Diagrams.Solve.Polynomial:quartForm from diagrams-solve-0.1, C

Percentage Accurate: 97.8% → 99.0%

Time: 10.8s

Alternatives: 14

Speedup: 1.5×

• 33.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (+ (- (+ (* x y) (/ (* z t) 16.0)) (/ (* a b) 4.0)) c))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	return (((x * y) + ((z * t) / 16.0)) - ((a * b) / 4.0)) + c;
}

Fortran

real(8) function code(x, y, z, t, a, b, c)
    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
    code = (((x * y) + ((z * t) / 16.0d0)) - ((a * b) / 4.0d0)) + c
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c) {
	return (((x * y) + ((z * t) / 16.0)) - ((a * b) / 4.0)) + c;
}

Python

def code(x, y, z, t, a, b, c):
	return (((x * y) + ((z * t) / 16.0)) - ((a * b) / 4.0)) + c

Julia

function code(x, y, z, t, a, b, c)
	return Float64(Float64(Float64(Float64(x * y) + Float64(Float64(z * t) / 16.0)) - Float64(Float64(a * b) / 4.0)) + c)
end

MATLAB

function tmp = code(x, y, z, t, a, b, c)
	tmp = (((x * y) + ((z * t) / 16.0)) - ((a * b) / 4.0)) + c;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := N[(N[(N[(N[(x * y), $MachinePrecision] + N[(N[(z * t), $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] - N[(N[(a * b), $MachinePrecision] / 4.0), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision]

Initial Program: 97.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (+ (- (+ (* x y) (/ (* z t) 16.0)) (/ (* a b) 4.0)) c))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	return (((x * y) + ((z * t) / 16.0)) - ((a * b) / 4.0)) + c;
}

Fortran

real(8) function code(x, y, z, t, a, b, c)
    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
    code = (((x * y) + ((z * t) / 16.0d0)) - ((a * b) / 4.0d0)) + c
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c) {
	return (((x * y) + ((z * t) / 16.0)) - ((a * b) / 4.0)) + c;
}

Python

def code(x, y, z, t, a, b, c):
	return (((x * y) + ((z * t) / 16.0)) - ((a * b) / 4.0)) + c

Julia

function code(x, y, z, t, a, b, c)
	return Float64(Float64(Float64(Float64(x * y) + Float64(Float64(z * t) / 16.0)) - Float64(Float64(a * b) / 4.0)) + c)
end

MATLAB

function tmp = code(x, y, z, t, a, b, c)
	tmp = (((x * y) + ((z * t) / 16.0)) - ((a * b) / 4.0)) + c;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := N[(N[(N[(N[(x * y), $MachinePrecision] + N[(N[(z * t), $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] - N[(N[(a * b), $MachinePrecision] / 4.0), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision]

Alternative 1: 99.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(t \cdot 0.0625, z, \mathsf{fma}\left(x, y, \left(a \cdot b\right) \cdot -0.25\right)\right) + c \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (+ (fma (* t 0.0625) z (fma x y (* (* a b) -0.25))) c))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	return fma((t * 0.0625), z, fma(x, y, ((a * b) * -0.25))) + c;
}

Julia

function code(x, y, z, t, a, b, c)
	return Float64(fma(Float64(t * 0.0625), z, fma(x, y, Float64(Float64(a * b) * -0.25))) + c)
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := N[(N[(N[(t * 0.0625), $MachinePrecision] * z + N[(x * y + N[(N[(a * b), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision]

Derivation

1. Initial program 97.6%

   \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \left(\left(\color{blue}{x \cdot y} + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]

   2. lift-*.f64 N/A

      \[\leadsto \left(\left(x \cdot y + \frac{\color{blue}{z \cdot t}}{16}\right) - \frac{a \cdot b}{4}\right) + c \]

   3. lift-/.f64 N/A

      \[\leadsto \left(\left(x \cdot y + \color{blue}{\frac{z \cdot t}{16}}\right) - \frac{a \cdot b}{4}\right) + c \]

   4. lift-+.f64 N/A

      \[\leadsto \left(\color{blue}{\left(x \cdot y + \frac{z \cdot t}{16}\right)} - \frac{a \cdot b}{4}\right) + c \]

   5. lift-*.f64 N/A

      \[\leadsto \left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{\color{blue}{a \cdot b}}{4}\right) + c \]

   6. lift-/.f64 N/A

      \[\leadsto \left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \color{blue}{\frac{a \cdot b}{4}}\right) + c \]

   7. sub-neg N/A

      \[\leadsto \color{blue}{\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right)} + c \]

   8. lift-+.f64 N/A

      \[\leadsto \left(\color{blue}{\left(x \cdot y + \frac{z \cdot t}{16}\right)} + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right) + c \]

   9. +-commutative N/A

      \[\leadsto \left(\color{blue}{\left(\frac{z \cdot t}{16} + x \cdot y\right)} + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right) + c \]

   10. associate-+l+ N/A

       \[\leadsto \color{blue}{\left(\frac{z \cdot t}{16} + \left(x \cdot y + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right)\right)} + c \]

   11. lift-/.f64 N/A

       \[\leadsto \left(\color{blue}{\frac{z \cdot t}{16}} + \left(x \cdot y + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right)\right) + c \]

   12. lift-*.f64 N/A

       \[\leadsto \left(\frac{\color{blue}{z \cdot t}}{16} + \left(x \cdot y + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right)\right) + c \]

   13. associate-/l* N/A

       \[\leadsto \left(\color{blue}{z \cdot \frac{t}{16}} + \left(x \cdot y + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right)\right) + c \]

   14. *-commutative N/A

       \[\leadsto \left(\color{blue}{\frac{t}{16} \cdot z} + \left(x \cdot y + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right)\right) + c \]

   15. lower-fma.f64 N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t}{16}, z, x \cdot y + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right)} + c \]

   16. div-inv N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{t \cdot \frac{1}{16}}, z, x \cdot y + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right) + c \]

   17. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{t \cdot \frac{1}{16}}, z, x \cdot y + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right) + c \]

   18. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(t \cdot \color{blue}{\frac{1}{16}}, z, x \cdot y + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right) + c \]

   19. lift-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(t \cdot \frac{1}{16}, z, \color{blue}{x \cdot y} + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right) + c \]

   20. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(t \cdot \frac{1}{16}, z, \color{blue}{\mathsf{fma}\left(x, y, \mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)}\right) + c \]

   21. lift-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(t \cdot \frac{1}{16}, z, \mathsf{fma}\left(x, y, \mathsf{neg}\left(\color{blue}{\frac{a \cdot b}{4}}\right)\right)\right) + c \]

   22. div-inv N/A

       \[\leadsto \mathsf{fma}\left(t \cdot \frac{1}{16}, z, \mathsf{fma}\left(x, y, \mathsf{neg}\left(\color{blue}{\left(a \cdot b\right) \cdot \frac{1}{4}}\right)\right)\right) + c \]

4. Applied rewrites 99.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(t \cdot 0.0625, z, \mathsf{fma}\left(x, y, \left(a \cdot b\right) \cdot -0.25\right)\right)} + c \]
5. Add Preprocessing

Alternative 2: 99.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(x \cdot y + \frac{t \cdot z}{16}\right) - \frac{a \cdot b}{4} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(t \cdot z, 0.0625, \mathsf{fma}\left(x, y, \mathsf{fma}\left(a, b \cdot -0.25, c\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, a \cdot \left(b \cdot -0.25\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (<= (- (+ (* x y) (/ (* t z) 16.0)) (/ (* a b) 4.0)) INFINITY)
   (fma (* t z) 0.0625 (fma x y (fma a (* b -0.25) c)))
   (fma y x (* a (* b -0.25)))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if ((((x * y) + ((t * z) / 16.0)) - ((a * b) / 4.0)) <= ((double) INFINITY)) {
		tmp = fma((t * z), 0.0625, fma(x, y, fma(a, (b * -0.25), c)));
	} else {
		tmp = fma(y, x, (a * (b * -0.25)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if (Float64(Float64(Float64(x * y) + Float64(Float64(t * z) / 16.0)) - Float64(Float64(a * b) / 4.0)) <= Inf)
		tmp = fma(Float64(t * z), 0.0625, fma(x, y, fma(a, Float64(b * -0.25), c)));
	else
		tmp = fma(y, x, Float64(a * Float64(b * -0.25)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[LessEqual[N[(N[(N[(x * y), $MachinePrecision] + N[(N[(t * z), $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] - N[(N[(a * b), $MachinePrecision] / 4.0), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(t * z), $MachinePrecision] * 0.0625 + N[(x * y + N[(a * N[(b * -0.25), $MachinePrecision] + c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * x + N[(a * N[(b * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 (*.f64 x y) (/.f64 (*.f64 z t) #s(literal 16 binary64))) (/.f64 (*.f64 a b) #s(literal 4 binary64))) < +inf.0

