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

Percentage Accurate: 97.6% → 98.8%

Time: 8.6s

Alternatives: 16

Speedup: 1.5×

• 32.9% 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: 97.6% 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: 98.8% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

      2. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. associate--l+ N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. div-inv N/A

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

      13. lower-*.f64 N/A

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

      14. metadata-eval N/A

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

      15. lower--.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. lower--.f64 100.0

          \[\leadsto \mathsf{fma}\left(t \cdot 0.0625, z, y \cdot x - \color{blue}{\left(\frac{a \cdot b}{4} - c\right)}\right) \]

   4. Applied rewrites 100.0%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate--r- N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. metadata-eval N/A

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

      8. lift-*.f64 N/A

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

      9. associate-+l+ N/A

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

      10. +-commutative N/A

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

      11. associate-+l+ N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-+.f64 100.0

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 100.0

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

   6. Applied rewrites 100.0%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 100.0

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

      7. lift-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 100.0

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 100.0

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 100.0

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

   8. Applied rewrites 100.0%

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

   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. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 71.4

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

   5. Applied rewrites 71.4%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 85.7%

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

4. Final simplification 99.6%

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

Alternative 2: 75.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, x, \left(z \cdot t\right) \cdot 0.0625\right)\\ t_2 := \frac{z \cdot t}{16} + x \cdot y\\ \mathbf{if}\;t\_2 \leq -6 \cdot 10^{+243}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq -5 \cdot 10^{+44}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot 0.0625, t, c\right)\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+70}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, b \cdot a, c\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (fma y x (* (* z t) 0.0625))) (t_2 (+ (/ (* z t) 16.0) (* x y))))
   (if (<= t_2 -6e+243)
     t_1
     (if (<= t_2 -5e+44)
       (fma (* z 0.0625) t c)
       (if (<= t_2 5e+70) (fma -0.25 (* b 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(y, x, ((z * t) * 0.0625));
	double t_2 = ((z * t) / 16.0) + (x * y);
	double tmp;
	if (t_2 <= -6e+243) {
		tmp = t_1;
	} else if (t_2 <= -5e+44) {
		tmp = fma((z * 0.0625), t, c);
	} else if (t_2 <= 5e+70) {
		tmp = fma(-0.25, (b * a), c);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = fma(y, x, Float64(Float64(z * t) * 0.0625))
	t_2 = Float64(Float64(Float64(z * t) / 16.0) + Float64(x * y))
	tmp = 0.0
	if (t_2 <= -6e+243)
		tmp = t_1;
	elseif (t_2 <= -5e+44)
		tmp = fma(Float64(z * 0.0625), t, c);
	elseif (t_2 <= 5e+70)
		tmp = fma(-0.25, Float64(b * a), 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[(N[(z * t), $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(z * t), $MachinePrecision] / 16.0), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -6e+243], t$95$1, If[LessEqual[t$95$2, -5e+44], N[(N[(z * 0.0625), $MachinePrecision] * t + c), $MachinePrecision], If[LessEqual[t$95$2, 5e+70], N[(-0.25 * N[(b * a), $MachinePrecision] + c), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (*.f64 x y) (/.f64 (*.f64 z t) #s(literal 16 binary64))) < -5.99999999999999969e243 or 5.0000000000000002e70 < (+.f64 (*.f64 x y) (/.f64 (*.f64 z t) #s(literal 16 binary64)))

   1. Initial program 93.6%

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

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 85.6

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

   5. Applied rewrites 85.6%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 79.3%

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

   10. Applied rewrites 80.3%

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

   if -5.99999999999999969e243 < (+.f64 (*.f64 x y) (/.f64 (*.f64 z t) #s(literal 16 binary64))) < -4.9999999999999996e44

   1. Initial program 99.9%

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

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 77.0

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

   5. Applied rewrites 77.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 69.4%

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

   10. Applied rewrites 69.4%

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

   if -4.9999999999999996e44 < (+.f64 (*.f64 x y) (/.f64 (*.f64 z t) #s(literal 16 binary64))) < 5.0000000000000002e70

   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 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. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 92.6

