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

Percentage Accurate: 89.8% → 95.0%

Time: 14.1s

Alternatives: 11

Speedup: 0.5×

• 47.2% 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: 89.8% 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: 95.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(c \cdot i\right) \cdot \left(\mathsf{fma}\left(c, b, a\right) \cdot -2\right)\\ t_2 := \left(c \cdot \left(a + b \cdot c\right)\right) \cdot i\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{+300}:\\ \;\;\;\;2 \cdot \left(\left(x \cdot y + z \cdot t\right) - i \cdot \left(c \cdot \mathsf{fma}\left(b, c, a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (* c i) (* (fma c b a) -2.0))) (t_2 (* (* c (+ a (* b c))) i)))
   (if (<= t_2 (- INFINITY))
     t_1
     (if (<= t_2 1e+300)
       (* 2.0 (- (+ (* x y) (* z t)) (* i (* c (fma b c a)))))
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (c * i) * (fma(c, b, a) * -2.0);
	double t_2 = (c * (a + (b * c))) * i;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_2 <= 1e+300) {
		tmp = 2.0 * (((x * y) + (z * t)) - (i * (c * fma(b, c, a))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(c * i) * Float64(fma(c, b, a) * -2.0))
	t_2 = Float64(Float64(c * Float64(a + Float64(b * c))) * i)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_2 <= 1e+300)
		tmp = Float64(2.0 * Float64(Float64(Float64(x * y) + Float64(z * t)) - Float64(i * Float64(c * fma(b, c, a)))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(c * i), $MachinePrecision] * N[(N[(c * b + a), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(c * N[(a + N[(b * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * i), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$1, If[LessEqual[t$95$2, 1e+300], N[(2.0 * N[(N[(N[(x * y), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision] - N[(i * N[(c * N[(b * c + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -inf.0 or 1.0000000000000001e300 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 74.0%

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

   3. Taylor expanded in a around 0

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

      1. unpow2 N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-in N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 74.0

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

   5. Simplified 74.0%

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

   6. Taylor expanded in i around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 90.2

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

   8. Simplified 90.2%

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

   if -inf.0 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 1.0000000000000001e300

   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 a around 0

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

      1. unpow2 N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-in N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 99.8

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

   5. Simplified 99.8%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq -\infty:\\ \;\;\;\;\left(c \cdot i\right) \cdot \left(\mathsf{fma}\left(c, b, a\right) \cdot -2\right)\\ \mathbf{elif}\;\left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq 10^{+300}:\\ \;\;\;\;2 \cdot \left(\left(x \cdot y + z \cdot t\right) - i \cdot \left(c \cdot \mathsf{fma}\left(b, c, a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot i\right) \cdot \left(\mathsf{fma}\left(c, b, a\right) \cdot -2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 73.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(c \cdot \left(a + b \cdot c\right)\right) \cdot i\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+288}:\\ \;\;\;\;-2 \cdot \left(c \cdot \left(\left(b \cdot c\right) \cdot i\right)\right)\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{+212}:\\ \;\;\;\;a \cdot \left(i \cdot \left(c \cdot -2\right)\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+183}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(t, z, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(-2 \cdot \left(c \cdot \left(c \cdot i\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (* c (+ a (* b c))) i)))
   (if (<= t_1 -5e+288)
     (* -2.0 (* c (* (* b c) i)))
     (if (<= t_1 -2e+212)
       (* a (* i (* c -2.0)))
       (if (<= t_1 2e+183)
         (* 2.0 (fma t z (* x y)))
         (* b (* -2.0 (* c (* c i)))))))))

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+288) {
		tmp = -2.0 * (c * ((b * c) * i));
	} else if (t_1 <= -2e+212) {
		tmp = a * (i * (c * -2.0));
	} else if (t_1 <= 2e+183) {
		tmp = 2.0 * fma(t, z, (x * y));
	} else {
		tmp = b * (-2.0 * (c * (c * i)));
	}
	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+288)
		tmp = Float64(-2.0 * Float64(c * Float64(Float64(b * c) * i)));
	elseif (t_1 <= -2e+212)
		tmp = Float64(a * Float64(i * Float64(c * -2.0)));
	elseif (t_1 <= 2e+183)
		tmp = Float64(2.0 * fma(t, z, Float64(x * y)));
	else
		tmp = Float64(b * Float64(-2.0 * Float64(c * Float64(c * i))));
	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+288], N[(-2.0 * N[(c * N[(N[(b * c), $MachinePrecision] * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -2e+212], N[(a * N[(i * N[(c * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+183], N[(2.0 * N[(t * z + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(b * N[(-2.0 * N[(c * N[(c * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -5.0000000000000003e288

