Diagrams.ThreeD.Shapes:frustum from diagrams-lib-1.3.0.3, A

Percentage Accurate: 89.6% → 94.4%

Time: 15.6s

Alternatives: 13

Speedup: 0.5×

• 46.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right) \end{array} \]

FPCore

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

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return 2.0 * (((x * y) + (z * t)) - (((a + (b * c)) * 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 = 2.0d0 * (((x * y) + (z * t)) - (((a + (b * c)) * 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 2.0 * (((x * y) + (z * t)) - (((a + (b * c)) * c) * i));
}

Python

def code(x, y, z, t, a, b, c, i):
	return 2.0 * (((x * y) + (z * t)) - (((a + (b * c)) * c) * i))

Julia

function code(x, y, z, t, a, b, c, i)
	return Float64(2.0 * Float64(Float64(Float64(x * y) + Float64(z * t)) - Float64(Float64(Float64(a + Float64(b * c)) * c) * i)))
end

MATLAB

function tmp = code(x, y, z, t, a, b, c, i)
	tmp = 2.0 * (((x * y) + (z * t)) - (((a + (b * c)) * c) * i));
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := N[(2.0 * N[(N[(N[(x * y), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(a + N[(b * c), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 89.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right) \end{array} \]

FPCore

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

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return 2.0 * (((x * y) + (z * t)) - (((a + (b * c)) * 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 = 2.0d0 * (((x * y) + (z * t)) - (((a + (b * c)) * 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 2.0 * (((x * y) + (z * t)) - (((a + (b * c)) * c) * i));
}

Python

def code(x, y, z, t, a, b, c, i):
	return 2.0 * (((x * y) + (z * t)) - (((a + (b * c)) * c) * i))

Julia

function code(x, y, z, t, a, b, c, i)
	return Float64(2.0 * Float64(Float64(Float64(x * y) + Float64(z * t)) - Float64(Float64(Float64(a + Float64(b * c)) * c) * i)))
end

MATLAB

function tmp = code(x, y, z, t, a, b, c, i)
	tmp = 2.0 * (((x * y) + (z * t)) - (((a + (b * c)) * c) * i));
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := N[(2.0 * N[(N[(N[(x * y), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(a + N[(b * c), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 94.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(c \cdot \left(a + b \cdot c\right)\right) \cdot i\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;2 \cdot \left(x \cdot y - c \cdot \left(i \cdot \mathsf{fma}\left(b, c, a\right)\right)\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+294}:\\ \;\;\;\;2 \cdot \left(\left(x \cdot y + z \cdot t\right) - t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(i \cdot \left(\mathsf{fma}\left(b, c, a\right) \cdot -2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (* c (+ a (* b c))) i)))
   (if (<= t_1 (- INFINITY))
     (* 2.0 (- (* x y) (* c (* i (fma b c a)))))
     (if (<= t_1 2e+294)
       (* 2.0 (- (+ (* x y) (* z t)) t_1))
       (* c (* i (* (fma b c a) -2.0)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (c * (a + (b * c))) * i;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = 2.0 * ((x * y) - (c * (i * fma(b, c, a))));
	} else if (t_1 <= 2e+294) {
		tmp = 2.0 * (((x * y) + (z * t)) - t_1);
	} else {
		tmp = c * (i * (fma(b, c, a) * -2.0));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(c * Float64(a + Float64(b * c))) * i)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(2.0 * Float64(Float64(x * y) - Float64(c * Float64(i * fma(b, c, a)))));
	elseif (t_1 <= 2e+294)
		tmp = Float64(2.0 * Float64(Float64(Float64(x * y) + Float64(z * t)) - t_1));
	else
		tmp = Float64(c * Float64(i * Float64(fma(b, c, a) * -2.0)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(c * N[(a + N[(b * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * i), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(2.0 * N[(N[(x * y), $MachinePrecision] - N[(c * N[(i * N[(b * c + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+294], N[(2.0 * N[(N[(N[(x * y), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision], N[(c * N[(i * N[(N[(b * c + a), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -inf.0

   1. Initial program 75.8%

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

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{2 \cdot \left(x \cdot y - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{2 \cdot \left(x \cdot y - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)} \]

      2. lower--.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\left(x \cdot y - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto 2 \cdot \left(\color{blue}{x \cdot y} - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto 2 \cdot \left(x \cdot y - \color{blue}{c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto 2 \cdot \left(x \cdot y - c \cdot \color{blue}{\left(\left(a + b \cdot c\right) \cdot i\right)}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto 2 \cdot \left(x \cdot y - c \cdot \color{blue}{\left(\left(a + b \cdot c\right) \cdot i\right)}\right) \]

      7. +-commutative N/A

         \[\leadsto 2 \cdot \left(x \cdot y - c \cdot \left(\color{blue}{\left(b \cdot c + a\right)} \cdot i\right)\right) \]

      8. lower-fma.f64 96.5

         \[\leadsto 2 \cdot \left(x \cdot y - c \cdot \left(\color{blue}{\mathsf{fma}\left(b, c, a\right)} \cdot i\right)\right) \]

   5. Simplified 96.5%

      \[\leadsto \color{blue}{2 \cdot \left(x \cdot y - c \cdot \left(\mathsf{fma}\left(b, c, a\right) \cdot i\right)\right)} \]

   if -inf.0 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 2.00000000000000013e294

   1. Initial program 99.3%

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

   if 2.00000000000000013e294 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 77.5%

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

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{-2 \cdot \left(c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto -2 \cdot \color{blue}{\left(\left(i \cdot \left(a + b \cdot c\right)\right) \cdot c\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-2 \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right) \cdot c} \]

      3. distribute-rgt-in N/A

         \[\leadsto \left(-2 \cdot \color{blue}{\left(a \cdot i + \left(b \cdot c\right) \cdot i\right)}\right) \cdot c \]

      4. associate-*r* N/A

         \[\leadsto \left(-2 \cdot \left(a \cdot i + \color{blue}{b \cdot \left(c \cdot i\right)}\right)\right) \cdot c \]

      5. distribute-lft-out N/A

         \[\leadsto \color{blue}{\left(-2 \cdot \left(a \cdot i\right) + -2 \cdot \left(b \cdot \left(c \cdot i\right)\right)\right)} \cdot c \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{c \cdot \left(-2 \cdot \left(a \cdot i\right) + -2 \cdot \left(b \cdot \left(c \cdot i\right)\right)\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{c \cdot \left(-2 \cdot \left(a \cdot i\right) + -2 \cdot \left(b \cdot \left(c \cdot i\right)\right)\right)} \]

      8. distribute-lft-out N/A

         \[\leadsto c \cdot \color{blue}{\left(-2 \cdot \left(a \cdot i + b \cdot \left(c \cdot i\right)\right)\right)} \]

      9. associate-*r* N/A

         \[\leadsto c \cdot \left(-2 \cdot \left(a \cdot i + \color{blue}{\left(b \cdot c\right) \cdot i}\right)\right) \]

      10. distribute-rgt-in N/A

          \[\leadsto c \cdot \left(-2 \cdot \color{blue}{\left(i \cdot \left(a + b \cdot c\right)\right)}\right) \]

      11. *-commutative N/A

          \[\leadsto c \cdot \left(-2 \cdot \color{blue}{\left(\left(a + b \cdot c\right) \cdot i\right)}\right) \]

      12. associate-*r* N/A

          \[\leadsto c \cdot \color{blue}{\left(\left(-2 \cdot \left(a + b \cdot c\right)\right) \cdot i\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto c \cdot \color{blue}{\left(\left(-2 \cdot \left(a + b \cdot c\right)\right) \cdot i\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto c \cdot \left(\color{blue}{\left(-2 \cdot \left(a + b \cdot c\right)\right)} \cdot i\right) \]

      15. +-commutative N/A

          \[\leadsto c \cdot \left(\left(-2 \cdot \color{blue}{\left(b \cdot c + a\right)}\right) \cdot i\right) \]

      16. lower-fma.f64 94.6

          \[\leadsto c \cdot \left(\left(-2 \cdot \color{blue}{\mathsf{fma}\left(b, c, a\right)}\right) \cdot i\right) \]

   5. Simplified 94.6%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq -\infty:\\ \;\;\;\;2 \cdot \left(x \cdot y - c \cdot \left(i \cdot \mathsf{fma}\left(b, c, a\right)\right)\right)\\ \mathbf{elif}\;\left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq 2 \cdot 10^{+294}:\\ \;\;\;\;2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(c \cdot \left(a + b \cdot c\right)\right) \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(i \cdot \left(\mathsf{fma}\left(b, c, a\right) \cdot -2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 82.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := c \cdot \left(i \cdot \mathsf{fma}\left(b, c, a\right)\right)\\ t_2 := \left(c \cdot \left(a + b \cdot c\right)\right) \cdot i\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+144}:\\ \;\;\;\;2 \cdot \left(x \cdot y - t\_1\right)\\ \mathbf{elif}\;t\_2 \leq -5 \cdot 10^{-98}:\\ \;\;\;\;2 \cdot \left(z \cdot t - t\_1\right)\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+294}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(z, t, \mathsf{fma}\left(x, y, i \cdot \left(b \cdot \left(c \cdot \left(-c\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(i \cdot \left(\mathsf{fma}\left(b, c, a\right) \cdot -2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* c (* i (fma b c a)))) (t_2 (* (* c (+ a (* b c))) i)))
   (if (<= t_2 -1e+144)
     (* 2.0 (- (* x y) t_1))
     (if (<= t_2 -5e-98)
       (* 2.0 (- (* z t) t_1))
       (if (<= t_2 2e+294)
         (* 2.0 (fma z t (fma x y (* i (* b (* c (- c)))))))
         (* c (* i (* (fma b c a) -2.0))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = c * (i * fma(b, c, a));
	double t_2 = (c * (a + (b * c))) * i;
	double tmp;
	if (t_2 <= -1e+144) {
		tmp = 2.0 * ((x * y) - t_1);
	} else if (t_2 <= -5e-98) {
		tmp = 2.0 * ((z * t) - t_1);
	} else if (t_2 <= 2e+294) {
		tmp = 2.0 * fma(z, t, fma(x, y, (i * (b * (c * -c)))));
	} else {
		tmp = c * (i * (fma(b, c, a) * -2.0));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(c * Float64(i * fma(b, c, a)))
	t_2 = Float64(Float64(c * Float64(a + Float64(b * c))) * i)
	tmp = 0.0
	if (t_2 <= -1e+144)
		tmp = Float64(2.0 * Float64(Float64(x * y) - t_1));
	elseif (t_2 <= -5e-98)
		tmp = Float64(2.0 * Float64(Float64(z * t) - t_1));
	elseif (t_2 <= 2e+294)
		tmp = Float64(2.0 * fma(z, t, fma(x, y, Float64(i * Float64(b * Float64(c * Float64(-c)))))));
	else
		tmp = Float64(c * Float64(i * Float64(fma(b, c, a) * -2.0)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(c * N[(i * N[(b * c + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(c * N[(a + N[(b * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * i), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+144], N[(2.0 * N[(N[(x * y), $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -5e-98], N[(2.0 * N[(N[(z * t), $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e+294], N[(2.0 * N[(z * t + N[(x * y + N[(i * N[(b * N[(c * (-c)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c * N[(i * N[(N[(b * c + a), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -1.00000000000000002e144

   1. Initial program 81.0%

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

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{2 \cdot \left(x \cdot y - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{2 \cdot \left(x \cdot y - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)} \]

      2. lower--.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\left(x \cdot y - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto 2 \cdot \left(\color{blue}{x \cdot y} - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto 2 \cdot \left(x \cdot y - \color{blue}{c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto 2 \cdot \left(x \cdot y - c \cdot \color{blue}{\left(\left(a + b \cdot c\right) \cdot i\right)}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto 2 \cdot \left(x \cdot y - c \cdot \color{blue}{\left(\left(a + b \cdot c\right) \cdot i\right)}\right) \]

