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

Percentage Accurate: 96.0% → 98.0%

Time: 10.2s

Alternatives: 14

Speedup: 1.3×

• 41.1% 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: 96.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.0% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 97.2%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-+l+ N/A

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

   7. lift-+.f64 N/A

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

   8. +-commutative N/A

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

   9. associate-+l+ N/A

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

   10. lift-*.f64 N/A

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

   11. lower-fma.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. lower-fma.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. lower-fma.f64 98.0

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

4. Applied rewrites 98.0%

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

Alternative 2: 62.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(i, c, a \cdot b\right)\\ \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+150}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;z \cdot t \leq -1 \cdot 10^{+130}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;z \cdot t \leq -0.01:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \cdot t \leq -1 \cdot 10^{-252}:\\ \;\;\;\;\mathsf{fma}\left(a, b, x \cdot y\right)\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{-235}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+95}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma i c (* a b))))
   (if (<= (* z t) -1e+150)
     (* z t)
     (if (<= (* z t) -1e+130)
       (* x y)
       (if (<= (* z t) -0.01)
         t_1
         (if (<= (* z t) -1e-252)
           (fma a b (* x y))
           (if (<= (* z t) 2e-235)
             t_1
             (if (<= (* z t) 2e+95) (fma y x (* c i)) (* z t)))))))))

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, (a * b));
	double tmp;
	if ((z * t) <= -1e+150) {
		tmp = z * t;
	} else if ((z * t) <= -1e+130) {
		tmp = x * y;
	} else if ((z * t) <= -0.01) {
		tmp = t_1;
	} else if ((z * t) <= -1e-252) {
		tmp = fma(a, b, (x * y));
	} else if ((z * t) <= 2e-235) {
		tmp = t_1;
	} else if ((z * t) <= 2e+95) {
		tmp = fma(y, x, (c * i));
	} else {
		tmp = z * t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(i, c, Float64(a * b))
	tmp = 0.0
	if (Float64(z * t) <= -1e+150)
		tmp = Float64(z * t);
	elseif (Float64(z * t) <= -1e+130)
		tmp = Float64(x * y);
	elseif (Float64(z * t) <= -0.01)
		tmp = t_1;
	elseif (Float64(z * t) <= -1e-252)
		tmp = fma(a, b, Float64(x * y));
	elseif (Float64(z * t) <= 2e-235)
		tmp = t_1;
	elseif (Float64(z * t) <= 2e+95)
		tmp = fma(y, x, Float64(c * i));
	else
		tmp = Float64(z * t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(i * c + N[(a * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(z * t), $MachinePrecision], -1e+150], N[(z * t), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], -1e+130], N[(x * y), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], -0.01], t$95$1, If[LessEqual[N[(z * t), $MachinePrecision], -1e-252], N[(a * b + N[(x * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 2e-235], t$95$1, If[LessEqual[N[(z * t), $MachinePrecision], 2e+95], N[(y * x + N[(c * i), $MachinePrecision]), $MachinePrecision], N[(z * t), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 z t) < -9.99999999999999981e149 or 2.00000000000000004e95 < (*.f64 z t)

   1. Initial program 94.2%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 67.5

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

   5. Applied rewrites 67.5%

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

   if -9.99999999999999981e149 < (*.f64 z t) < -1.0000000000000001e130

   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 inf

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

      1. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -1.0000000000000001e130 < (*.f64 z t) < -0.0100000000000000002 or -9.99999999999999943e-253 < (*.f64 z t) < 1.9999999999999999e-235

   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 inf

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

      1. lower-*.f64 77.5

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

   5. Applied rewrites 77.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 77.5

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

   7. Applied rewrites 77.5%

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

   if -0.0100000000000000002 < (*.f64 z t) < -9.99999999999999943e-253

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-+l+ N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-fma.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 85.1

