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

Percentage Accurate: 90.1% → 95.2%

Time: 14.4s

Alternatives: 15

Speedup: 1.1×

• 46.7% 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: 90.1% 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: 95.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z, t, \mathsf{fma}\left(y, x, \left(-c\right) \cdot \left(\mathsf{fma}\left(c, b, a\right) \cdot i\right)\right)\right) \cdot 2\\ \mathbf{if}\;c \leq -1.95 \cdot 10^{-87}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;c \leq -4 \cdot 10^{-288}:\\ \;\;\;\;\left(\left(x \cdot y + t \cdot z\right) - \left(\left(b \cdot c + a\right) \cdot c\right) \cdot i\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (fma z t (fma y x (* (- c) (* (fma c b a) i)))) 2.0)))
   (if (<= c -1.95e-87)
     t_1
     (if (<= c -4e-288)
       (* (- (+ (* x y) (* t z)) (* (* (+ (* b c) a) c) i)) 2.0)
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma(z, t, fma(y, x, (-c * (fma(c, b, a) * i)))) * 2.0;
	double tmp;
	if (c <= -1.95e-87) {
		tmp = t_1;
	} else if (c <= -4e-288) {
		tmp = (((x * y) + (t * z)) - ((((b * c) + a) * c) * i)) * 2.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(fma(z, t, fma(y, x, Float64(Float64(-c) * Float64(fma(c, b, a) * i)))) * 2.0)
	tmp = 0.0
	if (c <= -1.95e-87)
		tmp = t_1;
	elseif (c <= -4e-288)
		tmp = Float64(Float64(Float64(Float64(x * y) + Float64(t * z)) - Float64(Float64(Float64(Float64(b * c) + a) * c) * i)) * 2.0);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(z * t + N[(y * x + N[((-c) * N[(N[(c * b + a), $MachinePrecision] * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]}, If[LessEqual[c, -1.95e-87], t$95$1, If[LessEqual[c, -4e-288], N[(N[(N[(N[(x * y), $MachinePrecision] + N[(t * z), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(N[(b * c), $MachinePrecision] + a), $MachinePrecision] * c), $MachinePrecision] * i), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if c < -1.9499999999999999e-87 or -4.00000000000000023e-288 < c

   1. Initial program 85.7%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. sub-neg N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*r* N/A

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

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

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

      16. lower-*.f64 N/A

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

   4. Applied rewrites 97.1%

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

   if -1.9499999999999999e-87 < c < -4.00000000000000023e-288

   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. Recombined 2 regimes into one program.

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.95 \cdot 10^{-87}:\\ \;\;\;\;\mathsf{fma}\left(z, t, \mathsf{fma}\left(y, x, \left(-c\right) \cdot \left(\mathsf{fma}\left(c, b, a\right) \cdot i\right)\right)\right) \cdot 2\\ \mathbf{elif}\;c \leq -4 \cdot 10^{-288}:\\ \;\;\;\;\left(\left(x \cdot y + t \cdot z\right) - \left(\left(b \cdot c + a\right) \cdot c\right) \cdot i\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t, \mathsf{fma}\left(y, x, \left(-c\right) \cdot \left(\mathsf{fma}\left(c, b, a\right) \cdot i\right)\right)\right) \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-i, \mathsf{fma}\left(c, b, a\right) \cdot c, x \cdot y\right) \cdot 2\\ t_2 := \left(\left(b \cdot c + a\right) \cdot c\right) \cdot i\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{-6}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{+51}:\\ \;\;\;\;\mathsf{fma}\left(y, x, t \cdot z\right) \cdot 2\\ \mathbf{elif}\;t\_2 \leq 10^{+242}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot c\right) \cdot -2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (fma (- i) (* (fma c b a) c) (* x y)) 2.0))
        (t_2 (* (* (+ (* b c) a) c) i)))
   (if (<= t_2 -1e-6)
     t_1
     (if (<= t_2 1e+51)
       (* (fma y x (* t z)) 2.0)
       (if (<= t_2 1e+242) t_1 (* (* (* (fma c b a) i) c) -2.0))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma(-i, (fma(c, b, a) * c), (x * y)) * 2.0;
	double t_2 = (((b * c) + a) * c) * i;
	double tmp;
	if (t_2 <= -1e-6) {
		tmp = t_1;
	} else if (t_2 <= 1e+51) {
		tmp = fma(y, x, (t * z)) * 2.0;
	} else if (t_2 <= 1e+242) {
		tmp = t_1;
	} else {
		tmp = ((fma(c, b, a) * i) * c) * -2.0;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(fma(Float64(-i), Float64(fma(c, b, a) * c), Float64(x * y)) * 2.0)
	t_2 = Float64(Float64(Float64(Float64(b * c) + a) * c) * i)
	tmp = 0.0
	if (t_2 <= -1e-6)
		tmp = t_1;
	elseif (t_2 <= 1e+51)
		tmp = Float64(fma(y, x, Float64(t * z)) * 2.0);
	elseif (t_2 <= 1e+242)
		tmp = t_1;
	else
		tmp = Float64(Float64(Float64(fma(c, b, a) * i) * c) * -2.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[((-i) * N[(N[(c * b + a), $MachinePrecision] * c), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(b * c), $MachinePrecision] + a), $MachinePrecision] * c), $MachinePrecision] * i), $MachinePrecision]}, If[LessEqual[t$95$2, -1e-6], t$95$1, If[LessEqual[t$95$2, 1e+51], N[(N[(y * x + N[(t * z), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], If[LessEqual[t$95$2, 1e+242], t$95$1, N[(N[(N[(N[(c * b + a), $MachinePrecision] * i), $MachinePrecision] * c), $MachinePrecision] * -2.0), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -9.99999999999999955e-7 or 1e51 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 1.00000000000000005e242

