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

Percentage Accurate: 89.9% → 93.9%

Time: 12.6s

Alternatives: 11

Speedup: 1.1×

• 47.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 89.9% 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: 93.9% 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 89.9%

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

   1. lift--.f64 N/A

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

   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 94.6%

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

5. Final simplification 94.6%

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

Alternative 2: 88.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(b \cdot c + a\right) \cdot c\right) \cdot i\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+264}:\\ \;\;\;\;-2 \cdot \left(\left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot c\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+157}:\\ \;\;\;\;\mathsf{fma}\left(\left(-a\right) \cdot c, i, \mathsf{fma}\left(y, x, t \cdot z\right)\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-i, \mathsf{fma}\left(c, b, a\right) \cdot c, x \cdot y\right) \cdot 2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (* (+ (* b c) a) c) i)))
   (if (<= t_1 -1e+264)
     (* -2.0 (* (* (fma c b a) i) c))
     (if (<= t_1 1e+157)
       (* (fma (* (- a) c) i (fma y x (* t z))) 2.0)
       (* (fma (- i) (* (fma c b a) 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+264) {
		tmp = -2.0 * ((fma(c, b, a) * i) * c);
	} else if (t_1 <= 1e+157) {
		tmp = fma((-a * c), i, fma(y, x, (t * z))) * 2.0;
	} else {
		tmp = fma(-i, (fma(c, b, a) * 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+264)
		tmp = Float64(-2.0 * Float64(Float64(fma(c, b, a) * i) * c));
	elseif (t_1 <= 1e+157)
		tmp = Float64(fma(Float64(Float64(-a) * c), i, fma(y, x, Float64(t * z))) * 2.0);
	else
		tmp = Float64(fma(Float64(-i), Float64(fma(c, b, a) * 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+264], N[(-2.0 * N[(N[(N[(c * b + a), $MachinePrecision] * i), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+157], N[(N[(N[((-a) * c), $MachinePrecision] * i + N[(y * x + N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], N[(N[((-i) * N[(N[(c * b + a), $MachinePrecision] * c), $MachinePrecision] + 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) < -1.00000000000000004e264

   1. Initial program 74.1%

      \[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right) \]
   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 89.3

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

   5. Applied rewrites 89.3%

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

   if -1.00000000000000004e264 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 9.99999999999999983e156

   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

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. neg-mul-1 N/A

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

      7. lower-neg.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 93.5

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

   5. Applied rewrites 93.5%

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

   if 9.99999999999999983e156 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 81.3%

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

   3. Taylor expanded in 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 86.1

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

   5. Applied rewrites 86.1%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(b \cdot c + a\right) \cdot c\right) \cdot i \leq -1 \cdot 10^{+264}:\\ \;\;\;\;-2 \cdot \left(\left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot c\right)\\ \mathbf{elif}\;\left(\left(b \cdot c + a\right) \cdot c\right) \cdot i \leq 10^{+157}:\\ \;\;\;\;\mathsf{fma}\left(\left(-a\right) \cdot c, i, \mathsf{fma}\left(y, x, t \cdot z\right)\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-i, \mathsf{fma}\left(c, b, a\right) \cdot c, x \cdot y\right) \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(c, b, a\right) \cdot c\\ t_2 := \left(\left(b \cdot c + a\right) \cdot c\right) \cdot i\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+61}:\\ \;\;\;\;\mathsf{fma}\left(-i, t\_1, x \cdot y\right) \cdot 2\\ \mathbf{elif}\;t\_2 \leq 10^{-169}:\\ \;\;\;\;\mathsf{fma}\left(y, x, t \cdot z\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-i, t\_1, t \cdot z\right) \cdot 2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (fma c b a) c)) (t_2 (* (* (+ (* b c) a) c) i)))
   (if (<= t_2 -1e+61)
     (* (fma (- i) t_1 (* x y)) 2.0)
     (if (<= t_2 1e-169)
       (* (fma y x (* t z)) 2.0)
       (* (fma (- i) t_1 (* t z)) 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(c, b, a) * c;
	double t_2 = (((b * c) + a) * c) * i;
	double tmp;
	if (t_2 <= -1e+61) {
		tmp = fma(-i, t_1, (x * y)) * 2.0;
	} else if (t_2 <= 1e-169) {
		tmp = fma(y, x, (t * z)) * 2.0;
	} else {
		tmp = fma(-i, t_1, (t * z)) * 2.0;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(fma(c, b, a) * c)
	t_2 = Float64(Float64(Float64(Float64(b * c) + a) * c) * i)
	tmp = 0.0
	if (t_2 <= -1e+61)
		tmp = Float64(fma(Float64(-i), t_1, Float64(x * y)) * 2.0);
	elseif (t_2 <= 1e-169)
		tmp = Float64(fma(y, x, Float64(t * z)) * 2.0);
	else
		tmp = Float64(fma(Float64(-i), t_1, Float64(t * z)) * 2.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(c * b + a), $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, -1e+61], N[(N[((-i) * t$95$1 + N[(x * y), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], If[LessEqual[t$95$2, 1e-169], N[(N[(y * x + N[(t * z), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], N[(N[((-i) * t$95$1 + N[(t * z), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -9.99999999999999949e60