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

   7. Applied rewrites 85.1%

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

   8. Taylor expanded in z around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 83.3

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

   10. Applied rewrites 83.3%

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

   if 1.9999999999999999e-235 < (*.f64 z t) < 2.00000000000000004e95

   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 x around inf

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

      1. lower-*.f64 68.1

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

   5. Applied rewrites 68.1%

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

      1. *-commutative N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 68.1

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

   7. Applied rewrites 68.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, c \cdot i\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+150}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;z \cdot t \leq -1 \cdot 10^{+130}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;z \cdot t \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(i, c, a \cdot b\right)\\ \mathbf{elif}\;z \cdot t \leq -1 \cdot 10^{-252}:\\ \;\;\;\;\mathsf{fma}\left(a, b, x \cdot y\right)\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{-235}:\\ \;\;\;\;\mathsf{fma}\left(i, c, a \cdot b\right)\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+95}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 62.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(a, b, x \cdot y\right)\\ t_2 := \mathsf{fma}\left(i, c, a \cdot b\right)\\ \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+150}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;z \cdot t \leq -1 \cdot 10^{+130}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;z \cdot t \leq -0.01:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \cdot t \leq -1 \cdot 10^{-252}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \cdot t \leq 10^{-246}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+95}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;z \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma a b (* x y))) (t_2 (fma i c (* a b))))
   (if (<= (* z t) -1e+150)
     (* z t)
     (if (<= (* z t) -1e+130)
       (* x y)
       (if (<= (* z t) -0.01)
         t_2
         (if (<= (* z t) -1e-252)
           t_1
           (if (<= (* z t) 1e-246)
             t_2
             (if (<= (* z t) 2e+95) t_1 (* z t)))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma(a, b, (x * y));
	double t_2 = fma(i, c, (a * b));
	double tmp;
	if ((z * t) <= -1e+150) {
		tmp = z * t;
	} else if ((z * t) <= -1e+130) {
		tmp = x * y;
	} else if ((z * t) <= -0.01) {
		tmp = t_2;
	} else if ((z * t) <= -1e-252) {
		tmp = t_1;
	} else if ((z * t) <= 1e-246) {
		tmp = t_2;
	} else if ((z * t) <= 2e+95) {
		tmp = t_1;
	} else {
		tmp = z * t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(a, b, Float64(x * y))
	t_2 = fma(i, c, Float64(a * b))
	tmp = 0.0
	if (Float64(z * t) <= -1e+150)
		tmp = Float64(z * t);
	elseif (Float64(z * t) <= -1e+130)
		tmp = Float64(x * y);
	elseif (Float64(z * t) <= -0.01)
		tmp = t_2;
	elseif (Float64(z * t) <= -1e-252)
		tmp = t_1;
	elseif (Float64(z * t) <= 1e-246)
		tmp = t_2;
	elseif (Float64(z * t) <= 2e+95)
		tmp = t_1;
	else
		tmp = Float64(z * t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(a * b + N[(x * y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(i * c + N[(a * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(z * t), $MachinePrecision], -1e+150], N[(z * t), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], -1e+130], N[(x * y), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], -0.01], t$95$2, If[LessEqual[N[(z * t), $MachinePrecision], -1e-252], t$95$1, If[LessEqual[N[(z * t), $MachinePrecision], 1e-246], t$95$2, If[LessEqual[N[(z * t), $MachinePrecision], 2e+95], t$95$1, N[(z * t), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 z t) < -9.99999999999999981e149 or 2.00000000000000004e95 < (*.f64 z t)

   1. Initial program 94.2%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 67.5

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

   5. Applied rewrites 67.5%

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

   if -9.99999999999999981e149 < (*.f64 z t) < -1.0000000000000001e130

   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 inf

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

      1. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -1.0000000000000001e130 < (*.f64 z t) < -0.0100000000000000002 or -9.99999999999999943e-253 < (*.f64 z t) < 9.99999999999999956e-247