   1. Initial program 100.0%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(\color{blue}{x \cdot y} + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\left(x \cdot y + \frac{\color{blue}{z \cdot t}}{16}\right) - \frac{a \cdot b}{4}\right) + c \]

      3. lift-/.f64 N/A

         \[\leadsto \left(\left(x \cdot y + \color{blue}{\frac{z \cdot t}{16}}\right) - \frac{a \cdot b}{4}\right) + c \]

      4. lift-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(x \cdot y + \frac{z \cdot t}{16}\right)} - \frac{a \cdot b}{4}\right) + c \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{\color{blue}{a \cdot b}}{4}\right) + c \]

      6. lift-/.f64 N/A

         \[\leadsto \left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \color{blue}{\frac{a \cdot b}{4}}\right) + c \]

      7. sub-neg N/A

         \[\leadsto \color{blue}{\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right)} + c \]

      8. lift-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(x \cdot y + \frac{z \cdot t}{16}\right)} + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right) + c \]

      9. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{z \cdot t}{16} + x \cdot y\right)} + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right) + c \]

      10. associate-+l+ N/A

          \[\leadsto \color{blue}{\left(\frac{z \cdot t}{16} + \left(x \cdot y + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right)\right)} + c \]

      11. lift-/.f64 N/A

          \[\leadsto \left(\color{blue}{\frac{z \cdot t}{16}} + \left(x \cdot y + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right)\right) + c \]

      12. lift-*.f64 N/A

          \[\leadsto \left(\frac{\color{blue}{z \cdot t}}{16} + \left(x \cdot y + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right)\right) + c \]

      13. associate-/l* N/A

          \[\leadsto \left(\color{blue}{z \cdot \frac{t}{16}} + \left(x \cdot y + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right)\right) + c \]

      14. *-commutative N/A

          \[\leadsto \left(\color{blue}{\frac{t}{16} \cdot z} + \left(x \cdot y + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right)\right) + c \]

      15. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t}{16}, z, x \cdot y + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right)} + c \]

      16. div-inv N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{t \cdot \frac{1}{16}}, z, x \cdot y + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right) + c \]

      17. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{t \cdot \frac{1}{16}}, z, x \cdot y + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right) + c \]

      18. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(t \cdot \color{blue}{\frac{1}{16}}, z, x \cdot y + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right) + c \]

      19. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(t \cdot \frac{1}{16}, z, \color{blue}{x \cdot y} + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right) + c \]

      20. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(t \cdot \frac{1}{16}, z, \color{blue}{\mathsf{fma}\left(x, y, \mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)}\right) + c \]

      21. lift-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(t \cdot \frac{1}{16}, z, \mathsf{fma}\left(x, y, \mathsf{neg}\left(\color{blue}{\frac{a \cdot b}{4}}\right)\right)\right) + c \]

      22. div-inv N/A

          \[\leadsto \mathsf{fma}\left(t \cdot \frac{1}{16}, z, \mathsf{fma}\left(x, y, \mathsf{neg}\left(\color{blue}{\left(a \cdot b\right) \cdot \frac{1}{4}}\right)\right)\right) + c \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t \cdot 0.0625, z, \mathsf{fma}\left(x, y, \left(a \cdot b\right) \cdot -0.25\right)\right)} + c \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left(t \cdot \frac{1}{16}\right)} \cdot z + \left(x \cdot y + \left(a \cdot b\right) \cdot \frac{-1}{4}\right)\right) + c \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\left(t \cdot \frac{1}{16}\right) \cdot z + \left(x \cdot y + \color{blue}{\left(a \cdot b\right)} \cdot \frac{-1}{4}\right)\right) + c \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\left(t \cdot \frac{1}{16}\right) \cdot z + \left(x \cdot y + \color{blue}{\left(a \cdot b\right) \cdot \frac{-1}{4}}\right)\right) + c \]

      4. lift-fma.f64 N/A

         \[\leadsto \left(\left(t \cdot \frac{1}{16}\right) \cdot z + \color{blue}{\mathsf{fma}\left(x, y, \left(a \cdot b\right) \cdot \frac{-1}{4}\right)}\right) + c \]

      5. associate-+l+ N/A

         \[\leadsto \color{blue}{\left(t \cdot \frac{1}{16}\right) \cdot z + \left(\mathsf{fma}\left(x, y, \left(a \cdot b\right) \cdot \frac{-1}{4}\right) + c\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(t \cdot \frac{1}{16}\right)} \cdot z + \left(\mathsf{fma}\left(x, y, \left(a \cdot b\right) \cdot \frac{-1}{4}\right) + c\right) \]

      7. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{16} \cdot t\right)} \cdot z + \left(\mathsf{fma}\left(x, y, \left(a \cdot b\right) \cdot \frac{-1}{4}\right) + c\right) \]

      8. associate-*r* N/A

         \[\leadsto \color{blue}{\frac{1}{16} \cdot \left(t \cdot z\right)} + \left(\mathsf{fma}\left(x, y, \left(a \cdot b\right) \cdot \frac{-1}{4}\right) + c\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{1}{16} \cdot \color{blue}{\left(t \cdot z\right)} + \left(\mathsf{fma}\left(x, y, \left(a \cdot b\right) \cdot \frac{-1}{4}\right) + c\right) \]

      10. *-commutative N/A

          \[\leadsto \color{blue}{\left(t \cdot z\right) \cdot \frac{1}{16}} + \left(\mathsf{fma}\left(x, y, \left(a \cdot b\right) \cdot \frac{-1}{4}\right) + c\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(t \cdot z, \frac{1}{16}, \mathsf{fma}\left(x, y, \left(a \cdot b\right) \cdot \frac{-1}{4}\right) + c\right)} \]

      12. lift-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(t \cdot z, \frac{1}{16}, \color{blue}{\left(x \cdot y + \left(a \cdot b\right) \cdot \frac{-1}{4}\right)} + c\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(t \cdot z, \frac{1}{16}, \left(\color{blue}{x \cdot y} + \left(a \cdot b\right) \cdot \frac{-1}{4}\right) + c\right) \]

      14. associate-+l+ N/A

          \[\leadsto \mathsf{fma}\left(t \cdot z, \frac{1}{16}, \color{blue}{x \cdot y + \left(\left(a \cdot b\right) \cdot \frac{-1}{4} + c\right)}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(t \cdot z, \frac{1}{16}, \color{blue}{x \cdot y} + \left(\left(a \cdot b\right) \cdot \frac{-1}{4} + c\right)\right) \]

      16. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(t \cdot z, \frac{1}{16}, \color{blue}{\mathsf{fma}\left(x, y, \left(a \cdot b\right) \cdot \frac{-1}{4} + c\right)}\right) \]

   6. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t \cdot z, 0.0625, \mathsf{fma}\left(x, y, \mathsf{fma}\left(a, b \cdot -0.25, c\right)\right)\right)} \]

   if +inf.0 < (-.f64 (+.f64 (*.f64 x y) (/.f64 (*.f64 z t) #s(literal 16 binary64))) (/.f64 (*.f64 a b) #s(literal 4 binary64)))

   1. Initial program 0.0%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{\left(c + x \cdot y\right) + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot \left(a \cdot b\right)} \]

      2. metadata-eval N/A

         \[\leadsto \left(c + x \cdot y\right) + \color{blue}{\frac{-1}{4}} \cdot \left(a \cdot b\right) \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\frac{-1}{4} \cdot \left(a \cdot b\right) + \left(c + x \cdot y\right)} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(a \cdot b\right) \cdot \frac{-1}{4}} + \left(c + x \cdot y\right) \]

      5. associate-*l* N/A

         \[\leadsto \color{blue}{a \cdot \left(b \cdot \frac{-1}{4}\right)} + \left(c + x \cdot y\right) \]

      6. *-commutative N/A

         \[\leadsto a \cdot \color{blue}{\left(\frac{-1}{4} \cdot b\right)} + \left(c + x \cdot y\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a, \frac{-1}{4} \cdot b, c + x \cdot y\right)} \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{b \cdot \frac{-1}{4}}, c + x \cdot y\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{b \cdot \frac{-1}{4}}, c + x \cdot y\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(a, b \cdot \frac{-1}{4}, \color{blue}{x \cdot y + c}\right) \]

      11. lower-fma.f64 50.0

          \[\leadsto \mathsf{fma}\left(a, b \cdot -0.25, \color{blue}{\mathsf{fma}\left(x, y, c\right)}\right) \]