   1. Initial program 85.0%

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

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 91.5

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

   5. Simplified 91.5%

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

   6. Taylor expanded in b around inf

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

      1. lower-*.f64 85.2

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

   8. Simplified 85.2%

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

   if -5.0000000000000003e288 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -1.9999999999999998e212

   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

   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

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 73.5

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

   5. Simplified 73.5%

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

   if -1.9999999999999998e212 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 1.99999999999999989e183

   1. Initial program 99.9%

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

   3. Taylor expanded in c around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 82.9

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

   5. Simplified 82.9%

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

   if 1.99999999999999989e183 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 70.5%

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

   3. Taylor expanded in 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. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 59.5

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

   5. Simplified 59.5%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq -5 \cdot 10^{+288}:\\ \;\;\;\;-2 \cdot \left(c \cdot \left(\left(b \cdot c\right) \cdot i\right)\right)\\ \mathbf{elif}\;\left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq -2 \cdot 10^{+212}:\\ \;\;\;\;a \cdot \left(i \cdot \left(c \cdot -2\right)\right)\\ \mathbf{elif}\;\left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq 2 \cdot 10^{+183}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(t, z, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(-2 \cdot \left(c \cdot \left(c \cdot i\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(c \cdot \left(a + b \cdot c\right)\right) \cdot i\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+288}:\\ \;\;\;\;-2 \cdot \left(c \cdot \left(b \cdot \left(c \cdot i\right)\right)\right)\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{+212}:\\ \;\;\;\;a \cdot \left(i \cdot \left(c \cdot -2\right)\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+183}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(t, z, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(-2 \cdot \left(c \cdot \left(c \cdot i\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (* c (+ a (* b c))) i)))
   (if (<= t_1 -5e+288)
     (* -2.0 (* c (* b (* c i))))
     (if (<= t_1 -2e+212)
       (* a (* i (* c -2.0)))
       (if (<= t_1 2e+183)
         (* 2.0 (fma t z (* x y)))
         (* b (* -2.0 (* c (* c i)))))))))

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+288) {
		tmp = -2.0 * (c * (b * (c * i)));
	} else if (t_1 <= -2e+212) {
		tmp = a * (i * (c * -2.0));
	} else if (t_1 <= 2e+183) {
		tmp = 2.0 * fma(t, z, (x * y));
	} else {
		tmp = b * (-2.0 * (c * (c * i)));
	}
	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+288)
		tmp = Float64(-2.0 * Float64(c * Float64(b * Float64(c * i))));
	elseif (t_1 <= -2e+212)
		tmp = Float64(a * Float64(i * Float64(c * -2.0)));
	elseif (t_1 <= 2e+183)
		tmp = Float64(2.0 * fma(t, z, Float64(x * y)));
	else
		tmp = Float64(b * Float64(-2.0 * Float64(c * Float64(c * i))));
	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+288], N[(-2.0 * N[(c * N[(b * N[(c * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -2e+212], N[(a * N[(i * N[(c * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+183], N[(2.0 * N[(t * z + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(b * N[(-2.0 * N[(c * N[(c * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -5.0000000000000003e288

   1. Initial program 85.0%

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

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 91.5

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

   5. Simplified 91.5%

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

   6. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 85.1

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

   8. Simplified 85.1%

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

   if -5.0000000000000003e288 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -1.9999999999999998e212

   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

   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

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 73.5

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

   5. Simplified 73.5%

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

   if -1.9999999999999998e212 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 1.99999999999999989e183

   1. Initial program 99.9%

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

   3. Taylor expanded in c around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 82.9

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

   5. Simplified 82.9%

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

   if 1.99999999999999989e183 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 70.5%

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

   3. Taylor expanded in 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. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 59.5