   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 inf

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

      1. lower-*.f64 77.2

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

   5. Applied rewrites 77.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 77.2

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

   7. Applied rewrites 77.2%

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

   if -0.0100000000000000002 < (*.f64 z t) < -9.99999999999999943e-253 or 9.99999999999999956e-247 < (*.f64 z t) < 2.00000000000000004e95

   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

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-+l+ N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-fma.f64 98.8

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

   4. Applied rewrites 98.8%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 81.0

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

   7. Applied rewrites 81.0%

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

   8. Taylor expanded in z around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 74.7

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

   10. Applied rewrites 74.7%

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

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+150}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;z \cdot t \leq -1 \cdot 10^{+130}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;z \cdot t \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(i, c, a \cdot b\right)\\ \mathbf{elif}\;z \cdot t \leq -1 \cdot 10^{-252}:\\ \;\;\;\;\mathsf{fma}\left(a, b, x \cdot y\right)\\ \mathbf{elif}\;z \cdot t \leq 10^{-246}:\\ \;\;\;\;\mathsf{fma}\left(i, c, a \cdot b\right)\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+95}:\\ \;\;\;\;\mathsf{fma}\left(a, b, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 43.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+150}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;z \cdot t \leq -1 \cdot 10^{+103}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;z \cdot t \leq -0.01:\\ \;\;\;\;c \cdot i\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{-235}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+95}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* z t) -1e+150)
   (* z t)
   (if (<= (* z t) -1e+103)
     (* x y)
     (if (<= (* z t) -0.01)
       (* c i)
       (if (<= (* z t) 2e-235)
         (* a b)
         (if (<= (* z t) 2e+95) (* x y) (* z t)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((z * t) <= -1e+150) {
		tmp = z * t;
	} else if ((z * t) <= -1e+103) {
		tmp = x * y;
	} else if ((z * t) <= -0.01) {
		tmp = c * i;
	} else if ((z * t) <= 2e-235) {
		tmp = a * b;
	} else if ((z * t) <= 2e+95) {
		tmp = x * y;
	} else {
		tmp = z * t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8) :: tmp
    if ((z * t) <= (-1d+150)) then
        tmp = z * t
    else if ((z * t) <= (-1d+103)) then
        tmp = x * y
    else if ((z * t) <= (-0.01d0)) then
        tmp = c * i
    else if ((z * t) <= 2d-235) then
        tmp = a * b
    else if ((z * t) <= 2d+95) then
        tmp = x * y
    else
        tmp = z * t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((z * t) <= -1e+150) {
		tmp = z * t;
	} else if ((z * t) <= -1e+103) {
		tmp = x * y;
	} else if ((z * t) <= -0.01) {
		tmp = c * i;
	} else if ((z * t) <= 2e-235) {
		tmp = a * b;
	} else if ((z * t) <= 2e+95) {
		tmp = x * y;
	} else {
		tmp = z * t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if (z * t) <= -1e+150:
		tmp = z * t
	elif (z * t) <= -1e+103:
		tmp = x * y
	elif (z * t) <= -0.01:
		tmp = c * i
	elif (z * t) <= 2e-235:
		tmp = a * b
	elif (z * t) <= 2e+95:
		tmp = x * y
	else:
		tmp = z * t
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(z * t) <= -1e+150)
		tmp = Float64(z * t);
	elseif (Float64(z * t) <= -1e+103)
		tmp = Float64(x * y);
	elseif (Float64(z * t) <= -0.01)
		tmp = Float64(c * i);
	elseif (Float64(z * t) <= 2e-235)
		tmp = Float64(a * b);
	elseif (Float64(z * t) <= 2e+95)
		tmp = Float64(x * y);
	else
		tmp = Float64(z * t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	tmp = 0.0;
	if ((z * t) <= -1e+150)
		tmp = z * t;
	elseif ((z * t) <= -1e+103)
		tmp = x * y;
	elseif ((z * t) <= -0.01)
		tmp = c * i;
	elseif ((z * t) <= 2e-235)
		tmp = a * b;
	elseif ((z * t) <= 2e+95)
		tmp = x * y;
	else
		tmp = z * t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(z * t), $MachinePrecision], -1e+150], N[(z * t), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], -1e+103], N[(x * y), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], -0.01], N[(c * i), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 2e-235], N[(a * b), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 2e+95], N[(x * y), $MachinePrecision], N[(z * t), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 z t) < -9.99999999999999981e149 or 2.00000000000000004e95 < (*.f64 z t)

