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

Percentage Accurate: 95.9% → 97.9%

Time: 10.8s

Alternatives: 20

Speedup: 0.5×

• 40.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 95.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.9% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-def 100.0%

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

      3. associate-+l+ 100.0%

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

      4. fma-def 100.0%

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

      5. fma-def 100.0%

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

   3. Simplified 100.0%

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

      1. fma-udef 100.0%

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

      2. fma-udef 100.0%

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

      3. associate-+r+ 100.0%

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

   5. Applied egg-rr 100.0%

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

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

   1. Initial program 0.0%

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

      1. associate-+l+ 0.0%

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

      2. associate-+l+ 0.0%

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

      3. fma-def 0.0%

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

      4. fma-def 14.3%

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

      5. fma-def 42.9%

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

   3. Simplified 42.9%

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

   4. Taylor expanded in a around 0 57.1%

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

   5. Taylor expanded in c around 0 57.7%

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

      1. fma-def 72.0%

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

   7. Simplified 72.0%

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

4. Final simplification 99.2%

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

Alternative 2: 97.9% accurate, 0.0× speedup

Math

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

FPCore

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

Julia

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

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := N[(x * y + N[(z * t + 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. Step-by-step derivation

   1. associate-+l+ 97.2%

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

   2. associate-+l+ 97.2%

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

   3. fma-def 97.2%

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

   4. fma-def 97.6%

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

   5. fma-def 98.4%

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

3. Simplified 98.4%

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

4. Final simplification 98.4%

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

Alternative 3: 98.2% accurate, 0.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   1. +-commutative 97.2%

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

   2. fma-def 98.0%

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

   3. associate-+l+ 98.0%

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

   4. fma-def 98.0%

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

   5. fma-def 98.4%

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

3. Simplified 98.4%

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

4. Final simplification 98.4%

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

Alternative 4: 97.7% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

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

   1. Initial program 0.0%

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

   2. Taylor expanded in z around inf 57.1%

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

      1. +-commutative 57.1%

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

      2. *-commutative 57.1%

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

      3. fma-def 71.4%

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

      4. *-commutative 71.4%

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

   4. Applied egg-rr 71.4%

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

4. Final simplification 99.2%

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

Alternative 5: 97.9% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

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

   1. Initial program 0.0%

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

      1. associate-+l+ 0.0%

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

      2. associate-+l+ 0.0%

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

      3. fma-def 0.0%

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

      4. fma-def 14.3%

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

      5. fma-def 42.9%

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

   3. Simplified 42.9%

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

   4. Taylor expanded in a around 0 57.1%

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

   5. Taylor expanded in c around 0 57.7%

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

      1. fma-def 72.0%

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

   7. Simplified 72.0%

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

4. Final simplification 99.2%

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

Alternative 6: 97.0% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   1. +-commutative 97.2%

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

   2. fma-def 98.0%

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

   3. associate-+l+ 98.0%

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

   4. fma-def 98.0%

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

   5. fma-def 98.4%

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

3. Simplified 98.4%

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

   1. fma-udef 98.4%

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

   2. fma-udef 98.0%

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

   3. associate-+l+ 98.0%

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

   4. +-commutative 98.0%

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

   5. associate-+r+ 98.0%

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

5. Applied egg-rr 98.0%

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

6. Final simplification 98.0%

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

Alternative 7: 42.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \cdot i \leq -7.5 \cdot 10^{+23}:\\ \;\;\;\;c \cdot i\\ \mathbf{elif}\;c \cdot i \leq -4.6 \cdot 10^{-77}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;c \cdot i \leq -4.7 \cdot 10^{-240}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;c \cdot i \leq -5 \cdot 10^{-321}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;c \cdot i \leq 1.7 \cdot 10^{-224}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;c \cdot i \leq 1.8 \cdot 10^{-28}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;c \cdot i \leq 9.8 \cdot 10^{+121}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;c \cdot i\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* c i) -7.5e+23)
   (* c i)
   (if (<= (* c i) -4.6e-77)
     (* a b)
     (if (<= (* c i) -4.7e-240)
       (* z t)
       (if (<= (* c i) -5e-321)
         (* a b)
         (if (<= (* c i) 1.7e-224)
           (* x y)
           (if (<= (* c i) 1.8e-28)
             (* a b)
             (if (<= (* c i) 9.8e+121) (* 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) <= -7.5e+23) {
		tmp = c * i;
	} else if ((c * i) <= -4.6e-77) {
		tmp = a * b;
	} else if ((c * i) <= -4.7e-240) {
		tmp = z * t;
	} else if ((c * i) <= -5e-321) {
		tmp = a * b;
	} else if ((c * i) <= 1.7e-224) {
		tmp = x * y;
	} else if ((c * i) <= 1.8e-28) {
		tmp = a * b;
	} else if ((c * i) <= 9.8e+121) {
		tmp = x * y;
	} 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) <= (-7.5d+23)) then
        tmp = c * i
    else if ((c * i) <= (-4.6d-77)) then
        tmp = a * b
    else if ((c * i) <= (-4.7d-240)) then
        tmp = z * t
    else if ((c * i) <= (-5d-321)) then
        tmp = a * b
    else if ((c * i) <= 1.7d-224) then
        tmp = x * y
    else if ((c * i) <= 1.8d-28) then
        tmp = a * b
    else if ((c * i) <= 9.8d+121) then
        tmp = x * y
    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) <= -7.5e+23) {
		tmp = c * i;
	} else if ((c * i) <= -4.6e-77) {
		tmp = a * b;
	} else if ((c * i) <= -4.7e-240) {
		tmp = z * t;
	} else if ((c * i) <= -5e-321) {
		tmp = a * b;
	} else if ((c * i) <= 1.7e-224) {
		tmp = x * y;
	} else if ((c * i) <= 1.8e-28) {
		tmp = a * b;
	} else if ((c * i) <= 9.8e+121) {
		tmp = x * y;
	} else {
		tmp = c * i;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if (c * i) <= -7.5e+23:
		tmp = c * i
	elif (c * i) <= -4.6e-77:
		tmp = a * b
	elif (c * i) <= -4.7e-240:
		tmp = z * t
	elif (c * i) <= -5e-321:
		tmp = a * b
	elif (c * i) <= 1.7e-224:
		tmp = x * y
	elif (c * i) <= 1.8e-28:
		tmp = a * b
	elif (c * i) <= 9.8e+121:
		tmp = 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) <= -7.5e+23)
		tmp = Float64(c * i);
	elseif (Float64(c * i) <= -4.6e-77)
		tmp = Float64(a * b);
	elseif (Float64(c * i) <= -4.7e-240)
		tmp = Float64(z * t);
	elseif (Float64(c * i) <= -5e-321)
		tmp = Float64(a * b);
	elseif (Float64(c * i) <= 1.7e-224)
		tmp = Float64(x * y);
	elseif (Float64(c * i) <= 1.8e-28)
		tmp = Float64(a * b);
	elseif (Float64(c * i) <= 9.8e+121)
		tmp = Float64(x * y);
	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) <= -7.5e+23)
		tmp = c * i;
	elseif ((c * i) <= -4.6e-77)
		tmp = a * b;
	elseif ((c * i) <= -4.7e-240)
		tmp = z * t;
	elseif ((c * i) <= -5e-321)
		tmp = a * b;
	elseif ((c * i) <= 1.7e-224)
		tmp = x * y;
	elseif ((c * i) <= 1.8e-28)
		tmp = a * b;
	elseif ((c * i) <= 9.8e+121)
		tmp = x * y;
	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], -7.5e+23], N[(c * i), $MachinePrecision], If[LessEqual[N[(c * i), $MachinePrecision], -4.6e-77], N[(a * b), $MachinePrecision], If[LessEqual[N[(c * i), $MachinePrecision], -4.7e-240], N[(z * t), $MachinePrecision], If[LessEqual[N[(c * i), $MachinePrecision], -5e-321], N[(a * b), $MachinePrecision], If[LessEqual[N[(c * i), $MachinePrecision], 1.7e-224], N[(x * y), $MachinePrecision], If[LessEqual[N[(c * i), $MachinePrecision], 1.8e-28], N[(a * b), $MachinePrecision], If[LessEqual[N[(c * i), $MachinePrecision], 9.8e+121], N[(x * y), $MachinePrecision], N[(c * i), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 c i) < -7.49999999999999987e23 or 9.7999999999999995e121 < (*.f64 c i)