   5. Applied rewrites 50.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, b \cdot -0.25, \mathsf{fma}\left(x, y, c\right)\right)} \]

   6. Taylor expanded in c around 0

      \[\leadsto \color{blue}{\frac{-1}{4} \cdot \left(a \cdot b\right) + x \cdot y} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{4} \cdot a\right) \cdot b} + x \cdot y \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{b \cdot \left(\frac{-1}{4} \cdot a\right)} + x \cdot y \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(b, \frac{-1}{4} \cdot a, x \cdot y\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(b, \color{blue}{\frac{-1}{4} \cdot a}, x \cdot y\right) \]

      5. lower-*.f64 50.0

         \[\leadsto \mathsf{fma}\left(b, -0.25 \cdot a, \color{blue}{x \cdot y}\right) \]

   8. Applied rewrites 50.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b, -0.25 \cdot a, x \cdot y\right)} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto b \cdot \color{blue}{\left(\frac{-1}{4} \cdot a\right)} + x \cdot y \]

      2. lift-*.f64 N/A

         \[\leadsto b \cdot \left(\frac{-1}{4} \cdot a\right) + \color{blue}{x \cdot y} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{x \cdot y + b \cdot \left(\frac{-1}{4} \cdot a\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot y} + b \cdot \left(\frac{-1}{4} \cdot a\right) \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{y \cdot x} + b \cdot \left(\frac{-1}{4} \cdot a\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, b \cdot \left(\frac{-1}{4} \cdot a\right)\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, x, b \cdot \color{blue}{\left(\frac{-1}{4} \cdot a\right)}\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(y, x, \color{blue}{\left(b \cdot \frac{-1}{4}\right) \cdot a}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, x, \color{blue}{\left(b \cdot \frac{-1}{4}\right)} \cdot a\right) \]

      10. lower-*.f64 66.7

          \[\leadsto \mathsf{fma}\left(y, x, \color{blue}{\left(b \cdot -0.25\right) \cdot a}\right) \]

   10. Applied rewrites 66.7%

       \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, \left(b \cdot -0.25\right) \cdot a\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot y + \frac{t \cdot z}{16}\right) - \frac{a \cdot b}{4} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(t \cdot z, 0.0625, \mathsf{fma}\left(x, y, \mathsf{fma}\left(a, b \cdot -0.25, c\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, a \cdot \left(b \cdot -0.25\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(t \cdot 0.0625, z, x \cdot y\right)\\ t_2 := x \cdot y + \frac{t \cdot z}{16}\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+73}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+21}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot -0.25, a, c\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (fma (* t 0.0625) z (* x y))) (t_2 (+ (* x y) (/ (* t z) 16.0))))
   (if (<= t_2 -2e+73) t_1 (if (<= t_2 2e+21) (fma (* b -0.25) a c) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = fma((t * 0.0625), z, (x * y));
	double t_2 = (x * y) + ((t * z) / 16.0);
	double tmp;
	if (t_2 <= -2e+73) {
		tmp = t_1;
	} else if (t_2 <= 2e+21) {
		tmp = fma((b * -0.25), a, c);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = fma(Float64(t * 0.0625), z, Float64(x * y))
	t_2 = Float64(Float64(x * y) + Float64(Float64(t * z) / 16.0))
	tmp = 0.0
	if (t_2 <= -2e+73)
		tmp = t_1;
	elseif (t_2 <= 2e+21)
		tmp = fma(Float64(b * -0.25), a, c);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(t * 0.0625), $MachinePrecision] * z + N[(x * y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * y), $MachinePrecision] + N[(N[(t * z), $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+73], t$95$1, If[LessEqual[t$95$2, 2e+21], N[(N[(b * -0.25), $MachinePrecision] * a + c), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 x y) (/.f64 (*.f64 z t) #s(literal 16 binary64))) < -1.99999999999999997e73 or 2e21 < (+.f64 (*.f64 x y) (/.f64 (*.f64 z t) #s(literal 16 binary64)))

   1. Initial program 96.5%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{c + \left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right)} \]
   4. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(c + \frac{1}{16} \cdot \left(t \cdot z\right)\right) + x \cdot y} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{16} \cdot \left(t \cdot z\right) + c\right)} + x \cdot y \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{\frac{1}{16} \cdot \left(t \cdot z\right) + \left(c + x \cdot y\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{16}, t \cdot z, c + x \cdot y\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{16}, \color{blue}{t \cdot z}, c + x \cdot y\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{16}, t \cdot z, \color{blue}{x \cdot y + c}\right) \]

      7. lower-fma.f64 81.5

         \[\leadsto \mathsf{fma}\left(0.0625, t \cdot z, \color{blue}{\mathsf{fma}\left(x, y, c\right)}\right) \]

   5. Applied rewrites 81.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0625, t \cdot z, \mathsf{fma}\left(x, y, c\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{16} \cdot t\right) \cdot z} + \left(x \cdot y + c\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(t \cdot \frac{1}{16}\right)} \cdot z + \left(x \cdot y + c\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(t \cdot \frac{1}{16}\right)} \cdot z + \left(x \cdot y + c\right) \]

      4. lift-fma.f64 N/A

         \[\leadsto \left(t \cdot \frac{1}{16}\right) \cdot z + \color{blue}{\mathsf{fma}\left(x, y, c\right)} \]

      5. lower-fma.f64 82.7

         \[\leadsto \color{blue}{\mathsf{fma}\left(t \cdot 0.0625, z, \mathsf{fma}\left(x, y, c\right)\right)} \]

   7. Applied rewrites 82.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t \cdot 0.0625, z, \mathsf{fma}\left(x, y, c\right)\right)} \]

   8. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(t \cdot \frac{1}{16}, z, \color{blue}{x \cdot y}\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 74.3

         \[\leadsto \mathsf{fma}\left(t \cdot 0.0625, z, \color{blue}{x \cdot y}\right) \]

   10. Applied rewrites 74.3%

       \[\leadsto \mathsf{fma}\left(t \cdot 0.0625, z, \color{blue}{x \cdot y}\right) \]

   if -1.99999999999999997e73 < (+.f64 (*.f64 x y) (/.f64 (*.f64 z t) #s(literal 16 binary64))) < 2e21

   1. Initial program 100.0%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(a \cdot b\right) \cdot \frac{-1}{4}} + c \]

      2. associate-*l* N/A

         \[\leadsto \color{blue}{a \cdot \left(b \cdot \frac{-1}{4}\right)} + c \]

      3. *-commutative N/A

         \[\leadsto a \cdot \color{blue}{\left(\frac{-1}{4} \cdot b\right)} + c \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{a \cdot \left(\frac{-1}{4} \cdot b\right)} + c \]

      5. *-commutative N/A

         \[\leadsto a \cdot \color{blue}{\left(b \cdot \frac{-1}{4}\right)} + c \]

      6. lower-*.f64 85.7

         \[\leadsto a \cdot \color{blue}{\left(b \cdot -0.25\right)} + c \]

   5. Applied rewrites 85.7%

      \[\leadsto \color{blue}{a \cdot \left(b \cdot -0.25\right)} + c \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto a \cdot \color{blue}{\left(b \cdot \frac{-1}{4}\right)} + c \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(b \cdot \frac{-1}{4}\right) \cdot a} + c \]

      3. lower-fma.f64 85.7

         \[\leadsto \color{blue}{\mathsf{fma}\left(b \cdot -0.25, a, c\right)} \]

   7. Applied rewrites 85.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b \cdot -0.25, a, c\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y + \frac{t \cdot z}{16} \leq -2 \cdot 10^{+73}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot 0.0625, z, x \cdot y\right)\\ \mathbf{elif}\;x \cdot y + \frac{t \cdot z}{16} \leq 2 \cdot 10^{+21}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot -0.25, a, c\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot 0.0625, z, x \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(0.0625, t \cdot z, x \cdot y\right)\\ t_2 := x \cdot y + \frac{t \cdot z}{16}\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+73}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+21}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot -0.25, a, c\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (fma 0.0625 (* t z) (* x y))) (t_2 (+ (* x y) (/ (* t z) 16.0))))
   (if (<= t_2 -2e+73) t_1 (if (<= t_2 2e+21) (fma (* b -0.25) a c) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = fma(0.0625, (t * z), (x * y));
	double t_2 = (x * y) + ((t * z) / 16.0);
	double tmp;
	if (t_2 <= -2e+73) {
		tmp = t_1;
	} else if (t_2 <= 2e+21) {
		tmp = fma((b * -0.25), a, c);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = fma(0.0625, Float64(t * z), Float64(x * y))
	t_2 = Float64(Float64(x * y) + Float64(Float64(t * z) / 16.0))
	tmp = 0.0
	if (t_2 <= -2e+73)
		tmp = t_1;
	elseif (t_2 <= 2e+21)
		tmp = fma(Float64(b * -0.25), a, c);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(0.0625 * N[(t * z), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * y), $MachinePrecision] + N[(N[(t * z), $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+73], t$95$1, If[LessEqual[t$95$2, 2e+21], N[(N[(b * -0.25), $MachinePrecision] * a + c), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 x y) (/.f64 (*.f64 z t) #s(literal 16 binary64))) < -1.99999999999999997e73 or 2e21 < (+.f64 (*.f64 x y) (/.f64 (*.f64 z t) #s(literal 16 binary64)))