   1. Initial program 82.7%

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

   3. Taylor expanded in 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 82.9

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

   5. Applied rewrites 82.9%

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

   if -9.99999999999999949e60 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 1.00000000000000002e-169

   1. Initial program 97.7%

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

   3. Taylor expanded in 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 95.6

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

   5. Applied rewrites 95.6%

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

   if 1.00000000000000002e-169 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 87.8%

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

   3. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

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

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

      7. mul-1-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 81.2

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

   5. Applied rewrites 81.2%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(b \cdot c + a\right) \cdot c\right) \cdot i \leq -1 \cdot 10^{+61}:\\ \;\;\;\;\mathsf{fma}\left(-i, \mathsf{fma}\left(c, b, a\right) \cdot c, x \cdot y\right) \cdot 2\\ \mathbf{elif}\;\left(\left(b \cdot c + a\right) \cdot c\right) \cdot i \leq 10^{-169}:\\ \;\;\;\;\mathsf{fma}\left(y, x, t \cdot z\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-i, \mathsf{fma}\left(c, b, a\right) \cdot c, t \cdot z\right) \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-i, \mathsf{fma}\left(c, b, a\right) \cdot c, x \cdot y\right) \cdot 2\\ t_2 := \left(\left(b \cdot c + a\right) \cdot c\right) \cdot i\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+61}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{-72}:\\ \;\;\;\;\mathsf{fma}\left(y, x, t \cdot z\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (fma (- i) (* (fma c b a) c) (* x y)) 2.0))
        (t_2 (* (* (+ (* b c) a) c) i)))
   (if (<= t_2 -1e+61) t_1 (if (<= t_2 5e-72) (* (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(-i, (fma(c, b, a) * c), (x * y)) * 2.0;
	double t_2 = (((b * c) + a) * c) * i;
	double tmp;
	if (t_2 <= -1e+61) {
		tmp = t_1;
	} else if (t_2 <= 5e-72) {
		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(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+61)
		tmp = t_1;
	elseif (t_2 <= 5e-72)
		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[((-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+61], t$95$1, If[LessEqual[t$95$2, 5e-72], 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) < -9.99999999999999949e60 or 4.9999999999999996e-72 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 85.1%

      \[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right) \]
   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 81.2

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

   5. Applied rewrites 81.2%

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

   if -9.99999999999999949e60 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 4.9999999999999996e-72

   1. Initial program 97.9%

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

   3. Taylor expanded in c around 0

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

      1. +-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 95.8

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

   5. Applied rewrites 95.8%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(b \cdot c + a\right) \cdot c\right) \cdot i \leq -1 \cdot 10^{+61}:\\ \;\;\;\;\mathsf{fma}\left(-i, \mathsf{fma}\left(c, b, a\right) \cdot c, x \cdot y\right) \cdot 2\\ \mathbf{elif}\;\left(\left(b \cdot c + a\right) \cdot c\right) \cdot i \leq 5 \cdot 10^{-72}:\\ \;\;\;\;\mathsf{fma}\left(y, x, t \cdot z\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-i, \mathsf{fma}\left(c, b, a\right) \cdot c, x \cdot y\right) \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\mathsf{fma}\left(c, b, a\right) \cdot c\right) \cdot \left(-i\right)\right) \cdot 2\\ t_2 := \left(\left(b \cdot c + a\right) \cdot c\right) \cdot i\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+126}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{+157}:\\ \;\;\;\;\mathsf{fma}\left(y, x, t \cdot z\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (* (* (fma c b a) c) (- i)) 2.0))
        (t_2 (* (* (+ (* b c) a) c) i)))
   (if (<= t_2 -2e+126)
     t_1
     (if (<= t_2 1e+157) (* (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) * c) * -i) * 2.0;
	double t_2 = (((b * c) + a) * c) * i;
	double tmp;
	if (t_2 <= -2e+126) {
		tmp = t_1;
	} else if (t_2 <= 1e+157) {
		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) * c) * Float64(-i)) * 2.0)
	t_2 = Float64(Float64(Float64(Float64(b * c) + a) * c) * i)
	tmp = 0.0
	if (t_2 <= -2e+126)
		tmp = t_1;
	elseif (t_2 <= 1e+157)
		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] * c), $MachinePrecision] * (-i)), $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, -2e+126], t$95$1, If[LessEqual[t$95$2, 1e+157], 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) < -1.99999999999999985e126 or 9.99999999999999983e156 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 80.7%