   1. Initial program 94.7%

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

   2. Taylor expanded in c around inf 69.1%

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

   if -7.49999999999999987e23 < (*.f64 c i) < -4.59999999999999997e-77 or -4.70000000000000012e-240 < (*.f64 c i) < -4.99994e-321 or 1.69999999999999996e-224 < (*.f64 c i) < 1.7999999999999999e-28

   1. Initial program 97.1%

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

   2. Taylor expanded in x around 0 74.3%

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

   3. Taylor expanded in t around 0 55.8%

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

   4. Taylor expanded in c around 0 52.6%

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

   if -4.59999999999999997e-77 < (*.f64 c i) < -4.70000000000000012e-240

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 84.5%

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

      1. +-commutative 84.5%

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

      2. *-commutative 84.5%

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

      3. fma-def 84.5%

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

   4. Applied egg-rr 84.5%

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

   5. Taylor expanded in c around 0 79.4%

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

      1. +-commutative 79.4%

         \[\leadsto \color{blue}{t \cdot z + a \cdot b} \]

      2. fma-udef 79.4%

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

   7. Simplified 79.4%

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

   8. Taylor expanded in t around inf 59.1%

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

   if -4.99994e-321 < (*.f64 c i) < 1.69999999999999996e-224 or 1.7999999999999999e-28 < (*.f64 c i) < 9.7999999999999995e121

   1. Initial program 99.9%

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

      1. associate-+l+ 99.9%

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

      2. associate-+l+ 99.9%

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

      3. fma-def 100.0%

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

      4. fma-def 100.0%

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

      5. fma-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in a around 0 85.1%

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

   5. Taylor expanded in x around inf 55.5%

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

4. Final simplification 60.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \cdot i \leq -7.5 \cdot 10^{+23}:\\ \;\;\;\;c \cdot i\\ \mathbf{elif}\;c \cdot i \leq -4.6 \cdot 10^{-77}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;c \cdot i \leq -4.7 \cdot 10^{-240}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;c \cdot i \leq -5 \cdot 10^{-321}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;c \cdot i \leq 1.7 \cdot 10^{-224}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;c \cdot i \leq 1.8 \cdot 10^{-28}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;c \cdot i \leq 9.8 \cdot 10^{+121}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;c \cdot i\\ \end{array} \]

Alternative 8: 65.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot b + x \cdot y\\ t_2 := a \cdot b + z \cdot t\\ t_3 := c \cdot i + a \cdot b\\ \mathbf{if}\;c \cdot i \leq -1.6 \cdot 10^{+103}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;c \cdot i \leq -1.8 \cdot 10^{-53}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \cdot i \leq -6.5 \cdot 10^{-308}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;c \cdot i \leq 2.6 \cdot 10^{-223}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \cdot i \leq 2.6 \cdot 10^{-165}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;c \cdot i \leq 6.4 \cdot 10^{+120}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ (* a b) (* x y)))
        (t_2 (+ (* a b) (* z t)))
        (t_3 (+ (* c i) (* a b))))
   (if (<= (* c i) -1.6e+103)
     t_3
     (if (<= (* c i) -1.8e-53)
       t_1
       (if (<= (* c i) -6.5e-308)
         t_2
         (if (<= (* c i) 2.6e-223)
           t_1
           (if (<= (* c i) 2.6e-165)
             t_2
             (if (<= (* c i) 6.4e+120) t_1 t_3))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (a * b) + (x * y);
	double t_2 = (a * b) + (z * t);
	double t_3 = (c * i) + (a * b);
	double tmp;
	if ((c * i) <= -1.6e+103) {
		tmp = t_3;
	} else if ((c * i) <= -1.8e-53) {
		tmp = t_1;
	} else if ((c * i) <= -6.5e-308) {
		tmp = t_2;
	} else if ((c * i) <= 2.6e-223) {
		tmp = t_1;
	} else if ((c * i) <= 2.6e-165) {
		tmp = t_2;
	} else if ((c * i) <= 6.4e+120) {
		tmp = t_1;
	} else {
		tmp = t_3;
	}
	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) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = (a * b) + (x * y)
    t_2 = (a * b) + (z * t)
    t_3 = (c * i) + (a * b)
    if ((c * i) <= (-1.6d+103)) then
        tmp = t_3
    else if ((c * i) <= (-1.8d-53)) then
        tmp = t_1
    else if ((c * i) <= (-6.5d-308)) then
        tmp = t_2
    else if ((c * i) <= 2.6d-223) then
        tmp = t_1
    else if ((c * i) <= 2.6d-165) then
        tmp = t_2
    else if ((c * i) <= 6.4d+120) then
        tmp = t_1
    else
        tmp = t_3
    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 t_1 = (a * b) + (x * y);
	double t_2 = (a * b) + (z * t);
	double t_3 = (c * i) + (a * b);
	double tmp;
	if ((c * i) <= -1.6e+103) {
		tmp = t_3;
	} else if ((c * i) <= -1.8e-53) {
		tmp = t_1;
	} else if ((c * i) <= -6.5e-308) {
		tmp = t_2;
	} else if ((c * i) <= 2.6e-223) {
		tmp = t_1;
	} else if ((c * i) <= 2.6e-165) {
		tmp = t_2;
	} else if ((c * i) <= 6.4e+120) {
		tmp = t_1;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	t_1 = (a * b) + (x * y)
	t_2 = (a * b) + (z * t)
	t_3 = (c * i) + (a * b)
	tmp = 0
	if (c * i) <= -1.6e+103:
		tmp = t_3
	elif (c * i) <= -1.8e-53:
		tmp = t_1
	elif (c * i) <= -6.5e-308:
		tmp = t_2
	elif (c * i) <= 2.6e-223:
		tmp = t_1
	elif (c * i) <= 2.6e-165:
		tmp = t_2
	elif (c * i) <= 6.4e+120:
		tmp = t_1
	else:
		tmp = t_3
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(a * b) + Float64(x * y))
	t_2 = Float64(Float64(a * b) + Float64(z * t))
	t_3 = Float64(Float64(c * i) + Float64(a * b))
	tmp = 0.0
	if (Float64(c * i) <= -1.6e+103)
		tmp = t_3;
	elseif (Float64(c * i) <= -1.8e-53)
		tmp = t_1;
	elseif (Float64(c * i) <= -6.5e-308)
		tmp = t_2;
	elseif (Float64(c * i) <= 2.6e-223)
		tmp = t_1;
	elseif (Float64(c * i) <= 2.6e-165)
		tmp = t_2;
	elseif (Float64(c * i) <= 6.4e+120)
		tmp = t_1;
	else
		tmp = t_3;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	t_1 = (a * b) + (x * y);
	t_2 = (a * b) + (z * t);
	t_3 = (c * i) + (a * b);
	tmp = 0.0;
	if ((c * i) <= -1.6e+103)
		tmp = t_3;
	elseif ((c * i) <= -1.8e-53)
		tmp = t_1;
	elseif ((c * i) <= -6.5e-308)
		tmp = t_2;
	elseif ((c * i) <= 2.6e-223)
		tmp = t_1;
	elseif ((c * i) <= 2.6e-165)
		tmp = t_2;
	elseif ((c * i) <= 6.4e+120)
		tmp = t_1;
	else
		tmp = t_3;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(a * b), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(a * b), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(c * i), $MachinePrecision] + N[(a * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(c * i), $MachinePrecision], -1.6e+103], t$95$3, If[LessEqual[N[(c * i), $MachinePrecision], -1.8e-53], t$95$1, If[LessEqual[N[(c * i), $MachinePrecision], -6.5e-308], t$95$2, If[LessEqual[N[(c * i), $MachinePrecision], 2.6e-223], t$95$1, If[LessEqual[N[(c * i), $MachinePrecision], 2.6e-165], t$95$2, If[LessEqual[N[(c * i), $MachinePrecision], 6.4e+120], t$95$1, t$95$3]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 c i) < -1.59999999999999996e103 or 6.39999999999999964e120 < (*.f64 c i)

   1. Initial program 93.8%

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

   2. Taylor expanded in a around inf 81.6%

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

   if -1.59999999999999996e103 < (*.f64 c i) < -1.7999999999999999e-53 or -6.4999999999999999e-308 < (*.f64 c i) < 2.6e-223 or 2.60000000000000007e-165 < (*.f64 c i) < 6.39999999999999964e120