   1. Initial program 96.5%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{c + \left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right)} \]
   4. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(c + \frac{1}{16} \cdot \left(t \cdot z\right)\right) + x \cdot y} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{16} \cdot \left(t \cdot z\right) + c\right)} + x \cdot y \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{\frac{1}{16} \cdot \left(t \cdot z\right) + \left(c + x \cdot y\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{16}, t \cdot z, c + x \cdot y\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{16}, \color{blue}{t \cdot z}, c + x \cdot y\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{16}, t \cdot z, \color{blue}{x \cdot y + c}\right) \]

      7. lower-fma.f64 81.5

         \[\leadsto \mathsf{fma}\left(0.0625, t \cdot z, \color{blue}{\mathsf{fma}\left(x, y, c\right)}\right) \]

   5. Applied rewrites 81.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0625, t \cdot z, \mathsf{fma}\left(x, y, c\right)\right)} \]

   6. Taylor expanded in c around 0

      \[\leadsto \color{blue}{\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y} \]
   7. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{16}, t \cdot z, x \cdot y\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{16}, \color{blue}{t \cdot z}, x \cdot y\right) \]

      3. lower-*.f64 73.1

         \[\leadsto \mathsf{fma}\left(0.0625, t \cdot z, \color{blue}{x \cdot y}\right) \]

   8. Applied rewrites 73.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0625, t \cdot z, x \cdot y\right)} \]

   if -1.99999999999999997e73 < (+.f64 (*.f64 x y) (/.f64 (*.f64 z t) #s(literal 16 binary64))) < 2e21

   1. Initial program 100.0%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(a \cdot b\right) \cdot \frac{-1}{4}} + c \]

      2. associate-*l* N/A

         \[\leadsto \color{blue}{a \cdot \left(b \cdot \frac{-1}{4}\right)} + c \]

      3. *-commutative N/A

         \[\leadsto a \cdot \color{blue}{\left(\frac{-1}{4} \cdot b\right)} + c \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{a \cdot \left(\frac{-1}{4} \cdot b\right)} + c \]

      5. *-commutative N/A

         \[\leadsto a \cdot \color{blue}{\left(b \cdot \frac{-1}{4}\right)} + c \]

      6. lower-*.f64 85.7

         \[\leadsto a \cdot \color{blue}{\left(b \cdot -0.25\right)} + c \]

   5. Applied rewrites 85.7%

      \[\leadsto \color{blue}{a \cdot \left(b \cdot -0.25\right)} + c \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto a \cdot \color{blue}{\left(b \cdot \frac{-1}{4}\right)} + c \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(b \cdot \frac{-1}{4}\right) \cdot a} + c \]

      3. lower-fma.f64 85.7

         \[\leadsto \color{blue}{\mathsf{fma}\left(b \cdot -0.25, a, c\right)} \]

   7. Applied rewrites 85.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b \cdot -0.25, a, c\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y + \frac{t \cdot z}{16} \leq -2 \cdot 10^{+73}:\\ \;\;\;\;\mathsf{fma}\left(0.0625, t \cdot z, x \cdot y\right)\\ \mathbf{elif}\;x \cdot y + \frac{t \cdot z}{16} \leq 2 \cdot 10^{+21}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot -0.25, a, c\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.0625, t \cdot z, x \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 64.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(b \cdot -0.25, a, c\right)\\ \mathbf{if}\;a \cdot b \leq -1 \cdot 10^{+192}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \cdot b \leq 0.0004:\\ \;\;\;\;\mathsf{fma}\left(t \cdot 0.0625, z, c\right)\\ \mathbf{elif}\;a \cdot b \leq 5 \cdot 10^{+108}:\\ \;\;\;\;\mathsf{fma}\left(x, y, c\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (fma (* b -0.25) a c)))
   (if (<= (* a b) -1e+192)
     t_1
     (if (<= (* a b) 0.0004)
       (fma (* t 0.0625) z c)
       (if (<= (* a b) 5e+108) (fma x y c) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = fma((b * -0.25), a, c);
	double tmp;
	if ((a * b) <= -1e+192) {
		tmp = t_1;
	} else if ((a * b) <= 0.0004) {
		tmp = fma((t * 0.0625), z, c);
	} else if ((a * b) <= 5e+108) {
		tmp = fma(x, y, c);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = fma(Float64(b * -0.25), a, c)
	tmp = 0.0
	if (Float64(a * b) <= -1e+192)
		tmp = t_1;
	elseif (Float64(a * b) <= 0.0004)
		tmp = fma(Float64(t * 0.0625), z, c);
	elseif (Float64(a * b) <= 5e+108)
		tmp = fma(x, y, c);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(b * -0.25), $MachinePrecision] * a + c), $MachinePrecision]}, If[LessEqual[N[(a * b), $MachinePrecision], -1e+192], t$95$1, If[LessEqual[N[(a * b), $MachinePrecision], 0.0004], N[(N[(t * 0.0625), $MachinePrecision] * z + c), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], 5e+108], N[(x * y + c), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 a b) < -1.00000000000000004e192 or 4.99999999999999991e108 < (*.f64 a b)

   1. Initial program 95.8%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(a \cdot b\right) \cdot \frac{-1}{4}} + c \]

      2. associate-*l* N/A

         \[\leadsto \color{blue}{a \cdot \left(b \cdot \frac{-1}{4}\right)} + c \]

      3. *-commutative N/A

         \[\leadsto a \cdot \color{blue}{\left(\frac{-1}{4} \cdot b\right)} + c \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{a \cdot \left(\frac{-1}{4} \cdot b\right)} + c \]

      5. *-commutative N/A

         \[\leadsto a \cdot \color{blue}{\left(b \cdot \frac{-1}{4}\right)} + c \]

      6. lower-*.f64 82.7

         \[\leadsto a \cdot \color{blue}{\left(b \cdot -0.25\right)} + c \]

   5. Applied rewrites 82.7%

      \[\leadsto \color{blue}{a \cdot \left(b \cdot -0.25\right)} + c \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto a \cdot \color{blue}{\left(b \cdot \frac{-1}{4}\right)} + c \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(b \cdot \frac{-1}{4}\right) \cdot a} + c \]

      3. lower-fma.f64 82.7

         \[\leadsto \color{blue}{\mathsf{fma}\left(b \cdot -0.25, a, c\right)} \]

   7. Applied rewrites 82.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b \cdot -0.25, a, c\right)} \]

   if -1.00000000000000004e192 < (*.f64 a b) < 4.00000000000000019e-4

   1. Initial program 98.1%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{1}{16} \cdot \left(t \cdot z\right)} + c \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{16} \cdot \left(t \cdot z\right)} + c \]

      2. lower-*.f64 65.7

         \[\leadsto 0.0625 \cdot \color{blue}{\left(t \cdot z\right)} + c \]

   5. Applied rewrites 65.7%

      \[\leadsto \color{blue}{0.0625 \cdot \left(t \cdot z\right)} + c \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{16} \cdot t\right) \cdot z} + c \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(t \cdot \frac{1}{16}\right)} \cdot z + c \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(t \cdot \frac{1}{16}\right)} \cdot z + c \]

      4. lower-fma.f64 65.7

         \[\leadsto \color{blue}{\mathsf{fma}\left(t \cdot 0.0625, z, c\right)} \]

   7. Applied rewrites 65.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t \cdot 0.0625, z, c\right)} \]

   if 4.00000000000000019e-4 < (*.f64 a b) < 4.99999999999999991e108

   1. Initial program 100.0%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{c + \left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right)} \]
   4. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(c + \frac{1}{16} \cdot \left(t \cdot z\right)\right) + x \cdot y} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{16} \cdot \left(t \cdot z\right) + c\right)} + x \cdot y \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{\frac{1}{16} \cdot \left(t \cdot z\right) + \left(c + x \cdot y\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{16}, t \cdot z, c + x \cdot y\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{16}, \color{blue}{t \cdot z}, c + x \cdot y\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{16}, t \cdot z, \color{blue}{x \cdot y + c}\right) \]