   1. Initial program 89.1%

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

   3. Taylor expanded in t around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

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

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

      7. mul-1-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 81.8

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

   5. Applied rewrites 81.8%

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

   if -9.99999999999999955e-7 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 1e51

   1. Initial program 98.1%

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

   3. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 95.7

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

   5. Applied rewrites 95.7%

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

   if 1.00000000000000005e242 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 67.0%

      \[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right) \]
   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. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 84.3

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

   5. Applied rewrites 84.3%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(b \cdot c + a\right) \cdot c\right) \cdot i \leq -1 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(-i, \mathsf{fma}\left(c, b, a\right) \cdot c, x \cdot y\right) \cdot 2\\ \mathbf{elif}\;\left(\left(b \cdot c + a\right) \cdot c\right) \cdot i \leq 10^{+51}:\\ \;\;\;\;\mathsf{fma}\left(y, x, t \cdot z\right) \cdot 2\\ \mathbf{elif}\;\left(\left(b \cdot c + a\right) \cdot c\right) \cdot i \leq 10^{+242}:\\ \;\;\;\;\mathsf{fma}\left(-i, \mathsf{fma}\left(c, b, a\right) \cdot c, x \cdot y\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot c\right) \cdot -2\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 72.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(b \cdot c + a\right) \cdot c\right) \cdot i\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\left(\left(\left(c \cdot c\right) \cdot i\right) \cdot -2\right) \cdot b\\ \mathbf{elif}\;t\_1 \leq 10^{+51}:\\ \;\;\;\;\mathsf{fma}\left(y, x, t \cdot z\right) \cdot 2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+197}:\\ \;\;\;\;-2 \cdot \left(\left(i \cdot c\right) \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(i \cdot c\right) \cdot c\right) \cdot b\right) \cdot -2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (* (+ (* b c) a) c) i)))
   (if (<= t_1 (- INFINITY))
     (* (* (* (* c c) i) -2.0) b)
     (if (<= t_1 1e+51)
       (* (fma y x (* t z)) 2.0)
       (if (<= t_1 2e+197)
         (* -2.0 (* (* i c) a))
         (* (* (* (* i c) c) b) -2.0))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (((b * c) + a) * c) * i;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (((c * c) * i) * -2.0) * b;
	} else if (t_1 <= 1e+51) {
		tmp = fma(y, x, (t * z)) * 2.0;
	} else if (t_1 <= 2e+197) {
		tmp = -2.0 * ((i * c) * a);
	} else {
		tmp = (((i * c) * c) * b) * -2.0;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(Float64(b * c) + a) * c) * i)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(Float64(c * c) * i) * -2.0) * b);
	elseif (t_1 <= 1e+51)
		tmp = Float64(fma(y, x, Float64(t * z)) * 2.0);
	elseif (t_1 <= 2e+197)
		tmp = Float64(-2.0 * Float64(Float64(i * c) * a));
	else
		tmp = Float64(Float64(Float64(Float64(i * c) * c) * b) * -2.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(N[(b * c), $MachinePrecision] + a), $MachinePrecision] * c), $MachinePrecision] * i), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(N[(c * c), $MachinePrecision] * i), $MachinePrecision] * -2.0), $MachinePrecision] * b), $MachinePrecision], If[LessEqual[t$95$1, 1e+51], N[(N[(y * x + N[(t * z), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], If[LessEqual[t$95$1, 2e+197], N[(-2.0 * N[(N[(i * c), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(i * c), $MachinePrecision] * c), $MachinePrecision] * b), $MachinePrecision] * -2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 74.8%

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

   3. Taylor expanded in c 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

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 70.4

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

   5. Applied rewrites 70.4%

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

   if -inf.0 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 1e51

   1. Initial program 98.5%

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

   3. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 87.2

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

   5. Applied rewrites 87.2%

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

   if 1e51 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 1.9999999999999999e197

   1. Initial program 99.6%

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

   3. Taylor expanded in 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. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 56.3

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

   5. Applied rewrites 56.3%

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

   if 1.9999999999999999e197 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 70.3%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 17.1