   1. Initial program 98.5%

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

   2. Taylor expanded in z around 0 77.4%

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

   3. Taylor expanded in c around 0 69.6%

      \[\leadsto \color{blue}{a \cdot b + y \cdot x} \]

   if -1.7999999999999999e-53 < (*.f64 c i) < -6.4999999999999999e-308 or 2.6e-223 < (*.f64 c i) < 2.60000000000000007e-165

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 91.5%

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

      1. +-commutative 91.5%

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

      2. *-commutative 91.5%

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

      3. fma-def 91.5%

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

   4. Applied egg-rr 91.5%

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

   5. Taylor expanded in c around 0 88.7%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \cdot i \leq -1.6 \cdot 10^{+103}:\\ \;\;\;\;c \cdot i + a \cdot b\\ \mathbf{elif}\;c \cdot i \leq -1.8 \cdot 10^{-53}:\\ \;\;\;\;a \cdot b + x \cdot y\\ \mathbf{elif}\;c \cdot i \leq -6.5 \cdot 10^{-308}:\\ \;\;\;\;a \cdot b + z \cdot t\\ \mathbf{elif}\;c \cdot i \leq 2.6 \cdot 10^{-223}:\\ \;\;\;\;a \cdot b + x \cdot y\\ \mathbf{elif}\;c \cdot i \leq 2.6 \cdot 10^{-165}:\\ \;\;\;\;a \cdot b + z \cdot t\\ \mathbf{elif}\;c \cdot i \leq 6.4 \cdot 10^{+120}:\\ \;\;\;\;a \cdot b + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;c \cdot i + a \cdot b\\ \end{array} \]

Alternative 9: 65.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot b + z \cdot t\\ t_2 := z \cdot t + x \cdot y\\ \mathbf{if}\;c \cdot i \leq -1.6 \cdot 10^{-10}:\\ \;\;\;\;c \cdot i + x \cdot y\\ \mathbf{elif}\;c \cdot i \leq -5 \cdot 10^{-321}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \cdot i \leq 3.5 \cdot 10^{-226}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;c \cdot i \leq 3.65 \cdot 10^{-159}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \cdot i \leq 10^{-28}:\\ \;\;\;\;a \cdot b + x \cdot y\\ \mathbf{elif}\;c \cdot i \leq 1.25 \cdot 10^{+122}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;c \cdot i + a \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ (* a b) (* z t))) (t_2 (+ (* z t) (* x y))))
   (if (<= (* c i) -1.6e-10)
     (+ (* c i) (* x y))
     (if (<= (* c i) -5e-321)
       t_1
       (if (<= (* c i) 3.5e-226)
         t_2
         (if (<= (* c i) 3.65e-159)
           t_1
           (if (<= (* c i) 1e-28)
             (+ (* a b) (* x y))
             (if (<= (* c i) 1.25e+122) t_2 (+ (* c i) (* a b))))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (a * b) + (z * t);
	double t_2 = (z * t) + (x * y);
	double tmp;
	if ((c * i) <= -1.6e-10) {
		tmp = (c * i) + (x * y);
	} else if ((c * i) <= -5e-321) {
		tmp = t_1;
	} else if ((c * i) <= 3.5e-226) {
		tmp = t_2;
	} else if ((c * i) <= 3.65e-159) {
		tmp = t_1;
	} else if ((c * i) <= 1e-28) {
		tmp = (a * b) + (x * y);
	} else if ((c * i) <= 1.25e+122) {
		tmp = t_2;
	} else {
		tmp = (c * i) + (a * b);
	}
	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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (a * b) + (z * t)
    t_2 = (z * t) + (x * y)
    if ((c * i) <= (-1.6d-10)) then
        tmp = (c * i) + (x * y)
    else if ((c * i) <= (-5d-321)) then
        tmp = t_1
    else if ((c * i) <= 3.5d-226) then
        tmp = t_2
    else if ((c * i) <= 3.65d-159) then
        tmp = t_1
    else if ((c * i) <= 1d-28) then
        tmp = (a * b) + (x * y)
    else if ((c * i) <= 1.25d+122) then
        tmp = t_2
    else
        tmp = (c * i) + (a * b)
    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 t_1 = (a * b) + (z * t);
	double t_2 = (z * t) + (x * y);
	double tmp;
	if ((c * i) <= -1.6e-10) {
		tmp = (c * i) + (x * y);
	} else if ((c * i) <= -5e-321) {
		tmp = t_1;
	} else if ((c * i) <= 3.5e-226) {
		tmp = t_2;
	} else if ((c * i) <= 3.65e-159) {
		tmp = t_1;
	} else if ((c * i) <= 1e-28) {
		tmp = (a * b) + (x * y);
	} else if ((c * i) <= 1.25e+122) {
		tmp = t_2;
	} else {
		tmp = (c * i) + (a * b);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	t_1 = (a * b) + (z * t)
	t_2 = (z * t) + (x * y)
	tmp = 0
	if (c * i) <= -1.6e-10:
		tmp = (c * i) + (x * y)
	elif (c * i) <= -5e-321:
		tmp = t_1
	elif (c * i) <= 3.5e-226:
		tmp = t_2
	elif (c * i) <= 3.65e-159:
		tmp = t_1
	elif (c * i) <= 1e-28:
		tmp = (a * b) + (x * y)
	elif (c * i) <= 1.25e+122:
		tmp = t_2
	else:
		tmp = (c * i) + (a * b)
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(a * b) + Float64(z * t))
	t_2 = Float64(Float64(z * t) + Float64(x * y))
	tmp = 0.0
	if (Float64(c * i) <= -1.6e-10)
		tmp = Float64(Float64(c * i) + Float64(x * y));
	elseif (Float64(c * i) <= -5e-321)
		tmp = t_1;
	elseif (Float64(c * i) <= 3.5e-226)
		tmp = t_2;
	elseif (Float64(c * i) <= 3.65e-159)
		tmp = t_1;
	elseif (Float64(c * i) <= 1e-28)
		tmp = Float64(Float64(a * b) + Float64(x * y));
	elseif (Float64(c * i) <= 1.25e+122)
		tmp = t_2;
	else
		tmp = Float64(Float64(c * i) + Float64(a * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	t_1 = (a * b) + (z * t);
	t_2 = (z * t) + (x * y);
	tmp = 0.0;
	if ((c * i) <= -1.6e-10)
		tmp = (c * i) + (x * y);
	elseif ((c * i) <= -5e-321)
		tmp = t_1;
	elseif ((c * i) <= 3.5e-226)
		tmp = t_2;
	elseif ((c * i) <= 3.65e-159)
		tmp = t_1;
	elseif ((c * i) <= 1e-28)
		tmp = (a * b) + (x * y);
	elseif ((c * i) <= 1.25e+122)
		tmp = t_2;
	else
		tmp = (c * i) + (a * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(a * b), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(z * t), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(c * i), $MachinePrecision], -1.6e-10], N[(N[(c * i), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(c * i), $MachinePrecision], -5e-321], t$95$1, If[LessEqual[N[(c * i), $MachinePrecision], 3.5e-226], t$95$2, If[LessEqual[N[(c * i), $MachinePrecision], 3.65e-159], t$95$1, If[LessEqual[N[(c * i), $MachinePrecision], 1e-28], N[(N[(a * b), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(c * i), $MachinePrecision], 1.25e+122], t$95$2, N[(N[(c * i), $MachinePrecision] + N[(a * b), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 c i) < -1.5999999999999999e-10

   1. Initial program 98.4%

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

   2. Taylor expanded in x around inf 76.0%

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

   if -1.5999999999999999e-10 < (*.f64 c i) < -4.99994e-321 or 3.5e-226 < (*.f64 c i) < 3.6499999999999998e-159

   1. Initial program 97.9%

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

   2. Taylor expanded in x around 0 86.0%

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

      1. +-commutative 86.0%

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

      2. *-commutative 86.0%

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

      3. fma-def 88.1%

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

   4. Applied egg-rr 88.1%

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

   5. Taylor expanded in c around 0 83.0%

      \[\leadsto \color{blue}{a \cdot b + t \cdot z} \]

   if -4.99994e-321 < (*.f64 c i) < 3.5e-226 or 9.99999999999999971e-29 < (*.f64 c i) < 1.24999999999999997e122

   1. Initial program 99.9%

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

      1. associate-+l+ 99.9%

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

      2. associate-+l+ 99.9%

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

      3. fma-def 100.0%

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

      4. fma-def 100.0%

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

      5. fma-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in a around 0 85.1%

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

   5. Taylor expanded in c around 0 79.8%

      \[\leadsto \color{blue}{y \cdot x + t \cdot z} \]

   if 3.6499999999999998e-159 < (*.f64 c i) < 9.99999999999999971e-29

   1. Initial program 96.8%

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

   2. Taylor expanded in z around 0 75.4%

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

   3. Taylor expanded in c around 0 70.6%

      \[\leadsto \color{blue}{a \cdot b + y \cdot x} \]

   if 1.24999999999999997e122 < (*.f64 c i)

   1. Initial program 90.2%

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

   2. Taylor expanded in a around inf 85.5%

      \[\leadsto \color{blue}{a \cdot b} + c \cdot i \]
3. Recombined 5 regimes into one program.

