10^{-237}:\\ \;\;\;\;t \cdot z\\ \mathbf{elif}\;c \cdot i \leq 2 \cdot 10^{+140}:\\ \;\;\;\;a \cdot b\\ \mathbf{else}:\\ \;\;\;\;c \cdot i\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* c i) -10.0)
   (* c i)
   (if (<= (* c i) -5e-237) (* t z) (if (<= (* c i) 2e+140) (* a b) (* c i)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((c * i) <= -10.0) {
		tmp = c * i;
	} else if ((c * i) <= -5e-237) {
		tmp = t * z;
	} else if ((c * i) <= 2e+140) {
		tmp = a * b;
	} else {
		tmp = c * i;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8) :: tmp
    if ((c * i) <= (-10.0d0)) then
        tmp = c * i
    else if ((c * i) <= (-5d-237)) then
        tmp = t * z
    else if ((c * i) <= 2d+140) then
        tmp = a * b
    else
        tmp = c * i
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((c * i) <= -10.0) {
		tmp = c * i;
	} else if ((c * i) <= -5e-237) {
		tmp = t * z;
	} else if ((c * i) <= 2e+140) {
		tmp = a * b;
	} else {
		tmp = c * i;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if (c * i) <= -10.0:
		tmp = c * i
	elif (c * i) <= -5e-237:
		tmp = t * z
	elif (c * i) <= 2e+140:
		tmp = a * b
	else:
		tmp = c * i
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(c * i) <= -10.0)
		tmp = Float64(c * i);
	elseif (Float64(c * i) <= -5e-237)
		tmp = Float64(t * z);
	elseif (Float64(c * i) <= 2e+140)
		tmp = Float64(a * b);
	else
		tmp = Float64(c * i);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	tmp = 0.0;
	if ((c * i) <= -10.0)
		tmp = c * i;
	elseif ((c * i) <= -5e-237)
		tmp = t * z;
	elseif ((c * i) <= 2e+140)
		tmp = a * b;
	else
		tmp = c * i;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(c * i), $MachinePrecision], -10.0], N[(c * i), $MachinePrecision], If[LessEqual[N[(c * i), $MachinePrecision], -5e-237], N[(t * z), $MachinePrecision], If[LessEqual[N[(c * i), $MachinePrecision], 2e+140], N[(a * b), $MachinePrecision], N[(c * i), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 c i) < -10 or 2.00000000000000012e140 < (*.f64 c i)

   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 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 62.6

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

   5. Applied rewrites 62.6%

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

   if -10 < (*.f64 c i) < -5.0000000000000002e-237

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 55.5

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

   5. Applied rewrites 55.5%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 50.7%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 16.4%

       \[\leadsto b \cdot a \]

   12. Taylor expanded in z around inf

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

       1. lower-*.f64 39.1

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

   14. Applied rewrites 39.1%

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

   if -5.0000000000000002e-237 < (*.f64 c i) < 2.00000000000000012e140

   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 x around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 68.4

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

   5. Applied rewrites 68.4%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 65.5%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 45.8%

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

4. Final simplification 51.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \cdot i \leq -10:\\ \;\;\;\;c \cdot i\\ \mathbf{elif}\;c \cdot i \leq -5 \cdot 10^{-237}:\\ \;\;\;\;t \cdot z\\ \mathbf{elif}\;c \cdot i \leq 2 \cdot 10^{+140}:\\ \;\;\;\;a \cdot b\\ \mathbf{else}:\\ \;\;\;\;c \cdot i\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 62.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \cdot i \leq -5 \cdot 10^{+26}:\\ \;\;\;\;c \cdot i\\ \mathbf{elif}\;c \cdot i \leq 2 \cdot 10^{+140}:\\ \;\;\;\;\mathsf{fma}\left(b, a, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot i\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* c i) -5e+26)
   (* c i)
   (if (<= (* c i) 2e+140) (fma b a (* x y)) (* c i))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((c * i) <= -5e+26) {
		tmp = c * i;
	} else if ((c * i) <= 2e+140) {
		tmp = fma(b, a, (x * y));
	} else {
		tmp = c * i;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(c * i) <= -5e+26)
		tmp = Float64(c * i);
	elseif (Float64(c * i) <= 2e+140)
		tmp = fma(b, a, Float64(x * y));
	else
		tmp = Float64(c * i);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(c * i), $MachinePrecision], -5e+26], N[(c * i), $MachinePrecision], If[LessEqual[N[(c * i), $MachinePrecision], 2e+140], N[(b * a + N[(x * y), $MachinePrecision]), $MachinePrecision], N[(c * i), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 c i) < -5.0000000000000001e26 or 2.00000000000000012e140 < (*.f64 c i)