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

   5. Simplified 59.5%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq -5 \cdot 10^{+288}:\\ \;\;\;\;-2 \cdot \left(c \cdot \left(b \cdot \left(c \cdot i\right)\right)\right)\\ \mathbf{elif}\;\left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq -2 \cdot 10^{+212}:\\ \;\;\;\;a \cdot \left(i \cdot \left(c \cdot -2\right)\right)\\ \mathbf{elif}\;\left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq 2 \cdot 10^{+183}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(t, z, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(-2 \cdot \left(c \cdot \left(c \cdot i\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(c \cdot \left(a + b \cdot c\right)\right) \cdot i\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+288}:\\ \;\;\;\;\left(c \cdot i\right) \cdot \left(\left(b \cdot c\right) \cdot -2\right)\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{+212}:\\ \;\;\;\;a \cdot \left(i \cdot \left(c \cdot -2\right)\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+183}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(t, z, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(-2 \cdot \left(c \cdot \left(c \cdot i\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (* c (+ a (* b c))) i)))
   (if (<= t_1 -5e+288)
     (* (* c i) (* (* b c) -2.0))
     (if (<= t_1 -2e+212)
       (* a (* i (* c -2.0)))
       (if (<= t_1 2e+183)
         (* 2.0 (fma t z (* x y)))
         (* b (* -2.0 (* c (* c i)))))))))

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+288) {
		tmp = (c * i) * ((b * c) * -2.0);
	} else if (t_1 <= -2e+212) {
		tmp = a * (i * (c * -2.0));
	} else if (t_1 <= 2e+183) {
		tmp = 2.0 * fma(t, z, (x * y));
	} else {
		tmp = b * (-2.0 * (c * (c * i)));
	}
	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+288)
		tmp = Float64(Float64(c * i) * Float64(Float64(b * c) * -2.0));
	elseif (t_1 <= -2e+212)
		tmp = Float64(a * Float64(i * Float64(c * -2.0)));
	elseif (t_1 <= 2e+183)
		tmp = Float64(2.0 * fma(t, z, Float64(x * y)));
	else
		tmp = Float64(b * Float64(-2.0 * Float64(c * Float64(c * i))));
	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+288], N[(N[(c * i), $MachinePrecision] * N[(N[(b * c), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -2e+212], N[(a * N[(i * N[(c * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+183], N[(2.0 * N[(t * z + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(b * N[(-2.0 * N[(c * N[(c * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -5.0000000000000003e288

   1. Initial program 85.0%

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

   3. Taylor expanded in a around 0

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

      1. unpow2 N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-in N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 85.0

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

   5. Simplified 85.0%

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

   6. Taylor expanded in i around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 91.5

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

   8. Simplified 91.5%

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

   9. Taylor expanded in c around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 83.1

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

   11. Simplified 83.1%

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

   if -5.0000000000000003e288 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -1.9999999999999998e212

   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

   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

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 73.5

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

   5. Simplified 73.5%

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

   if -1.9999999999999998e212 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 1.99999999999999989e183

   1. Initial program 99.9%

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

   3. Taylor expanded in c around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 82.9

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

   5. Simplified 82.9%

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

   if 1.99999999999999989e183 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 70.5%

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

   3. Taylor expanded in 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. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 59.5

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

   5. Simplified 59.5%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq -5 \cdot 10^{+288}:\\ \;\;\;\;\left(c \cdot i\right) \cdot \left(\left(b \cdot c\right) \cdot -2\right)\\ \mathbf{elif}\;\left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq -2 \cdot 10^{+212}:\\ \;\;\;\;a \cdot \left(i \cdot \left(c \cdot -2\right)\right)\\ \mathbf{elif}\;\left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq 2 \cdot 10^{+183}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(t, z, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(-2 \cdot \left(c \cdot \left(c \cdot i\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := b \cdot \left(-2 \cdot \left(c \cdot \left(c \cdot i\right)\right)\right)\\ t_2 := \left(c \cdot \left(a + b \cdot c\right)\right) \cdot i\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+288}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq -2 \cdot 10^{+212}:\\ \;\;\;\;a \cdot \left(i \cdot \left(c \cdot -2\right)\right)\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+183}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(t, z, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* b (* -2.0 (* c (* c i))))) (t_2 (* (* c (+ a (* b c))) i)))
   (if (<= t_2 -5e+288)
     t_1
     (if (<= t_2 -2e+212)
       (* a (* i (* c -2.0)))
       (if (<= t_2 2e+183) (* 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 * (-2.0 * (c * (c * i)));
	double t_2 = (c * (a + (b * c))) * i;
	double tmp;
	if (t_2 <= -5e+288) {
		tmp = t_1;
	} else if (t_2 <= -2e+212) {
		tmp = a * (i * (c * -2.0));
	} else if (t_2 <= 2e+183) {
		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(-2.0 * Float64(c * Float64(c * i))))
	t_2 = Float64(Float64(c * Float64(a + Float64(b * c))) * i)
	tmp = 0.0
	if (t_2 <= -5e+288)
		tmp = t_1;
	elseif (t_2 <= -2e+212)
		tmp = Float64(a * Float64(i * Float64(c * -2.0)));
	elseif (t_2 <= 2e+183)
		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[(-2.0 * N[(c * N[(c * i), $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, -5e+288], t$95$1, If[LessEqual[t$95$2, -2e+212], N[(a * N[(i * N[(c * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e+183], 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) < -5.0000000000000003e288 or 1.99999999999999989e183 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 77.0%