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \cdot i \leq -1.6 \cdot 10^{-10}:\\ \;\;\;\;c \cdot i + x \cdot y\\ \mathbf{elif}\;c \cdot i \leq -5 \cdot 10^{-321}:\\ \;\;\;\;a \cdot b + z \cdot t\\ \mathbf{elif}\;c \cdot i \leq 3.5 \cdot 10^{-226}:\\ \;\;\;\;z \cdot t + x \cdot y\\ \mathbf{elif}\;c \cdot i \leq 3.65 \cdot 10^{-159}:\\ \;\;\;\;a \cdot b + z \cdot t\\ \mathbf{elif}\;c \cdot i \leq 10^{-28}:\\ \;\;\;\;a \cdot b + x \cdot y\\ \mathbf{elif}\;c \cdot i \leq 1.25 \cdot 10^{+122}:\\ \;\;\;\;z \cdot t + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;c \cdot i + a \cdot b\\ \end{array} \]

Alternative 10: 97.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot t + x \cdot y\\ t_2 := c \cdot i + \left(a \cdot b + t_1\right)\\ \mathbf{if}\;t_2 \leq \infty:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

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

C

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

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (z * t) + (x * y);
	double t_2 = (c * i) + ((a * b) + t_1);
	double tmp;
	if (t_2 <= Double.POSITIVE_INFINITY) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	t_1 = (z * t) + (x * y)
	t_2 = (c * i) + ((a * b) + t_1)
	tmp = 0
	if t_2 <= math.inf:
		tmp = t_2
	else:
		tmp = t_1
	return tmp

Julia

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

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	t_1 = (z * t) + (x * y);
	t_2 = (c * i) + ((a * b) + t_1);
	tmp = 0.0;
	if (t_2 <= Inf)
		tmp = t_2;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

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

   1. Initial program 0.0%

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

      1. associate-+l+ 0.0%

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

      2. associate-+l+ 0.0%

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

      3. fma-def 0.0%

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

      4. fma-def 14.3%

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

      5. fma-def 42.9%

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

   3. Simplified 42.9%

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

   4. Taylor expanded in a around 0 57.1%

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

   5. Taylor expanded in c around 0 57.7%

      \[\leadsto \color{blue}{y \cdot x + t \cdot z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.8%

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

Alternative 11: 42.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \cdot i \leq -1.15 \cdot 10^{+25}:\\ \;\;\;\;c \cdot i\\ \mathbf{elif}\;c \cdot i \leq -5 \cdot 10^{-77}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;c \cdot i \leq 1.45 \cdot 10^{-179}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;c \cdot i \leq 9.5 \cdot 10^{-22}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;c \cdot i \leq 1.35 \cdot 10^{+69}:\\ \;\;\;\;z \cdot t\\ \mathbf{else}:\\ \;\;\;\;c \cdot i\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* c i) -1.15e+25)
   (* c i)
   (if (<= (* c i) -5e-77)
     (* a b)
     (if (<= (* c i) 1.45e-179)
       (* z t)
       (if (<= (* c i) 9.5e-22)
         (* a b)
         (if (<= (* c i) 1.35e+69) (* 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) <= -1.15e+25) {
		tmp = c * i;
	} else if ((c * i) <= -5e-77) {
		tmp = a * b;
	} else if ((c * i) <= 1.45e-179) {
		tmp = z * t;
	} else if ((c * i) <= 9.5e-22) {
		tmp = a * b;
	} else if ((c * i) <= 1.35e+69) {
		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) <= (-1.15d+25)) then
        tmp = c * i
    else if ((c * i) <= (-5d-77)) then
        tmp = a * b
    else if ((c * i) <= 1.45d-179) then
        tmp = z * t
    else if ((c * i) <= 9.5d-22) then
        tmp = a * b
    else if ((c * i) <= 1.35d+69) 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) <= -1.15e+25) {
		tmp = c * i;
	} else if ((c * i) <= -5e-77) {
		tmp = a * b;
	} else if ((c * i) <= 1.45e-179) {
		tmp = z * t;
	} else if ((c * i) <= 9.5e-22) {
		tmp = a * b;
	} else if ((c * i) <= 1.35e+69) {
		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) <= -1.15e+25:
		tmp = c * i
	elif (c * i) <= -5e-77:
		tmp = a * b
	elif (c * i) <= 1.45e-179:
		tmp = z * t
	elif (c * i) <= 9.5e-22:
		tmp = a * b
	elif (c * i) <= 1.35e+69:
		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) <= -1.15e+25)
		tmp = Float64(c * i);
	elseif (Float64(c * i) <= -5e-77)
		tmp = Float64(a * b);
	elseif (Float64(c * i) <= 1.45e-179)
		tmp = Float64(z * t);
	elseif (Float64(c * i) <= 9.5e-22)
		tmp = Float64(a * b);
	elseif (Float64(c * i) <= 1.35e+69)
		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) <= -1.15e+25)
		tmp = c * i;
	elseif ((c * i) <= -5e-77)
		tmp = a * b;
	elseif ((c * i) <= 1.45e-179)
		tmp = z * t;
	elseif ((c * i) <= 9.5e-22)
		tmp = a * b;
	elseif ((c * i) <= 1.35e+69)
		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], -1.15e+25], N[(c * i), $MachinePrecision], If[LessEqual[N[(c * i), $MachinePrecision], -5e-77], N[(a * b), $MachinePrecision], If[LessEqual[N[(c * i), $MachinePrecision], 1.45e-179], N[(z * t), $MachinePrecision], If[LessEqual[N[(c * i), $MachinePrecision], 9.5e-22], N[(a * b), $MachinePrecision], If[LessEqual[N[(c * i), $MachinePrecision], 1.35e+69], N[(z * t), $MachinePrecision], N[(c * i), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 c i) < -1.1499999999999999e25 or 1.3499999999999999e69 < (*.f64 c i)

   1. Initial program 95.3%

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

   2. Taylor expanded in c around inf 62.6%

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

   if -1.1499999999999999e25 < (*.f64 c i) < -4.99999999999999963e-77 or 1.4499999999999999e-179 < (*.f64 c i) < 9.4999999999999994e-22

   1. Initial program 96.7%

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

   2. Taylor expanded in x around 0 69.0%

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

   3. Taylor expanded in t around 0 54.0%

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

   4. Taylor expanded in c around 0 50.4%

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

   if -4.99999999999999963e-77 < (*.f64 c i) < 1.4499999999999999e-179 or 9.4999999999999994e-22 < (*.f64 c i) < 1.3499999999999999e69

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 62.8%

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

      1. +-commutative 62.8%

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

      2. *-commutative 62.8%

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

      3. fma-def 62.8%

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

   4. Applied egg-rr 62.8%

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

   5. Taylor expanded in c around 0 58.3%

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

      1. +-commutative 58.3%

         \[\leadsto \color{blue}{t \cdot z + a \cdot b} \]

      2. fma-udef 58.4%

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

   7. Simplified 58.4%

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

   8. Taylor expanded in t around inf 38.7%

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

4. Final simplification 51.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \cdot i \leq -1.15 \cdot 10^{+25}:\\ \;\;\;\;c \cdot i\\ \mathbf{elif}\;c \cdot i \leq -5 \cdot 10^{-77}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;c \cdot i \leq 1.45 \cdot 10^{-179}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;c \cdot i \leq 9.5 \cdot 10^{-22}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;c \cdot i \leq 1.35 \cdot 10^{+69}:\\ \;\;\;\;z \cdot t\\ \mathbf{else}:\\ \;\;\;\;c \cdot i\\ \end{array} \]