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

   5. Applied rewrites 92.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 83.2%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot t}{16} + x \cdot y \leq -6 \cdot 10^{+243}:\\ \;\;\;\;\mathsf{fma}\left(y, x, \left(z \cdot t\right) \cdot 0.0625\right)\\ \mathbf{elif}\;\frac{z \cdot t}{16} + x \cdot y \leq -5 \cdot 10^{+44}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot 0.0625, t, c\right)\\ \mathbf{elif}\;\frac{z \cdot t}{16} + x \cdot y \leq 5 \cdot 10^{+70}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, b \cdot a, c\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, \left(z \cdot t\right) \cdot 0.0625\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 64.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-0.25, b \cdot a, c\right)\\ \mathbf{if}\;b \cdot a \leq -5 \cdot 10^{+155}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \cdot a \leq 2000:\\ \;\;\;\;\mathsf{fma}\left(z \cdot 0.0625, t, c\right)\\ \mathbf{elif}\;b \cdot a \leq 2 \cdot 10^{+134}:\\ \;\;\;\;\mathsf{fma}\left(x, y, c\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (fma -0.25 (* b a) c)))
   (if (<= (* b a) -5e+155)
     t_1
     (if (<= (* b a) 2000.0)
       (fma (* z 0.0625) t c)
       (if (<= (* b a) 2e+134) (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(-0.25, (b * a), c);
	double tmp;
	if ((b * a) <= -5e+155) {
		tmp = t_1;
	} else if ((b * a) <= 2000.0) {
		tmp = fma((z * 0.0625), t, c);
	} else if ((b * a) <= 2e+134) {
		tmp = fma(x, y, c);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = fma(-0.25, Float64(b * a), c)
	tmp = 0.0
	if (Float64(b * a) <= -5e+155)
		tmp = t_1;
	elseif (Float64(b * a) <= 2000.0)
		tmp = fma(Float64(z * 0.0625), t, c);
	elseif (Float64(b * a) <= 2e+134)
		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[(-0.25 * N[(b * a), $MachinePrecision] + c), $MachinePrecision]}, If[LessEqual[N[(b * a), $MachinePrecision], -5e+155], t$95$1, If[LessEqual[N[(b * a), $MachinePrecision], 2000.0], N[(N[(z * 0.0625), $MachinePrecision] * t + c), $MachinePrecision], If[LessEqual[N[(b * a), $MachinePrecision], 2e+134], N[(x * y + c), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 a b) < -4.9999999999999999e155 or 1.99999999999999984e134 < (*.f64 a b)

   1. Initial program 96.3%

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

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

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 91.8

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

   5. Applied rewrites 91.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 85.2%

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

   if -4.9999999999999999e155 < (*.f64 a b) < 2e3

   1. Initial program 97.3%

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

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 93.8

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

   5. Applied rewrites 93.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 68.9%

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

   10. Applied rewrites 68.9%

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

   if 2e3 < (*.f64 a b) < 1.99999999999999984e134

   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

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 86.1

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

   5. Applied rewrites 86.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 36.7%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 71.7%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot a \leq -5 \cdot 10^{+155}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, b \cdot a, c\right)\\ \mathbf{elif}\;b \cdot a \leq 2000:\\ \;\;\;\;\mathsf{fma}\left(z \cdot 0.0625, t, c\right)\\ \mathbf{elif}\;b \cdot a \leq 2 \cdot 10^{+134}:\\ \;\;\;\;\mathsf{fma}\left(x, y, c\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, b \cdot a, c\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 88.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot a \leq -1 \cdot 10^{+25}:\\ \;\;\;\;\mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(-0.25, b \cdot a, c\right)\right)\\ \mathbf{elif}\;b \cdot a \leq 4 \cdot 10^{+161}:\\ \;\;\;\;\mathsf{fma}\left(y, x, \mathsf{fma}\left(z \cdot 0.0625, t, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot -0.25, b, x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (<= (* b a) -1e+25)
   (fma (* 0.0625 t) z (fma -0.25 (* b a) c))
   (if (<= (* b a) 4e+161)
     (fma y x (fma (* z 0.0625) t c))
     (fma (* a -0.25) b (* x y)))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if ((b * a) <= -1e+25) {
		tmp = fma((0.0625 * t), z, fma(-0.25, (b * a), c));
	} else if ((b * a) <= 4e+161) {
		tmp = fma(y, x, fma((z * 0.0625), t, c));
	} else {
		tmp = fma((a * -0.25), b, (x * y));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if (Float64(b * a) <= -1e+25)
		tmp = fma(Float64(0.0625 * t), z, fma(-0.25, Float64(b * a), c));
	elseif (Float64(b * a) <= 4e+161)
		tmp = fma(y, x, fma(Float64(z * 0.0625), t, c));
	else
		tmp = fma(Float64(a * -0.25), b, Float64(x * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[LessEqual[N[(b * a), $MachinePrecision], -1e+25], N[(N[(0.0625 * t), $MachinePrecision] * z + N[(-0.25 * N[(b * a), $MachinePrecision] + c), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(b * a), $MachinePrecision], 4e+161], N[(y * x + N[(N[(z * 0.0625), $MachinePrecision] * t + c), $MachinePrecision]), $MachinePrecision], N[(N[(a * -0.25), $MachinePrecision] * b + N[(x * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 a b) < -1.00000000000000009e25