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

   5. Applied rewrites 17.1%

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

   6. Taylor expanded in c around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 57.4

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

   8. Applied rewrites 57.4%

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

   10. Applied rewrites 64.5%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(b \cdot c + a\right) \cdot c\right) \cdot i \leq -\infty:\\ \;\;\;\;\left(\left(\left(c \cdot c\right) \cdot i\right) \cdot -2\right) \cdot b\\ \mathbf{elif}\;\left(\left(b \cdot c + a\right) \cdot c\right) \cdot i \leq 10^{+51}:\\ \;\;\;\;\mathsf{fma}\left(y, x, t \cdot z\right) \cdot 2\\ \mathbf{elif}\;\left(\left(b \cdot c + a\right) \cdot c\right) \cdot i \leq 2 \cdot 10^{+197}:\\ \;\;\;\;-2 \cdot \left(\left(i \cdot c\right) \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(i \cdot c\right) \cdot c\right) \cdot b\right) \cdot -2\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 72.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(b \cdot c + a\right) \cdot c\right) \cdot i\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\left(\left(\left(c \cdot c\right) \cdot i\right) \cdot -2\right) \cdot b\\ \mathbf{elif}\;t\_1 \leq 10^{+51}:\\ \;\;\;\;\mathsf{fma}\left(y, x, t \cdot z\right) \cdot 2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+230}:\\ \;\;\;\;-2 \cdot \left(\left(i \cdot c\right) \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-2 \cdot b\right) \cdot \left(i \cdot c\right)\right) \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (* (+ (* b c) a) c) i)))
   (if (<= t_1 (- INFINITY))
     (* (* (* (* c c) i) -2.0) b)
     (if (<= t_1 1e+51)
       (* (fma y x (* t z)) 2.0)
       (if (<= t_1 2e+230)
         (* -2.0 (* (* i c) a))
         (* (* (* -2.0 b) (* i c)) c))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (((b * c) + a) * c) * i;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (((c * c) * i) * -2.0) * b;
	} else if (t_1 <= 1e+51) {
		tmp = fma(y, x, (t * z)) * 2.0;
	} else if (t_1 <= 2e+230) {
		tmp = -2.0 * ((i * c) * a);
	} else {
		tmp = ((-2.0 * b) * (i * c)) * c;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(Float64(b * c) + a) * c) * i)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(Float64(c * c) * i) * -2.0) * b);
	elseif (t_1 <= 1e+51)
		tmp = Float64(fma(y, x, Float64(t * z)) * 2.0);
	elseif (t_1 <= 2e+230)
		tmp = Float64(-2.0 * Float64(Float64(i * c) * a));
	else
		tmp = Float64(Float64(Float64(-2.0 * b) * Float64(i * c)) * c);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(N[(b * c), $MachinePrecision] + a), $MachinePrecision] * c), $MachinePrecision] * i), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(N[(c * c), $MachinePrecision] * i), $MachinePrecision] * -2.0), $MachinePrecision] * b), $MachinePrecision], If[LessEqual[t$95$1, 1e+51], N[(N[(y * x + N[(t * z), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], If[LessEqual[t$95$1, 2e+230], N[(-2.0 * N[(N[(i * c), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-2.0 * b), $MachinePrecision] * N[(i * c), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 74.8%

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

   3. Taylor expanded in c 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

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 70.4

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

   5. Applied rewrites 70.4%

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

   if -inf.0 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 1e51

   1. Initial program 98.5%

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

   3. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 87.2

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

   5. Applied rewrites 87.2%

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

   if 1e51 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 2.0000000000000002e230

   1. Initial program 99.5%

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

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 49.6

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

   5. Applied rewrites 49.6%

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

   if 2.0000000000000002e230 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 67.6%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 14.9

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

   5. Applied rewrites 14.9%

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

   6. Taylor expanded in c around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 60.1

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

   8. Applied rewrites 60.1%

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

   10. Applied rewrites 67.9%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(b \cdot c + a\right) \cdot c\right) \cdot i \leq -\infty:\\ \;\;\;\;\left(\left(\left(c \cdot c\right) \cdot i\right) \cdot -2\right) \cdot b\\ \mathbf{elif}\;\left(\left(b \cdot c + a\right) \cdot c\right) \cdot i \leq 10^{+51}:\\ \;\;\;\;\mathsf{fma}\left(y, x, t \cdot z\right) \cdot 2\\ \mathbf{elif}\;\left(\left(b \cdot c + a\right) \cdot c\right) \cdot i \leq 2 \cdot 10^{+230}:\\ \;\;\;\;-2 \cdot \left(\left(i \cdot c\right) \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-2 \cdot b\right) \cdot \left(i \cdot c\right)\right) \cdot c\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 71.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(b \cdot c + a\right) \cdot c\right) \cdot i\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\left(\left(\left(c \cdot c\right) \cdot b\right) \cdot i\right) \cdot -2\\ \mathbf{elif}\;t\_1 \leq 10^{+51}:\\ \;\;\;\;\mathsf{fma}\left(y, x, t \cdot z\right) \cdot 2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+230}:\\ \;\;\;\;-2 \cdot \left(\left(i \cdot c\right) \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-2 \cdot b\right) \cdot \left(i \cdot c\right)\right) \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (* (+ (* b c) a) c) i)))
   (if (<= t_1 (- INFINITY))
     (* (* (* (* c c) b) i) -2.0)
     (if (<= t_1 1e+51)
       (* (fma y x (* t z)) 2.0)
       (if (<= t_1 2e+230)
         (* -2.0 (* (* i c) a))
         (* (* (* -2.0 b) (* i c)) c))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (((b * c) + a) * c) * i;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (((c * c) * b) * i) * -2.0;
	} else if (t_1 <= 1e+51) {
		tmp = fma(y, x, (t * z)) * 2.0;
	} else if (t_1 <= 2e+230) {
		tmp = -2.0 * ((i * c) * a);
	} else {
		tmp = ((-2.0 * b) * (i * c)) * c;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(Float64(b * c) + a) * c) * i)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(Float64(c * c) * b) * i) * -2.0);
	elseif (t_1 <= 1e+51)
		tmp = Float64(fma(y, x, Float64(t * z)) * 2.0);
	elseif (t_1 <= 2e+230)
		tmp = Float64(-2.0 * Float64(Float64(i * c) * a));
	else
		tmp = Float64(Float64(Float64(-2.0 * b) * Float64(i * c)) * c);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(N[(b * c), $MachinePrecision] + a), $MachinePrecision] * c), $MachinePrecision] * i), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(N[(c * c), $MachinePrecision] * b), $MachinePrecision] * i), $MachinePrecision] * -2.0), $MachinePrecision], If[LessEqual[t$95$1, 1e+51], N[(N[(y * x + N[(t * z), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], If[LessEqual[t$95$1, 2e+230], N[(-2.0 * N[(N[(i * c), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-2.0 * b), $MachinePrecision] * N[(i * c), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 74.8%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 21.2