Alternative 12: 76.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := c \cdot i + \left(a \cdot b + z \cdot t\right)\\ t_2 := z \cdot t + x \cdot y\\ \mathbf{if}\;y \leq -3:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 6.4 \cdot 10^{+77}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{+114}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 4.7 \cdot 10^{+222}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;a \cdot b + x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ (* c i) (+ (* a b) (* z t)))) (t_2 (+ (* z t) (* x y))))
   (if (<= y -3.0)
     t_2
     (if (<= y 6.4e+77)
       t_1
       (if (<= y 2.6e+114)
         t_2
         (if (<= y 4.7e+222) t_1 (+ (* a b) (* x y))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (c * i) + ((a * b) + (z * t));
	double t_2 = (z * t) + (x * y);
	double tmp;
	if (y <= -3.0) {
		tmp = t_2;
	} else if (y <= 6.4e+77) {
		tmp = t_1;
	} else if (y <= 2.6e+114) {
		tmp = t_2;
	} else if (y <= 4.7e+222) {
		tmp = t_1;
	} else {
		tmp = (a * b) + (x * y);
	}
	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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (c * i) + ((a * b) + (z * t))
    t_2 = (z * t) + (x * y)
    if (y <= (-3.0d0)) then
        tmp = t_2
    else if (y <= 6.4d+77) then
        tmp = t_1
    else if (y <= 2.6d+114) then
        tmp = t_2
    else if (y <= 4.7d+222) then
        tmp = t_1
    else
        tmp = (a * b) + (x * y)
    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 t_1 = (c * i) + ((a * b) + (z * t));
	double t_2 = (z * t) + (x * y);
	double tmp;
	if (y <= -3.0) {
		tmp = t_2;
	} else if (y <= 6.4e+77) {
		tmp = t_1;
	} else if (y <= 2.6e+114) {
		tmp = t_2;
	} else if (y <= 4.7e+222) {
		tmp = t_1;
	} else {
		tmp = (a * b) + (x * y);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	t_1 = (c * i) + ((a * b) + (z * t))
	t_2 = (z * t) + (x * y)
	tmp = 0
	if y <= -3.0:
		tmp = t_2
	elif y <= 6.4e+77:
		tmp = t_1
	elif y <= 2.6e+114:
		tmp = t_2
	elif y <= 4.7e+222:
		tmp = t_1
	else:
		tmp = (a * b) + (x * y)
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(c * i) + Float64(Float64(a * b) + Float64(z * t)))
	t_2 = Float64(Float64(z * t) + Float64(x * y))
	tmp = 0.0
	if (y <= -3.0)
		tmp = t_2;
	elseif (y <= 6.4e+77)
		tmp = t_1;
	elseif (y <= 2.6e+114)
		tmp = t_2;
	elseif (y <= 4.7e+222)
		tmp = t_1;
	else
		tmp = Float64(Float64(a * b) + Float64(x * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	t_1 = (c * i) + ((a * b) + (z * t));
	t_2 = (z * t) + (x * y);
	tmp = 0.0;
	if (y <= -3.0)
		tmp = t_2;
	elseif (y <= 6.4e+77)
		tmp = t_1;
	elseif (y <= 2.6e+114)
		tmp = t_2;
	elseif (y <= 4.7e+222)
		tmp = t_1;
	else
		tmp = (a * b) + (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(c * i), $MachinePrecision] + N[(N[(a * b), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(z * t), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -3.0], t$95$2, If[LessEqual[y, 6.4e+77], t$95$1, If[LessEqual[y, 2.6e+114], t$95$2, If[LessEqual[y, 4.7e+222], t$95$1, N[(N[(a * b), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -3 or 6.4000000000000003e77 < y < 2.6e114

   1. Initial program 96.0%

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

      1. associate-+l+ 96.0%

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

      2. associate-+l+ 96.0%

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

      3. fma-def 96.1%

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

      4. fma-def 97.4%

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

      5. fma-def 98.7%

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

   3. Simplified 98.7%

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

   4. Taylor expanded in a around 0 82.7%

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

   5. Taylor expanded in c around 0 68.3%

      \[\leadsto \color{blue}{y \cdot x + t \cdot z} \]

   if -3 < y < 6.4000000000000003e77 or 2.6e114 < y < 4.6999999999999999e222

   1. Initial program 98.1%

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

   2. Taylor expanded in x around 0 87.6%

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

   if 4.6999999999999999e222 < y

   1. Initial program 95.8%

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

   2. Taylor expanded in z around 0 95.8%

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

   3. Taylor expanded in c around 0 100.0%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3:\\ \;\;\;\;z \cdot t + x \cdot y\\ \mathbf{elif}\;y \leq 6.4 \cdot 10^{+77}:\\ \;\;\;\;c \cdot i + \left(a \cdot b + z \cdot t\right)\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{+114}:\\ \;\;\;\;z \cdot t + x \cdot y\\ \mathbf{elif}\;y \leq 4.7 \cdot 10^{+222}:\\ \;\;\;\;c \cdot i + \left(a \cdot b + z \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot b + x \cdot y\\ \end{array} \]

Alternative 13: 77.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot t + x \cdot y\\ \mathbf{if}\;z \leq -2.9 \cdot 10^{+233}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -5.8 \cdot 10^{+204}:\\ \;\;\;\;c \cdot i + \left(a \cdot b + z \cdot t\right)\\ \mathbf{elif}\;z \leq -2.5 \cdot 10^{+125}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1700000000000:\\ \;\;\;\;c \cdot i + \left(a \cdot b + x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot i + z \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ (* z t) (* x y))))
   (if (<= z -2.9e+233)
     t_1
     (if (<= z -5.8e+204)
       (+ (* c i) (+ (* a b) (* z t)))
       (if (<= z -2.5e+125)
         t_1
         (if (<= z 1700000000000.0)
           (+ (* c i) (+ (* a b) (* x y)))
           (+ (* 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 = (z * t) + (x * y);
	double tmp;
	if (z <= -2.9e+233) {
		tmp = t_1;
	} else if (z <= -5.8e+204) {
		tmp = (c * i) + ((a * b) + (z * t));
	} else if (z <= -2.5e+125) {
		tmp = t_1;
	} else if (z <= 1700000000000.0) {
		tmp = (c * i) + ((a * b) + (x * y));
	} else {
		tmp = (c * i) + (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) :: t_1
    real(8) :: tmp
    t_1 = (z * t) + (x * y)
    if (z <= (-2.9d+233)) then
        tmp = t_1
    else if (z <= (-5.8d+204)) then
        tmp = (c * i) + ((a * b) + (z * t))
    else if (z <= (-2.5d+125)) then
        tmp = t_1
    else if (z <= 1700000000000.0d0) then
        tmp = (c * i) + ((a * b) + (x * y))
    else
        tmp = (c * i) + (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 t_1 = (z * t) + (x * y);
	double tmp;
	if (z <= -2.9e+233) {
		tmp = t_1;
	} else if (z <= -5.8e+204) {
		tmp = (c * i) + ((a * b) + (z * t));
	} else if (z <= -2.5e+125) {
		tmp = t_1;
	} else if (z <= 1700000000000.0) {
		tmp = (c * i) + ((a * b) + (x * y));
	} else {
		tmp = (c * i) + (z * t);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	t_1 = (z * t) + (x * y)
	tmp = 0
	if z <= -2.9e+233:
		tmp = t_1
	elif z <= -5.8e+204:
		tmp = (c * i) + ((a * b) + (z * t))
	elif z <= -2.5e+125:
		tmp = t_1
	elif z <= 1700000000000.0:
		tmp = (c * i) + ((a * b) + (x * y))
	else:
		tmp = (c * i) + (z * t)
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(z * t) + Float64(x * y))
	tmp = 0.0
	if (z <= -2.9e+233)
		tmp = t_1;
	elseif (z <= -5.8e+204)
		tmp = Float64(Float64(c * i) + Float64(Float64(a * b) + Float64(z * t)));
	elseif (z <= -2.5e+125)
		tmp = t_1;
	elseif (z <= 1700000000000.0)
		tmp = Float64(Float64(c * i) + Float64(Float64(a * b) + Float64(x * y)));
	else
		tmp = Float64(Float64(c * i) + Float64(z * t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	t_1 = (z * t) + (x * y);
	tmp = 0.0;
	if (z <= -2.9e+233)
		tmp = t_1;
	elseif (z <= -5.8e+204)
		tmp = (c * i) + ((a * b) + (z * t));
	elseif (z <= -2.5e+125)
		tmp = t_1;
	elseif (z <= 1700000000000.0)
		tmp = (c * i) + ((a * b) + (x * y));
	else
		tmp = (c * i) + (z * t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(z * t), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -2.9e+233], t$95$1, If[LessEqual[z, -5.8e+204], N[(N[(c * i), $MachinePrecision] + N[(N[(a * b), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, -2.5e+125], t$95$1, If[LessEqual[z, 1700000000000.0], N[(N[(c * i), $MachinePrecision] + N[(N[(a * b), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c * i), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -2.90000000000000012e233 or -5.80000000000000007e204 < z < -2.49999999999999981e125