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

   3. Taylor expanded in 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. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 69.3

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

   5. Simplified 69.3%

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

   if -5.0000000000000003e288 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -1.9999999999999998e212

   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

   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

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 73.5

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

   5. Simplified 73.5%

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

   if -1.9999999999999998e212 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 1.99999999999999989e183

   1. Initial program 99.9%

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

   3. Taylor expanded in c around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 82.9

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

   5. Simplified 82.9%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq -5 \cdot 10^{+288}:\\ \;\;\;\;b \cdot \left(-2 \cdot \left(c \cdot \left(c \cdot i\right)\right)\right)\\ \mathbf{elif}\;\left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq -2 \cdot 10^{+212}:\\ \;\;\;\;a \cdot \left(i \cdot \left(c \cdot -2\right)\right)\\ \mathbf{elif}\;\left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq 2 \cdot 10^{+183}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(t, z, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(-2 \cdot \left(c \cdot \left(c \cdot i\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 88.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(c \cdot i\right) \cdot \left(\mathsf{fma}\left(c, b, a\right) \cdot -2\right)\\ t_2 := \left(c \cdot \left(a + b \cdot c\right)\right) \cdot i\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+187}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{+188}:\\ \;\;\;\;2 \cdot \left(\left(x \cdot y + z \cdot t\right) - i \cdot \left(a \cdot c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (* c i) (* (fma c b a) -2.0))) (t_2 (* (* c (+ a (* b c))) i)))
   (if (<= t_2 -1e+187)
     t_1
     (if (<= t_2 4e+188) (* 2.0 (- (+ (* x y) (* z t)) (* i (* a c)))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (c * i) * (fma(c, b, a) * -2.0);
	double t_2 = (c * (a + (b * c))) * i;
	double tmp;
	if (t_2 <= -1e+187) {
		tmp = t_1;
	} else if (t_2 <= 4e+188) {
		tmp = 2.0 * (((x * y) + (z * t)) - (i * (a * c)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(c * i) * Float64(fma(c, b, a) * -2.0))
	t_2 = Float64(Float64(c * Float64(a + Float64(b * c))) * i)
	tmp = 0.0
	if (t_2 <= -1e+187)
		tmp = t_1;
	elseif (t_2 <= 4e+188)
		tmp = Float64(2.0 * Float64(Float64(Float64(x * y) + Float64(z * t)) - Float64(i * Float64(a * c))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(c * i), $MachinePrecision] * N[(N[(c * b + a), $MachinePrecision] * -2.0), $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+187], t$95$1, If[LessEqual[t$95$2, 4e+188], N[(2.0 * N[(N[(N[(x * y), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision] - N[(i * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -9.99999999999999907e186 or 4.0000000000000001e188 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 79.2%

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

   3. Taylor expanded in a around 0

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

      1. unpow2 N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-in N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 79.2

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

   5. Simplified 79.2%

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

   6. Taylor expanded in i around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 87.9

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

   8. Simplified 87.9%

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

   if -9.99999999999999907e186 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 4.0000000000000001e188