   1. Initial program 94.2%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 67.5

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

   5. Applied rewrites 67.5%

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

   if -9.99999999999999981e149 < (*.f64 z t) < -1e103 or 1.9999999999999999e-235 < (*.f64 z t) < 2.00000000000000004e95

   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 x around inf

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

      1. lower-*.f64 49.5

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

   5. Applied rewrites 49.5%

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

   if -1e103 < (*.f64 z t) < -0.0100000000000000002

   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 c around inf

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

      1. lower-*.f64 49.6

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

   5. Applied rewrites 49.6%

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

   if -0.0100000000000000002 < (*.f64 z t) < 1.9999999999999999e-235

   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 a around inf

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

      1. lower-*.f64 42.5

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

   5. Applied rewrites 42.5%

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

4. Final simplification 53.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+150}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;z \cdot t \leq -1 \cdot 10^{+103}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;z \cdot t \leq -0.01:\\ \;\;\;\;c \cdot i\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{-235}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+95}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z, t, x \cdot y\right)\\ t_2 := x \cdot y + z \cdot t\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+95}:\\ \;\;\;\;\mathsf{fma}\left(i, c, a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma z t (* x y))) (t_2 (+ (* x y) (* z t))))
   (if (<= t_2 (- INFINITY)) t_1 (if (<= t_2 2e+95) (fma i c (* a b)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma(z, t, (x * y));
	double t_2 = (x * y) + (z * t);
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_2 <= 2e+95) {
		tmp = fma(i, c, (a * b));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(z, t, Float64(x * y))
	t_2 = Float64(Float64(x * y) + Float64(z * t))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_2 <= 2e+95)
		tmp = fma(i, c, Float64(a * b));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 x y) (*.f64 z t)) < -inf.0 or 2.00000000000000004e95 < (+.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 95.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-+l+ N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-fma.f64 96.3

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

   4. Applied rewrites 96.3%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 84.1

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

   7. Applied rewrites 84.1%

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

   if -inf.0 < (+.f64 (*.f64 x y) (*.f64 z t)) < 2.00000000000000004e95

   1. Initial program 98.6%

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

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 70.6

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

   5. Applied rewrites 70.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 71.3

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

   7. Applied rewrites 71.3%

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

Alternative 6: 89.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \cdot i \leq -1 \cdot 10^{+60}:\\ \;\;\;\;\mathsf{fma}\left(a, b, \mathsf{fma}\left(c, i, x \cdot y\right)\right)\\ \mathbf{elif}\;c \cdot i \leq 10^{+168}:\\ \;\;\;\;\mathsf{fma}\left(z, t, \mathsf{fma}\left(x, y, a \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, b, \mathsf{fma}\left(x, y, c \cdot i\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* c i) -1e+60)
   (fma a b (fma c i (* x y)))
   (if (<= (* c i) 1e+168)
     (fma z t (fma x y (* a b)))
     (fma a b (fma 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) <= -1e+60) {
		tmp = fma(a, b, fma(c, i, (x * y)));
	} else if ((c * i) <= 1e+168) {
		tmp = fma(z, t, fma(x, y, (a * b)));
	} else {
		tmp = fma(a, b, fma(x, y, (c * i)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(c * i) <= -1e+60)
		tmp = fma(a, b, fma(c, i, Float64(x * y)));
	elseif (Float64(c * i) <= 1e+168)
		tmp = fma(z, t, fma(x, y, Float64(a * b)));
	else
		tmp = fma(a, b, fma(x, y, Float64(c * i)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(c * i), $MachinePrecision], -1e+60], N[(a * b + N[(c * i + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(c * i), $MachinePrecision], 1e+168], N[(z * t + N[(x * y + N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * b + N[(x * y + N[(c * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 c i) < -9.9999999999999995e59