      7. lower-fma.f64 85.1

         \[\leadsto \mathsf{fma}\left(0.0625, t \cdot z, \color{blue}{\mathsf{fma}\left(x, y, c\right)}\right) \]

   5. Applied rewrites 85.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0625, t \cdot z, \mathsf{fma}\left(x, y, c\right)\right)} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{c + x \cdot y} \]
   7. Step-by-step derivation

      1. +-commutative N/A

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

      2. lower-fma.f64 72.4

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

   8. Applied rewrites 72.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, c\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 89.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, x, \mathsf{fma}\left(a, b \cdot -0.25, c\right)\right)\\ \mathbf{if}\;a \cdot b \leq -1 \cdot 10^{+192}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \cdot b \leq 10^{+18}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot 0.0625, z, \mathsf{fma}\left(x, y, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (fma y x (fma a (* b -0.25) c))))
   (if (<= (* a b) -1e+192)
     t_1
     (if (<= (* a b) 1e+18) (fma (* t 0.0625) z (fma x y c)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = fma(y, x, fma(a, (b * -0.25), c));
	double tmp;
	if ((a * b) <= -1e+192) {
		tmp = t_1;
	} else if ((a * b) <= 1e+18) {
		tmp = fma((t * 0.0625), z, fma(x, y, c));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = fma(y, x, fma(a, Float64(b * -0.25), c))
	tmp = 0.0
	if (Float64(a * b) <= -1e+192)
		tmp = t_1;
	elseif (Float64(a * b) <= 1e+18)
		tmp = fma(Float64(t * 0.0625), z, fma(x, y, c));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(y * x + N[(a * N[(b * -0.25), $MachinePrecision] + c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(a * b), $MachinePrecision], -1e+192], t$95$1, If[LessEqual[N[(a * b), $MachinePrecision], 1e+18], N[(N[(t * 0.0625), $MachinePrecision] * z + N[(x * y + c), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a b) < -1.00000000000000004e192 or 1e18 < (*.f64 a b)

   1. Initial program 96.7%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{\left(c + x \cdot y\right) + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot \left(a \cdot b\right)} \]

      2. metadata-eval N/A

         \[\leadsto \left(c + x \cdot y\right) + \color{blue}{\frac{-1}{4}} \cdot \left(a \cdot b\right) \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\frac{-1}{4} \cdot \left(a \cdot b\right) + \left(c + x \cdot y\right)} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(a \cdot b\right) \cdot \frac{-1}{4}} + \left(c + x \cdot y\right) \]

      5. associate-*l* N/A

         \[\leadsto \color{blue}{a \cdot \left(b \cdot \frac{-1}{4}\right)} + \left(c + x \cdot y\right) \]

      6. *-commutative N/A

         \[\leadsto a \cdot \color{blue}{\left(\frac{-1}{4} \cdot b\right)} + \left(c + x \cdot y\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a, \frac{-1}{4} \cdot b, c + x \cdot y\right)} \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{b \cdot \frac{-1}{4}}, c + x \cdot y\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{b \cdot \frac{-1}{4}}, c + x \cdot y\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(a, b \cdot \frac{-1}{4}, \color{blue}{x \cdot y + c}\right) \]

      11. lower-fma.f64 91.5

          \[\leadsto \mathsf{fma}\left(a, b \cdot -0.25, \color{blue}{\mathsf{fma}\left(x, y, c\right)}\right) \]

   5. Applied rewrites 91.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, b \cdot -0.25, \mathsf{fma}\left(x, y, c\right)\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto a \cdot \color{blue}{\left(b \cdot \frac{-1}{4}\right)} + \left(x \cdot y + c\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{a \cdot \left(b \cdot \frac{-1}{4}\right)} + \left(x \cdot y + c\right) \]

      3. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(a \cdot \left(b \cdot \frac{-1}{4}\right) + x \cdot y\right) + c} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot y + a \cdot \left(b \cdot \frac{-1}{4}\right)\right)} + c \]

      5. associate-+r+ N/A

         \[\leadsto \color{blue}{x \cdot y + \left(a \cdot \left(b \cdot \frac{-1}{4}\right) + c\right)} \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{y \cdot x} + \left(a \cdot \left(b \cdot \frac{-1}{4}\right) + c\right) \]

      7. lift-*.f64 N/A

         \[\leadsto y \cdot x + \left(\color{blue}{a \cdot \left(b \cdot \frac{-1}{4}\right)} + c\right) \]

      8. lift-fma.f64 N/A

         \[\leadsto y \cdot x + \color{blue}{\mathsf{fma}\left(a, b \cdot \frac{-1}{4}, c\right)} \]

      9. lower-fma.f64 92.6

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, \mathsf{fma}\left(a, b \cdot -0.25, c\right)\right)} \]

   7. Applied rewrites 92.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, \mathsf{fma}\left(a, b \cdot -0.25, c\right)\right)} \]

   if -1.00000000000000004e192 < (*.f64 a b) < 1e18

   1. Initial program 98.2%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{c + \left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right)} \]
   4. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(c + \frac{1}{16} \cdot \left(t \cdot z\right)\right) + x \cdot y} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{16} \cdot \left(t \cdot z\right) + c\right)} + x \cdot y \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{\frac{1}{16} \cdot \left(t \cdot z\right) + \left(c + x \cdot y\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{16}, t \cdot z, c + x \cdot y\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{16}, \color{blue}{t \cdot z}, c + x \cdot y\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{16}, t \cdot z, \color{blue}{x \cdot y + c}\right) \]

      7. lower-fma.f64 92.5

         \[\leadsto \mathsf{fma}\left(0.0625, t \cdot z, \color{blue}{\mathsf{fma}\left(x, y, c\right)}\right) \]

   5. Applied rewrites 92.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0625, t \cdot z, \mathsf{fma}\left(x, y, c\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{16} \cdot t\right) \cdot z} + \left(x \cdot y + c\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(t \cdot \frac{1}{16}\right)} \cdot z + \left(x \cdot y + c\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(t \cdot \frac{1}{16}\right)} \cdot z + \left(x \cdot y + c\right) \]

      4. lift-fma.f64 N/A

         \[\leadsto \left(t \cdot \frac{1}{16}\right) \cdot z + \color{blue}{\mathsf{fma}\left(x, y, c\right)} \]

      5. lower-fma.f64 93.7

         \[\leadsto \color{blue}{\mathsf{fma}\left(t \cdot 0.0625, z, \mathsf{fma}\left(x, y, c\right)\right)} \]

   7. Applied rewrites 93.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t \cdot 0.0625, z, \mathsf{fma}\left(x, y, c\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 88.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, x, \mathsf{fma}\left(a, b \cdot -0.25, c\right)\right)\\ \mathbf{if}\;a \cdot b \leq -1 \cdot 10^{+192}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \cdot b \leq 10^{+18}:\\ \;\;\;\;\mathsf{fma}\left(0.0625, t \cdot z, \mathsf{fma}\left(x, y, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (fma y x (fma a (* b -0.25) c))))
   (if (<= (* a b) -1e+192)
     t_1
     (if (<= (* a b) 1e+18) (fma 0.0625 (* t z) (fma x y c)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = fma(y, x, fma(a, (b * -0.25), c));
	double tmp;
	if ((a * b) <= -1e+192) {
		tmp = t_1;
	} else if ((a * b) <= 1e+18) {
		tmp = fma(0.0625, (t * z), fma(x, y, c));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = fma(y, x, fma(a, Float64(b * -0.25), c))
	tmp = 0.0
	if (Float64(a * b) <= -1e+192)
		tmp = t_1;
	elseif (Float64(a * b) <= 1e+18)
		tmp = fma(0.0625, Float64(t * z), fma(x, y, c));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(y * x + N[(a * N[(b * -0.25), $MachinePrecision] + c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(a * b), $MachinePrecision], -1e+192], t$95$1, If[LessEqual[N[(a * b), $MachinePrecision], 1e+18], N[(0.0625 * N[(t * z), $MachinePrecision] + N[(x * y + c), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a b) < -1.00000000000000004e192 or 1e18 < (*.f64 a b)

   1. Initial program 96.7%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{\left(c + x \cdot y\right) + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot \left(a \cdot b\right)} \]

      2. metadata-eval N/A

         \[\leadsto \left(c + x \cdot y\right) + \color{blue}{\frac{-1}{4}} \cdot \left(a \cdot b\right) \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\frac{-1}{4} \cdot \left(a \cdot b\right) + \left(c + x \cdot y\right)} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(a \cdot b\right) \cdot \frac{-1}{4}} + \left(c + x \cdot y\right) \]