   1. Initial program 92.5%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. associate--l+ N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. div-inv N/A

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

      13. lower-*.f64 N/A

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

      14. metadata-eval N/A

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

      15. lower--.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. lower--.f64 98.5

          \[\leadsto \mathsf{fma}\left(t \cdot 0.0625, z, y \cdot x - \color{blue}{\left(\frac{a \cdot b}{4} - c\right)}\right) \]

   4. Applied rewrites 98.5%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(t \cdot \frac{1}{16}, z, \color{blue}{c - \frac{1}{4} \cdot \left(a \cdot b\right)}\right) \]
   6. Step-by-step derivation

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 91.2

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

   7. Applied rewrites 91.2%

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

   if -1.00000000000000009e25 < (*.f64 a b) < 4.0000000000000002e161

   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 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

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 95.0

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

   5. Applied rewrites 95.0%

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

   7. Applied rewrites 95.0%

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

   if 4.0000000000000002e161 < (*.f64 a b)

   1. Initial program 97.3%

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

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

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 89.5

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

   5. Applied rewrites 89.5%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 89.7%

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

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot a \leq -1 \cdot 10^{+25}:\\ \;\;\;\;\mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(-0.25, b \cdot a, c\right)\right)\\ \mathbf{elif}\;b \cdot a \leq 4 \cdot 10^{+161}:\\ \;\;\;\;\mathsf{fma}\left(y, x, \mathsf{fma}\left(z \cdot 0.0625, t, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot -0.25, b, x \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 88.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot a \leq -5 \cdot 10^{+155}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, b \cdot a, \mathsf{fma}\left(y, x, c\right)\right)\\ \mathbf{elif}\;b \cdot a \leq 4 \cdot 10^{+161}:\\ \;\;\;\;\mathsf{fma}\left(y, x, \mathsf{fma}\left(z \cdot 0.0625, t, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot -0.25, b, x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (<= (* b a) -5e+155)
   (fma -0.25 (* b a) (fma y x c))
   (if (<= (* b a) 4e+161)
     (fma y x (fma (* z 0.0625) t c))
     (fma (* a -0.25) b (* x y)))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if ((b * a) <= -5e+155) {
		tmp = fma(-0.25, (b * a), fma(y, x, c));
	} else if ((b * a) <= 4e+161) {
		tmp = fma(y, x, fma((z * 0.0625), t, c));
	} else {
		tmp = fma((a * -0.25), b, (x * y));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if (Float64(b * a) <= -5e+155)
		tmp = fma(-0.25, Float64(b * a), fma(y, x, c));
	elseif (Float64(b * a) <= 4e+161)
		tmp = fma(y, x, fma(Float64(z * 0.0625), t, c));
	else
		tmp = fma(Float64(a * -0.25), b, Float64(x * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[LessEqual[N[(b * a), $MachinePrecision], -5e+155], N[(-0.25 * N[(b * a), $MachinePrecision] + N[(y * x + c), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(b * a), $MachinePrecision], 4e+161], N[(y * x + N[(N[(z * 0.0625), $MachinePrecision] * t + c), $MachinePrecision]), $MachinePrecision], N[(N[(a * -0.25), $MachinePrecision] * b + N[(x * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 a b) < -4.9999999999999999e155