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

   3. Taylor expanded in i around inf

      \[\leadsto 2 \cdot \color{blue}{\left(-1 \cdot \left(c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

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

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

      6. mul-1-neg N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 81.8

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

   5. Applied rewrites 81.8%

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

   if -1.99999999999999985e126 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 9.99999999999999983e156

   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 85.9

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

   5. Applied rewrites 85.9%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(b \cdot c + a\right) \cdot c\right) \cdot i \leq -2 \cdot 10^{+126}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(c, b, a\right) \cdot c\right) \cdot \left(-i\right)\right) \cdot 2\\ \mathbf{elif}\;\left(\left(b \cdot c + a\right) \cdot c\right) \cdot i \leq 10^{+157}:\\ \;\;\;\;\mathsf{fma}\left(y, x, t \cdot z\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(c, b, a\right) \cdot c\right) \cdot \left(-i\right)\right) \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 82.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -2 \cdot \left(\left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot c\right)\\ t_2 := \left(\left(b \cdot c + a\right) \cdot c\right) \cdot i\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+126}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{+157}:\\ \;\;\;\;\mathsf{fma}\left(y, x, t \cdot z\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* -2.0 (* (* (fma c b a) i) c))) (t_2 (* (* (+ (* b c) a) c) i)))
   (if (<= t_2 -2e+126)
     t_1
     (if (<= t_2 1e+157) (* (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 = -2.0 * ((fma(c, b, a) * i) * c);
	double t_2 = (((b * c) + a) * c) * i;
	double tmp;
	if (t_2 <= -2e+126) {
		tmp = t_1;
	} else if (t_2 <= 1e+157) {
		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(-2.0 * Float64(Float64(fma(c, b, a) * i) * c))
	t_2 = Float64(Float64(Float64(Float64(b * c) + a) * c) * i)
	tmp = 0.0
	if (t_2 <= -2e+126)
		tmp = t_1;
	elseif (t_2 <= 1e+157)
		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[(-2.0 * N[(N[(N[(c * b + a), $MachinePrecision] * i), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(b * c), $MachinePrecision] + a), $MachinePrecision] * c), $MachinePrecision] * i), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+126], t$95$1, If[LessEqual[t$95$2, 1e+157], 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) < -1.99999999999999985e126 or 9.99999999999999983e156 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 80.7%

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

   3. Taylor expanded in 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 81.7

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

   5. Applied rewrites 81.7%

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

   if -1.99999999999999985e126 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 9.99999999999999983e156

   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 85.9

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

   5. Applied rewrites 85.9%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(b \cdot c + a\right) \cdot c\right) \cdot i \leq -2 \cdot 10^{+126}:\\ \;\;\;\;-2 \cdot \left(\left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot c\right)\\ \mathbf{elif}\;\left(\left(b \cdot c + a\right) \cdot c\right) \cdot i \leq 10^{+157}:\\ \;\;\;\;\mathsf{fma}\left(y, x, t \cdot z\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot c\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 73.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(b \cdot c + a\right) \cdot c\right) \cdot i\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+264}:\\ \;\;\;\;\left(\left(\left(b \cdot c\right) \cdot c\right) \cdot i\right) \cdot -2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+253}:\\ \;\;\;\;\mathsf{fma}\left(y, x, t \cdot z\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(b \cdot i\right) \cdot c\right) \cdot c\right) \cdot -2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (* (+ (* b c) a) c) i)))
   (if (<= t_1 -1e+264)
     (* (* (* (* b c) c) i) -2.0)
     (if (<= t_1 5e+253)
       (* (fma y x (* t z)) 2.0)
       (* (* (* (* b i) c) 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 <= -1e+264) {
		tmp = (((b * c) * c) * i) * -2.0;
	} else if (t_1 <= 5e+253) {
		tmp = fma(y, x, (t * z)) * 2.0;
	} else {
		tmp = (((b * i) * c) * 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 <= -1e+264)
		tmp = Float64(Float64(Float64(Float64(b * c) * c) * i) * -2.0);
	elseif (t_1 <= 5e+253)
		tmp = Float64(fma(y, x, Float64(t * z)) * 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(b * i) * c) * 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, -1e+264], N[(N[(N[(N[(b * c), $MachinePrecision] * c), $MachinePrecision] * i), $MachinePrecision] * -2.0), $MachinePrecision], If[LessEqual[t$95$1, 5e+253], N[(N[(y * x + N[(t * z), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(N[(N[(b * i), $MachinePrecision] * c), $MachinePrecision] * c), $MachinePrecision] * -2.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -1.00000000000000004e264