   1. Initial program 92.3%

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

      1. associate-+l+ 92.3%

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

      2. associate-+l+ 92.3%

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

      3. fma-def 92.3%

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

      4. fma-def 92.3%

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

      5. fma-def 96.2%

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

   3. Simplified 96.2%

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

   4. Taylor expanded in a around 0 96.4%

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

   5. Taylor expanded in c around 0 85.1%

      \[\leadsto \color{blue}{y \cdot x + t \cdot z} \]

   if -2.90000000000000012e233 < z < -5.80000000000000007e204

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 100.0%

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

   if -2.49999999999999981e125 < z < 1.7e12

   1. Initial program 98.1%

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

   2. Taylor expanded in z around 0 88.5%

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

   if 1.7e12 < z

   1. Initial program 96.9%

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

   2. Taylor expanded in z around inf 52.5%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.9 \cdot 10^{+233}:\\ \;\;\;\;z \cdot t + x \cdot y\\ \mathbf{elif}\;z \leq -5.8 \cdot 10^{+204}:\\ \;\;\;\;c \cdot i + \left(a \cdot b + z \cdot t\right)\\ \mathbf{elif}\;z \leq -2.5 \cdot 10^{+125}:\\ \;\;\;\;z \cdot t + x \cdot y\\ \mathbf{elif}\;z \leq 1700000000000:\\ \;\;\;\;c \cdot i + \left(a \cdot b + x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot i + z \cdot t\\ \end{array} \]

Alternative 14: 88.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot b \leq -6.2 \cdot 10^{+100} \lor \neg \left(a \cdot b \leq 2.8 \cdot 10^{+137}\right):\\ \;\;\;\;c \cdot i + \left(a \cdot b + z \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot i + \left(z \cdot t + x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (or (<= (* a b) -6.2e+100) (not (<= (* a b) 2.8e+137)))
   (+ (* c i) (+ (* a b) (* z t)))
   (+ (* c i) (+ (* 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 (((a * b) <= -6.2e+100) || !((a * b) <= 2.8e+137)) {
		tmp = (c * i) + ((a * b) + (z * t));
	} else {
		tmp = (c * i) + ((z * t) + (x * y));
	}
	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 (((a * b) <= (-6.2d+100)) .or. (.not. ((a * b) <= 2.8d+137))) then
        tmp = (c * i) + ((a * b) + (z * t))
    else
        tmp = (c * i) + ((z * t) + (x * y))
    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 (((a * b) <= -6.2e+100) || !((a * b) <= 2.8e+137)) {
		tmp = (c * i) + ((a * b) + (z * t));
	} else {
		tmp = (c * i) + ((z * t) + (x * y));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if ((a * b) <= -6.2e+100) or not ((a * b) <= 2.8e+137):
		tmp = (c * i) + ((a * b) + (z * t))
	else:
		tmp = (c * i) + ((z * t) + (x * y))
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if ((Float64(a * b) <= -6.2e+100) || !(Float64(a * b) <= 2.8e+137))
		tmp = Float64(Float64(c * i) + Float64(Float64(a * b) + Float64(z * t)));
	else
		tmp = Float64(Float64(c * i) + Float64(Float64(z * t) + Float64(x * y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	tmp = 0.0;
	if (((a * b) <= -6.2e+100) || ~(((a * b) <= 2.8e+137)))
		tmp = (c * i) + ((a * b) + (z * t));
	else
		tmp = (c * i) + ((z * t) + (x * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[Or[LessEqual[N[(a * b), $MachinePrecision], -6.2e+100], N[Not[LessEqual[N[(a * b), $MachinePrecision], 2.8e+137]], $MachinePrecision]], N[(N[(c * i), $MachinePrecision] + N[(N[(a * b), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c * i), $MachinePrecision] + N[(N[(z * t), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a b) < -6.20000000000000014e100 or 2.80000000000000001e137 < (*.f64 a b)

   1. Initial program 94.3%

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

   2. Taylor expanded in x around 0 90.0%

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

   if -6.20000000000000014e100 < (*.f64 a b) < 2.80000000000000001e137

   1. Initial program 98.8%

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

   2. Taylor expanded in a around 0 92.4%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b \leq -6.2 \cdot 10^{+100} \lor \neg \left(a \cdot b \leq 2.8 \cdot 10^{+137}\right):\\ \;\;\;\;c \cdot i + \left(a \cdot b + z \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot i + \left(z \cdot t + x \cdot y\right)\\ \end{array} \]

Alternative 15: 52.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := c \cdot i + a \cdot b\\ \mathbf{if}\;y \leq -0.49:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{+51}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{+125}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{+223}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ (* c i) (* a b))))
   (if (<= y -0.49)
     (* x y)
     (if (<= y 1.1e+51)
       t_1
       (if (<= y 6.5e+125) (* z t) (if (<= y 1.9e+223) t_1 (* x y)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (c * i) + (a * b);
	double tmp;
	if (y <= -0.49) {
		tmp = x * y;
	} else if (y <= 1.1e+51) {
		tmp = t_1;
	} else if (y <= 6.5e+125) {
		tmp = z * t;
	} else if (y <= 1.9e+223) {
		tmp = t_1;
	} else {
		tmp = x * y;
	}
	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) :: t_1
    real(8) :: tmp
    t_1 = (c * i) + (a * b)
    if (y <= (-0.49d0)) then
        tmp = x * y
    else if (y <= 1.1d+51) then
        tmp = t_1
    else if (y <= 6.5d+125) then
        tmp = z * t
    else if (y <= 1.9d+223) then
        tmp = t_1
    else
        tmp = x * y
    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 t_1 = (c * i) + (a * b);
	double tmp;
	if (y <= -0.49) {
		tmp = x * y;
	} else if (y <= 1.1e+51) {
		tmp = t_1;
	} else if (y <= 6.5e+125) {
		tmp = z * t;
	} else if (y <= 1.9e+223) {
		tmp = t_1;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	t_1 = (c * i) + (a * b)
	tmp = 0
	if y <= -0.49:
		tmp = x * y
	elif y <= 1.1e+51:
		tmp = t_1
	elif y <= 6.5e+125:
		tmp = z * t
	elif y <= 1.9e+223:
		tmp = t_1
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(c * i) + Float64(a * b))
	tmp = 0.0
	if (y <= -0.49)
		tmp = Float64(x * y);
	elseif (y <= 1.1e+51)
		tmp = t_1;
	elseif (y <= 6.5e+125)
		tmp = Float64(z * t);
	elseif (y <= 1.9e+223)
		tmp = t_1;
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	t_1 = (c * i) + (a * b);
	tmp = 0.0;
	if (y <= -0.49)
		tmp = x * y;
	elseif (y <= 1.1e+51)
		tmp = t_1;
	elseif (y <= 6.5e+125)
		tmp = z * t;
	elseif (y <= 1.9e+223)
		tmp = t_1;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(c * i), $MachinePrecision] + N[(a * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.49], N[(x * y), $MachinePrecision], If[LessEqual[y, 1.1e+51], t$95$1, If[LessEqual[y, 6.5e+125], N[(z * t), $MachinePrecision], If[LessEqual[y, 1.9e+223], t$95$1, N[(x * y), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.48999999999999999 or 1.9e223 < y

   1. Initial program 95.7%

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

      1. associate-+l+ 95.7%

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

      2. associate-+l+ 95.7%

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

      3. fma-def 95.8%

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

      4. fma-def 96.8%

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

      5. fma-def 97.9%

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

   3. Simplified 97.9%

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

   4. Taylor expanded in a around 0 80.7%

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

   5. Taylor expanded in x around inf 56.4%

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

   if -0.48999999999999999 < y < 1.09999999999999996e51 or 6.4999999999999999e125 < y < 1.9e223