   1. Initial program 99.9%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 94.2

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

   5. Simplified 94.2%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq -1 \cdot 10^{+187}:\\ \;\;\;\;\left(c \cdot i\right) \cdot \left(\mathsf{fma}\left(c, b, a\right) \cdot -2\right)\\ \mathbf{elif}\;\left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq 4 \cdot 10^{+188}:\\ \;\;\;\;2 \cdot \left(\left(x \cdot y + z \cdot t\right) - i \cdot \left(a \cdot c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot i\right) \cdot \left(\mathsf{fma}\left(c, b, a\right) \cdot -2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 88.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(c \cdot i\right) \cdot \left(\mathsf{fma}\left(c, b, a\right) \cdot -2\right)\\ t_2 := \left(c \cdot \left(a + b \cdot c\right)\right) \cdot i\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+187}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+183}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(a, -c \cdot i, \mathsf{fma}\left(t, z, x \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (* c i) (* (fma c b a) -2.0))) (t_2 (* (* c (+ a (* b c))) i)))
   (if (<= t_2 -1e+187)
     t_1
     (if (<= t_2 2e+183) (* 2.0 (fma a (- (* c i)) (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(c, b, a) * -2.0);
	double t_2 = (c * (a + (b * c))) * i;
	double tmp;
	if (t_2 <= -1e+187) {
		tmp = t_1;
	} else if (t_2 <= 2e+183) {
		tmp = 2.0 * fma(a, -(c * i), 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(Float64(c * i) * Float64(fma(c, b, a) * -2.0))
	t_2 = Float64(Float64(c * Float64(a + Float64(b * c))) * i)
	tmp = 0.0
	if (t_2 <= -1e+187)
		tmp = t_1;
	elseif (t_2 <= 2e+183)
		tmp = Float64(2.0 * fma(a, Float64(-Float64(c * i)), 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[(N[(c * i), $MachinePrecision] * N[(N[(c * b + a), $MachinePrecision] * -2.0), $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+187], t$95$1, If[LessEqual[t$95$2, 2e+183], N[(2.0 * N[(a * (-N[(c * i), $MachinePrecision]) + N[(t * z + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -9.99999999999999907e186 or 1.99999999999999989e183 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 79.4%

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

   3. Taylor expanded in a around 0

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

      1. unpow2 N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-in N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 79.4

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

   5. Simplified 79.4%

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

   6. Taylor expanded in i around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 87.9

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

   8. Simplified 87.9%

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

   if -9.99999999999999907e186 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 1.99999999999999989e183

   1. Initial program 99.9%

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

   3. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

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

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

      4. mul-1-neg N/A

         \[\leadsto 2 \cdot \left(a \cdot \color{blue}{\left(-1 \cdot \left(c \cdot i\right)\right)} + \left(t \cdot z + x \cdot y\right)\right) \]

      5. lower-fma.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 91.5

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

   5. Simplified 91.5%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq -1 \cdot 10^{+187}:\\ \;\;\;\;\left(c \cdot i\right) \cdot \left(\mathsf{fma}\left(c, b, a\right) \cdot -2\right)\\ \mathbf{elif}\;\left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq 2 \cdot 10^{+183}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(a, -c \cdot i, \mathsf{fma}\left(t, z, x \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot i\right) \cdot \left(\mathsf{fma}\left(c, b, a\right) \cdot -2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 83.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(c \cdot i\right) \cdot \left(\mathsf{fma}\left(c, b, a\right) \cdot -2\right)\\ t_2 := \left(c \cdot \left(a + b \cdot c\right)\right) \cdot i\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+187}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{+62}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(t, z, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (* c i) (* (fma c b a) -2.0))) (t_2 (* (* c (+ a (* b c))) i)))
   (if (<= t_2 -1e+187)
     t_1
     (if (<= t_2 4e+62) (* 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(c, b, a) * -2.0);
	double t_2 = (c * (a + (b * c))) * i;
	double tmp;
	if (t_2 <= -1e+187) {
		tmp = t_1;
	} else if (t_2 <= 4e+62) {
		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(Float64(c * i) * Float64(fma(c, b, a) * -2.0))
	t_2 = Float64(Float64(c * Float64(a + Float64(b * c))) * i)
	tmp = 0.0
	if (t_2 <= -1e+187)
		tmp = t_1;
	elseif (t_2 <= 4e+62)
		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[(N[(c * i), $MachinePrecision] * N[(N[(c * b + a), $MachinePrecision] * -2.0), $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+187], t$95$1, If[LessEqual[t$95$2, 4e+62], 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) < -9.99999999999999907e186 or 4.00000000000000014e62 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 81.3%

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

   3. Taylor expanded in a around 0

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

      1. unpow2 N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-in N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 81.3

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

   5. Simplified 81.3%

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

   6. Taylor expanded in i around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 84.5

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

   8. Simplified 84.5%

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

   if -9.99999999999999907e186 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 4.00000000000000014e62