      7. +-commutative N/A

         \[\leadsto 2 \cdot \left(x \cdot y - c \cdot \left(\color{blue}{\left(b \cdot c + a\right)} \cdot i\right)\right) \]

      8. lower-fma.f64 93.2

         \[\leadsto 2 \cdot \left(x \cdot y - c \cdot \left(\color{blue}{\mathsf{fma}\left(b, c, a\right)} \cdot i\right)\right) \]

   5. Simplified 93.2%

      \[\leadsto \color{blue}{2 \cdot \left(x \cdot y - c \cdot \left(\mathsf{fma}\left(b, c, a\right) \cdot i\right)\right)} \]

   if -1.00000000000000002e144 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -5.00000000000000018e-98

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{2 \cdot \left(t \cdot z - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{2 \cdot \left(t \cdot z - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)} \]

      2. lower--.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\left(t \cdot z - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto 2 \cdot \left(\color{blue}{t \cdot z} - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto 2 \cdot \left(t \cdot z - \color{blue}{c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto 2 \cdot \left(t \cdot z - c \cdot \color{blue}{\left(\left(a + b \cdot c\right) \cdot i\right)}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto 2 \cdot \left(t \cdot z - c \cdot \color{blue}{\left(\left(a + b \cdot c\right) \cdot i\right)}\right) \]

      7. +-commutative N/A

         \[\leadsto 2 \cdot \left(t \cdot z - c \cdot \left(\color{blue}{\left(b \cdot c + a\right)} \cdot i\right)\right) \]

      8. lower-fma.f64 85.4

         \[\leadsto 2 \cdot \left(t \cdot z - c \cdot \left(\color{blue}{\mathsf{fma}\left(b, c, a\right)} \cdot i\right)\right) \]

   5. Simplified 85.4%

      \[\leadsto \color{blue}{2 \cdot \left(t \cdot z - c \cdot \left(\mathsf{fma}\left(b, c, a\right) \cdot i\right)\right)} \]

   if -5.00000000000000018e-98 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 2.00000000000000013e294

   1. Initial program 99.0%

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

   3. Taylor expanded in a around 0

      \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \color{blue}{b \cdot \left({c}^{2} \cdot i\right)}\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \color{blue}{\left({c}^{2} \cdot i\right) \cdot b}\right) \]

      2. unpow2 N/A

         \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\color{blue}{\left(c \cdot c\right)} \cdot i\right) \cdot b\right) \]

      3. associate-*l* N/A

         \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \color{blue}{\left(c \cdot \left(c \cdot i\right)\right)} \cdot b\right) \]

      4. associate-*r* N/A

         \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \color{blue}{c \cdot \left(\left(c \cdot i\right) \cdot b\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - c \cdot \color{blue}{\left(b \cdot \left(c \cdot i\right)\right)}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \color{blue}{c \cdot \left(b \cdot \left(c \cdot i\right)\right)}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - c \cdot \color{blue}{\left(b \cdot \left(c \cdot i\right)\right)}\right) \]

      8. lower-*.f64 92.0

         \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - c \cdot \left(b \cdot \color{blue}{\left(c \cdot i\right)}\right)\right) \]

   5. Simplified 92.0%

      \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \color{blue}{c \cdot \left(b \cdot \left(c \cdot i\right)\right)}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto 2 \cdot \left(\left(\color{blue}{x \cdot y} + z \cdot t\right) - c \cdot \left(b \cdot \left(c \cdot i\right)\right)\right) \]

      2. lift-*.f64 N/A

         \[\leadsto 2 \cdot \left(\left(x \cdot y + \color{blue}{z \cdot t}\right) - c \cdot \left(b \cdot \left(c \cdot i\right)\right)\right) \]

      3. lift-*.f64 N/A

         \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - c \cdot \left(b \cdot \color{blue}{\left(c \cdot i\right)}\right)\right) \]

      4. lift-*.f64 N/A

         \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - c \cdot \color{blue}{\left(b \cdot \left(c \cdot i\right)\right)}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \color{blue}{c \cdot \left(b \cdot \left(c \cdot i\right)\right)}\right) \]

      6. +-commutative N/A

         \[\leadsto 2 \cdot \left(\color{blue}{\left(z \cdot t + x \cdot y\right)} - c \cdot \left(b \cdot \left(c \cdot i\right)\right)\right) \]

      7. associate--l+ N/A

         \[\leadsto 2 \cdot \color{blue}{\left(z \cdot t + \left(x \cdot y - c \cdot \left(b \cdot \left(c \cdot i\right)\right)\right)\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto 2 \cdot \left(\color{blue}{z \cdot t} + \left(x \cdot y - c \cdot \left(b \cdot \left(c \cdot i\right)\right)\right)\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\mathsf{fma}\left(z, t, x \cdot y - c \cdot \left(b \cdot \left(c \cdot i\right)\right)\right)} \]

      10. sub-neg N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(z, t, \color{blue}{x \cdot y + \left(\mathsf{neg}\left(c \cdot \left(b \cdot \left(c \cdot i\right)\right)\right)\right)}\right) \]

      11. lift-*.f64 N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(z, t, \color{blue}{x \cdot y} + \left(\mathsf{neg}\left(c \cdot \left(b \cdot \left(c \cdot i\right)\right)\right)\right)\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(z, t, \color{blue}{\mathsf{fma}\left(x, y, \mathsf{neg}\left(c \cdot \left(b \cdot \left(c \cdot i\right)\right)\right)\right)}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(z, t, \mathsf{fma}\left(x, y, \mathsf{neg}\left(\color{blue}{c \cdot \left(b \cdot \left(c \cdot i\right)\right)}\right)\right)\right) \]

      14. distribute-lft-neg-in N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(z, t, \mathsf{fma}\left(x, y, \color{blue}{\left(\mathsf{neg}\left(c\right)\right) \cdot \left(b \cdot \left(c \cdot i\right)\right)}\right)\right) \]

      15. lift-neg.f64 N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(z, t, \mathsf{fma}\left(x, y, \color{blue}{\left(\mathsf{neg}\left(c\right)\right)} \cdot \left(b \cdot \left(c \cdot i\right)\right)\right)\right) \]

      16. *-commutative N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(z, t, \mathsf{fma}\left(x, y, \color{blue}{\left(b \cdot \left(c \cdot i\right)\right) \cdot \left(\mathsf{neg}\left(c\right)\right)}\right)\right) \]

      17. lift-*.f64 N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(z, t, \mathsf{fma}\left(x, y, \color{blue}{\left(b \cdot \left(c \cdot i\right)\right)} \cdot \left(\mathsf{neg}\left(c\right)\right)\right)\right) \]

      18. lift-*.f64 N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(z, t, \mathsf{fma}\left(x, y, \left(b \cdot \color{blue}{\left(c \cdot i\right)}\right) \cdot \left(\mathsf{neg}\left(c\right)\right)\right)\right) \]

      19. *-commutative N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(z, t, \mathsf{fma}\left(x, y, \left(b \cdot \color{blue}{\left(i \cdot c\right)}\right) \cdot \left(\mathsf{neg}\left(c\right)\right)\right)\right) \]

      20. associate-*r* N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(z, t, \mathsf{fma}\left(x, y, \color{blue}{\left(\left(b \cdot i\right) \cdot c\right)} \cdot \left(\mathsf{neg}\left(c\right)\right)\right)\right) \]

      21. *-commutative N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(z, t, \mathsf{fma}\left(x, y, \left(\color{blue}{\left(i \cdot b\right)} \cdot c\right) \cdot \left(\mathsf{neg}\left(c\right)\right)\right)\right) \]

   7. Applied egg-rr 93.7%

      \[\leadsto 2 \cdot \color{blue}{\mathsf{fma}\left(z, t, \mathsf{fma}\left(x, y, i \cdot \left(b \cdot \left(c \cdot \left(-c\right)\right)\right)\right)\right)} \]

   if 2.00000000000000013e294 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 77.5%

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

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{-2 \cdot \left(c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto -2 \cdot \color{blue}{\left(\left(i \cdot \left(a + b \cdot c\right)\right) \cdot c\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-2 \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right) \cdot c} \]

      3. distribute-rgt-in N/A

         \[\leadsto \left(-2 \cdot \color{blue}{\left(a \cdot i + \left(b \cdot c\right) \cdot i\right)}\right) \cdot c \]

      4. associate-*r* N/A

         \[\leadsto \left(-2 \cdot \left(a \cdot i + \color{blue}{b \cdot \left(c \cdot i\right)}\right)\right) \cdot c \]

      5. distribute-lft-out N/A

         \[\leadsto \color{blue}{\left(-2 \cdot \left(a \cdot i\right) + -2 \cdot \left(b \cdot \left(c \cdot i\right)\right)\right)} \cdot c \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{c \cdot \left(-2 \cdot \left(a \cdot i\right) + -2 \cdot \left(b \cdot \left(c \cdot i\right)\right)\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{c \cdot \left(-2 \cdot \left(a \cdot i\right) + -2 \cdot \left(b \cdot \left(c \cdot i\right)\right)\right)} \]

      8. distribute-lft-out N/A

         \[\leadsto c \cdot \color{blue}{\left(-2 \cdot \left(a \cdot i + b \cdot \left(c \cdot i\right)\right)\right)} \]

      9. associate-*r* N/A

         \[\leadsto c \cdot \left(-2 \cdot \left(a \cdot i + \color{blue}{\left(b \cdot c\right) \cdot i}\right)\right) \]

      10. distribute-rgt-in N/A

          \[\leadsto c \cdot \left(-2 \cdot \color{blue}{\left(i \cdot \left(a + b \cdot c\right)\right)}\right) \]

      11. *-commutative N/A

          \[\leadsto c \cdot \left(-2 \cdot \color{blue}{\left(\left(a + b \cdot c\right) \cdot i\right)}\right) \]

      12. associate-*r* N/A

          \[\leadsto c \cdot \color{blue}{\left(\left(-2 \cdot \left(a + b \cdot c\right)\right) \cdot i\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto c \cdot \color{blue}{\left(\left(-2 \cdot \left(a + b \cdot c\right)\right) \cdot i\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto c \cdot \left(\color{blue}{\left(-2 \cdot \left(a + b \cdot c\right)\right)} \cdot i\right) \]

      15. +-commutative N/A

          \[\leadsto c \cdot \left(\left(-2 \cdot \color{blue}{\left(b \cdot c + a\right)}\right) \cdot i\right) \]

      16. lower-fma.f64 94.6

          \[\leadsto c \cdot \left(\left(-2 \cdot \color{blue}{\mathsf{fma}\left(b, c, a\right)}\right) \cdot i\right) \]