   1. Initial program 98.6%

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

   2. Taylor expanded in a around inf 65.0%

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

   if 1.09999999999999996e51 < y < 6.4999999999999999e125

   1. Initial program 93.3%

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

   2. Taylor expanded in x around 0 62.7%

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

      1. +-commutative 62.7%

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

      2. *-commutative 62.7%

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

      3. fma-def 62.7%

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

   4. Applied egg-rr 62.7%

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

   5. Taylor expanded in c around 0 49.3%

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

      1. +-commutative 49.3%

         \[\leadsto \color{blue}{t \cdot z + a \cdot b} \]

      2. fma-udef 49.3%

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

   7. Simplified 49.3%

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

   8. Taylor expanded in t around inf 42.3%

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

4. Final simplification 60.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.49:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{+51}:\\ \;\;\;\;c \cdot i + a \cdot b\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{+125}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{+223}:\\ \;\;\;\;c \cdot i + a \cdot b\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 16: 58.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot b + x \cdot y\\ \mathbf{if}\;y \leq -9 \cdot 10^{-105}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 8 \cdot 10^{-224}:\\ \;\;\;\;a \cdot b + z \cdot t\\ \mathbf{elif}\;y \leq 0.0185:\\ \;\;\;\;c \cdot i + a \cdot b\\ \mathbf{elif}\;y \leq 2.25 \cdot 10^{+223}:\\ \;\;\;\;c \cdot i + z \cdot t\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ (* a b) (* x y))))
   (if (<= y -9e-105)
     t_1
     (if (<= y 8e-224)
       (+ (* a b) (* z t))
       (if (<= y 0.0185)
         (+ (* c i) (* a b))
         (if (<= y 2.25e+223) (+ (* c i) (* z t)) t_1))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (a * b) + (x * y);
	double tmp;
	if (y <= -9e-105) {
		tmp = t_1;
	} else if (y <= 8e-224) {
		tmp = (a * b) + (z * t);
	} else if (y <= 0.0185) {
		tmp = (c * i) + (a * b);
	} else if (y <= 2.25e+223) {
		tmp = (c * i) + (z * t);
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_1 = (a * b) + (x * y)
    if (y <= (-9d-105)) then
        tmp = t_1
    else if (y <= 8d-224) then
        tmp = (a * b) + (z * t)
    else if (y <= 0.0185d0) then
        tmp = (c * i) + (a * b)
    else if (y <= 2.25d+223) then
        tmp = (c * i) + (z * t)
    else
        tmp = t_1
    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 t_1 = (a * b) + (x * y);
	double tmp;
	if (y <= -9e-105) {
		tmp = t_1;
	} else if (y <= 8e-224) {
		tmp = (a * b) + (z * t);
	} else if (y <= 0.0185) {
		tmp = (c * i) + (a * b);
	} else if (y <= 2.25e+223) {
		tmp = (c * i) + (z * t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	t_1 = (a * b) + (x * y)
	tmp = 0
	if y <= -9e-105:
		tmp = t_1
	elif y <= 8e-224:
		tmp = (a * b) + (z * t)
	elif y <= 0.0185:
		tmp = (c * i) + (a * b)
	elif y <= 2.25e+223:
		tmp = (c * i) + (z * t)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(a * b) + Float64(x * y))
	tmp = 0.0
	if (y <= -9e-105)
		tmp = t_1;
	elseif (y <= 8e-224)
		tmp = Float64(Float64(a * b) + Float64(z * t));
	elseif (y <= 0.0185)
		tmp = Float64(Float64(c * i) + Float64(a * b));
	elseif (y <= 2.25e+223)
		tmp = Float64(Float64(c * i) + Float64(z * t));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	t_1 = (a * b) + (x * y);
	tmp = 0.0;
	if (y <= -9e-105)
		tmp = t_1;
	elseif (y <= 8e-224)
		tmp = (a * b) + (z * t);
	elseif (y <= 0.0185)
		tmp = (c * i) + (a * b);
	elseif (y <= 2.25e+223)
		tmp = (c * i) + (z * t);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(a * b), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -9e-105], t$95$1, If[LessEqual[y, 8e-224], N[(N[(a * b), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.0185], N[(N[(c * i), $MachinePrecision] + N[(a * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.25e+223], N[(N[(c * i), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -8.9999999999999995e-105 or 2.25e223 < y

   1. Initial program 96.5%

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

   2. Taylor expanded in z around 0 84.5%

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

   3. Taylor expanded in c around 0 71.0%

      \[\leadsto \color{blue}{a \cdot b + y \cdot x} \]

   if -8.9999999999999995e-105 < y < 8.0000000000000002e-224

   1. Initial program 98.0%

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

   2. Taylor expanded in x around 0 92.6%

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

      1. +-commutative 92.6%

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

      2. *-commutative 92.6%

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

      3. fma-def 92.6%

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

   4. Applied egg-rr 92.6%

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

   5. Taylor expanded in c around 0 58.4%

      \[\leadsto \color{blue}{a \cdot b + t \cdot z} \]

   if 8.0000000000000002e-224 < y < 0.0184999999999999991

   1. Initial program 100.0%

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

   2. Taylor expanded in a around inf 64.6%

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

   if 0.0184999999999999991 < y < 2.25e223

   1. Initial program 95.5%

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

   2. Taylor expanded in z around inf 68.5%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9 \cdot 10^{-105}:\\ \;\;\;\;a \cdot b + x \cdot y\\ \mathbf{elif}\;y \leq 8 \cdot 10^{-224}:\\ \;\;\;\;a \cdot b + z \cdot t\\ \mathbf{elif}\;y \leq 0.0185:\\ \;\;\;\;c \cdot i + a \cdot b\\ \mathbf{elif}\;y \leq 2.25 \cdot 10^{+223}:\\ \;\;\;\;c \cdot i + z \cdot t\\ \mathbf{else}:\\ \;\;\;\;a \cdot b + x \cdot y\\ \end{array} \]

Alternative 17: 57.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.85 \cdot 10^{-88}:\\ \;\;\;\;c \cdot i + x \cdot y\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{-225}:\\ \;\;\;\;a \cdot b + z \cdot t\\ \mathbf{elif}\;y \leq 0.000106:\\ \;\;\;\;c \cdot i + a \cdot b\\ \mathbf{elif}\;y \leq 4.7 \cdot 10^{+222}:\\ \;\;\;\;c \cdot i + z \cdot t\\ \mathbf{else}:\\ \;\;\;\;a \cdot b + x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= y -1.85e-88)
   (+ (* c i) (* x y))
   (if (<= y 3.2e-225)
     (+ (* a b) (* z t))
     (if (<= y 0.000106)
       (+ (* c i) (* a b))
       (if (<= y 4.7e+222) (+ (* c i) (* z t)) (+ (* a b) (* x y)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (y <= -1.85e-88) {
		tmp = (c * i) + (x * y);
	} else if (y <= 3.2e-225) {
		tmp = (a * b) + (z * t);
	} else if (y <= 0.000106) {
		tmp = (c * i) + (a * b);
	} else if (y <= 4.7e+222) {
		tmp = (c * i) + (z * t);
	} else {
		tmp = (a * b) + (x * y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8) :: tmp
    if (y <= (-1.85d-88)) then
        tmp = (c * i) + (x * y)
    else if (y <= 3.2d-225) then
        tmp = (a * b) + (z * t)
    else if (y <= 0.000106d0) then
        tmp = (c * i) + (a * b)
    else if (y <= 4.7d+222) then
        tmp = (c * i) + (z * t)
    else
        tmp = (a * b) + (x * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (y <= -1.85e-88) {
		tmp = (c * i) + (x * y);
	} else if (y <= 3.2e-225) {
		tmp = (a * b) + (z * t);
	} else if (y <= 0.000106) {
		tmp = (c * i) + (a * b);
	} else if (y <= 4.7e+222) {
		tmp = (c * i) + (z * t);
	} else {
		tmp = (a * b) + (x * y);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if y <= -1.85e-88:
		tmp = (c * i) + (x * y)
	elif y <= 3.2e-225:
		tmp = (a * b) + (z * t)
	elif y <= 0.000106:
		tmp = (c * i) + (a * b)
	elif y <= 4.7e+222:
		tmp = (c * i) + (z * t)
	else:
		tmp = (a * b) + (x * y)
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (y <= -1.85e-88)
		tmp = Float64(Float64(c * i) + Float64(x * y));
	elseif (y <= 3.2e-225)
		tmp = Float64(Float64(a * b) + Float64(z * t));
	elseif (y <= 0.000106)
		tmp = Float64(Float64(c * i) + Float64(a * b));
	elseif (y <= 4.7e+222)
		tmp = Float64(Float64(c * i) + Float64(z * t));
	else
		tmp = Float64(Float64(a * b) + Float64(x * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	tmp = 0.0;
	if (y <= -1.85e-88)
		tmp = (c * i) + (x * y);
	elseif (y <= 3.2e-225)
		tmp = (a * b) + (z * t);
	elseif (y <= 0.000106)
		tmp = (c * i) + (a * b);
	elseif (y <= 4.7e+222)
		tmp = (c * i) + (z * t);
	else
		tmp = (a * b) + (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[y, -1.85e-88], N[(N[(c * i), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.2e-225], N[(N[(a * b), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.000106], N[(N[(c * i), $MachinePrecision] + N[(a * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.7e+222], N[(N[(c * i), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision], N[(N[(a * b), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if y < -1.8499999999999999e-88