   1. Initial program 95.3%

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

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

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 97.7

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

   5. Applied rewrites 97.7%

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

   if -4.9999999999999999e155 < (*.f64 a b) < 4.0000000000000002e161

   1. Initial program 97.7%

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

   3. Taylor expanded in 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

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 92.9

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

   5. Applied rewrites 92.9%

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

   7. Applied rewrites 92.9%

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

   if 4.0000000000000002e161 < (*.f64 a b)

   1. Initial program 97.3%

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

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

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 89.5

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

   5. Applied rewrites 89.5%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 89.7%

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

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot a \leq -5 \cdot 10^{+155}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, b \cdot a, \mathsf{fma}\left(y, x, c\right)\right)\\ \mathbf{elif}\;b \cdot a \leq 4 \cdot 10^{+161}:\\ \;\;\;\;\mathsf{fma}\left(y, x, \mathsf{fma}\left(z \cdot 0.0625, t, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot -0.25, b, x \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 88.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot a \leq -5 \cdot 10^{+155}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, b \cdot a, \mathsf{fma}\left(y, x, c\right)\right)\\ \mathbf{elif}\;b \cdot a \leq 4 \cdot 10^{+161}:\\ \;\;\;\;\mathsf{fma}\left(y, x, \mathsf{fma}\left(z \cdot t, 0.0625, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot -0.25, b, x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (<= (* b a) -5e+155)
   (fma -0.25 (* b a) (fma y x c))
   (if (<= (* b a) 4e+161)
     (fma y x (fma (* z t) 0.0625 c))
     (fma (* a -0.25) b (* x y)))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if ((b * a) <= -5e+155) {
		tmp = fma(-0.25, (b * a), fma(y, x, c));
	} else if ((b * a) <= 4e+161) {
		tmp = fma(y, x, fma((z * t), 0.0625, c));
	} else {
		tmp = fma((a * -0.25), b, (x * y));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if (Float64(b * a) <= -5e+155)
		tmp = fma(-0.25, Float64(b * a), fma(y, x, c));
	elseif (Float64(b * a) <= 4e+161)
		tmp = fma(y, x, fma(Float64(z * t), 0.0625, c));
	else
		tmp = fma(Float64(a * -0.25), b, Float64(x * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[LessEqual[N[(b * a), $MachinePrecision], -5e+155], N[(-0.25 * N[(b * a), $MachinePrecision] + N[(y * x + c), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(b * a), $MachinePrecision], 4e+161], N[(y * x + N[(N[(z * t), $MachinePrecision] * 0.0625 + c), $MachinePrecision]), $MachinePrecision], N[(N[(a * -0.25), $MachinePrecision] * b + N[(x * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 a b) < -4.9999999999999999e155

   1. Initial program 95.3%

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

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

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 97.7

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

   5. Applied rewrites 97.7%

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

   if -4.9999999999999999e155 < (*.f64 a b) < 4.0000000000000002e161

   1. Initial program 97.7%

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

   3. Taylor expanded in 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

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 92.9

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

   5. Applied rewrites 92.9%

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

   if 4.0000000000000002e161 < (*.f64 a b)

   1. Initial program 97.3%

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

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

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 89.5

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

   5. Applied rewrites 89.5%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 89.7%

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

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot a \leq -5 \cdot 10^{+155}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, b \cdot a, \mathsf{fma}\left(y, x, c\right)\right)\\ \mathbf{elif}\;b \cdot a \leq 4 \cdot 10^{+161}:\\ \;\;\;\;\mathsf{fma}\left(y, x, \mathsf{fma}\left(z \cdot t, 0.0625, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot -0.25, b, x \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 86.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+175}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot 0.0625, t, c\right)\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+84}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, b \cdot a, \mathsf{fma}\left(y, x, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, \left(z \cdot t\right) \cdot 0.0625\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (<= (* z t) -2e+175)
   (fma (* z 0.0625) t c)
   (if (<= (* z t) 2e+84)
     (fma -0.25 (* b a) (fma y x c))
     (fma y x (* (* z t) 0.0625)))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if ((z * t) <= -2e+175) {
		tmp = fma((z * 0.0625), t, c);
	} else if ((z * t) <= 2e+84) {
		tmp = fma(-0.25, (b * a), fma(y, x, c));
	} else {
		tmp = fma(y, x, ((z * t) * 0.0625));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if (Float64(z * t) <= -2e+175)
		tmp = fma(Float64(z * 0.0625), t, c);
	elseif (Float64(z * t) <= 2e+84)
		tmp = fma(-0.25, Float64(b * a), fma(y, x, c));
	else
		tmp = fma(y, x, Float64(Float64(z * t) * 0.0625));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[LessEqual[N[(z * t), $MachinePrecision], -2e+175], N[(N[(z * 0.0625), $MachinePrecision] * t + c), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 2e+84], N[(-0.25 * N[(b * a), $MachinePrecision] + N[(y * x + c), $MachinePrecision]), $MachinePrecision], N[(y * x + N[(N[(z * t), $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -1.9999999999999999e175