   5. Simplified 94.6%

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

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq -1 \cdot 10^{+144}:\\ \;\;\;\;2 \cdot \left(x \cdot y - c \cdot \left(i \cdot \mathsf{fma}\left(b, c, a\right)\right)\right)\\ \mathbf{elif}\;\left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq -5 \cdot 10^{-98}:\\ \;\;\;\;2 \cdot \left(z \cdot t - c \cdot \left(i \cdot \mathsf{fma}\left(b, c, a\right)\right)\right)\\ \mathbf{elif}\;\left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq 2 \cdot 10^{+294}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(z, t, \mathsf{fma}\left(x, y, i \cdot \left(b \cdot \left(c \cdot \left(-c\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(i \cdot \left(\mathsf{fma}\left(b, c, a\right) \cdot -2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := c \cdot \left(i \cdot \mathsf{fma}\left(b, c, a\right)\right)\\ t_2 := \left(c \cdot \left(a + b \cdot c\right)\right) \cdot i\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+144}:\\ \;\;\;\;2 \cdot \left(x \cdot y - t\_1\right)\\ \mathbf{elif}\;t\_2 \leq -5 \cdot 10^{-98}:\\ \;\;\;\;2 \cdot \left(z \cdot t - t\_1\right)\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+294}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(x, y, \mathsf{fma}\left(t, z, i \cdot \left(b \cdot \left(c \cdot \left(-c\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(i \cdot \left(\mathsf{fma}\left(b, c, a\right) \cdot -2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* c (* i (fma b c a)))) (t_2 (* (* c (+ a (* b c))) i)))
   (if (<= t_2 -1e+144)
     (* 2.0 (- (* x y) t_1))
     (if (<= t_2 -5e-98)
       (* 2.0 (- (* z t) t_1))
       (if (<= t_2 2e+294)
         (* 2.0 (fma x y (fma t z (* i (* b (* c (- c)))))))
         (* c (* i (* (fma b c a) -2.0))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = c * (i * fma(b, c, a));
	double t_2 = (c * (a + (b * c))) * i;
	double tmp;
	if (t_2 <= -1e+144) {
		tmp = 2.0 * ((x * y) - t_1);
	} else if (t_2 <= -5e-98) {
		tmp = 2.0 * ((z * t) - t_1);
	} else if (t_2 <= 2e+294) {
		tmp = 2.0 * fma(x, y, fma(t, z, (i * (b * (c * -c)))));
	} else {
		tmp = c * (i * (fma(b, c, a) * -2.0));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(c * Float64(i * fma(b, c, a)))
	t_2 = Float64(Float64(c * Float64(a + Float64(b * c))) * i)
	tmp = 0.0
	if (t_2 <= -1e+144)
		tmp = Float64(2.0 * Float64(Float64(x * y) - t_1));
	elseif (t_2 <= -5e-98)
		tmp = Float64(2.0 * Float64(Float64(z * t) - t_1));
	elseif (t_2 <= 2e+294)
		tmp = Float64(2.0 * fma(x, y, fma(t, z, Float64(i * Float64(b * Float64(c * Float64(-c)))))));
	else
		tmp = Float64(c * Float64(i * Float64(fma(b, c, a) * -2.0)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(c * N[(i * N[(b * c + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(c * N[(a + N[(b * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * i), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+144], N[(2.0 * N[(N[(x * y), $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -5e-98], N[(2.0 * N[(N[(z * t), $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e+294], N[(2.0 * N[(x * y + N[(t * z + N[(i * N[(b * N[(c * (-c)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c * N[(i * N[(N[(b * c + a), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -1.00000000000000002e144

   1. Initial program 81.0%

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

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{2 \cdot \left(x \cdot y - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{2 \cdot \left(x \cdot y - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)} \]

      2. lower--.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\left(x \cdot y - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto 2 \cdot \left(\color{blue}{x \cdot y} - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto 2 \cdot \left(x \cdot y - \color{blue}{c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto 2 \cdot \left(x \cdot y - c \cdot \color{blue}{\left(\left(a + b \cdot c\right) \cdot i\right)}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto 2 \cdot \left(x \cdot y - c \cdot \color{blue}{\left(\left(a + b \cdot c\right) \cdot i\right)}\right) \]

      7. +-commutative N/A

         \[\leadsto 2 \cdot \left(x \cdot y - c \cdot \left(\color{blue}{\left(b \cdot c + a\right)} \cdot i\right)\right) \]

      8. lower-fma.f64 93.2

         \[\leadsto 2 \cdot \left(x \cdot y - c \cdot \left(\color{blue}{\mathsf{fma}\left(b, c, a\right)} \cdot i\right)\right) \]

   5. Simplified 93.2%

      \[\leadsto \color{blue}{2 \cdot \left(x \cdot y - c \cdot \left(\mathsf{fma}\left(b, c, a\right) \cdot i\right)\right)} \]

   if -1.00000000000000002e144 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -5.00000000000000018e-98

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{2 \cdot \left(t \cdot z - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{2 \cdot \left(t \cdot z - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)} \]

      2. lower--.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\left(t \cdot z - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto 2 \cdot \left(\color{blue}{t \cdot z} - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto 2 \cdot \left(t \cdot z - \color{blue}{c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto 2 \cdot \left(t \cdot z - c \cdot \color{blue}{\left(\left(a + b \cdot c\right) \cdot i\right)}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto 2 \cdot \left(t \cdot z - c \cdot \color{blue}{\left(\left(a + b \cdot c\right) \cdot i\right)}\right) \]

      7. +-commutative N/A

         \[\leadsto 2 \cdot \left(t \cdot z - c \cdot \left(\color{blue}{\left(b \cdot c + a\right)} \cdot i\right)\right) \]

      8. lower-fma.f64 85.4

         \[\leadsto 2 \cdot \left(t \cdot z - c \cdot \left(\color{blue}{\mathsf{fma}\left(b, c, a\right)} \cdot i\right)\right) \]

   5. Simplified 85.4%

      \[\leadsto \color{blue}{2 \cdot \left(t \cdot z - c \cdot \left(\mathsf{fma}\left(b, c, a\right) \cdot i\right)\right)} \]

   if -5.00000000000000018e-98 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 2.00000000000000013e294

   1. Initial program 99.0%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{2 \cdot \left(\left(t \cdot z + x \cdot y\right) - b \cdot \left({c}^{2} \cdot i\right)\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{2 \cdot \left(\left(t \cdot z + x \cdot y\right) - b \cdot \left({c}^{2} \cdot i\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto 2 \cdot \color{blue}{\left(\left(t \cdot z + x \cdot y\right) + \left(\mathsf{neg}\left(b \cdot \left({c}^{2} \cdot i\right)\right)\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto 2 \cdot \color{blue}{\left(\left(\mathsf{neg}\left(b \cdot \left({c}^{2} \cdot i\right)\right)\right) + \left(t \cdot z + x \cdot y\right)\right)} \]

      4. associate-+r+ N/A

         \[\leadsto 2 \cdot \color{blue}{\left(\left(\left(\mathsf{neg}\left(b \cdot \left({c}^{2} \cdot i\right)\right)\right) + t \cdot z\right) + x \cdot y\right)} \]

      5. +-commutative N/A

         \[\leadsto 2 \cdot \color{blue}{\left(x \cdot y + \left(\left(\mathsf{neg}\left(b \cdot \left({c}^{2} \cdot i\right)\right)\right) + t \cdot z\right)\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\mathsf{fma}\left(x, y, \left(\mathsf{neg}\left(b \cdot \left({c}^{2} \cdot i\right)\right)\right) + t \cdot z\right)} \]

      7. +-commutative N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(x, y, \color{blue}{t \cdot z + \left(\mathsf{neg}\left(b \cdot \left({c}^{2} \cdot i\right)\right)\right)}\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(x, y, \color{blue}{\mathsf{fma}\left(t, z, \mathsf{neg}\left(b \cdot \left({c}^{2} \cdot i\right)\right)\right)}\right) \]

      9. *-commutative N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(x, y, \mathsf{fma}\left(t, z, \mathsf{neg}\left(\color{blue}{\left({c}^{2} \cdot i\right) \cdot b}\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(x, y, \mathsf{fma}\left(t, z, \mathsf{neg}\left(\color{blue}{\left(i \cdot {c}^{2}\right)} \cdot b\right)\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(x, y, \mathsf{fma}\left(t, z, \mathsf{neg}\left(\color{blue}{i \cdot \left({c}^{2} \cdot b\right)}\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(x, y, \mathsf{fma}\left(t, z, \mathsf{neg}\left(i \cdot \color{blue}{\left(b \cdot {c}^{2}\right)}\right)\right)\right) \]

      13. distribute-rgt-neg-in N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(x, y, \mathsf{fma}\left(t, z, \color{blue}{i \cdot \left(\mathsf{neg}\left(b \cdot {c}^{2}\right)\right)}\right)\right) \]

      14. mul-1-neg N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(x, y, \mathsf{fma}\left(t, z, i \cdot \color{blue}{\left(-1 \cdot \left(b \cdot {c}^{2}\right)\right)}\right)\right) \]

      15. lower-*.f64 N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(x, y, \mathsf{fma}\left(t, z, \color{blue}{i \cdot \left(-1 \cdot \left(b \cdot {c}^{2}\right)\right)}\right)\right) \]

      16. mul-1-neg N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(x, y, \mathsf{fma}\left(t, z, i \cdot \color{blue}{\left(\mathsf{neg}\left(b \cdot {c}^{2}\right)\right)}\right)\right) \]

      17. distribute-rgt-neg-out N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(x, y, \mathsf{fma}\left(t, z, i \cdot \color{blue}{\left(b \cdot \left(\mathsf{neg}\left({c}^{2}\right)\right)\right)}\right)\right) \]

      18. mul-1-neg N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(x, y, \mathsf{fma}\left(t, z, i \cdot \left(b \cdot \color{blue}{\left(-1 \cdot {c}^{2}\right)}\right)\right)\right) \]

      19. lower-*.f64 N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(x, y, \mathsf{fma}\left(t, z, i \cdot \color{blue}{\left(b \cdot \left(-1 \cdot {c}^{2}\right)\right)}\right)\right) \]

      20. mul-1-neg N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(x, y, \mathsf{fma}\left(t, z, i \cdot \left(b \cdot \color{blue}{\left(\mathsf{neg}\left({c}^{2}\right)\right)}\right)\right)\right) \]

      21. unpow2 N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(x, y, \mathsf{fma}\left(t, z, i \cdot \left(b \cdot \left(\mathsf{neg}\left(\color{blue}{c \cdot c}\right)\right)\right)\right)\right) \]

      22. distribute-rgt-neg-in N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(x, y, \mathsf{fma}\left(t, z, i \cdot \left(b \cdot \color{blue}{\left(c \cdot \left(\mathsf{neg}\left(c\right)\right)\right)}\right)\right)\right) \]

      23. mul-1-neg N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(x, y, \mathsf{fma}\left(t, z, i \cdot \left(b \cdot \left(c \cdot \color{blue}{\left(-1 \cdot c\right)}\right)\right)\right)\right) \]

   5. Simplified 92.8%

      \[\leadsto \color{blue}{2 \cdot \mathsf{fma}\left(x, y, \mathsf{fma}\left(t, z, i \cdot \left(b \cdot \left(c \cdot \left(-c\right)\right)\right)\right)\right)} \]

   if 2.00000000000000013e294 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 77.5%

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

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{-2 \cdot \left(c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto -2 \cdot \color{blue}{\left(\left(i \cdot \left(a + b \cdot c\right)\right) \cdot c\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-2 \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right) \cdot c} \]

      3. distribute-rgt-in N/A

         \[\leadsto \left(-2 \cdot \color{blue}{\left(a \cdot i + \left(b \cdot c\right) \cdot i\right)}\right) \cdot c \]

      4. associate-*r* N/A

         \[\leadsto \left(-2 \cdot \left(a \cdot i + \color{blue}{b \cdot \left(c \cdot i\right)}\right)\right) \cdot c \]

      5. distribute-lft-out N/A

         \[\leadsto \color{blue}{\left(-2 \cdot \left(a \cdot i\right) + -2 \cdot \left(b \cdot \left(c \cdot i\right)\right)\right)} \cdot c \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{c \cdot \left(-2 \cdot \left(a \cdot i\right) + -2 \cdot \left(b \cdot \left(c \cdot i\right)\right)\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{c \cdot \left(-2 \cdot \left(a \cdot i\right) + -2 \cdot \left(b \cdot \left(c \cdot i\right)\right)\right)} \]

      8. distribute-lft-out N/A

         \[\leadsto c \cdot \color{blue}{\left(-2 \cdot \left(a \cdot i + b \cdot \left(c \cdot i\right)\right)\right)} \]

      9. associate-*r* N/A

         \[\leadsto c \cdot \left(-2 \cdot \left(a \cdot i + \color{blue}{\left(b \cdot c\right) \cdot i}\right)\right) \]

      10. distribute-rgt-in N/A

          \[\leadsto c \cdot \left(-2 \cdot \color{blue}{\left(i \cdot \left(a + b \cdot c\right)\right)}\right) \]

      11. *-commutative N/A

          \[\leadsto c \cdot \left(-2 \cdot \color{blue}{\left(\left(a + b \cdot c\right) \cdot i\right)}\right) \]