   1. Initial program 96.4%

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

   2. Taylor expanded in x around inf 63.6%

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

   if -1.8499999999999999e-88 < y < 3.19999999999999975e-225

   1. Initial program 98.2%

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

   2. Taylor expanded in x around 0 91.7%

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

      1. +-commutative 91.7%

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

      2. *-commutative 91.7%

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

      3. fma-def 91.7%

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

   4. Applied egg-rr 91.7%

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

   5. Taylor expanded in c around 0 61.0%

      \[\leadsto \color{blue}{a \cdot b + t \cdot z} \]

   if 3.19999999999999975e-225 < y < 1.06e-4

   1. Initial program 100.0%

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

   2. Taylor expanded in a around inf 64.6%

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

   if 1.06e-4 < y < 4.6999999999999999e222

   1. Initial program 95.5%

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

   2. Taylor expanded in z around inf 68.5%

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

   if 4.6999999999999999e222 < y

   1. Initial program 95.8%

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

   2. Taylor expanded in z around 0 95.8%

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

   3. Taylor expanded in c around 0 100.0%

      \[\leadsto \color{blue}{a \cdot b + y \cdot x} \]
3. Recombined 5 regimes into one program.

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.85 \cdot 10^{-88}:\\ \;\;\;\;c \cdot i + x \cdot y\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{-225}:\\ \;\;\;\;a \cdot b + z \cdot t\\ \mathbf{elif}\;y \leq 0.000106:\\ \;\;\;\;c \cdot i + a \cdot b\\ \mathbf{elif}\;y \leq 4.7 \cdot 10^{+222}:\\ \;\;\;\;c \cdot i + z \cdot t\\ \mathbf{else}:\\ \;\;\;\;a \cdot b + x \cdot y\\ \end{array} \]

Alternative 18: 53.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.8:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{+33}:\\ \;\;\;\;c \cdot i + a \cdot b\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{+222}:\\ \;\;\;\;a \cdot b + z \cdot t\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= y -4.8)
   (* x y)
   (if (<= y 1.9e+33)
     (+ (* c i) (* a b))
     (if (<= y 4.8e+222) (+ (* a b) (* 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 (y <= -4.8) {
		tmp = x * y;
	} else if (y <= 1.9e+33) {
		tmp = (c * i) + (a * b);
	} else if (y <= 4.8e+222) {
		tmp = (a * b) + (z * t);
	} else {
		tmp = x * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8) :: tmp
    if (y <= (-4.8d0)) then
        tmp = x * y
    else if (y <= 1.9d+33) then
        tmp = (c * i) + (a * b)
    else if (y <= 4.8d+222) then
        tmp = (a * b) + (z * t)
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (y <= -4.8) {
		tmp = x * y;
	} else if (y <= 1.9e+33) {
		tmp = (c * i) + (a * b);
	} else if (y <= 4.8e+222) {
		tmp = (a * b) + (z * t);
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if y <= -4.8:
		tmp = x * y
	elif y <= 1.9e+33:
		tmp = (c * i) + (a * b)
	elif y <= 4.8e+222:
		tmp = (a * b) + (z * t)
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (y <= -4.8)
		tmp = Float64(x * y);
	elseif (y <= 1.9e+33)
		tmp = Float64(Float64(c * i) + Float64(a * b));
	elseif (y <= 4.8e+222)
		tmp = Float64(Float64(a * b) + Float64(z * t));
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	tmp = 0.0;
	if (y <= -4.8)
		tmp = x * y;
	elseif (y <= 1.9e+33)
		tmp = (c * i) + (a * b);
	elseif (y <= 4.8e+222)
		tmp = (a * b) + (z * t);
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[y, -4.8], N[(x * y), $MachinePrecision], If[LessEqual[y, 1.9e+33], N[(N[(c * i), $MachinePrecision] + N[(a * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.8e+222], N[(N[(a * b), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.79999999999999982 or 4.8000000000000002e222 < y

   1. Initial program 95.7%

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

      1. associate-+l+ 95.7%

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

      2. associate-+l+ 95.7%

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

      3. fma-def 95.8%

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

      4. fma-def 96.8%

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

      5. fma-def 97.9%

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

   3. Simplified 97.9%

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

   4. Taylor expanded in a around 0 80.7%

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

   5. Taylor expanded in x around inf 56.4%

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

   if -4.79999999999999982 < y < 1.90000000000000001e33

   1. Initial program 99.2%

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

   2. Taylor expanded in a around inf 65.9%

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

   if 1.90000000000000001e33 < y < 4.8000000000000002e222

   1. Initial program 94.1%

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

   2. Taylor expanded in x around 0 74.9%

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

      1. +-commutative 74.9%

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

      2. *-commutative 74.9%

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

      3. fma-def 74.9%

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

   4. Applied egg-rr 74.9%

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

   5. Taylor expanded in c around 0 49.3%

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

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.8:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{+33}:\\ \;\;\;\;c \cdot i + a \cdot b\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{+222}:\\ \;\;\;\;a \cdot b + z \cdot t\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 19: 42.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \cdot i \leq -2.6 \cdot 10^{+25}:\\ \;\;\;\;c \cdot i\\ \mathbf{elif}\;c \cdot i \leq 6 \cdot 10^{+126}:\\ \;\;\;\;a \cdot b\\ \mathbf{else}:\\ \;\;\;\;c \cdot i\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* c i) -2.6e+25) (* c i) (if (<= (* c i) 6e+126) (* 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) <= -2.6e+25) {
		tmp = c * i;
	} else if ((c * i) <= 6e+126) {
		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) <= (-2.6d+25)) then
        tmp = c * i
    else if ((c * i) <= 6d+126) 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) <= -2.6e+25) {
		tmp = c * i;
	} else if ((c * i) <= 6e+126) {
		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) <= -2.6e+25:
		tmp = c * i
	elif (c * i) <= 6e+126:
		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) <= -2.6e+25)
		tmp = Float64(c * i);
	elseif (Float64(c * i) <= 6e+126)
		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) <= -2.6e+25)
		tmp = c * i;
	elseif ((c * i) <= 6e+126)
		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], -2.6e+25], N[(c * i), $MachinePrecision], If[LessEqual[N[(c * i), $MachinePrecision], 6e+126], N[(a * b), $MachinePrecision], N[(c * i), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 c i) < -2.5999999999999998e25 or 6.0000000000000005e126 < (*.f64 c i)

   1. Initial program 94.6%

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

   2. Taylor expanded in c around inf 69.9%

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

   if -2.5999999999999998e25 < (*.f64 c i) < 6.0000000000000005e126

   1. Initial program 98.7%

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

   2. Taylor expanded in x around 0 63.4%

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

   3. Taylor expanded in t around 0 38.0%

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

   4. Taylor expanded in c around 0 33.7%

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

4. Final simplification 46.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \cdot i \leq -2.6 \cdot 10^{+25}:\\ \;\;\;\;c \cdot i\\ \mathbf{elif}\;c \cdot i \leq 6 \cdot 10^{+126}:\\ \;\;\;\;a \cdot b\\ \mathbf{else}:\\ \;\;\;\;c \cdot i\\ \end{array} \]

Alternative 20: 27.5% 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. Taylor expanded in x around 0 71.3%

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

3. Taylor expanded in t around 0 51.2%

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

4. Taylor expanded in c around 0 25.0%

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

5. Final simplification 25.0%

   \[\leadsto a \cdot b \]

Reproduce

herbie shell --seed 2023207 
(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)))