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

   5. Applied rewrites 21.2%

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

   6. Taylor expanded in c around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 70.4

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

   8. Applied rewrites 70.4%

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

   10. Applied rewrites 68.4%

       \[\leadsto \left(\left(b \cdot \left(c \cdot c\right)\right) \cdot i\right) \cdot -2 \]

   if -inf.0 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 1e51

   1. Initial program 98.5%

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

   3. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 87.2

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

   5. Applied rewrites 87.2%

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

   if 1e51 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 2.0000000000000002e230

   1. Initial program 99.5%

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

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 49.6

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

   5. Applied rewrites 49.6%

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

   if 2.0000000000000002e230 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 67.6%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 14.9

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

   5. Applied rewrites 14.9%

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

   6. Taylor expanded in c around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 60.1

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

   8. Applied rewrites 60.1%

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

   10. Applied rewrites 67.9%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(b \cdot c + a\right) \cdot c\right) \cdot i \leq -\infty:\\ \;\;\;\;\left(\left(\left(c \cdot c\right) \cdot b\right) \cdot i\right) \cdot -2\\ \mathbf{elif}\;\left(\left(b \cdot c + a\right) \cdot c\right) \cdot i \leq 10^{+51}:\\ \;\;\;\;\mathsf{fma}\left(y, x, t \cdot z\right) \cdot 2\\ \mathbf{elif}\;\left(\left(b \cdot c + a\right) \cdot c\right) \cdot i \leq 2 \cdot 10^{+230}:\\ \;\;\;\;-2 \cdot \left(\left(i \cdot c\right) \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-2 \cdot b\right) \cdot \left(i \cdot c\right)\right) \cdot c\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 72.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(-2 \cdot b\right) \cdot \left(i \cdot c\right)\right) \cdot c\\ t_2 := \left(\left(b \cdot c + a\right) \cdot c\right) \cdot i\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{+51}:\\ \;\;\;\;\mathsf{fma}\left(y, x, t \cdot z\right) \cdot 2\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+230}:\\ \;\;\;\;-2 \cdot \left(\left(i \cdot c\right) \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (* (* -2.0 b) (* i c)) c)) (t_2 (* (* (+ (* b c) a) c) i)))
   (if (<= t_2 (- INFINITY))
     t_1
     (if (<= t_2 1e+51)
       (* (fma y x (* t z)) 2.0)
       (if (<= t_2 2e+230) (* -2.0 (* (* i 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 = ((-2.0 * b) * (i * c)) * c;
	double t_2 = (((b * c) + a) * c) * i;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_2 <= 1e+51) {
		tmp = fma(y, x, (t * z)) * 2.0;
	} else if (t_2 <= 2e+230) {
		tmp = -2.0 * ((i * c) * a);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(-2.0 * b) * Float64(i * c)) * c)
	t_2 = Float64(Float64(Float64(Float64(b * c) + a) * c) * i)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_2 <= 1e+51)
		tmp = Float64(fma(y, x, Float64(t * z)) * 2.0);
	elseif (t_2 <= 2e+230)
		tmp = Float64(-2.0 * Float64(Float64(i * 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[(N[(-2.0 * b), $MachinePrecision] * N[(i * c), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(b * c), $MachinePrecision] + a), $MachinePrecision] * c), $MachinePrecision] * i), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$1, If[LessEqual[t$95$2, 1e+51], N[(N[(y * x + N[(t * z), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], If[LessEqual[t$95$2, 2e+230], N[(-2.0 * N[(N[(i * c), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -inf.0 or 2.0000000000000002e230 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 70.8%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 17.7

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

   5. Applied rewrites 17.7%

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

   6. Taylor expanded in c around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 64.7

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

   8. Applied rewrites 64.7%

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

   10. Applied rewrites 67.1%

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

   if -inf.0 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 1e51

   1. Initial program 98.5%

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

   3. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 87.2

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

   5. Applied rewrites 87.2%

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

   if 1e51 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 2.0000000000000002e230