      12. associate-*r* N/A

          \[\leadsto c \cdot \color{blue}{\left(\left(-2 \cdot \left(a + b \cdot c\right)\right) \cdot i\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto c \cdot \color{blue}{\left(\left(-2 \cdot \left(a + b \cdot c\right)\right) \cdot i\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto c \cdot \left(\color{blue}{\left(-2 \cdot \left(a + b \cdot c\right)\right)} \cdot i\right) \]

      15. +-commutative N/A

          \[\leadsto c \cdot \left(\left(-2 \cdot \color{blue}{\left(b \cdot c + a\right)}\right) \cdot i\right) \]

      16. lower-fma.f64 94.6

          \[\leadsto c \cdot \left(\left(-2 \cdot \color{blue}{\mathsf{fma}\left(b, c, a\right)}\right) \cdot i\right) \]

   5. Simplified 94.6%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq -1 \cdot 10^{+144}:\\ \;\;\;\;2 \cdot \left(x \cdot y - c \cdot \left(i \cdot \mathsf{fma}\left(b, c, a\right)\right)\right)\\ \mathbf{elif}\;\left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq -5 \cdot 10^{-98}:\\ \;\;\;\;2 \cdot \left(z \cdot t - c \cdot \left(i \cdot \mathsf{fma}\left(b, c, a\right)\right)\right)\\ \mathbf{elif}\;\left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq 2 \cdot 10^{+294}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(x, y, \mathsf{fma}\left(t, z, i \cdot \left(b \cdot \left(c \cdot \left(-c\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(i \cdot \left(\mathsf{fma}\left(b, c, a\right) \cdot -2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := c \cdot \left(i \cdot \mathsf{fma}\left(b, c, a\right)\right)\\ t_2 := \left(c \cdot \left(a + b \cdot c\right)\right) \cdot i\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+144}:\\ \;\;\;\;2 \cdot \left(x \cdot y - t\_1\right)\\ \mathbf{elif}\;t\_2 \leq -5 \cdot 10^{-98}:\\ \;\;\;\;2 \cdot \left(z \cdot t - t\_1\right)\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+164}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(t, z, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(i \cdot \left(\mathsf{fma}\left(b, c, a\right) \cdot -2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* c (* i (fma b c a)))) (t_2 (* (* c (+ a (* b c))) i)))
   (if (<= t_2 -1e+144)
     (* 2.0 (- (* x y) t_1))
     (if (<= t_2 -5e-98)
       (* 2.0 (- (* z t) t_1))
       (if (<= t_2 2e+164)
         (* 2.0 (fma t z (* x y)))
         (* c (* i (* (fma b c a) -2.0))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = c * (i * fma(b, c, a));
	double t_2 = (c * (a + (b * c))) * i;
	double tmp;
	if (t_2 <= -1e+144) {
		tmp = 2.0 * ((x * y) - t_1);
	} else if (t_2 <= -5e-98) {
		tmp = 2.0 * ((z * t) - t_1);
	} else if (t_2 <= 2e+164) {
		tmp = 2.0 * fma(t, z, (x * y));
	} else {
		tmp = c * (i * (fma(b, c, a) * -2.0));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(c * Float64(i * fma(b, c, a)))
	t_2 = Float64(Float64(c * Float64(a + Float64(b * c))) * i)
	tmp = 0.0
	if (t_2 <= -1e+144)
		tmp = Float64(2.0 * Float64(Float64(x * y) - t_1));
	elseif (t_2 <= -5e-98)
		tmp = Float64(2.0 * Float64(Float64(z * t) - t_1));
	elseif (t_2 <= 2e+164)
		tmp = Float64(2.0 * fma(t, z, Float64(x * y)));
	else
		tmp = Float64(c * Float64(i * Float64(fma(b, c, a) * -2.0)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(c * N[(i * N[(b * c + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(c * N[(a + N[(b * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * i), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+144], N[(2.0 * N[(N[(x * y), $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -5e-98], N[(2.0 * N[(N[(z * t), $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e+164], N[(2.0 * N[(t * z + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c * N[(i * N[(N[(b * c + a), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -1.00000000000000002e144

   1. Initial program 81.0%

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

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{2 \cdot \left(x \cdot y - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{2 \cdot \left(x \cdot y - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)} \]

      2. lower--.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\left(x \cdot y - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto 2 \cdot \left(\color{blue}{x \cdot y} - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto 2 \cdot \left(x \cdot y - \color{blue}{c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto 2 \cdot \left(x \cdot y - c \cdot \color{blue}{\left(\left(a + b \cdot c\right) \cdot i\right)}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto 2 \cdot \left(x \cdot y - c \cdot \color{blue}{\left(\left(a + b \cdot c\right) \cdot i\right)}\right) \]

      7. +-commutative N/A

         \[\leadsto 2 \cdot \left(x \cdot y - c \cdot \left(\color{blue}{\left(b \cdot c + a\right)} \cdot i\right)\right) \]

      8. lower-fma.f64 93.2

         \[\leadsto 2 \cdot \left(x \cdot y - c \cdot \left(\color{blue}{\mathsf{fma}\left(b, c, a\right)} \cdot i\right)\right) \]

   5. Simplified 93.2%

      \[\leadsto \color{blue}{2 \cdot \left(x \cdot y - c \cdot \left(\mathsf{fma}\left(b, c, a\right) \cdot i\right)\right)} \]

   if -1.00000000000000002e144 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -5.00000000000000018e-98

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{2 \cdot \left(t \cdot z - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{2 \cdot \left(t \cdot z - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)} \]

      2. lower--.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\left(t \cdot z - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto 2 \cdot \left(\color{blue}{t \cdot z} - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto 2 \cdot \left(t \cdot z - \color{blue}{c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto 2 \cdot \left(t \cdot z - c \cdot \color{blue}{\left(\left(a + b \cdot c\right) \cdot i\right)}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto 2 \cdot \left(t \cdot z - c \cdot \color{blue}{\left(\left(a + b \cdot c\right) \cdot i\right)}\right) \]

      7. +-commutative N/A

         \[\leadsto 2 \cdot \left(t \cdot z - c \cdot \left(\color{blue}{\left(b \cdot c + a\right)} \cdot i\right)\right) \]

      8. lower-fma.f64 85.4

         \[\leadsto 2 \cdot \left(t \cdot z - c \cdot \left(\color{blue}{\mathsf{fma}\left(b, c, a\right)} \cdot i\right)\right) \]

   5. Simplified 85.4%

      \[\leadsto \color{blue}{2 \cdot \left(t \cdot z - c \cdot \left(\mathsf{fma}\left(b, c, a\right) \cdot i\right)\right)} \]

   if -5.00000000000000018e-98 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 2e164

   1. Initial program 99.0%

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

   3. Taylor expanded in c around 0

      \[\leadsto \color{blue}{2 \cdot \left(t \cdot z + x \cdot y\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(t \cdot z + x \cdot y\right) \cdot 2} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(t \cdot z + x \cdot y\right) \cdot 2} \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(t, z, x \cdot y\right)} \cdot 2 \]

      4. lower-*.f64 91.6

         \[\leadsto \mathsf{fma}\left(t, z, \color{blue}{x \cdot y}\right) \cdot 2 \]

   5. Simplified 91.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, z, x \cdot y\right) \cdot 2} \]

   if 2e164 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 80.0%

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

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{-2 \cdot \left(c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto -2 \cdot \color{blue}{\left(\left(i \cdot \left(a + b \cdot c\right)\right) \cdot c\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-2 \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right) \cdot c} \]

      3. distribute-rgt-in N/A

         \[\leadsto \left(-2 \cdot \color{blue}{\left(a \cdot i + \left(b \cdot c\right) \cdot i\right)}\right) \cdot c \]

      4. associate-*r* N/A

         \[\leadsto \left(-2 \cdot \left(a \cdot i + \color{blue}{b \cdot \left(c \cdot i\right)}\right)\right) \cdot c \]

      5. distribute-lft-out N/A

         \[\leadsto \color{blue}{\left(-2 \cdot \left(a \cdot i\right) + -2 \cdot \left(b \cdot \left(c \cdot i\right)\right)\right)} \cdot c \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{c \cdot \left(-2 \cdot \left(a \cdot i\right) + -2 \cdot \left(b \cdot \left(c \cdot i\right)\right)\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{c \cdot \left(-2 \cdot \left(a \cdot i\right) + -2 \cdot \left(b \cdot \left(c \cdot i\right)\right)\right)} \]

      8. distribute-lft-out N/A

         \[\leadsto c \cdot \color{blue}{\left(-2 \cdot \left(a \cdot i + b \cdot \left(c \cdot i\right)\right)\right)} \]

      9. associate-*r* N/A

         \[\leadsto c \cdot \left(-2 \cdot \left(a \cdot i + \color{blue}{\left(b \cdot c\right) \cdot i}\right)\right) \]

      10. distribute-rgt-in N/A

          \[\leadsto c \cdot \left(-2 \cdot \color{blue}{\left(i \cdot \left(a + b \cdot c\right)\right)}\right) \]

      11. *-commutative N/A

          \[\leadsto c \cdot \left(-2 \cdot \color{blue}{\left(\left(a + b \cdot c\right) \cdot i\right)}\right) \]

      12. associate-*r* N/A

          \[\leadsto c \cdot \color{blue}{\left(\left(-2 \cdot \left(a + b \cdot c\right)\right) \cdot i\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto c \cdot \color{blue}{\left(\left(-2 \cdot \left(a + b \cdot c\right)\right) \cdot i\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto c \cdot \left(\color{blue}{\left(-2 \cdot \left(a + b \cdot c\right)\right)} \cdot i\right) \]

      15. +-commutative N/A

          \[\leadsto c \cdot \left(\left(-2 \cdot \color{blue}{\left(b \cdot c + a\right)}\right) \cdot i\right) \]

      16. lower-fma.f64 89.9

          \[\leadsto c \cdot \left(\left(-2 \cdot \color{blue}{\mathsf{fma}\left(b, c, a\right)}\right) \cdot i\right) \]

   5. Simplified 89.9%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq -1 \cdot 10^{+144}:\\ \;\;\;\;2 \cdot \left(x \cdot y - c \cdot \left(i \cdot \mathsf{fma}\left(b, c, a\right)\right)\right)\\ \mathbf{elif}\;\left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq -5 \cdot 10^{-98}:\\ \;\;\;\;2 \cdot \left(z \cdot t - c \cdot \left(i \cdot \mathsf{fma}\left(b, c, a\right)\right)\right)\\ \mathbf{elif}\;\left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq 2 \cdot 10^{+164}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(t, z, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(i \cdot \left(\mathsf{fma}\left(b, c, a\right) \cdot -2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := c \cdot \left(i \cdot \left(\mathsf{fma}\left(b, c, a\right) \cdot -2\right)\right)\\ t_2 := \left(c \cdot \left(a + b \cdot c\right)\right) \cdot i\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+290}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq -5 \cdot 10^{-98}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(t, z, -a \cdot \left(c \cdot i\right)\right)\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+164}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(t, z, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* c (* i (* (fma b c a) -2.0)))) (t_2 (* (* c (+ a (* b c))) i)))
   (if (<= t_2 -1e+290)
     t_1
     (if (<= t_2 -5e-98)
       (* 2.0 (fma t z (- (* a (* c i)))))
       (if (<= t_2 2e+164) (* 2.0 (fma t z (* x y))) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = c * (i * (fma(b, c, a) * -2.0));
	double t_2 = (c * (a + (b * c))) * i;
	double tmp;
	if (t_2 <= -1e+290) {
		tmp = t_1;
	} else if (t_2 <= -5e-98) {
		tmp = 2.0 * fma(t, z, -(a * (c * i)));
	} else if (t_2 <= 2e+164) {
		tmp = 2.0 * fma(t, z, (x * y));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(c * Float64(i * Float64(fma(b, c, a) * -2.0)))
	t_2 = Float64(Float64(c * Float64(a + Float64(b * c))) * i)
	tmp = 0.0
	if (t_2 <= -1e+290)
		tmp = t_1;
	elseif (t_2 <= -5e-98)
		tmp = Float64(2.0 * fma(t, z, Float64(-Float64(a * Float64(c * i)))));
	elseif (t_2 <= 2e+164)
		tmp = Float64(2.0 * fma(t, z, Float64(x * y)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(c * N[(i * N[(N[(b * c + a), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(c * N[(a + N[(b * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * i), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+290], t$95$1, If[LessEqual[t$95$2, -5e-98], N[(2.0 * N[(t * z + (-N[(a * N[(c * i), $MachinePrecision]), $MachinePrecision])), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e+164], N[(2.0 * N[(t * z + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -1.00000000000000006e290 or 2e164 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 78.6%