   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 z around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 91.4

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

   5. Applied rewrites 91.4%

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

   if -9.9999999999999995e59 < (*.f64 c i) < 9.9999999999999993e167

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-+l+ N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-fma.f64 99.4

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 93.3

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

   7. Applied rewrites 93.3%

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

   if 9.9999999999999993e167 < (*.f64 c i)

   1. Initial program 94.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-+l+ N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-fma.f64 97.4

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

   4. Applied rewrites 97.4%

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

   5. Taylor expanded in z around 0

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

      1. lower-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 94.8

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

   7. Applied rewrites 94.8%

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

Alternative 7: 89.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \cdot i \leq -1 \cdot 10^{+60}:\\ \;\;\;\;\mathsf{fma}\left(a, b, \mathsf{fma}\left(c, i, x \cdot y\right)\right)\\ \mathbf{elif}\;c \cdot i \leq 10^{+168}:\\ \;\;\;\;\mathsf{fma}\left(x, y, \mathsf{fma}\left(a, b, z \cdot t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, b, \mathsf{fma}\left(x, y, c \cdot i\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* c i) -1e+60)
   (fma a b (fma c i (* x y)))
   (if (<= (* c i) 1e+168)
     (fma x y (fma a b (* z t)))
     (fma a b (fma 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) <= -1e+60) {
		tmp = fma(a, b, fma(c, i, (x * y)));
	} else if ((c * i) <= 1e+168) {
		tmp = fma(x, y, fma(a, b, (z * t)));
	} else {
		tmp = fma(a, b, fma(x, y, (c * i)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(c * i) <= -1e+60)
		tmp = fma(a, b, fma(c, i, Float64(x * y)));
	elseif (Float64(c * i) <= 1e+168)
		tmp = fma(x, y, fma(a, b, Float64(z * t)));
	else
		tmp = fma(a, b, fma(x, y, Float64(c * i)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(c * i), $MachinePrecision], -1e+60], N[(a * b + N[(c * i + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(c * i), $MachinePrecision], 1e+168], N[(x * y + N[(a * b + N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * b + N[(x * y + N[(c * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 c i) < -9.9999999999999995e59

   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 z around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 91.4

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

   5. Applied rewrites 91.4%

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

   if -9.9999999999999995e59 < (*.f64 c i) < 9.9999999999999993e167

   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 c around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 93.3

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

   5. Applied rewrites 93.3%

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

   if 9.9999999999999993e167 < (*.f64 c i)

   1. Initial program 94.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-+l+ N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-fma.f64 97.4

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

   4. Applied rewrites 97.4%

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

   5. Taylor expanded in z around 0

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

      1. lower-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 94.8

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

   7. Applied rewrites 94.8%

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

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \cdot i \leq -1 \cdot 10^{+60}:\\ \;\;\;\;\mathsf{fma}\left(a, b, \mathsf{fma}\left(c, i, x \cdot y\right)\right)\\ \mathbf{elif}\;c \cdot i \leq 10^{+168}:\\ \;\;\;\;\mathsf{fma}\left(x, y, \mathsf{fma}\left(a, b, z \cdot t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, b, \mathsf{fma}\left(x, y, c \cdot i\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 89.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(c, i, \mathsf{fma}\left(a, b, z \cdot t\right)\right)\\ \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+150}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+68}:\\ \;\;\;\;\mathsf{fma}\left(a, b, \mathsf{fma}\left(x, y, c \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma c i (fma a b (* z t)))))
   (if (<= (* z t) -1e+150)
     t_1
     (if (<= (* z t) 5e+68) (fma a b (fma x y (* 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(c, i, fma(a, b, (z * t)));
	double tmp;
	if ((z * t) <= -1e+150) {
		tmp = t_1;
	} else if ((z * t) <= 5e+68) {
		tmp = fma(a, b, fma(x, y, (c * i)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(c, i, fma(a, b, Float64(z * t)))
	tmp = 0.0
	if (Float64(z * t) <= -1e+150)
		tmp = t_1;
	elseif (Float64(z * t) <= 5e+68)
		tmp = fma(a, b, fma(x, y, 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[(c * i + N[(a * b + N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(z * t), $MachinePrecision], -1e+150], t$95$1, If[LessEqual[N[(z * t), $MachinePrecision], 5e+68], N[(a * b + N[(x * y + N[(c * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -9.99999999999999981e149 or 5.0000000000000004e68 < (*.f64 z t)