   1. Initial program 99.5%

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

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 49.6

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

   5. Applied rewrites 49.6%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(b \cdot c + a\right) \cdot c\right) \cdot i \leq -\infty:\\ \;\;\;\;\left(\left(-2 \cdot b\right) \cdot \left(i \cdot c\right)\right) \cdot c\\ \mathbf{elif}\;\left(\left(b \cdot c + a\right) \cdot c\right) \cdot i \leq 10^{+51}:\\ \;\;\;\;\mathsf{fma}\left(y, x, t \cdot z\right) \cdot 2\\ \mathbf{elif}\;\left(\left(b \cdot c + a\right) \cdot c\right) \cdot i \leq 2 \cdot 10^{+230}:\\ \;\;\;\;-2 \cdot \left(\left(i \cdot c\right) \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-2 \cdot b\right) \cdot \left(i \cdot c\right)\right) \cdot c\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(b \cdot c + a\right) \cdot c\right) \cdot i\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\left(-2 \cdot i\right) \cdot \left(\mathsf{fma}\left(b, c, a\right) \cdot c\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+77}:\\ \;\;\;\;\mathsf{fma}\left(z, t, \mathsf{fma}\left(y, x, \left(a \cdot i\right) \cdot \left(-c\right)\right)\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, c, a\right) \cdot i, -c, x \cdot y\right) \cdot 2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (* (+ (* b c) a) c) i)))
   (if (<= t_1 (- INFINITY))
     (* (* -2.0 i) (* (fma b c a) c))
     (if (<= t_1 5e+77)
       (* (fma z t (fma y x (* (* a i) (- c)))) 2.0)
       (* (fma (* (fma b c a) i) (- c) (* x y)) 2.0)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (((b * c) + a) * c) * i;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (-2.0 * i) * (fma(b, c, a) * c);
	} else if (t_1 <= 5e+77) {
		tmp = fma(z, t, fma(y, x, ((a * i) * -c))) * 2.0;
	} else {
		tmp = fma((fma(b, c, a) * i), -c, (x * y)) * 2.0;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(Float64(b * c) + a) * c) * i)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(-2.0 * i) * Float64(fma(b, c, a) * c));
	elseif (t_1 <= 5e+77)
		tmp = Float64(fma(z, t, fma(y, x, Float64(Float64(a * i) * Float64(-c)))) * 2.0);
	else
		tmp = Float64(fma(Float64(fma(b, c, a) * i), Float64(-c), Float64(x * y)) * 2.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(N[(b * c), $MachinePrecision] + a), $MachinePrecision] * c), $MachinePrecision] * i), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(-2.0 * i), $MachinePrecision] * N[(N[(b * c + a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+77], N[(N[(z * t + N[(y * x + N[(N[(a * i), $MachinePrecision] * (-c)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(N[(N[(b * c + a), $MachinePrecision] * i), $MachinePrecision] * (-c) + N[(x * y), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 74.8%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

   4. Applied rewrites 81.5%

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

   5. Taylor expanded in i around inf

      \[\leadsto \color{blue}{-2 \cdot \left(c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)} \]
   6. 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. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 79.8

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

   7. Applied rewrites 79.8%

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

   9. Applied rewrites 84.2%

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

   if -inf.0 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 5.00000000000000004e77

   1. Initial program 98.5%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. sub-neg N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*r* N/A

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

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

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

      16. lower-*.f64 N/A

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

   4. Applied rewrites 97.1%

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

   5. Taylor expanded in c around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 90.0

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

   7. Applied rewrites 90.0%

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

   8. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 92.7

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

   10. Applied rewrites 92.7%

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

   if 5.00000000000000004e77 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   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 x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 18.0

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

   5. Applied rewrites 18.0%

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

   6. Taylor expanded in t around 0

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 89.0

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

   8. Applied rewrites 89.0%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(b \cdot c + a\right) \cdot c\right) \cdot i \leq -\infty:\\ \;\;\;\;\left(-2 \cdot i\right) \cdot \left(\mathsf{fma}\left(b, c, a\right) \cdot c\right)\\ \mathbf{elif}\;\left(\left(b \cdot c + a\right) \cdot c\right) \cdot i \leq 5 \cdot 10^{+77}:\\ \;\;\;\;\mathsf{fma}\left(z, t, \mathsf{fma}\left(y, x, \left(a \cdot i\right) \cdot \left(-c\right)\right)\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, c, a\right) \cdot i, -c, x \cdot y\right) \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 83.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(b \cdot c + a\right) \cdot c\right) \cdot i\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(-i, \mathsf{fma}\left(c, b, a\right) \cdot c, x \cdot y\right) \cdot 2\\ \mathbf{elif}\;t\_1 \leq 10^{+51}:\\ \;\;\;\;\mathsf{fma}\left(y, x, t \cdot z\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, c, a\right) \cdot i, -c, x \cdot y\right) \cdot 2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (* (+ (* b c) a) c) i)))
   (if (<= t_1 -1e-6)
     (* (fma (- i) (* (fma c b a) c) (* x y)) 2.0)
     (if (<= t_1 1e+51)
       (* (fma y x (* t z)) 2.0)
       (* (fma (* (fma b c a) i) (- c) (* x y)) 2.0)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (((b * c) + a) * c) * i;
	double tmp;
	if (t_1 <= -1e-6) {
		tmp = fma(-i, (fma(c, b, a) * c), (x * y)) * 2.0;
	} else if (t_1 <= 1e+51) {
		tmp = fma(y, x, (t * z)) * 2.0;
	} else {
		tmp = fma((fma(b, c, a) * i), -c, (x * y)) * 2.0;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(Float64(b * c) + a) * c) * i)
	tmp = 0.0
	if (t_1 <= -1e-6)
		tmp = Float64(fma(Float64(-i), Float64(fma(c, b, a) * c), Float64(x * y)) * 2.0);
	elseif (t_1 <= 1e+51)
		tmp = Float64(fma(y, x, Float64(t * z)) * 2.0);
	else
		tmp = Float64(fma(Float64(fma(b, c, a) * i), Float64(-c), Float64(x * y)) * 2.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(N[(b * c), $MachinePrecision] + a), $MachinePrecision] * c), $MachinePrecision] * i), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-6], N[(N[((-i) * N[(N[(c * b + a), $MachinePrecision] * c), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], If[LessEqual[t$95$1, 1e+51], N[(N[(y * x + N[(t * z), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(N[(N[(b * c + a), $MachinePrecision] * i), $MachinePrecision] * (-c) + N[(x * y), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -9.99999999999999955e-7