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

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{-2 \cdot \left(c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto -2 \cdot \color{blue}{\left(\left(i \cdot \left(a + b \cdot c\right)\right) \cdot c\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-2 \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right) \cdot c} \]

      3. distribute-rgt-in N/A

         \[\leadsto \left(-2 \cdot \color{blue}{\left(a \cdot i + \left(b \cdot c\right) \cdot i\right)}\right) \cdot c \]

      4. associate-*r* N/A

         \[\leadsto \left(-2 \cdot \left(a \cdot i + \color{blue}{b \cdot \left(c \cdot i\right)}\right)\right) \cdot c \]

      5. distribute-lft-out N/A

         \[\leadsto \color{blue}{\left(-2 \cdot \left(a \cdot i\right) + -2 \cdot \left(b \cdot \left(c \cdot i\right)\right)\right)} \cdot c \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{c \cdot \left(-2 \cdot \left(a \cdot i\right) + -2 \cdot \left(b \cdot \left(c \cdot i\right)\right)\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{c \cdot \left(-2 \cdot \left(a \cdot i\right) + -2 \cdot \left(b \cdot \left(c \cdot i\right)\right)\right)} \]

      8. distribute-lft-out N/A

         \[\leadsto c \cdot \color{blue}{\left(-2 \cdot \left(a \cdot i + b \cdot \left(c \cdot i\right)\right)\right)} \]

      9. associate-*r* N/A

         \[\leadsto c \cdot \left(-2 \cdot \left(a \cdot i + \color{blue}{\left(b \cdot c\right) \cdot i}\right)\right) \]

      10. distribute-rgt-in N/A

          \[\leadsto c \cdot \left(-2 \cdot \color{blue}{\left(i \cdot \left(a + b \cdot c\right)\right)}\right) \]

      11. *-commutative N/A

          \[\leadsto c \cdot \left(-2 \cdot \color{blue}{\left(\left(a + b \cdot c\right) \cdot i\right)}\right) \]

      12. associate-*r* N/A

          \[\leadsto c \cdot \color{blue}{\left(\left(-2 \cdot \left(a + b \cdot c\right)\right) \cdot i\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto c \cdot \color{blue}{\left(\left(-2 \cdot \left(a + b \cdot c\right)\right) \cdot i\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto c \cdot \left(\color{blue}{\left(-2 \cdot \left(a + b \cdot c\right)\right)} \cdot i\right) \]

      15. +-commutative N/A

          \[\leadsto c \cdot \left(\left(-2 \cdot \color{blue}{\left(b \cdot c + a\right)}\right) \cdot i\right) \]

      16. lower-fma.f64 86.7

          \[\leadsto c \cdot \left(\left(-2 \cdot \color{blue}{\mathsf{fma}\left(b, c, a\right)}\right) \cdot i\right) \]

   5. Simplified 86.7%

      \[\leadsto \color{blue}{c \cdot \left(\left(-2 \cdot \mathsf{fma}\left(b, c, a\right)\right) \cdot i\right)} \]

   if -1.00000000000000006e290 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -5.00000000000000018e-98

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{2 \cdot \left(t \cdot z - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{2 \cdot \left(t \cdot z - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)} \]

      2. lower--.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\left(t \cdot z - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto 2 \cdot \left(\color{blue}{t \cdot z} - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto 2 \cdot \left(t \cdot z - \color{blue}{c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto 2 \cdot \left(t \cdot z - c \cdot \color{blue}{\left(\left(a + b \cdot c\right) \cdot i\right)}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto 2 \cdot \left(t \cdot z - c \cdot \color{blue}{\left(\left(a + b \cdot c\right) \cdot i\right)}\right) \]

      7. +-commutative N/A

         \[\leadsto 2 \cdot \left(t \cdot z - c \cdot \left(\color{blue}{\left(b \cdot c + a\right)} \cdot i\right)\right) \]

      8. lower-fma.f64 77.6

         \[\leadsto 2 \cdot \left(t \cdot z - c \cdot \left(\color{blue}{\mathsf{fma}\left(b, c, a\right)} \cdot i\right)\right) \]

   5. Simplified 77.6%

      \[\leadsto \color{blue}{2 \cdot \left(t \cdot z - c \cdot \left(\mathsf{fma}\left(b, c, a\right) \cdot i\right)\right)} \]

   6. Taylor expanded in b around 0

      \[\leadsto \color{blue}{2 \cdot \left(t \cdot z - a \cdot \left(c \cdot i\right)\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{2 \cdot \left(t \cdot z - a \cdot \left(c \cdot i\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto 2 \cdot \color{blue}{\left(t \cdot z + \left(\mathsf{neg}\left(a \cdot \left(c \cdot i\right)\right)\right)\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\mathsf{fma}\left(t, z, \mathsf{neg}\left(a \cdot \left(c \cdot i\right)\right)\right)} \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(t, z, \color{blue}{a \cdot \left(\mathsf{neg}\left(c \cdot i\right)\right)}\right) \]

      5. mul-1-neg N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(t, z, a \cdot \color{blue}{\left(-1 \cdot \left(c \cdot i\right)\right)}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(t, z, \color{blue}{a \cdot \left(-1 \cdot \left(c \cdot i\right)\right)}\right) \]

      7. mul-1-neg N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(t, z, a \cdot \color{blue}{\left(\mathsf{neg}\left(c \cdot i\right)\right)}\right) \]

      8. *-commutative N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(t, z, a \cdot \left(\mathsf{neg}\left(\color{blue}{i \cdot c}\right)\right)\right) \]

      9. distribute-rgt-neg-in N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(t, z, a \cdot \color{blue}{\left(i \cdot \left(\mathsf{neg}\left(c\right)\right)\right)}\right) \]

      10. mul-1-neg N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(t, z, a \cdot \left(i \cdot \color{blue}{\left(-1 \cdot c\right)}\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(t, z, a \cdot \color{blue}{\left(i \cdot \left(-1 \cdot c\right)\right)}\right) \]

      12. mul-1-neg N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(t, z, a \cdot \left(i \cdot \color{blue}{\left(\mathsf{neg}\left(c\right)\right)}\right)\right) \]

      13. lower-neg.f64 70.0

          \[\leadsto 2 \cdot \mathsf{fma}\left(t, z, a \cdot \left(i \cdot \color{blue}{\left(-c\right)}\right)\right) \]

   8. Simplified 70.0%

      \[\leadsto \color{blue}{2 \cdot \mathsf{fma}\left(t, z, a \cdot \left(i \cdot \left(-c\right)\right)\right)} \]

   if -5.00000000000000018e-98 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 2e164

   1. Initial program 99.0%

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

   3. Taylor expanded in c around 0

      \[\leadsto \color{blue}{2 \cdot \left(t \cdot z + x \cdot y\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(t \cdot z + x \cdot y\right) \cdot 2} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(t \cdot z + x \cdot y\right) \cdot 2} \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(t, z, x \cdot y\right)} \cdot 2 \]

      4. lower-*.f64 91.6

         \[\leadsto \mathsf{fma}\left(t, z, \color{blue}{x \cdot y}\right) \cdot 2 \]

   5. Simplified 91.6%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq -1 \cdot 10^{+290}:\\ \;\;\;\;c \cdot \left(i \cdot \left(\mathsf{fma}\left(b, c, a\right) \cdot -2\right)\right)\\ \mathbf{elif}\;\left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq -5 \cdot 10^{-98}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(t, z, -a \cdot \left(c \cdot i\right)\right)\\ \mathbf{elif}\;\left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq 2 \cdot 10^{+164}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(t, z, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(i \cdot \left(\mathsf{fma}\left(b, c, a\right) \cdot -2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(c \cdot \left(a + b \cdot c\right)\right) \cdot i\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-98}:\\ \;\;\;\;2 \cdot \left(z \cdot t - c \cdot \left(i \cdot \mathsf{fma}\left(b, c, a\right)\right)\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+164}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(t, z, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(i \cdot \left(\mathsf{fma}\left(b, c, a\right) \cdot -2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (* c (+ a (* b c))) i)))
   (if (<= t_1 -5e-98)
     (* 2.0 (- (* z t) (* c (* i (fma b c a)))))
     (if (<= t_1 2e+164)
       (* 2.0 (fma t z (* x y)))
       (* c (* i (* (fma b c a) -2.0)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (c * (a + (b * c))) * i;
	double tmp;
	if (t_1 <= -5e-98) {
		tmp = 2.0 * ((z * t) - (c * (i * fma(b, c, a))));
	} else if (t_1 <= 2e+164) {
		tmp = 2.0 * fma(t, z, (x * y));
	} else {
		tmp = c * (i * (fma(b, c, a) * -2.0));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(c * Float64(a + Float64(b * c))) * i)
	tmp = 0.0
	if (t_1 <= -5e-98)
		tmp = Float64(2.0 * Float64(Float64(z * t) - Float64(c * Float64(i * fma(b, c, a)))));
	elseif (t_1 <= 2e+164)
		tmp = Float64(2.0 * fma(t, z, Float64(x * y)));
	else
		tmp = Float64(c * Float64(i * Float64(fma(b, c, a) * -2.0)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(c * N[(a + N[(b * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * i), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-98], N[(2.0 * N[(N[(z * t), $MachinePrecision] - N[(c * N[(i * N[(b * c + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+164], N[(2.0 * N[(t * z + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c * N[(i * N[(N[(b * c + a), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -5.00000000000000018e-98

   1. Initial program 86.6%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{2 \cdot \left(t \cdot z - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{2 \cdot \left(t \cdot z - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)} \]

      2. lower--.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\left(t \cdot z - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto 2 \cdot \left(\color{blue}{t \cdot z} - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto 2 \cdot \left(t \cdot z - \color{blue}{c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto 2 \cdot \left(t \cdot z - c \cdot \color{blue}{\left(\left(a + b \cdot c\right) \cdot i\right)}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto 2 \cdot \left(t \cdot z - c \cdot \color{blue}{\left(\left(a + b \cdot c\right) \cdot i\right)}\right) \]

      7. +-commutative N/A

         \[\leadsto 2 \cdot \left(t \cdot z - c \cdot \left(\color{blue}{\left(b \cdot c + a\right)} \cdot i\right)\right) \]

      8. lower-fma.f64 80.2

         \[\leadsto 2 \cdot \left(t \cdot z - c \cdot \left(\color{blue}{\mathsf{fma}\left(b, c, a\right)} \cdot i\right)\right) \]

   5. Simplified 80.2%

      \[\leadsto \color{blue}{2 \cdot \left(t \cdot z - c \cdot \left(\mathsf{fma}\left(b, c, a\right) \cdot i\right)\right)} \]

   if -5.00000000000000018e-98 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 2e164

   1. Initial program 99.0%

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

   3. Taylor expanded in c around 0

      \[\leadsto \color{blue}{2 \cdot \left(t \cdot z + x \cdot y\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(t \cdot z + x \cdot y\right) \cdot 2} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(t \cdot z + x \cdot y\right) \cdot 2} \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(t, z, x \cdot y\right)} \cdot 2 \]

      4. lower-*.f64 91.6

         \[\leadsto \mathsf{fma}\left(t, z, \color{blue}{x \cdot y}\right) \cdot 2 \]

   5. Simplified 91.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, z, x \cdot y\right) \cdot 2} \]

   if 2e164 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 80.0%

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

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{-2 \cdot \left(c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto -2 \cdot \color{blue}{\left(\left(i \cdot \left(a + b \cdot c\right)\right) \cdot c\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-2 \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right) \cdot c} \]