      5. associate-*l* N/A

         \[\leadsto \color{blue}{a \cdot \left(b \cdot \frac{-1}{4}\right)} + \left(c + x \cdot y\right) \]

      6. *-commutative N/A

         \[\leadsto a \cdot \color{blue}{\left(\frac{-1}{4} \cdot b\right)} + \left(c + x \cdot y\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a, \frac{-1}{4} \cdot b, c + x \cdot y\right)} \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{b \cdot \frac{-1}{4}}, c + x \cdot y\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{b \cdot \frac{-1}{4}}, c + x \cdot y\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(a, b \cdot \frac{-1}{4}, \color{blue}{x \cdot y + c}\right) \]

      11. lower-fma.f64 91.5

          \[\leadsto \mathsf{fma}\left(a, b \cdot -0.25, \color{blue}{\mathsf{fma}\left(x, y, c\right)}\right) \]

   5. Applied rewrites 91.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, b \cdot -0.25, \mathsf{fma}\left(x, y, c\right)\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto a \cdot \color{blue}{\left(b \cdot \frac{-1}{4}\right)} + \left(x \cdot y + c\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{a \cdot \left(b \cdot \frac{-1}{4}\right)} + \left(x \cdot y + c\right) \]

      3. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(a \cdot \left(b \cdot \frac{-1}{4}\right) + x \cdot y\right) + c} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot y + a \cdot \left(b \cdot \frac{-1}{4}\right)\right)} + c \]

      5. associate-+r+ N/A

         \[\leadsto \color{blue}{x \cdot y + \left(a \cdot \left(b \cdot \frac{-1}{4}\right) + c\right)} \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{y \cdot x} + \left(a \cdot \left(b \cdot \frac{-1}{4}\right) + c\right) \]

      7. lift-*.f64 N/A

         \[\leadsto y \cdot x + \left(\color{blue}{a \cdot \left(b \cdot \frac{-1}{4}\right)} + c\right) \]

      8. lift-fma.f64 N/A

         \[\leadsto y \cdot x + \color{blue}{\mathsf{fma}\left(a, b \cdot \frac{-1}{4}, c\right)} \]

      9. lower-fma.f64 92.6

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, \mathsf{fma}\left(a, b \cdot -0.25, c\right)\right)} \]

   7. Applied rewrites 92.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, \mathsf{fma}\left(a, b \cdot -0.25, c\right)\right)} \]

   if -1.00000000000000004e192 < (*.f64 a b) < 1e18

   1. Initial program 98.2%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{c + \left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right)} \]
   4. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(c + \frac{1}{16} \cdot \left(t \cdot z\right)\right) + x \cdot y} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{16} \cdot \left(t \cdot z\right) + c\right)} + x \cdot y \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{\frac{1}{16} \cdot \left(t \cdot z\right) + \left(c + x \cdot y\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{16}, t \cdot z, c + x \cdot y\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{16}, \color{blue}{t \cdot z}, c + x \cdot y\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{16}, t \cdot z, \color{blue}{x \cdot y + c}\right) \]

      7. lower-fma.f64 92.5

         \[\leadsto \mathsf{fma}\left(0.0625, t \cdot z, \color{blue}{\mathsf{fma}\left(x, y, c\right)}\right) \]

   5. Applied rewrites 92.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0625, t \cdot z, \mathsf{fma}\left(x, y, c\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 88.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(a, b \cdot -0.25, \mathsf{fma}\left(x, y, c\right)\right)\\ \mathbf{if}\;a \cdot b \leq -1 \cdot 10^{+192}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \cdot b \leq 10^{+18}:\\ \;\;\;\;\mathsf{fma}\left(0.0625, t \cdot z, \mathsf{fma}\left(x, y, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (fma a (* b -0.25) (fma x y c))))
   (if (<= (* a b) -1e+192)
     t_1
     (if (<= (* a b) 1e+18) (fma 0.0625 (* t z) (fma x y c)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = fma(a, (b * -0.25), fma(x, y, c));
	double tmp;
	if ((a * b) <= -1e+192) {
		tmp = t_1;
	} else if ((a * b) <= 1e+18) {
		tmp = fma(0.0625, (t * z), fma(x, y, c));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = fma(a, Float64(b * -0.25), fma(x, y, c))
	tmp = 0.0
	if (Float64(a * b) <= -1e+192)
		tmp = t_1;
	elseif (Float64(a * b) <= 1e+18)
		tmp = fma(0.0625, Float64(t * z), fma(x, y, c));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(a * N[(b * -0.25), $MachinePrecision] + N[(x * y + c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(a * b), $MachinePrecision], -1e+192], t$95$1, If[LessEqual[N[(a * b), $MachinePrecision], 1e+18], N[(0.0625 * N[(t * z), $MachinePrecision] + N[(x * y + c), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a b) < -1.00000000000000004e192 or 1e18 < (*.f64 a b)

   1. Initial program 96.7%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{\left(c + x \cdot y\right) + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot \left(a \cdot b\right)} \]

      2. metadata-eval N/A

         \[\leadsto \left(c + x \cdot y\right) + \color{blue}{\frac{-1}{4}} \cdot \left(a \cdot b\right) \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\frac{-1}{4} \cdot \left(a \cdot b\right) + \left(c + x \cdot y\right)} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(a \cdot b\right) \cdot \frac{-1}{4}} + \left(c + x \cdot y\right) \]

      5. associate-*l* N/A

         \[\leadsto \color{blue}{a \cdot \left(b \cdot \frac{-1}{4}\right)} + \left(c + x \cdot y\right) \]

      6. *-commutative N/A

         \[\leadsto a \cdot \color{blue}{\left(\frac{-1}{4} \cdot b\right)} + \left(c + x \cdot y\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a, \frac{-1}{4} \cdot b, c + x \cdot y\right)} \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{b \cdot \frac{-1}{4}}, c + x \cdot y\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{b \cdot \frac{-1}{4}}, c + x \cdot y\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(a, b \cdot \frac{-1}{4}, \color{blue}{x \cdot y + c}\right) \]

      11. lower-fma.f64 91.5

          \[\leadsto \mathsf{fma}\left(a, b \cdot -0.25, \color{blue}{\mathsf{fma}\left(x, y, c\right)}\right) \]

   5. Applied rewrites 91.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, b \cdot -0.25, \mathsf{fma}\left(x, y, c\right)\right)} \]

   if -1.00000000000000004e192 < (*.f64 a b) < 1e18

   1. Initial program 98.2%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{c + \left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right)} \]
   4. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(c + \frac{1}{16} \cdot \left(t \cdot z\right)\right) + x \cdot y} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{16} \cdot \left(t \cdot z\right) + c\right)} + x \cdot y \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{\frac{1}{16} \cdot \left(t \cdot z\right) + \left(c + x \cdot y\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{16}, t \cdot z, c + x \cdot y\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{16}, \color{blue}{t \cdot z}, c + x \cdot y\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{16}, t \cdot z, \color{blue}{x \cdot y + c}\right) \]

      7. lower-fma.f64 92.5

         \[\leadsto \mathsf{fma}\left(0.0625, t \cdot z, \color{blue}{\mathsf{fma}\left(x, y, c\right)}\right) \]

   5. Applied rewrites 92.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0625, t \cdot z, \mathsf{fma}\left(x, y, c\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 86.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, x, a \cdot \left(b \cdot -0.25\right)\right)\\ \mathbf{if}\;a \cdot b \leq -1 \cdot 10^{+192}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \cdot b \leq 2 \cdot 10^{+87}:\\ \;\;\;\;\mathsf{fma}\left(0.0625, t \cdot z, \mathsf{fma}\left(x, y, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (fma y x (* a (* b -0.25)))))
   (if (<= (* a b) -1e+192)
     t_1
     (if (<= (* a b) 2e+87) (fma 0.0625 (* t z) (fma x y c)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = fma(y, x, (a * (b * -0.25)));
	double tmp;
	if ((a * b) <= -1e+192) {
		tmp = t_1;
	} else if ((a * b) <= 2e+87) {
		tmp = fma(0.0625, (t * z), fma(x, y, c));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = fma(y, x, Float64(a * Float64(b * -0.25)))
	tmp = 0.0
	if (Float64(a * b) <= -1e+192)
		tmp = t_1;
	elseif (Float64(a * b) <= 2e+87)
		tmp = fma(0.0625, Float64(t * z), fma(x, y, c));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(y * x + N[(a * N[(b * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(a * b), $MachinePrecision], -1e+192], t$95$1, If[LessEqual[N[(a * b), $MachinePrecision], 2e+87], N[(0.0625 * N[(t * z), $MachinePrecision] + N[(x * y + c), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a b) < -1.00000000000000004e192 or 1.9999999999999999e87 < (*.f64 a b)