   1. Initial program 74.1%

      \[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right) \]
   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 12.1

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

   5. Applied rewrites 12.1%

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

   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 74.9

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

   8. Applied rewrites 74.9%

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

   10. Applied rewrites 74.6%

       \[\leadsto \left(\left(b \cdot c\right) \cdot \left(c \cdot i\right)\right) \cdot -2 \]
   11. Step-by-step derivation

   12. Applied rewrites 76.5%

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

   if -1.00000000000000004e264 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 4.9999999999999997e253

   1. Initial program 98.6%

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

   3. Taylor expanded in 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 80.2

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

   5. Applied rewrites 80.2%

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

   if 4.9999999999999997e253 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 80.1%

      \[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right) \]
   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 13.8

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

   5. Applied rewrites 13.8%

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

   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 66.6

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

   8. Applied rewrites 66.6%

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

   10. Applied rewrites 71.6%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(b \cdot c + a\right) \cdot c\right) \cdot i \leq -1 \cdot 10^{+264}:\\ \;\;\;\;\left(\left(\left(b \cdot c\right) \cdot c\right) \cdot i\right) \cdot -2\\ \mathbf{elif}\;\left(\left(b \cdot c + a\right) \cdot c\right) \cdot i \leq 5 \cdot 10^{+253}:\\ \;\;\;\;\mathsf{fma}\left(y, x, t \cdot z\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(b \cdot i\right) \cdot c\right) \cdot c\right) \cdot -2\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\left(b \cdot i\right) \cdot c\right) \cdot c\right) \cdot -2\\ t_2 := \left(\left(b \cdot c + a\right) \cdot c\right) \cdot i\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+264}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+253}:\\ \;\;\;\;\mathsf{fma}\left(y, x, t \cdot z\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (* (* (* b i) c) c) -2.0)) (t_2 (* (* (+ (* b c) a) c) i)))
   (if (<= t_2 -1e+264)
     t_1
     (if (<= t_2 5e+253) (* (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 = (((b * i) * c) * c) * -2.0;
	double t_2 = (((b * c) + a) * c) * i;
	double tmp;
	if (t_2 <= -1e+264) {
		tmp = t_1;
	} else if (t_2 <= 5e+253) {
		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(Float64(b * i) * c) * c) * -2.0)
	t_2 = Float64(Float64(Float64(Float64(b * c) + a) * c) * i)
	tmp = 0.0
	if (t_2 <= -1e+264)
		tmp = t_1;
	elseif (t_2 <= 5e+253)
		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[(b * i), $MachinePrecision] * c), $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, -1e+264], t$95$1, If[LessEqual[t$95$2, 5e+253], 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) < -1.00000000000000004e264 or 4.9999999999999997e253 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 77.5%

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

   3. Taylor expanded in 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 13.1

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

   5. Applied rewrites 13.1%

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

   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 70.2

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

   8. Applied rewrites 70.2%

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

   10. Applied rewrites 73.7%

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

   if -1.00000000000000004e264 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 4.9999999999999997e253