   1. Initial program 85.3%

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

   3. Taylor expanded in t around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

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

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

      7. mul-1-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 77.9

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

   5. Applied rewrites 77.9%

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

   if -9.99999999999999955e-7 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 1e51

   1. Initial program 98.1%

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

   3. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 95.7

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

   5. Applied rewrites 95.7%

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

   if 1e51 < (*.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 x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 17.0

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

   5. Applied rewrites 17.0%

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

   6. Taylor expanded in t around 0

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 87.3

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

   8. Applied rewrites 87.3%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(b \cdot c + a\right) \cdot c\right) \cdot i \leq -1 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(-i, \mathsf{fma}\left(c, b, a\right) \cdot c, x \cdot y\right) \cdot 2\\ \mathbf{elif}\;\left(\left(b \cdot c + a\right) \cdot c\right) \cdot i \leq 10^{+51}:\\ \;\;\;\;\mathsf{fma}\left(y, x, t \cdot z\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, c, a\right) \cdot i, -c, x \cdot y\right) \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 79.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(b \cdot c + a\right) \cdot c\right) \cdot i\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+248}:\\ \;\;\;\;\left(-2 \cdot i\right) \cdot \left(\mathsf{fma}\left(b, c, a\right) \cdot c\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+51}:\\ \;\;\;\;\mathsf{fma}\left(y, x, t \cdot z\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot c\right) \cdot -2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (* (+ (* b c) a) c) i)))
   (if (<= t_1 -4e+248)
     (* (* -2.0 i) (* (fma b c a) c))
     (if (<= t_1 1e+51)
       (* (fma y x (* t z)) 2.0)
       (* (* (* (fma c b a) i) c) -2.0)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (((b * c) + a) * c) * i;
	double tmp;
	if (t_1 <= -4e+248) {
		tmp = (-2.0 * i) * (fma(b, c, a) * c);
	} else if (t_1 <= 1e+51) {
		tmp = fma(y, x, (t * z)) * 2.0;
	} else {
		tmp = ((fma(c, b, a) * i) * c) * -2.0;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(Float64(b * c) + a) * c) * i)
	tmp = 0.0
	if (t_1 <= -4e+248)
		tmp = Float64(Float64(-2.0 * i) * Float64(fma(b, c, a) * c));
	elseif (t_1 <= 1e+51)
		tmp = Float64(fma(y, x, Float64(t * z)) * 2.0);
	else
		tmp = Float64(Float64(Float64(fma(c, b, a) * i) * c) * -2.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(N[(b * c), $MachinePrecision] + a), $MachinePrecision] * c), $MachinePrecision] * i), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+248], N[(N[(-2.0 * i), $MachinePrecision] * N[(N[(b * c + a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+51], N[(N[(y * x + N[(t * z), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(N[(N[(c * b + a), $MachinePrecision] * i), $MachinePrecision] * c), $MachinePrecision] * -2.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -4.00000000000000018e248

   1. Initial program 76.9%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

   4. Applied rewrites 83.1%

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

   5. Taylor expanded in i around inf

      \[\leadsto \color{blue}{-2 \cdot \left(c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)} \]
   6. 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. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 77.8

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

   7. Applied rewrites 77.8%

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

   9. Applied rewrites 81.7%

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

   if -4.00000000000000018e248 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 1e51

   1. Initial program 98.5%

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

   3. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 89.1

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

   5. Applied rewrites 89.1%

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

   if 1e51 < (*.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 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. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 75.2