   1. Initial program 96.0%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{\left(c + x \cdot y\right) + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot \left(a \cdot b\right)} \]

      2. metadata-eval N/A

         \[\leadsto \left(c + x \cdot y\right) + \color{blue}{\frac{-1}{4}} \cdot \left(a \cdot b\right) \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\frac{-1}{4} \cdot \left(a \cdot b\right) + \left(c + x \cdot y\right)} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(a \cdot b\right) \cdot \frac{-1}{4}} + \left(c + x \cdot y\right) \]

      5. associate-*l* N/A

         \[\leadsto \color{blue}{a \cdot \left(b \cdot \frac{-1}{4}\right)} + \left(c + x \cdot y\right) \]

      6. *-commutative N/A

         \[\leadsto a \cdot \color{blue}{\left(\frac{-1}{4} \cdot b\right)} + \left(c + x \cdot y\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a, \frac{-1}{4} \cdot b, c + x \cdot y\right)} \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{b \cdot \frac{-1}{4}}, c + x \cdot y\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{b \cdot \frac{-1}{4}}, c + x \cdot y\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(a, b \cdot \frac{-1}{4}, \color{blue}{x \cdot y + c}\right) \]

      11. lower-fma.f64 92.3

          \[\leadsto \mathsf{fma}\left(a, b \cdot -0.25, \color{blue}{\mathsf{fma}\left(x, y, c\right)}\right) \]

   5. Applied rewrites 92.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, b \cdot -0.25, \mathsf{fma}\left(x, y, c\right)\right)} \]

   6. Taylor expanded in c around 0

      \[\leadsto \color{blue}{\frac{-1}{4} \cdot \left(a \cdot b\right) + x \cdot y} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{4} \cdot a\right) \cdot b} + x \cdot y \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{b \cdot \left(\frac{-1}{4} \cdot a\right)} + x \cdot y \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(b, \frac{-1}{4} \cdot a, x \cdot y\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(b, \color{blue}{\frac{-1}{4} \cdot a}, x \cdot y\right) \]

      5. lower-*.f64 89.4

         \[\leadsto \mathsf{fma}\left(b, -0.25 \cdot a, \color{blue}{x \cdot y}\right) \]

   8. Applied rewrites 89.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b, -0.25 \cdot a, x \cdot y\right)} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto b \cdot \color{blue}{\left(\frac{-1}{4} \cdot a\right)} + x \cdot y \]

      2. lift-*.f64 N/A

         \[\leadsto b \cdot \left(\frac{-1}{4} \cdot a\right) + \color{blue}{x \cdot y} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{x \cdot y + b \cdot \left(\frac{-1}{4} \cdot a\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot y} + b \cdot \left(\frac{-1}{4} \cdot a\right) \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{y \cdot x} + b \cdot \left(\frac{-1}{4} \cdot a\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, b \cdot \left(\frac{-1}{4} \cdot a\right)\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, x, b \cdot \color{blue}{\left(\frac{-1}{4} \cdot a\right)}\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(y, x, \color{blue}{\left(b \cdot \frac{-1}{4}\right) \cdot a}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, x, \color{blue}{\left(b \cdot \frac{-1}{4}\right)} \cdot a\right) \]

      10. lower-*.f64 90.8

          \[\leadsto \mathsf{fma}\left(y, x, \color{blue}{\left(b \cdot -0.25\right) \cdot a}\right) \]

   10. Applied rewrites 90.8%

       \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, \left(b \cdot -0.25\right) \cdot a\right)} \]

   if -1.00000000000000004e192 < (*.f64 a b) < 1.9999999999999999e87

   1. Initial program 98.3%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{c + \left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right)} \]
   4. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(c + \frac{1}{16} \cdot \left(t \cdot z\right)\right) + x \cdot y} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{16} \cdot \left(t \cdot z\right) + c\right)} + x \cdot y \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{\frac{1}{16} \cdot \left(t \cdot z\right) + \left(c + x \cdot y\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{16}, t \cdot z, c + x \cdot y\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{16}, \color{blue}{t \cdot z}, c + x \cdot y\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{16}, t \cdot z, \color{blue}{x \cdot y + c}\right) \]

      7. lower-fma.f64 91.5

         \[\leadsto \mathsf{fma}\left(0.0625, t \cdot z, \color{blue}{\mathsf{fma}\left(x, y, c\right)}\right) \]

   5. Applied rewrites 91.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0625, t \cdot z, \mathsf{fma}\left(x, y, c\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b \leq -1 \cdot 10^{+192}:\\ \;\;\;\;\mathsf{fma}\left(y, x, a \cdot \left(b \cdot -0.25\right)\right)\\ \mathbf{elif}\;a \cdot b \leq 2 \cdot 10^{+87}:\\ \;\;\;\;\mathsf{fma}\left(0.0625, t \cdot z, \mathsf{fma}\left(x, y, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, a \cdot \left(b \cdot -0.25\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 65.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{+86}:\\ \;\;\;\;\mathsf{fma}\left(x, y, c\right)\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+69}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot -0.25, a, c\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, y, c\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (<= (* x y) -5e+86)
   (fma x y c)
   (if (<= (* x y) 5e+69) (fma (* b -0.25) a c) (fma x y c))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if ((x * y) <= -5e+86) {
		tmp = fma(x, y, c);
	} else if ((x * y) <= 5e+69) {
		tmp = fma((b * -0.25), a, c);
	} else {
		tmp = fma(x, y, c);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if (Float64(x * y) <= -5e+86)
		tmp = fma(x, y, c);
	elseif (Float64(x * y) <= 5e+69)
		tmp = fma(Float64(b * -0.25), a, c);
	else
		tmp = fma(x, y, c);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[LessEqual[N[(x * y), $MachinePrecision], -5e+86], N[(x * y + c), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 5e+69], N[(N[(b * -0.25), $MachinePrecision] * a + c), $MachinePrecision], N[(x * y + c), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -4.9999999999999998e86 or 5.00000000000000036e69 < (*.f64 x y)

   1. Initial program 94.9%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{c + \left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right)} \]
   4. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(c + \frac{1}{16} \cdot \left(t \cdot z\right)\right) + x \cdot y} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{16} \cdot \left(t \cdot z\right) + c\right)} + x \cdot y \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{\frac{1}{16} \cdot \left(t \cdot z\right) + \left(c + x \cdot y\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{16}, t \cdot z, c + x \cdot y\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{16}, \color{blue}{t \cdot z}, c + x \cdot y\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{16}, t \cdot z, \color{blue}{x \cdot y + c}\right) \]

      7. lower-fma.f64 80.5

         \[\leadsto \mathsf{fma}\left(0.0625, t \cdot z, \color{blue}{\mathsf{fma}\left(x, y, c\right)}\right) \]

   5. Applied rewrites 80.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0625, t \cdot z, \mathsf{fma}\left(x, y, c\right)\right)} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{c + x \cdot y} \]
   7. Step-by-step derivation

      1. +-commutative N/A

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

      2. lower-fma.f64 68.5

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

   8. Applied rewrites 68.5%

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

   if -4.9999999999999998e86 < (*.f64 x y) < 5.00000000000000036e69

   1. Initial program 99.3%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(a \cdot b\right) \cdot \frac{-1}{4}} + c \]

      2. associate-*l* N/A

         \[\leadsto \color{blue}{a \cdot \left(b \cdot \frac{-1}{4}\right)} + c \]

      3. *-commutative N/A

         \[\leadsto a \cdot \color{blue}{\left(\frac{-1}{4} \cdot b\right)} + c \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{a \cdot \left(\frac{-1}{4} \cdot b\right)} + c \]

      5. *-commutative N/A

         \[\leadsto a \cdot \color{blue}{\left(b \cdot \frac{-1}{4}\right)} + c \]

      6. lower-*.f64 59.8

         \[\leadsto a \cdot \color{blue}{\left(b \cdot -0.25\right)} + c \]

   5. Applied rewrites 59.8%

      \[\leadsto \color{blue}{a \cdot \left(b \cdot -0.25\right)} + c \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto a \cdot \color{blue}{\left(b \cdot \frac{-1}{4}\right)} + c \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(b \cdot \frac{-1}{4}\right) \cdot a} + c \]

      3. lower-fma.f64 59.8

         \[\leadsto \color{blue}{\mathsf{fma}\left(b \cdot -0.25, a, c\right)} \]