      3. distribute-rgt-in N/A

         \[\leadsto \left(-2 \cdot \color{blue}{\left(a \cdot i + \left(b \cdot c\right) \cdot i\right)}\right) \cdot c \]

      4. associate-*r* N/A

         \[\leadsto \left(-2 \cdot \left(a \cdot i + \color{blue}{b \cdot \left(c \cdot i\right)}\right)\right) \cdot c \]

      5. distribute-lft-out N/A

         \[\leadsto \color{blue}{\left(-2 \cdot \left(a \cdot i\right) + -2 \cdot \left(b \cdot \left(c \cdot i\right)\right)\right)} \cdot c \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{c \cdot \left(-2 \cdot \left(a \cdot i\right) + -2 \cdot \left(b \cdot \left(c \cdot i\right)\right)\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{c \cdot \left(-2 \cdot \left(a \cdot i\right) + -2 \cdot \left(b \cdot \left(c \cdot i\right)\right)\right)} \]

      8. distribute-lft-out N/A

         \[\leadsto c \cdot \color{blue}{\left(-2 \cdot \left(a \cdot i + b \cdot \left(c \cdot i\right)\right)\right)} \]

      9. associate-*r* N/A

         \[\leadsto c \cdot \left(-2 \cdot \left(a \cdot i + \color{blue}{\left(b \cdot c\right) \cdot i}\right)\right) \]

      10. distribute-rgt-in N/A

          \[\leadsto c \cdot \left(-2 \cdot \color{blue}{\left(i \cdot \left(a + b \cdot c\right)\right)}\right) \]

      11. *-commutative N/A

          \[\leadsto c \cdot \left(-2 \cdot \color{blue}{\left(\left(a + b \cdot c\right) \cdot i\right)}\right) \]

      12. associate-*r* N/A

          \[\leadsto c \cdot \color{blue}{\left(\left(-2 \cdot \left(a + b \cdot c\right)\right) \cdot i\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto c \cdot \color{blue}{\left(\left(-2 \cdot \left(a + b \cdot c\right)\right) \cdot i\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto c \cdot \left(\color{blue}{\left(-2 \cdot \left(a + b \cdot c\right)\right)} \cdot i\right) \]

      15. +-commutative N/A

          \[\leadsto c \cdot \left(\left(-2 \cdot \color{blue}{\left(b \cdot c + a\right)}\right) \cdot i\right) \]

      16. lower-fma.f64 89.9

          \[\leadsto c \cdot \left(\left(-2 \cdot \color{blue}{\mathsf{fma}\left(b, c, a\right)}\right) \cdot i\right) \]

   5. Simplified 89.9%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq -5 \cdot 10^{-98}:\\ \;\;\;\;2 \cdot \left(z \cdot t - c \cdot \left(i \cdot \mathsf{fma}\left(b, c, a\right)\right)\right)\\ \mathbf{elif}\;\left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq 2 \cdot 10^{+164}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(t, z, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(i \cdot \left(\mathsf{fma}\left(b, c, a\right) \cdot -2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 80.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := c \cdot \left(i \cdot \left(\mathsf{fma}\left(b, c, a\right) \cdot -2\right)\right)\\ t_2 := \left(c \cdot \left(a + b \cdot c\right)\right) \cdot i\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+57}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+164}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(t, z, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* c (* i (* (fma b c a) -2.0)))) (t_2 (* (* c (+ a (* b c))) i)))
   (if (<= t_2 -1e+57)
     t_1
     (if (<= t_2 2e+164) (* 2.0 (fma t z (* x y))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = c * (i * (fma(b, c, a) * -2.0));
	double t_2 = (c * (a + (b * c))) * i;
	double tmp;
	if (t_2 <= -1e+57) {
		tmp = t_1;
	} else if (t_2 <= 2e+164) {
		tmp = 2.0 * fma(t, z, (x * y));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(c * Float64(i * Float64(fma(b, c, a) * -2.0)))
	t_2 = Float64(Float64(c * Float64(a + Float64(b * c))) * i)
	tmp = 0.0
	if (t_2 <= -1e+57)
		tmp = t_1;
	elseif (t_2 <= 2e+164)
		tmp = Float64(2.0 * fma(t, z, Float64(x * y)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(c * N[(i * N[(N[(b * c + a), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(c * N[(a + N[(b * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * i), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+57], t$95$1, If[LessEqual[t$95$2, 2e+164], N[(2.0 * N[(t * z + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -1.00000000000000005e57 or 2e164 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 82.2%

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

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{-2 \cdot \left(c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto -2 \cdot \color{blue}{\left(\left(i \cdot \left(a + b \cdot c\right)\right) \cdot c\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-2 \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right) \cdot c} \]

      3. distribute-rgt-in N/A

         \[\leadsto \left(-2 \cdot \color{blue}{\left(a \cdot i + \left(b \cdot c\right) \cdot i\right)}\right) \cdot c \]

      4. associate-*r* N/A

         \[\leadsto \left(-2 \cdot \left(a \cdot i + \color{blue}{b \cdot \left(c \cdot i\right)}\right)\right) \cdot c \]

      5. distribute-lft-out N/A

         \[\leadsto \color{blue}{\left(-2 \cdot \left(a \cdot i\right) + -2 \cdot \left(b \cdot \left(c \cdot i\right)\right)\right)} \cdot c \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{c \cdot \left(-2 \cdot \left(a \cdot i\right) + -2 \cdot \left(b \cdot \left(c \cdot i\right)\right)\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{c \cdot \left(-2 \cdot \left(a \cdot i\right) + -2 \cdot \left(b \cdot \left(c \cdot i\right)\right)\right)} \]

      8. distribute-lft-out N/A

         \[\leadsto c \cdot \color{blue}{\left(-2 \cdot \left(a \cdot i + b \cdot \left(c \cdot i\right)\right)\right)} \]

      9. associate-*r* N/A

         \[\leadsto c \cdot \left(-2 \cdot \left(a \cdot i + \color{blue}{\left(b \cdot c\right) \cdot i}\right)\right) \]

      10. distribute-rgt-in N/A

          \[\leadsto c \cdot \left(-2 \cdot \color{blue}{\left(i \cdot \left(a + b \cdot c\right)\right)}\right) \]

      11. *-commutative N/A

          \[\leadsto c \cdot \left(-2 \cdot \color{blue}{\left(\left(a + b \cdot c\right) \cdot i\right)}\right) \]

      12. associate-*r* N/A

          \[\leadsto c \cdot \color{blue}{\left(\left(-2 \cdot \left(a + b \cdot c\right)\right) \cdot i\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto c \cdot \color{blue}{\left(\left(-2 \cdot \left(a + b \cdot c\right)\right) \cdot i\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto c \cdot \left(\color{blue}{\left(-2 \cdot \left(a + b \cdot c\right)\right)} \cdot i\right) \]

      15. +-commutative N/A

          \[\leadsto c \cdot \left(\left(-2 \cdot \color{blue}{\left(b \cdot c + a\right)}\right) \cdot i\right) \]

      16. lower-fma.f64 81.1

          \[\leadsto c \cdot \left(\left(-2 \cdot \color{blue}{\mathsf{fma}\left(b, c, a\right)}\right) \cdot i\right) \]

   5. Simplified 81.1%

      \[\leadsto \color{blue}{c \cdot \left(\left(-2 \cdot \mathsf{fma}\left(b, c, a\right)\right) \cdot i\right)} \]

   if -1.00000000000000005e57 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 2e164

   1. Initial program 99.1%

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

   3. Taylor expanded in c around 0

      \[\leadsto \color{blue}{2 \cdot \left(t \cdot z + x \cdot y\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(t \cdot z + x \cdot y\right) \cdot 2} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(t \cdot z + x \cdot y\right) \cdot 2} \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(t, z, x \cdot y\right)} \cdot 2 \]

      4. lower-*.f64 87.6

         \[\leadsto \mathsf{fma}\left(t, z, \color{blue}{x \cdot y}\right) \cdot 2 \]

   5. Simplified 87.6%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq -1 \cdot 10^{+57}:\\ \;\;\;\;c \cdot \left(i \cdot \left(\mathsf{fma}\left(b, c, a\right) \cdot -2\right)\right)\\ \mathbf{elif}\;\left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq 2 \cdot 10^{+164}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(t, z, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(i \cdot \left(\mathsf{fma}\left(b, c, a\right) \cdot -2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 72.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(c \cdot \left(a + b \cdot c\right)\right) \cdot i\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+298}:\\ \;\;\;\;-2 \cdot \left(c \cdot \left(b \cdot \left(c \cdot i\right)\right)\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+175}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(t, z, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(i \cdot \left(-2 \cdot \left(c \cdot c\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (* c (+ a (* b c))) i)))
   (if (<= t_1 -1e+298)
     (* -2.0 (* c (* b (* c i))))
     (if (<= t_1 5e+175)
       (* 2.0 (fma t z (* x y)))
       (* b (* i (* -2.0 (* c c))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (c * (a + (b * c))) * i;
	double tmp;
	if (t_1 <= -1e+298) {
		tmp = -2.0 * (c * (b * (c * i)));
	} else if (t_1 <= 5e+175) {
		tmp = 2.0 * fma(t, z, (x * y));
	} else {
		tmp = b * (i * (-2.0 * (c * c)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(c * Float64(a + Float64(b * c))) * i)
	tmp = 0.0
	if (t_1 <= -1e+298)
		tmp = Float64(-2.0 * Float64(c * Float64(b * Float64(c * i))));
	elseif (t_1 <= 5e+175)
		tmp = Float64(2.0 * fma(t, z, Float64(x * y)));
	else
		tmp = Float64(b * Float64(i * Float64(-2.0 * Float64(c * c))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(c * N[(a + N[(b * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * i), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+298], N[(-2.0 * N[(c * N[(b * N[(c * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+175], N[(2.0 * N[(t * z + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(b * N[(i * N[(-2.0 * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -9.9999999999999996e297

   1. Initial program 76.7%

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

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{-2 \cdot \left(b \cdot \left({c}^{2} \cdot i\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(b \cdot \left({c}^{2} \cdot i\right)\right) \cdot -2} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{b \cdot \left(\left({c}^{2} \cdot i\right) \cdot -2\right)} \]

      3. *-commutative N/A

         \[\leadsto b \cdot \color{blue}{\left(-2 \cdot \left({c}^{2} \cdot i\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{b \cdot \left(-2 \cdot \left({c}^{2} \cdot i\right)\right)} \]

      5. associate-*r* N/A

         \[\leadsto b \cdot \color{blue}{\left(\left(-2 \cdot {c}^{2}\right) \cdot i\right)} \]

      6. *-commutative N/A

         \[\leadsto b \cdot \color{blue}{\left(i \cdot \left(-2 \cdot {c}^{2}\right)\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto b \cdot \color{blue}{\left(i \cdot \left(-2 \cdot {c}^{2}\right)\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto b \cdot \left(i \cdot \color{blue}{\left(-2 \cdot {c}^{2}\right)}\right) \]

      9. unpow2 N/A

         \[\leadsto b \cdot \left(i \cdot \left(-2 \cdot \color{blue}{\left(c \cdot c\right)}\right)\right) \]

      10. lower-*.f64 62.7

          \[\leadsto b \cdot \left(i \cdot \left(-2 \cdot \color{blue}{\left(c \cdot c\right)}\right)\right) \]

   5. Simplified 62.7%

      \[\leadsto \color{blue}{b \cdot \left(i \cdot \left(-2 \cdot \left(c \cdot c\right)\right)\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto b \cdot \left(i \cdot \left(-2 \cdot \color{blue}{\left(c \cdot c\right)}\right)\right) \]

      2. lift-*.f64 N/A

         \[\leadsto b \cdot \left(i \cdot \color{blue}{\left(-2 \cdot \left(c \cdot c\right)\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(b \cdot i\right) \cdot \left(-2 \cdot \left(c \cdot c\right)\right)} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(i \cdot b\right)} \cdot \left(-2 \cdot \left(c \cdot c\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left(-2 \cdot \left(c \cdot c\right)\right) \cdot \left(i \cdot b\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(-2 \cdot \left(c \cdot c\right)\right)} \cdot \left(i \cdot b\right) \]