   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 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

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

      2. associate-+l+ N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 90.9

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

   5. Applied rewrites 90.9%

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

   if -9.99999999999999981e149 < (*.f64 z t) < 5.0000000000000004e68

   1. Initial program 99.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-+l+ N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-fma.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in z around 0

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

      1. lower-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 93.7

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

   7. Applied rewrites 93.7%

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

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+150}:\\ \;\;\;\;\mathsf{fma}\left(c, i, \mathsf{fma}\left(a, b, z \cdot t\right)\right)\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+68}:\\ \;\;\;\;\mathsf{fma}\left(a, b, \mathsf{fma}\left(x, y, c \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(c, i, \mathsf{fma}\left(a, b, z \cdot t\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 84.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+150}:\\ \;\;\;\;\mathsf{fma}\left(z, t, c \cdot i\right)\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+38}:\\ \;\;\;\;\mathsf{fma}\left(a, b, \mathsf{fma}\left(c, i, x \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t, x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* z t) -1e+150)
   (fma z t (* c i))
   (if (<= (* z t) 2e+38) (fma a b (fma c i (* x y))) (fma z t (* x y)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((z * t) <= -1e+150) {
		tmp = fma(z, t, (c * i));
	} else if ((z * t) <= 2e+38) {
		tmp = fma(a, b, fma(c, i, (x * y)));
	} else {
		tmp = fma(z, t, (x * y));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(z * t) <= -1e+150)
		tmp = fma(z, t, Float64(c * i));
	elseif (Float64(z * t) <= 2e+38)
		tmp = fma(a, b, fma(c, i, Float64(x * y)));
	else
		tmp = fma(z, t, Float64(x * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(z * t), $MachinePrecision], -1e+150], N[(z * t + N[(c * i), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 2e+38], N[(a * b + N[(c * i + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z * t + N[(x * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -9.99999999999999981e149

   1. Initial program 95.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-+l+ N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-fma.f64 95.0

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

   4. Applied rewrites 95.0%

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

   5. Taylor expanded in c around inf

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

      1. lower-*.f64 85.5

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

   7. Applied rewrites 85.5%

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

   if -9.99999999999999981e149 < (*.f64 z t) < 1.99999999999999995e38

   1. Initial program 99.4%

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

   3. Taylor expanded in z around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 94.1

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

   5. Applied rewrites 94.1%

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

   if 1.99999999999999995e38 < (*.f64 z t)

   1. Initial program 92.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-+l+ N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-fma.f64 94.6