   1. Initial program 99.9%

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

   3. Taylor expanded in c around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 86.5

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

   5. Simplified 86.5%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq -1 \cdot 10^{+187}:\\ \;\;\;\;\left(c \cdot i\right) \cdot \left(\mathsf{fma}\left(c, b, a\right) \cdot -2\right)\\ \mathbf{elif}\;\left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq 4 \cdot 10^{+62}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(t, z, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot i\right) \cdot \left(\mathsf{fma}\left(c, b, a\right) \cdot -2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 63.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot \left(i \cdot \left(c \cdot -2\right)\right)\\ t_2 := \left(c \cdot \left(a + b \cdot c\right)\right) \cdot i\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+212}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+183}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(t, z, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* a (* i (* c -2.0)))) (t_2 (* (* c (+ a (* b c))) i)))
   (if (<= t_2 -2e+212)
     t_1
     (if (<= t_2 2e+183) (* 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 = a * (i * (c * -2.0));
	double t_2 = (c * (a + (b * c))) * i;
	double tmp;
	if (t_2 <= -2e+212) {
		tmp = t_1;
	} else if (t_2 <= 2e+183) {
		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(a * Float64(i * Float64(c * -2.0)))
	t_2 = Float64(Float64(c * Float64(a + Float64(b * c))) * i)
	tmp = 0.0
	if (t_2 <= -2e+212)
		tmp = t_1;
	elseif (t_2 <= 2e+183)
		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[(a * N[(i * N[(c * -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, -2e+212], t$95$1, If[LessEqual[t$95$2, 2e+183], 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) < -1.9999999999999998e212 or 1.99999999999999989e183 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 79.2%

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

   3. Taylor expanded in 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

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 42.7

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

   5. Simplified 42.7%

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

   if -1.9999999999999998e212 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 1.99999999999999989e183

   1. Initial program 99.9%

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

   3. Taylor expanded in c around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 82.9

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

   5. Simplified 82.9%

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

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq -2 \cdot 10^{+212}:\\ \;\;\;\;a \cdot \left(i \cdot \left(c \cdot -2\right)\right)\\ \mathbf{elif}\;\left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq 2 \cdot 10^{+183}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(t, z, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(i \cdot \left(c \cdot -2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 42.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 \cdot \left(x \cdot y\right)\\ \mathbf{if}\;x \cdot y \leq -1.22 \cdot 10^{-135}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \cdot y \leq 7.4:\\ \;\;\;\;2 \cdot \left(z \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* 2.0 (* x y))))
   (if (<= (* x y) -1.22e-135) t_1 (if (<= (* x y) 7.4) (* 2.0 (* z t)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = 2.0 * (x * y);
	double tmp;
	if ((x * y) <= -1.22e-135) {
		tmp = t_1;
	} else if ((x * y) <= 7.4) {
		tmp = 2.0 * (z * t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 2.0d0 * (x * y)
    if ((x * y) <= (-1.22d-135)) then
        tmp = t_1
    else if ((x * y) <= 7.4d0) then
        tmp = 2.0d0 * (z * t)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = 2.0 * (x * y);
	double tmp;
	if ((x * y) <= -1.22e-135) {
		tmp = t_1;
	} else if ((x * y) <= 7.4) {
		tmp = 2.0 * (z * t);
	} 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) <= -1.22e-135:
		tmp = t_1
	elif (x * y) <= 7.4:
		tmp = 2.0 * (z * t)
	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) <= -1.22e-135)
		tmp = t_1;
	elseif (Float64(x * y) <= 7.4)
		tmp = Float64(2.0 * Float64(z * t));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	t_1 = 2.0 * (x * y);
	tmp = 0.0;
	if ((x * y) <= -1.22e-135)
		tmp = t_1;
	elseif ((x * y) <= 7.4)
		tmp = 2.0 * (z * t);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(2.0 * N[(x * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * y), $MachinePrecision], -1.22e-135], t$95$1, If[LessEqual[N[(x * y), $MachinePrecision], 7.4], N[(2.0 * N[(z * t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -1.22e-135 or 7.4000000000000004 < (*.f64 x y)

   1. Initial program 89.2%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 48.8

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

   5. Simplified 48.8%

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

   if -1.22e-135 < (*.f64 x y) < 7.4000000000000004

   1. Initial program 92.6%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 44.8

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

   5. Simplified 44.8%

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

4. Final simplification 46.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1.22 \cdot 10^{-135}:\\ \;\;\;\;2 \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;x \cdot y \leq 7.4:\\ \;\;\;\;2 \cdot \left(z \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(x \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 29.0% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

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 * (z * t)
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 * (z * t);
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.8%

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

3. Taylor expanded in z around inf

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

   1. lower-*.f64 28.9

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

5. Simplified 28.9%

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

6. Final simplification 28.9%

   \[\leadsto 2 \cdot \left(z \cdot t\right) \]
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 2024215 
(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))))