      7. associate-*l* N/A

         \[\leadsto \color{blue}{-2 \cdot \left(\left(c \cdot c\right) \cdot \left(i \cdot b\right)\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto -2 \cdot \left(\color{blue}{\left(c \cdot c\right)} \cdot \left(i \cdot b\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto -2 \cdot \color{blue}{\left(c \cdot \left(c \cdot \left(i \cdot b\right)\right)\right)} \]

      10. associate-*l* N/A

          \[\leadsto -2 \cdot \left(c \cdot \color{blue}{\left(\left(c \cdot i\right) \cdot b\right)}\right) \]

      11. lift-*.f64 N/A

          \[\leadsto -2 \cdot \left(c \cdot \left(\color{blue}{\left(c \cdot i\right)} \cdot b\right)\right) \]

      12. *-commutative N/A

          \[\leadsto -2 \cdot \left(c \cdot \color{blue}{\left(b \cdot \left(c \cdot i\right)\right)}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto -2 \cdot \left(c \cdot \color{blue}{\left(b \cdot \left(c \cdot i\right)\right)}\right) \]

      14. lift-*.f64 N/A

          \[\leadsto -2 \cdot \color{blue}{\left(c \cdot \left(b \cdot \left(c \cdot i\right)\right)\right)} \]

      15. *-commutative N/A

          \[\leadsto \color{blue}{\left(c \cdot \left(b \cdot \left(c \cdot i\right)\right)\right) \cdot -2} \]

      16. lower-*.f64 65.6

          \[\leadsto \color{blue}{\left(c \cdot \left(b \cdot \left(c \cdot i\right)\right)\right) \cdot -2} \]

   7. Applied egg-rr 65.6%

      \[\leadsto \color{blue}{\left(c \cdot \left(b \cdot \left(c \cdot i\right)\right)\right) \cdot -2} \]

   if -9.9999999999999996e297 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 5e175

   1. Initial program 99.2%

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

   3. Taylor expanded in c around 0

      \[\leadsto \color{blue}{2 \cdot \left(t \cdot z + x \cdot y\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(t \cdot z + x \cdot y\right) \cdot 2} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(t \cdot z + x \cdot y\right) \cdot 2} \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(t, z, x \cdot y\right)} \cdot 2 \]

      4. lower-*.f64 79.6

         \[\leadsto \mathsf{fma}\left(t, z, \color{blue}{x \cdot y}\right) \cdot 2 \]

   5. Simplified 79.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, z, x \cdot y\right) \cdot 2} \]

   if 5e175 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 79.6%

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

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{-2 \cdot \left(b \cdot \left({c}^{2} \cdot i\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(b \cdot \left({c}^{2} \cdot i\right)\right) \cdot -2} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{b \cdot \left(\left({c}^{2} \cdot i\right) \cdot -2\right)} \]

      3. *-commutative N/A

         \[\leadsto b \cdot \color{blue}{\left(-2 \cdot \left({c}^{2} \cdot i\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{b \cdot \left(-2 \cdot \left({c}^{2} \cdot i\right)\right)} \]

      5. associate-*r* N/A

         \[\leadsto b \cdot \color{blue}{\left(\left(-2 \cdot {c}^{2}\right) \cdot i\right)} \]

      6. *-commutative N/A

         \[\leadsto b \cdot \color{blue}{\left(i \cdot \left(-2 \cdot {c}^{2}\right)\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto b \cdot \color{blue}{\left(i \cdot \left(-2 \cdot {c}^{2}\right)\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto b \cdot \left(i \cdot \color{blue}{\left(-2 \cdot {c}^{2}\right)}\right) \]

      9. unpow2 N/A

         \[\leadsto b \cdot \left(i \cdot \left(-2 \cdot \color{blue}{\left(c \cdot c\right)}\right)\right) \]

      10. lower-*.f64 77.6

          \[\leadsto b \cdot \left(i \cdot \left(-2 \cdot \color{blue}{\left(c \cdot c\right)}\right)\right) \]

   5. Simplified 77.6%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq -1 \cdot 10^{+298}:\\ \;\;\;\;-2 \cdot \left(c \cdot \left(b \cdot \left(c \cdot i\right)\right)\right)\\ \mathbf{elif}\;\left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq 5 \cdot 10^{+175}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(t, z, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(i \cdot \left(-2 \cdot \left(c \cdot c\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 72.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := b \cdot \left(i \cdot \left(-2 \cdot \left(c \cdot c\right)\right)\right)\\ t_2 := \left(c \cdot \left(a + b \cdot c\right)\right) \cdot i\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+298}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+175}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(t, z, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* b (* i (* -2.0 (* c c))))) (t_2 (* (* c (+ a (* b c))) i)))
   (if (<= t_2 -1e+298)
     t_1
     (if (<= t_2 5e+175) (* 2.0 (fma t z (* x y))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = b * (i * (-2.0 * (c * c)));
	double t_2 = (c * (a + (b * c))) * i;
	double tmp;
	if (t_2 <= -1e+298) {
		tmp = t_1;
	} else if (t_2 <= 5e+175) {
		tmp = 2.0 * fma(t, z, (x * y));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(b * Float64(i * Float64(-2.0 * Float64(c * c))))
	t_2 = Float64(Float64(c * Float64(a + Float64(b * c))) * i)
	tmp = 0.0
	if (t_2 <= -1e+298)
		tmp = t_1;
	elseif (t_2 <= 5e+175)
		tmp = Float64(2.0 * fma(t, z, Float64(x * y)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(b * N[(i * N[(-2.0 * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(c * N[(a + N[(b * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * i), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+298], t$95$1, If[LessEqual[t$95$2, 5e+175], N[(2.0 * N[(t * z + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -9.9999999999999996e297 or 5e175 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 78.1%

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

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{-2 \cdot \left(b \cdot \left({c}^{2} \cdot i\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(b \cdot \left({c}^{2} \cdot i\right)\right) \cdot -2} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{b \cdot \left(\left({c}^{2} \cdot i\right) \cdot -2\right)} \]

      3. *-commutative N/A

         \[\leadsto b \cdot \color{blue}{\left(-2 \cdot \left({c}^{2} \cdot i\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{b \cdot \left(-2 \cdot \left({c}^{2} \cdot i\right)\right)} \]

      5. associate-*r* N/A

         \[\leadsto b \cdot \color{blue}{\left(\left(-2 \cdot {c}^{2}\right) \cdot i\right)} \]

      6. *-commutative N/A

         \[\leadsto b \cdot \color{blue}{\left(i \cdot \left(-2 \cdot {c}^{2}\right)\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto b \cdot \color{blue}{\left(i \cdot \left(-2 \cdot {c}^{2}\right)\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto b \cdot \left(i \cdot \color{blue}{\left(-2 \cdot {c}^{2}\right)}\right) \]

      9. unpow2 N/A

         \[\leadsto b \cdot \left(i \cdot \left(-2 \cdot \color{blue}{\left(c \cdot c\right)}\right)\right) \]

      10. lower-*.f64 69.8

          \[\leadsto b \cdot \left(i \cdot \left(-2 \cdot \color{blue}{\left(c \cdot c\right)}\right)\right) \]

   5. Simplified 69.8%

      \[\leadsto \color{blue}{b \cdot \left(i \cdot \left(-2 \cdot \left(c \cdot c\right)\right)\right)} \]

   if -9.9999999999999996e297 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 5e175

   1. Initial program 99.2%

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

   3. Taylor expanded in c around 0

      \[\leadsto \color{blue}{2 \cdot \left(t \cdot z + x \cdot y\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(t \cdot z + x \cdot y\right) \cdot 2} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(t \cdot z + x \cdot y\right) \cdot 2} \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(t, z, x \cdot y\right)} \cdot 2 \]

      4. lower-*.f64 79.6

         \[\leadsto \mathsf{fma}\left(t, z, \color{blue}{x \cdot y}\right) \cdot 2 \]

   5. Simplified 79.6%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq -1 \cdot 10^{+298}:\\ \;\;\;\;b \cdot \left(i \cdot \left(-2 \cdot \left(c \cdot c\right)\right)\right)\\ \mathbf{elif}\;\left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq 5 \cdot 10^{+175}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(t, z, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(i \cdot \left(-2 \cdot \left(c \cdot c\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 43.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 \cdot \left(x \cdot y\right)\\ \mathbf{if}\;x \cdot y \leq -3.8 \cdot 10^{+52}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \cdot y \leq -2.45 \cdot 10^{-152}:\\ \;\;\;\;c \cdot \left(a \cdot \left(i \cdot -2\right)\right)\\ \mathbf{elif}\;x \cdot y \leq 5.5 \cdot 10^{+111}:\\ \;\;\;\;t \cdot \left(2 \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* 2.0 (* x y))))
   (if (<= (* x y) -3.8e+52)
     t_1
     (if (<= (* x y) -2.45e-152)
       (* c (* a (* i -2.0)))
       (if (<= (* x y) 5.5e+111) (* t (* 2.0 z)) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = 2.0 * (x * y);
	double tmp;
	if ((x * y) <= -3.8e+52) {
		tmp = t_1;
	} else if ((x * y) <= -2.45e-152) {
		tmp = c * (a * (i * -2.0));
	} else if ((x * y) <= 5.5e+111) {
		tmp = t * (2.0 * z);
	} 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 = 2.0d0 * (x * y)
    if ((x * y) <= (-3.8d+52)) then
        tmp = t_1
    else if ((x * y) <= (-2.45d-152)) then
        tmp = c * (a * (i * (-2.0d0)))
    else if ((x * y) <= 5.5d+111) then
        tmp = t * (2.0d0 * z)
    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 = 2.0 * (x * y);
	double tmp;
	if ((x * y) <= -3.8e+52) {
		tmp = t_1;
	} else if ((x * y) <= -2.45e-152) {
		tmp = c * (a * (i * -2.0));
	} else if ((x * y) <= 5.5e+111) {
		tmp = t * (2.0 * z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	t_1 = 2.0 * (x * y)
	tmp = 0
	if (x * y) <= -3.8e+52:
		tmp = t_1
	elif (x * y) <= -2.45e-152:
		tmp = c * (a * (i * -2.0))
	elif (x * y) <= 5.5e+111:
		tmp = t * (2.0 * z)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(2.0 * Float64(x * y))
	tmp = 0.0
	if (Float64(x * y) <= -3.8e+52)
		tmp = t_1;
	elseif (Float64(x * y) <= -2.45e-152)
		tmp = Float64(c * Float64(a * Float64(i * -2.0)));
	elseif (Float64(x * y) <= 5.5e+111)
		tmp = Float64(t * Float64(2.0 * z));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	t_1 = 2.0 * (x * y);
	tmp = 0.0;
	if ((x * y) <= -3.8e+52)
		tmp = t_1;
	elseif ((x * y) <= -2.45e-152)
		tmp = c * (a * (i * -2.0));
	elseif ((x * y) <= 5.5e+111)
		tmp = t * (2.0 * z);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(2.0 * N[(x * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * y), $MachinePrecision], -3.8e+52], t$95$1, If[LessEqual[N[(x * y), $MachinePrecision], -2.45e-152], N[(c * N[(a * N[(i * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 5.5e+111], N[(t * N[(2.0 * z), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -3.8e52 or 5.4999999999999998e111 < (*.f64 x y)

   1. Initial program 84.0%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{2 \cdot \left(x \cdot y\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot y\right) \cdot 2} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot y\right) \cdot 2} \]

      3. lower-*.f64 64.5

         \[\leadsto \color{blue}{\left(x \cdot y\right)} \cdot 2 \]