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

   5. Applied rewrites 75.2%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(b \cdot c + a\right) \cdot c\right) \cdot i \leq -4 \cdot 10^{+248}:\\ \;\;\;\;\left(-2 \cdot i\right) \cdot \left(\mathsf{fma}\left(b, c, a\right) \cdot c\right)\\ \mathbf{elif}\;\left(\left(b \cdot c + a\right) \cdot c\right) \cdot i \leq 10^{+51}:\\ \;\;\;\;\mathsf{fma}\left(y, x, t \cdot z\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot c\right) \cdot -2\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 80.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot c\right) \cdot -2\\ t_2 := \left(\left(b \cdot c + a\right) \cdot c\right) \cdot i\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+212}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{+51}:\\ \;\;\;\;\mathsf{fma}\left(y, x, t \cdot z\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (* (* (fma c b a) i) c) -2.0)) (t_2 (* (* (+ (* b c) a) c) i)))
   (if (<= t_2 -5e+212)
     t_1
     (if (<= t_2 1e+51) (* (fma y x (* t z)) 2.0) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = ((fma(c, b, a) * i) * c) * -2.0;
	double t_2 = (((b * c) + a) * c) * i;
	double tmp;
	if (t_2 <= -5e+212) {
		tmp = t_1;
	} else if (t_2 <= 1e+51) {
		tmp = fma(y, x, (t * z)) * 2.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(fma(c, b, a) * i) * c) * -2.0)
	t_2 = Float64(Float64(Float64(Float64(b * c) + a) * c) * i)
	tmp = 0.0
	if (t_2 <= -5e+212)
		tmp = t_1;
	elseif (t_2 <= 1e+51)
		tmp = Float64(fma(y, x, Float64(t * z)) * 2.0);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(N[(c * b + a), $MachinePrecision] * i), $MachinePrecision] * c), $MachinePrecision] * -2.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(b * c), $MachinePrecision] + a), $MachinePrecision] * c), $MachinePrecision] * i), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+212], t$95$1, If[LessEqual[t$95$2, 1e+51], N[(N[(y * x + N[(t * z), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -4.99999999999999992e212 or 1e51 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 78.2%

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

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

      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. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 76.0

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

   5. Applied rewrites 76.0%

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

   if -4.99999999999999992e212 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 1e51

   1. Initial program 98.4%

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

   3. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 89.6

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

   5. Applied rewrites 89.6%

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

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(b \cdot c + a\right) \cdot c\right) \cdot i \leq -5 \cdot 10^{+212}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot c\right) \cdot -2\\ \mathbf{elif}\;\left(\left(b \cdot c + a\right) \cdot c\right) \cdot i \leq 10^{+51}:\\ \;\;\;\;\mathsf{fma}\left(y, x, t \cdot z\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot c\right) \cdot -2\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 86.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.6 \cdot 10^{+83}:\\ \;\;\;\;\mathsf{fma}\left(z, t, \mathsf{fma}\left(y, x, \left(\left(i \cdot c\right) \cdot b\right) \cdot \left(-c\right)\right)\right) \cdot 2\\ \mathbf{elif}\;c \leq 5000000:\\ \;\;\;\;\left(\left(x \cdot y + t \cdot z\right) - \left(i \cdot c\right) \cdot a\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, c, a\right) \cdot i, -c, x \cdot y\right) \cdot 2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= c -1.6e+83)
   (* (fma z t (fma y x (* (* (* i c) b) (- c)))) 2.0)
   (if (<= c 5000000.0)
     (* (- (+ (* x y) (* t z)) (* (* i c) a)) 2.0)
     (* (fma (* (fma b c a) i) (- c) (* x y)) 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 <= -1.6e+83) {
		tmp = fma(z, t, fma(y, x, (((i * c) * b) * -c))) * 2.0;
	} else if (c <= 5000000.0) {
		tmp = (((x * y) + (t * z)) - ((i * c) * a)) * 2.0;
	} else {
		tmp = fma((fma(b, c, a) * i), -c, (x * y)) * 2.0;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (c <= -1.6e+83)
		tmp = Float64(fma(z, t, fma(y, x, Float64(Float64(Float64(i * c) * b) * Float64(-c)))) * 2.0);
	elseif (c <= 5000000.0)
		tmp = Float64(Float64(Float64(Float64(x * y) + Float64(t * z)) - Float64(Float64(i * c) * a)) * 2.0);
	else
		tmp = Float64(fma(Float64(fma(b, c, a) * i), Float64(-c), Float64(x * y)) * 2.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[c, -1.6e+83], N[(N[(z * t + N[(y * x + N[(N[(N[(i * c), $MachinePrecision] * b), $MachinePrecision] * (-c)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], If[LessEqual[c, 5000000.0], N[(N[(N[(N[(x * y), $MachinePrecision] + N[(t * z), $MachinePrecision]), $MachinePrecision] - N[(N[(i * c), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(N[(N[(b * c + a), $MachinePrecision] * i), $MachinePrecision] * (-c) + N[(x * y), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if c < -1.5999999999999999e83

   1. Initial program 70.2%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. sub-neg N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*r* N/A