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

   4. Applied rewrites 94.6%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 75.4

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

   7. Applied rewrites 75.4%

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

Alternative 10: 42.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \cdot i \leq -1.05 \cdot 10^{+138}:\\ \;\;\;\;c \cdot i\\ \mathbf{elif}\;c \cdot i \leq -2.2 \cdot 10^{-82}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;c \cdot i \leq 7.2 \cdot 10^{+168}:\\ \;\;\;\;z \cdot t\\ \mathbf{else}:\\ \;\;\;\;c \cdot i\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* c i) -1.05e+138)
   (* c i)
   (if (<= (* c i) -2.2e-82)
     (* a b)
     (if (<= (* c i) 7.2e+168) (* z t) (* c i)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((c * i) <= -1.05e+138) {
		tmp = c * i;
	} else if ((c * i) <= -2.2e-82) {
		tmp = a * b;
	} else if ((c * i) <= 7.2e+168) {
		tmp = z * t;
	} 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) <= (-1.05d+138)) then
        tmp = c * i
    else if ((c * i) <= (-2.2d-82)) then
        tmp = a * b
    else if ((c * i) <= 7.2d+168) then
        tmp = z * t
    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) <= -1.05e+138) {
		tmp = c * i;
	} else if ((c * i) <= -2.2e-82) {
		tmp = a * b;
	} else if ((c * i) <= 7.2e+168) {
		tmp = z * t;
	} else {
		tmp = c * i;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if (c * i) <= -1.05e+138:
		tmp = c * i
	elif (c * i) <= -2.2e-82:
		tmp = a * b
	elif (c * i) <= 7.2e+168:
		tmp = z * t
	else:
		tmp = c * i
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(c * i) <= -1.05e+138)
		tmp = Float64(c * i);
	elseif (Float64(c * i) <= -2.2e-82)
		tmp = Float64(a * b);
	elseif (Float64(c * i) <= 7.2e+168)
		tmp = Float64(z * t);
	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) <= -1.05e+138)
		tmp = c * i;
	elseif ((c * i) <= -2.2e-82)
		tmp = a * b;
	elseif ((c * i) <= 7.2e+168)
		tmp = z * t;
	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], -1.05e+138], N[(c * i), $MachinePrecision], If[LessEqual[N[(c * i), $MachinePrecision], -2.2e-82], N[(a * b), $MachinePrecision], If[LessEqual[N[(c * i), $MachinePrecision], 7.2e+168], N[(z * t), $MachinePrecision], N[(c * i), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 c i) < -1.05000000000000003e138 or 7.1999999999999999e168 < (*.f64 c i)

   1. Initial program 92.4%

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

   3. Taylor expanded in c around inf

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

      1. lower-*.f64 77.5

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

   5. Applied rewrites 77.5%

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

   if -1.05000000000000003e138 < (*.f64 c i) < -2.19999999999999986e-82

   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 inf

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

      1. lower-*.f64 36.2

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

   5. Applied rewrites 36.2%

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

   if -2.19999999999999986e-82 < (*.f64 c i) < 7.1999999999999999e168

   1. Initial program 98.6%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 39.2

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

   5. Applied rewrites 39.2%

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

4. Final simplification 48.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \cdot i \leq -1.05 \cdot 10^{+138}:\\ \;\;\;\;c \cdot i\\ \mathbf{elif}\;c \cdot i \leq -2.2 \cdot 10^{-82}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;c \cdot i \leq 7.2 \cdot 10^{+168}:\\ \;\;\;\;z \cdot t\\ \mathbf{else}:\\ \;\;\;\;c \cdot i\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 66.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, x, c \cdot i\right)\\ \mathbf{if}\;c \cdot i \leq -1 \cdot 10^{+60}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;c \cdot i \leq 2 \cdot 10^{+146}:\\ \;\;\;\;\mathsf{fma}\left(z, t, a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma y x (* c i))))
   (if (<= (* c i) -1e+60)
     t_1
     (if (<= (* c i) 2e+146) (fma z t (* a b)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma(y, x, (c * i));
	double tmp;
	if ((c * i) <= -1e+60) {
		tmp = t_1;
	} else if ((c * i) <= 2e+146) {
		tmp = fma(z, t, (a * b));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(y, x, Float64(c * i))
	tmp = 0.0
	if (Float64(c * i) <= -1e+60)
		tmp = t_1;
	elseif (Float64(c * i) <= 2e+146)
		tmp = fma(z, t, Float64(a * b));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(y * x + N[(c * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(c * i), $MachinePrecision], -1e+60], t$95$1, If[LessEqual[N[(c * i), $MachinePrecision], 2e+146], N[(z * t + N[(a * b), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 c i) < -9.9999999999999995e59 or 1.99999999999999987e146 < (*.f64 c i)