   7. Applied rewrites 59.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b \cdot -0.25, a, c\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 62.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot \left(b \cdot -0.25\right)\\ \mathbf{if}\;a \cdot b \leq -1 \cdot 10^{+162}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \cdot b \leq 5 \cdot 10^{+108}:\\ \;\;\;\;\mathsf{fma}\left(x, y, c\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (* a (* b -0.25))))
   (if (<= (* a b) -1e+162) t_1 (if (<= (* a b) 5e+108) (fma x y c) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = a * (b * -0.25);
	double tmp;
	if ((a * b) <= -1e+162) {
		tmp = t_1;
	} else if ((a * b) <= 5e+108) {
		tmp = fma(x, y, c);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(a * Float64(b * -0.25))
	tmp = 0.0
	if (Float64(a * b) <= -1e+162)
		tmp = t_1;
	elseif (Float64(a * b) <= 5e+108)
		tmp = fma(x, y, c);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(a * N[(b * -0.25), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(a * b), $MachinePrecision], -1e+162], t$95$1, If[LessEqual[N[(a * b), $MachinePrecision], 5e+108], N[(x * y + c), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a b) < -9.9999999999999994e161 or 4.99999999999999991e108 < (*.f64 a b)

   1. Initial program 96.0%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(a \cdot b\right) \cdot \frac{-1}{4}} \]

      2. associate-*l* N/A

         \[\leadsto \color{blue}{a \cdot \left(b \cdot \frac{-1}{4}\right)} \]

      3. *-commutative N/A

         \[\leadsto a \cdot \color{blue}{\left(\frac{-1}{4} \cdot b\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{a \cdot \left(\frac{-1}{4} \cdot b\right)} \]

      5. *-commutative N/A

         \[\leadsto a \cdot \color{blue}{\left(b \cdot \frac{-1}{4}\right)} \]

      6. lower-*.f64 78.1

         \[\leadsto a \cdot \color{blue}{\left(b \cdot -0.25\right)} \]

   5. Applied rewrites 78.1%

      \[\leadsto \color{blue}{a \cdot \left(b \cdot -0.25\right)} \]

   if -9.9999999999999994e161 < (*.f64 a b) < 4.99999999999999991e108

   1. Initial program 98.3%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{c + \left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right)} \]
   4. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(c + \frac{1}{16} \cdot \left(t \cdot z\right)\right) + x \cdot y} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{16} \cdot \left(t \cdot z\right) + c\right)} + x \cdot y \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{\frac{1}{16} \cdot \left(t \cdot z\right) + \left(c + x \cdot y\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{16}, t \cdot z, c + x \cdot y\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{16}, \color{blue}{t \cdot z}, c + x \cdot y\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{16}, t \cdot z, \color{blue}{x \cdot y + c}\right) \]

      7. lower-fma.f64 91.7

         \[\leadsto \mathsf{fma}\left(0.0625, t \cdot z, \color{blue}{\mathsf{fma}\left(x, y, c\right)}\right) \]

   5. Applied rewrites 91.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0625, t \cdot z, \mathsf{fma}\left(x, y, c\right)\right)} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{c + x \cdot y} \]
   7. Step-by-step derivation

      1. +-commutative N/A

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

      2. lower-fma.f64 55.5

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

   8. Applied rewrites 55.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, c\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 62.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 0.0625 \cdot \left(t \cdot z\right)\\ \mathbf{if}\;t \cdot z \leq -1 \cdot 10^{+158}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \cdot z \leq 2 \cdot 10^{+90}:\\ \;\;\;\;\mathsf{fma}\left(x, y, c\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (* 0.0625 (* t z))))
   (if (<= (* t z) -1e+158) t_1 (if (<= (* t z) 2e+90) (fma x y c) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = 0.0625 * (t * z);
	double tmp;
	if ((t * z) <= -1e+158) {
		tmp = t_1;
	} else if ((t * z) <= 2e+90) {
		tmp = fma(x, y, c);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(0.0625 * Float64(t * z))
	tmp = 0.0
	if (Float64(t * z) <= -1e+158)
		tmp = t_1;
	elseif (Float64(t * z) <= 2e+90)
		tmp = fma(x, y, c);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(0.0625 * N[(t * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t * z), $MachinePrecision], -1e+158], t$95$1, If[LessEqual[N[(t * z), $MachinePrecision], 2e+90], N[(x * y + c), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -9.99999999999999953e157 or 1.99999999999999993e90 < (*.f64 z t)

   1. Initial program 92.9%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{1}{16} \cdot \left(t \cdot z\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{16} \cdot \left(t \cdot z\right)} \]

      2. lower-*.f64 68.6

         \[\leadsto 0.0625 \cdot \color{blue}{\left(t \cdot z\right)} \]

   5. Applied rewrites 68.6%

      \[\leadsto \color{blue}{0.0625 \cdot \left(t \cdot z\right)} \]

   if -9.99999999999999953e157 < (*.f64 z t) < 1.99999999999999993e90

   1. Initial program 100.0%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{c + \left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right)} \]
   4. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(c + \frac{1}{16} \cdot \left(t \cdot z\right)\right) + x \cdot y} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{16} \cdot \left(t \cdot z\right) + c\right)} + x \cdot y \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{\frac{1}{16} \cdot \left(t \cdot z\right) + \left(c + x \cdot y\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{16}, t \cdot z, c + x \cdot y\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{16}, \color{blue}{t \cdot z}, c + x \cdot y\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{16}, t \cdot z, \color{blue}{x \cdot y + c}\right) \]

      7. lower-fma.f64 67.4

         \[\leadsto \mathsf{fma}\left(0.0625, t \cdot z, \color{blue}{\mathsf{fma}\left(x, y, c\right)}\right) \]

   5. Applied rewrites 67.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0625, t \cdot z, \mathsf{fma}\left(x, y, c\right)\right)} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{c + x \cdot y} \]
   7. Step-by-step derivation

      1. +-commutative N/A

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

      2. lower-fma.f64 56.4

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

   8. Applied rewrites 56.4%

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

4. Final simplification 60.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot z \leq -1 \cdot 10^{+158}:\\ \;\;\;\;0.0625 \cdot \left(t \cdot z\right)\\ \mathbf{elif}\;t \cdot z \leq 2 \cdot 10^{+90}:\\ \;\;\;\;\mathsf{fma}\left(x, y, c\right)\\ \mathbf{else}:\\ \;\;\;\;0.0625 \cdot \left(t \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 49.1% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x, y, c\right) \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 97.6%

   \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
2. Add Preprocessing

3. Taylor expanded in a around 0

   \[\leadsto \color{blue}{c + \left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right)} \]
4. Step-by-step derivation

   1. associate-+r+ N/A

      \[\leadsto \color{blue}{\left(c + \frac{1}{16} \cdot \left(t \cdot z\right)\right) + x \cdot y} \]

   2. +-commutative N/A

      \[\leadsto \color{blue}{\left(\frac{1}{16} \cdot \left(t \cdot z\right) + c\right)} + x \cdot y \]

   3. associate-+l+ N/A

      \[\leadsto \color{blue}{\frac{1}{16} \cdot \left(t \cdot z\right) + \left(c + x \cdot y\right)} \]

   4. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{16}, t \cdot z, c + x \cdot y\right)} \]

   5. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{16}, \color{blue}{t \cdot z}, c + x \cdot y\right) \]

   6. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{16}, t \cdot z, \color{blue}{x \cdot y + c}\right) \]

   7. lower-fma.f64 72.1

      \[\leadsto \mathsf{fma}\left(0.0625, t \cdot z, \color{blue}{\mathsf{fma}\left(x, y, c\right)}\right) \]

5. Applied rewrites 72.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.0625, t \cdot z, \mathsf{fma}\left(x, y, c\right)\right)} \]

6. Taylor expanded in t around 0

   \[\leadsto \color{blue}{c + x \cdot y} \]
7. Step-by-step derivation

   1. +-commutative N/A

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

   2. lower-fma.f64 43.5

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

8. Applied rewrites 43.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, c\right)} \]
9. Add Preprocessing

Alternative 14: 28.7% accurate, 7.8× speedup

Math

\[\begin{array}{l} \\ x \cdot y \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t, a, b, c)
    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
    code = x * y
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.6%

   \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
2. Add Preprocessing

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{x \cdot y} \]
4. Step-by-step derivation

   1. lower-*.f64 26.7

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

5. Applied rewrites 26.7%

   \[\leadsto \color{blue}{x \cdot y} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024216 
(FPCore (x y z t a b c)
  :name "Diagrams.Solve.Polynomial:quartForm  from diagrams-solve-0.1, C"
  :precision binary64
  (+ (- (+ (* x y) (/ (* z t) 16.0)) (/ (* a b) 4.0)) c))