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

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

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

      16. lower-*.f64 N/A

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

   4. Applied rewrites 95.4%

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

   5. Taylor expanded in c around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 95.3

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

   7. Applied rewrites 95.3%

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

   if -1.5999999999999999e83 < c < 5e6

   1. Initial program 97.2%

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

   3. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 91.3

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

   5. Applied rewrites 91.3%

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

   if 5e6 < c

   1. Initial program 80.5%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 34.2

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

   5. Applied rewrites 34.2%

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

   6. Taylor expanded in t around 0

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 85.9

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

   8. Applied rewrites 85.9%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.6 \cdot 10^{+83}:\\ \;\;\;\;\mathsf{fma}\left(z, t, \mathsf{fma}\left(y, x, \left(\left(i \cdot c\right) \cdot b\right) \cdot \left(-c\right)\right)\right) \cdot 2\\ \mathbf{elif}\;c \leq 5000000:\\ \;\;\;\;\left(\left(x \cdot y + t \cdot z\right) - \left(i \cdot c\right) \cdot a\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, c, a\right) \cdot i, -c, x \cdot y\right) \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 57.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(b \cdot c + a\right) \cdot c\right) \cdot i \leq 10^{+51}:\\ \;\;\;\;\mathsf{fma}\left(y, x, t \cdot z\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\left(i \cdot c\right) \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* (* (+ (* b c) a) c) i) 1e+51)
   (* (fma y x (* t z)) 2.0)
   (* -2.0 (* (* i c) a))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (((((b * c) + a) * c) * i) <= 1e+51) {
		tmp = fma(y, x, (t * z)) * 2.0;
	} else {
		tmp = -2.0 * ((i * c) * a);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(b * c) + a) * c) * i) <= 1e+51)
		tmp = Float64(fma(y, x, Float64(t * z)) * 2.0);
	else
		tmp = Float64(-2.0 * Float64(Float64(i * c) * a));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(N[(N[(N[(b * c), $MachinePrecision] + a), $MachinePrecision] * c), $MachinePrecision] * i), $MachinePrecision], 1e+51], N[(N[(y * x + N[(t * z), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], N[(-2.0 * N[(N[(i * c), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 1e51

   1. Initial program 92.8%

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

   3. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 75.0

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

   5. Applied rewrites 75.0%

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

   if 1e51 < (*.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 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. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 42.9

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

   5. Applied rewrites 42.9%

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

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(b \cdot c + a\right) \cdot c\right) \cdot i \leq 10^{+51}:\\ \;\;\;\;\mathsf{fma}\left(y, x, t \cdot z\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\left(i \cdot c\right) \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 94.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 88.2%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. associate--l+ N/A

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

   5. lift-*.f64 N/A

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

   6. lower-fma.f64 N/A

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

   7. sub-neg N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lift-*.f64 N/A

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

   14. associate-*r* N/A

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

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

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

   16. lower-*.f64 N/A

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

4. Applied rewrites 94.2%

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

5. Final simplification 94.2%

   \[\leadsto \mathsf{fma}\left(z, t, \mathsf{fma}\left(y, x, \left(-c\right) \cdot \left(\mathsf{fma}\left(c, b, a\right) \cdot i\right)\right)\right) \cdot 2 \]
6. Add Preprocessing

Alternative 14: 44.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t \cdot z\right) \cdot 2\\ \mathbf{if}\;t \cdot z \leq -1.2 \cdot 10^{+50}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \cdot z \leq 1.36 \cdot 10^{+75}:\\ \;\;\;\;\left(x \cdot y\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (* t z) 2.0)))
   (if (<= (* t z) -1.2e+50)
     t_1
     (if (<= (* t z) 1.36e+75) (* (* x y) 2.0) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (t * z) * 2.0;
	double tmp;
	if ((t * z) <= -1.2e+50) {
		tmp = t_1;
	} else if ((t * z) <= 1.36e+75) {
		tmp = (x * y) * 2.0;
	} 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 = (t * z) * 2.0d0
    if ((t * z) <= (-1.2d+50)) then
        tmp = t_1
    else if ((t * z) <= 1.36d+75) then
        tmp = (x * y) * 2.0d0
    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 = (t * z) * 2.0;
	double tmp;
	if ((t * z) <= -1.2e+50) {
		tmp = t_1;
	} else if ((t * z) <= 1.36e+75) {
		tmp = (x * y) * 2.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	t_1 = (t * z) * 2.0
	tmp = 0
	if (t * z) <= -1.2e+50:
		tmp = t_1
	elif (t * z) <= 1.36e+75:
		tmp = (x * y) * 2.0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(t * z) * 2.0)
	tmp = 0.0
	if (Float64(t * z) <= -1.2e+50)
		tmp = t_1;
	elseif (Float64(t * z) <= 1.36e+75)
		tmp = Float64(Float64(x * y) * 2.0);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	t_1 = (t * z) * 2.0;
	tmp = 0.0;
	if ((t * z) <= -1.2e+50)
		tmp = t_1;
	elseif ((t * z) <= 1.36e+75)
		tmp = (x * y) * 2.0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(t * z), $MachinePrecision] * 2.0), $MachinePrecision]}, If[LessEqual[N[(t * z), $MachinePrecision], -1.2e+50], t$95$1, If[LessEqual[N[(t * z), $MachinePrecision], 1.36e+75], N[(N[(x * y), $MachinePrecision] * 2.0), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -1.2000000000000001e50 or 1.36e75 < (*.f64 z t)

   1. Initial program 88.4%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 63.7

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

   5. Applied rewrites 63.7%

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

   if -1.2000000000000001e50 < (*.f64 z t) < 1.36e75

   1. Initial program 88.1%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 46.8

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

   5. Applied rewrites 46.8%

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

4. Final simplification 53.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot z \leq -1.2 \cdot 10^{+50}:\\ \;\;\;\;\left(t \cdot z\right) \cdot 2\\ \mathbf{elif}\;t \cdot z \leq 1.36 \cdot 10^{+75}:\\ \;\;\;\;\left(x \cdot y\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(t \cdot z\right) \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 29.3% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \left(x \cdot y\right) \cdot 2 \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t, a, b, c, i)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    code = (x * y) * 2.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.2%

   \[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right) \]
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. *-commutative N/A

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

   2. lower-*.f64 35.8

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

5. Applied rewrites 35.8%

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

6. Final simplification 35.8%

   \[\leadsto \left(x \cdot y\right) \cdot 2 \]
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 2024268 
(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))))