   1. Initial program 93.9%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 82.6

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

   5. Applied rewrites 82.6%

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

      1. *-commutative N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 82.6

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

   7. Applied rewrites 82.6%

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

   if -9.9999999999999995e59 < (*.f64 c i) < 1.99999999999999987e146

   1. Initial program 98.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-+l+ N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-fma.f64 99.4

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 69.0

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

   7. Applied rewrites 69.0%

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

Alternative 12: 62.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+150}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+95}:\\ \;\;\;\;\mathsf{fma}\left(a, b, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* z t) -1e+150)
   (* z t)
   (if (<= (* z t) 2e+95) (fma a b (* x y)) (* z t))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((z * t) <= -1e+150) {
		tmp = z * t;
	} else if ((z * t) <= 2e+95) {
		tmp = fma(a, b, (x * y));
	} else {
		tmp = z * t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(z * t) <= -1e+150)
		tmp = Float64(z * t);
	elseif (Float64(z * t) <= 2e+95)
		tmp = fma(a, b, Float64(x * y));
	else
		tmp = Float64(z * t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(z * t), $MachinePrecision], -1e+150], N[(z * t), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 2e+95], N[(a * b + N[(x * y), $MachinePrecision]), $MachinePrecision], N[(z * t), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -9.99999999999999981e149 or 2.00000000000000004e95 < (*.f64 z t)

   1. Initial program 94.2%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 67.5

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

   5. Applied rewrites 67.5%

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

   if -9.99999999999999981e149 < (*.f64 z t) < 2.00000000000000004e95

   1. Initial program 98.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-+l+ N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-fma.f64 99.4

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 71.0

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

   7. Applied rewrites 71.0%

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

   8. Taylor expanded in z around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 64.4

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

   10. Applied rewrites 64.4%

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

4. Final simplification 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+150}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+95}:\\ \;\;\;\;\mathsf{fma}\left(a, b, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 42.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \cdot i \leq -1.05 \cdot 10^{+138}:\\ \;\;\;\;c \cdot i\\ \mathbf{elif}\;c \cdot i \leq 4 \cdot 10^{+146}:\\ \;\;\;\;a \cdot b\\ \mathbf{else}:\\ \;\;\;\;c \cdot i\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* c i) -1.05e+138) (* c i) (if (<= (* c i) 4e+146) (* 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) <= -1.05e+138) {
		tmp = c * i;
	} else if ((c * i) <= 4e+146) {
		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) <= (-1.05d+138)) then
        tmp = c * i
    else if ((c * i) <= 4d+146) 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) <= -1.05e+138) {
		tmp = c * i;
	} else if ((c * i) <= 4e+146) {
		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) <= -1.05e+138:
		tmp = c * i
	elif (c * i) <= 4e+146:
		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) <= -1.05e+138)
		tmp = Float64(c * i);
	elseif (Float64(c * i) <= 4e+146)
		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) <= -1.05e+138)
		tmp = c * i;
	elseif ((c * i) <= 4e+146)
		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], -1.05e+138], N[(c * i), $MachinePrecision], If[LessEqual[N[(c * i), $MachinePrecision], 4e+146], N[(a * b), $MachinePrecision], N[(c * i), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 c i) < -1.05000000000000003e138 or 3.99999999999999973e146 < (*.f64 c i)

   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 c around inf

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

      1. lower-*.f64 72.9

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

   5. Applied rewrites 72.9%

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

   if -1.05000000000000003e138 < (*.f64 c i) < 3.99999999999999973e146

   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 a around inf

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

      1. lower-*.f64 33.4

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

   5. Applied rewrites 33.4%

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

Alternative 14: 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 97.2%

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

3. Taylor expanded in a around inf

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

   1. lower-*.f64 26.4

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

5. Applied rewrites 26.4%

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

Reproduce

herbie shell --seed 2024219 
(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)))