   1. Initial program 98.6%

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

   3. Taylor expanded in 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 80.2

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

   5. Applied rewrites 80.2%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(b \cdot c + a\right) \cdot c\right) \cdot i \leq -1 \cdot 10^{+264}:\\ \;\;\;\;\left(\left(\left(b \cdot i\right) \cdot c\right) \cdot c\right) \cdot -2\\ \mathbf{elif}\;\left(\left(b \cdot c + a\right) \cdot c\right) \cdot i \leq 5 \cdot 10^{+253}:\\ \;\;\;\;\mathsf{fma}\left(y, x, t \cdot z\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(b \cdot i\right) \cdot c\right) \cdot c\right) \cdot -2\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 62.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(c \cdot i\right) \cdot a\right) \cdot -2\\ t_2 := \left(\left(b \cdot c + a\right) \cdot c\right) \cdot i\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+292}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{+157}:\\ \;\;\;\;\mathsf{fma}\left(y, x, t \cdot z\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (* (* c i) a) -2.0)) (t_2 (* (* (+ (* b c) a) c) i)))
   (if (<= t_2 -1e+292)
     t_1
     (if (<= t_2 1e+157) (* (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 = ((c * i) * a) * -2.0;
	double t_2 = (((b * c) + a) * c) * i;
	double tmp;
	if (t_2 <= -1e+292) {
		tmp = t_1;
	} else if (t_2 <= 1e+157) {
		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(c * i) * a) * -2.0)
	t_2 = Float64(Float64(Float64(Float64(b * c) + a) * c) * i)
	tmp = 0.0
	if (t_2 <= -1e+292)
		tmp = t_1;
	elseif (t_2 <= 1e+157)
		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[(c * i), $MachinePrecision] * a), $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+292], t$95$1, If[LessEqual[t$95$2, 1e+157], 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) < -1e292 or 9.99999999999999983e156 < (*.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 41.5

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

   5. Applied rewrites 41.5%

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

   if -1e292 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 9.99999999999999983e156

   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 81.1

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

   5. Applied rewrites 81.1%

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

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(b \cdot c + a\right) \cdot c\right) \cdot i \leq -1 \cdot 10^{+292}:\\ \;\;\;\;\left(\left(c \cdot i\right) \cdot a\right) \cdot -2\\ \mathbf{elif}\;\left(\left(b \cdot c + a\right) \cdot c\right) \cdot i \leq 10^{+157}:\\ \;\;\;\;\mathsf{fma}\left(y, x, t \cdot z\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\left(c \cdot i\right) \cdot a\right) \cdot -2\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 45.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x \cdot y\right) \cdot 2\\ \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{+52}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \cdot y \leq 10^{+44}:\\ \;\;\;\;\left(t \cdot z\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (* x y) 2.0)))
   (if (<= (* x y) -1e+52) t_1 (if (<= (* x y) 1e+44) (* (* 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 = (x * y) * 2.0;
	double tmp;
	if ((x * y) <= -1e+52) {
		tmp = t_1;
	} else if ((x * y) <= 1e+44) {
		tmp = (t * z) * 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 = (x * y) * 2.0d0
    if ((x * y) <= (-1d+52)) then
        tmp = t_1
    else if ((x * y) <= 1d+44) then
        tmp = (t * z) * 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 = (x * y) * 2.0;
	double tmp;
	if ((x * y) <= -1e+52) {
		tmp = t_1;
	} else if ((x * y) <= 1e+44) {
		tmp = (t * z) * 2.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	t_1 = (x * y) * 2.0
	tmp = 0
	if (x * y) <= -1e+52:
		tmp = t_1
	elif (x * y) <= 1e+44:
		tmp = (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(x * y) * 2.0)
	tmp = 0.0
	if (Float64(x * y) <= -1e+52)
		tmp = t_1;
	elseif (Float64(x * y) <= 1e+44)
		tmp = Float64(Float64(t * z) * 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 = (x * y) * 2.0;
	tmp = 0.0;
	if ((x * y) <= -1e+52)
		tmp = t_1;
	elseif ((x * y) <= 1e+44)
		tmp = (t * z) * 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[(x * y), $MachinePrecision] * 2.0), $MachinePrecision]}, If[LessEqual[N[(x * y), $MachinePrecision], -1e+52], t$95$1, If[LessEqual[N[(x * y), $MachinePrecision], 1e+44], N[(N[(t * z), $MachinePrecision] * 2.0), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -9.9999999999999999e51 or 1.0000000000000001e44 < (*.f64 x y)

   1. Initial program 83.2%

      \[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right) \]
   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 52.7

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

   5. Applied rewrites 52.7%

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

   if -9.9999999999999999e51 < (*.f64 x y) < 1.0000000000000001e44

   1. Initial program 95.6%

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

   3. Taylor expanded in 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 40.3

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

   5. Applied rewrites 40.3%

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

4. Final simplification 46.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{+52}:\\ \;\;\;\;\left(x \cdot y\right) \cdot 2\\ \mathbf{elif}\;x \cdot y \leq 10^{+44}:\\ \;\;\;\;\left(t \cdot z\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 30.3% 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 89.9%

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

3. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 28.2

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

5. Applied rewrites 28.2%

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

6. Final simplification 28.2%

   \[\leadsto \left(x \cdot y\right) \cdot 2 \]
7. Add Preprocessing

Developer Target 1: 94.2% 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 2024273 
(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))))