   5. Simplified 64.5%

      \[\leadsto \color{blue}{\left(x \cdot y\right) \cdot 2} \]

   if -3.8e52 < (*.f64 x y) < -2.44999999999999992e-152

   1. Initial program 93.7%

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

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{-2 \cdot \left(a \cdot \left(c \cdot i\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto -2 \cdot \left(a \cdot \color{blue}{\left(i \cdot c\right)}\right) \]

      2. associate-*r* N/A

         \[\leadsto -2 \cdot \color{blue}{\left(\left(a \cdot i\right) \cdot c\right)} \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left(-2 \cdot \left(a \cdot i\right)\right) \cdot c} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{c \cdot \left(-2 \cdot \left(a \cdot i\right)\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{c \cdot \left(-2 \cdot \left(a \cdot i\right)\right)} \]

      6. *-commutative N/A

         \[\leadsto c \cdot \color{blue}{\left(\left(a \cdot i\right) \cdot -2\right)} \]

      7. associate-*l* N/A

         \[\leadsto c \cdot \color{blue}{\left(a \cdot \left(i \cdot -2\right)\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto c \cdot \color{blue}{\left(a \cdot \left(i \cdot -2\right)\right)} \]

      9. lower-*.f64 32.5

         \[\leadsto c \cdot \left(a \cdot \color{blue}{\left(i \cdot -2\right)}\right) \]

   5. Simplified 32.5%

      \[\leadsto \color{blue}{c \cdot \left(a \cdot \left(i \cdot -2\right)\right)} \]

   if -2.44999999999999992e-152 < (*.f64 x y) < 5.4999999999999998e111

   1. Initial program 93.6%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{2 \cdot \left(t \cdot z\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(t \cdot z\right) \cdot 2} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{t \cdot \left(z \cdot 2\right)} \]

      3. *-commutative N/A

         \[\leadsto t \cdot \color{blue}{\left(2 \cdot z\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{t \cdot \left(2 \cdot z\right)} \]

      5. *-commutative N/A

         \[\leadsto t \cdot \color{blue}{\left(z \cdot 2\right)} \]

      6. lower-*.f64 42.5

         \[\leadsto t \cdot \color{blue}{\left(z \cdot 2\right)} \]

   5. Simplified 42.5%

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

4. Final simplification 48.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -3.8 \cdot 10^{+52}:\\ \;\;\;\;2 \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;x \cdot y \leq -2.45 \cdot 10^{-152}:\\ \;\;\;\;c \cdot \left(a \cdot \left(i \cdot -2\right)\right)\\ \mathbf{elif}\;x \cdot y \leq 5.5 \cdot 10^{+111}:\\ \;\;\;\;t \cdot \left(2 \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(x \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 57.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq 2 \cdot 10^{+294}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(t, z, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(a \cdot \left(i \cdot -2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* (* c (+ a (* b c))) i) 2e+294)
   (* 2.0 (fma t z (* x y)))
   (* c (* a (* i -2.0)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (((c * (a + (b * c))) * i) <= 2e+294) {
		tmp = 2.0 * fma(t, z, (x * y));
	} else {
		tmp = c * (a * (i * -2.0));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(Float64(c * Float64(a + Float64(b * c))) * i) <= 2e+294)
		tmp = Float64(2.0 * fma(t, z, Float64(x * y)));
	else
		tmp = Float64(c * Float64(a * Float64(i * -2.0)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(N[(c * N[(a + N[(b * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * i), $MachinePrecision], 2e+294], N[(2.0 * N[(t * z + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c * N[(a * N[(i * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 2.00000000000000013e294

   1. Initial program 93.1%

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

   3. Taylor expanded in c around 0

      \[\leadsto \color{blue}{2 \cdot \left(t \cdot z + x \cdot y\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(t \cdot z + x \cdot y\right) \cdot 2} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(t \cdot z + x \cdot y\right) \cdot 2} \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(t, z, x \cdot y\right)} \cdot 2 \]

      4. lower-*.f64 65.7

         \[\leadsto \mathsf{fma}\left(t, z, \color{blue}{x \cdot y}\right) \cdot 2 \]

   5. Simplified 65.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, z, x \cdot y\right) \cdot 2} \]

   if 2.00000000000000013e294 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 77.5%

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

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{-2 \cdot \left(a \cdot \left(c \cdot i\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto -2 \cdot \left(a \cdot \color{blue}{\left(i \cdot c\right)}\right) \]

      2. associate-*r* N/A

         \[\leadsto -2 \cdot \color{blue}{\left(\left(a \cdot i\right) \cdot c\right)} \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left(-2 \cdot \left(a \cdot i\right)\right) \cdot c} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{c \cdot \left(-2 \cdot \left(a \cdot i\right)\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{c \cdot \left(-2 \cdot \left(a \cdot i\right)\right)} \]

      6. *-commutative N/A

         \[\leadsto c \cdot \color{blue}{\left(\left(a \cdot i\right) \cdot -2\right)} \]

      7. associate-*l* N/A

         \[\leadsto c \cdot \color{blue}{\left(a \cdot \left(i \cdot -2\right)\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto c \cdot \color{blue}{\left(a \cdot \left(i \cdot -2\right)\right)} \]

      9. lower-*.f64 46.3

         \[\leadsto c \cdot \left(a \cdot \color{blue}{\left(i \cdot -2\right)}\right) \]

   5. Simplified 46.3%

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

4. Final simplification 62.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq 2 \cdot 10^{+294}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(t, z, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(a \cdot \left(i \cdot -2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 44.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 \cdot \left(x \cdot y\right)\\ \mathbf{if}\;x \cdot y \leq -9.6 \cdot 10^{+100}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \cdot y \leq 1.18 \cdot 10^{+111}:\\ \;\;\;\;t \cdot \left(2 \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* 2.0 (* x y))))
   (if (<= (* x y) -9.6e+100)
     t_1
     (if (<= (* x y) 1.18e+111) (* t (* 2.0 z)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = 2.0 * (x * y);
	double tmp;
	if ((x * y) <= -9.6e+100) {
		tmp = t_1;
	} else if ((x * y) <= 1.18e+111) {
		tmp = t * (2.0 * z);
	} 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 = 2.0d0 * (x * y)
    if ((x * y) <= (-9.6d+100)) then
        tmp = t_1
    else if ((x * y) <= 1.18d+111) then
        tmp = t * (2.0d0 * z)
    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 = 2.0 * (x * y);
	double tmp;
	if ((x * y) <= -9.6e+100) {
		tmp = t_1;
	} else if ((x * y) <= 1.18e+111) {
		tmp = t * (2.0 * z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	t_1 = 2.0 * (x * y)
	tmp = 0
	if (x * y) <= -9.6e+100:
		tmp = t_1
	elif (x * y) <= 1.18e+111:
		tmp = t * (2.0 * z)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(2.0 * Float64(x * y))
	tmp = 0.0
	if (Float64(x * y) <= -9.6e+100)
		tmp = t_1;
	elseif (Float64(x * y) <= 1.18e+111)
		tmp = Float64(t * Float64(2.0 * z));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	t_1 = 2.0 * (x * y);
	tmp = 0.0;
	if ((x * y) <= -9.6e+100)
		tmp = t_1;
	elseif ((x * y) <= 1.18e+111)
		tmp = t * (2.0 * z);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(2.0 * N[(x * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * y), $MachinePrecision], -9.6e+100], t$95$1, If[LessEqual[N[(x * y), $MachinePrecision], 1.18e+111], N[(t * N[(2.0 * z), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -9.60000000000000046e100 or 1.1799999999999999e111 < (*.f64 x y)

   1. Initial program 82.9%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{2 \cdot \left(x \cdot y\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot y\right) \cdot 2} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot y\right) \cdot 2} \]

      3. lower-*.f64 67.7

         \[\leadsto \color{blue}{\left(x \cdot y\right)} \cdot 2 \]

   5. Simplified 67.7%

      \[\leadsto \color{blue}{\left(x \cdot y\right) \cdot 2} \]

   if -9.60000000000000046e100 < (*.f64 x y) < 1.1799999999999999e111

   1. Initial program 93.9%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{2 \cdot \left(t \cdot z\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(t \cdot z\right) \cdot 2} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{t \cdot \left(z \cdot 2\right)} \]

      3. *-commutative N/A

         \[\leadsto t \cdot \color{blue}{\left(2 \cdot z\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{t \cdot \left(2 \cdot z\right)} \]

      5. *-commutative N/A

         \[\leadsto t \cdot \color{blue}{\left(z \cdot 2\right)} \]

      6. lower-*.f64 36.7

         \[\leadsto t \cdot \color{blue}{\left(z \cdot 2\right)} \]

   5. Simplified 36.7%

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

4. Final simplification 47.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -9.6 \cdot 10^{+100}:\\ \;\;\;\;2 \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;x \cdot y \leq 1.18 \cdot 10^{+111}:\\ \;\;\;\;t \cdot \left(2 \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(x \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 29.2% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ t \cdot \left(2 \cdot z\right) \end{array} \]

FPCore

(FPCore (x y z t a b c i) :precision binary64 (* t (* 2.0 z)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return t * (2.0 * z);
}

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 = t * (2.0d0 * z)
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return t * (2.0 * z);
}

Python

def code(x, y, z, t, a, b, c, i):
	return t * (2.0 * z)

Julia

function code(x, y, z, t, a, b, c, i)
	return Float64(t * Float64(2.0 * z))
end

MATLAB

function tmp = code(x, y, z, t, a, b, c, i)
	tmp = t * (2.0 * z);
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := N[(t * N[(2.0 * z), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 90.2%

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

3. Taylor expanded in z around inf

   \[\leadsto \color{blue}{2 \cdot \left(t \cdot z\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(t \cdot z\right) \cdot 2} \]

   2. associate-*r* N/A

      \[\leadsto \color{blue}{t \cdot \left(z \cdot 2\right)} \]

   3. *-commutative N/A

      \[\leadsto t \cdot \color{blue}{\left(2 \cdot z\right)} \]

   4. lower-*.f64 N/A

      \[\leadsto \color{blue}{t \cdot \left(2 \cdot z\right)} \]

   5. *-commutative N/A

      \[\leadsto t \cdot \color{blue}{\left(z \cdot 2\right)} \]

   6. lower-*.f64 27.9

      \[\leadsto t \cdot \color{blue}{\left(z \cdot 2\right)} \]

5. Simplified 27.9%

   \[\leadsto \color{blue}{t \cdot \left(z \cdot 2\right)} \]

6. Final simplification 27.9%

   \[\leadsto t \cdot \left(2 \cdot z\right) \]
7. Add Preprocessing

Developer Target 1: 94.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(a + b \cdot c\right) \cdot \left(c \cdot i\right)\right) \end{array} \]

FPCore

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

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return 2.0 * (((x * y) + (z * t)) - ((a + (b * c)) * (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 = 2.0d0 * (((x * y) + (z * t)) - ((a + (b * c)) * (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 2.0 * (((x * y) + (z * t)) - ((a + (b * c)) * (c * i)));
}

Python

def code(x, y, z, t, a, b, c, i):
	return 2.0 * (((x * y) + (z * t)) - ((a + (b * c)) * (c * i)))

Julia

function code(x, y, z, t, a, b, c, i)
	return Float64(2.0 * Float64(Float64(Float64(x * y) + Float64(z * t)) - Float64(Float64(a + Float64(b * c)) * Float64(c * i))))
end

MATLAB

function tmp = code(x, y, z, t, a, b, c, i)
	tmp = 2.0 * (((x * y) + (z * t)) - ((a + (b * c)) * (c * i)));
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := N[(2.0 * N[(N[(N[(x * y), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision] - N[(N[(a + N[(b * c), $MachinePrecision]), $MachinePrecision] * N[(c * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024207 
(FPCore (x y z t a b c i)
  :name "Diagrams.ThreeD.Shapes:frustum from diagrams-lib-1.3.0.3, A"
  :precision binary64

  :alt
  (! :herbie-platform default (* 2 (- (+ (* x y) (* z t)) (* (+ a (* b c)) (* c i)))))

  (* 2.0 (- (+ (* x y) (* z t)) (* (* (+ a (* b c)) c) i))))