   1. Initial program 95.9%

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

   3. Taylor expanded in 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 64.7

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

   5. Applied rewrites 64.7%

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

   if -5.0000000000000001e26 < (*.f64 c i) < 2.00000000000000012e140

   1. Initial program 98.7%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 77.4

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

   5. Applied rewrites 77.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 41.5%

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

   9. Taylor expanded in c around 0

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

   11. Applied rewrites 73.1%

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

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \cdot i \leq -5 \cdot 10^{+26}:\\ \;\;\;\;c \cdot i\\ \mathbf{elif}\;c \cdot i \leq 2 \cdot 10^{+140}:\\ \;\;\;\;\mathsf{fma}\left(b, a, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot i\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 60.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \cdot i \leq -50000000000:\\ \;\;\;\;c \cdot i\\ \mathbf{elif}\;c \cdot i \leq 2 \cdot 10^{+147}:\\ \;\;\;\;\mathsf{fma}\left(a, b, t \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot i\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* c i) -50000000000.0)
   (* c i)
   (if (<= (* c i) 2e+147) (fma a b (* t z)) (* c i))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((c * i) <= -50000000000.0) {
		tmp = c * i;
	} else if ((c * i) <= 2e+147) {
		tmp = fma(a, b, (t * z));
	} else {
		tmp = c * i;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(c * i) <= -50000000000.0)
		tmp = Float64(c * i);
	elseif (Float64(c * i) <= 2e+147)
		tmp = fma(a, b, Float64(t * z));
	else
		tmp = Float64(c * i);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(c * i), $MachinePrecision], -50000000000.0], N[(c * i), $MachinePrecision], If[LessEqual[N[(c * i), $MachinePrecision], 2e+147], N[(a * b + N[(t * z), $MachinePrecision]), $MachinePrecision], N[(c * i), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 c i) < -5e10 or 2e147 < (*.f64 c i)

   1. Initial program 96.0%

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

   3. Taylor expanded in c around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 63.4

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

   5. Applied rewrites 63.4%

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

   if -5e10 < (*.f64 c i) < 2e147

   1. Initial program 98.7%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 65.7

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

   5. Applied rewrites 65.7%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 61.8%

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

4. Final simplification 62.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \cdot i \leq -50000000000:\\ \;\;\;\;c \cdot i\\ \mathbf{elif}\;c \cdot i \leq 2 \cdot 10^{+147}:\\ \;\;\;\;\mathsf{fma}\left(a, b, t \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot i\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 42.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \cdot i \leq -50000000000:\\ \;\;\;\;c \cdot i\\ \mathbf{elif}\;c \cdot i \leq 2 \cdot 10^{+140}:\\ \;\;\;\;a \cdot b\\ \mathbf{else}:\\ \;\;\;\;c \cdot i\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* c i) -50000000000.0)
   (* c i)
   (if (<= (* c i) 2e+140) (* a b) (* c i))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((c * i) <= -50000000000.0) {
		tmp = c * i;
	} else if ((c * i) <= 2e+140) {
		tmp = a * b;
	} else {
		tmp = c * i;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8) :: tmp
    if ((c * i) <= (-50000000000.0d0)) then
        tmp = c * i
    else if ((c * i) <= 2d+140) then
        tmp = a * b
    else
        tmp = c * i
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((c * i) <= -50000000000.0) {
		tmp = c * i;
	} else if ((c * i) <= 2e+140) {
		tmp = a * b;
	} else {
		tmp = c * i;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if (c * i) <= -50000000000.0:
		tmp = c * i
	elif (c * i) <= 2e+140:
		tmp = a * b
	else:
		tmp = c * i
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(c * i) <= -50000000000.0)
		tmp = Float64(c * i);
	elseif (Float64(c * i) <= 2e+140)
		tmp = Float64(a * b);
	else
		tmp = Float64(c * i);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	tmp = 0.0;
	if ((c * i) <= -50000000000.0)
		tmp = c * i;
	elseif ((c * i) <= 2e+140)
		tmp = a * b;
	else
		tmp = c * i;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(c * i), $MachinePrecision], -50000000000.0], N[(c * i), $MachinePrecision], If[LessEqual[N[(c * i), $MachinePrecision], 2e+140], N[(a * b), $MachinePrecision], N[(c * i), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 c i) < -5e10 or 2.00000000000000012e140 < (*.f64 c i)

   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 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 62.8

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

   5. Applied rewrites 62.8%

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

   if -5e10 < (*.f64 c i) < 2.00000000000000012e140

   1. Initial program 98.7%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 65.4

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

   5. Applied rewrites 65.4%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 61.5%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 38.3%

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

4. Final simplification 48.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \cdot i \leq -50000000000:\\ \;\;\;\;c \cdot i\\ \mathbf{elif}\;c \cdot i \leq 2 \cdot 10^{+140}:\\ \;\;\;\;a \cdot b\\ \mathbf{else}:\\ \;\;\;\;c \cdot i\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 27.3% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 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 x around 0

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

   1. *-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-*.f64 74.0

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

5. Applied rewrites 74.0%

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

6. Taylor expanded in c around 0

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

8. Applied rewrites 49.4%

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

9. Taylor expanded in z around 0

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

11. Applied rewrites 28.8%

    \[\leadsto b \cdot a \]

12. Final simplification 28.8%

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

Reproduce

herbie shell --seed 2024298 
(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)))