   1. Initial program 87.7%

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

   3. Taylor expanded in 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

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 79.6

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

   5. Applied rewrites 79.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 74.9%

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

   10. Applied rewrites 75.0%

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

   if -1.9999999999999999e175 < (*.f64 z t) < 2.00000000000000012e84

   1. Initial program 99.4%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   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. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 90.1

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

   5. Applied rewrites 90.1%

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

   if 2.00000000000000012e84 < (*.f64 z t)

   1. Initial program 97.4%

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

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 87.9

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

   5. Applied rewrites 87.9%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 79.0%

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

   10. Applied rewrites 81.6%

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

Alternative 8: 65.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot a \leq -5 \cdot 10^{+155}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, b \cdot a, c\right)\\ \mathbf{elif}\;b \cdot a \leq 2000:\\ \;\;\;\;\mathsf{fma}\left(z \cdot 0.0625, t, c\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot -0.25, b, x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (<= (* b a) -5e+155)
   (fma -0.25 (* b a) c)
   (if (<= (* b a) 2000.0)
     (fma (* z 0.0625) t c)
     (fma (* a -0.25) b (* x y)))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if ((b * a) <= -5e+155) {
		tmp = fma(-0.25, (b * a), c);
	} else if ((b * a) <= 2000.0) {
		tmp = fma((z * 0.0625), t, c);
	} else {
		tmp = fma((a * -0.25), b, (x * y));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if (Float64(b * a) <= -5e+155)
		tmp = fma(-0.25, Float64(b * a), c);
	elseif (Float64(b * a) <= 2000.0)
		tmp = fma(Float64(z * 0.0625), t, c);
	else
		tmp = fma(Float64(a * -0.25), b, Float64(x * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[LessEqual[N[(b * a), $MachinePrecision], -5e+155], N[(-0.25 * N[(b * a), $MachinePrecision] + c), $MachinePrecision], If[LessEqual[N[(b * a), $MachinePrecision], 2000.0], N[(N[(z * 0.0625), $MachinePrecision] * t + c), $MachinePrecision], N[(N[(a * -0.25), $MachinePrecision] * b + N[(x * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 a b) < -4.9999999999999999e155

   1. Initial program 95.3%

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

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

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 97.7

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

   5. Applied rewrites 97.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 95.4%

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

   if -4.9999999999999999e155 < (*.f64 a b) < 2e3

   1. Initial program 97.3%

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

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 93.8

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

   5. Applied rewrites 93.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 68.9%

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

   10. Applied rewrites 68.9%

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

   if 2e3 < (*.f64 a b)

   1. Initial program 98.4%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   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. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 84.6

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

   5. Applied rewrites 84.6%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 75.5%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot a \leq -5 \cdot 10^{+155}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, b \cdot a, c\right)\\ \mathbf{elif}\;b \cdot a \leq 2000:\\ \;\;\;\;\mathsf{fma}\left(z \cdot 0.0625, t, c\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot -0.25, b, x \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 65.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot a \leq -5 \cdot 10^{+155}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, b \cdot a, c\right)\\ \mathbf{elif}\;b \cdot a \leq 2000:\\ \;\;\;\;\mathsf{fma}\left(z \cdot 0.0625, t, c\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, \left(a \cdot -0.25\right) \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (<= (* b a) -5e+155)
   (fma -0.25 (* b a) c)
   (if (<= (* b a) 2000.0)
     (fma (* z 0.0625) t c)
     (fma y x (* (* a -0.25) b)))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if ((b * a) <= -5e+155) {
		tmp = fma(-0.25, (b * a), c);
	} else if ((b * a) <= 2000.0) {
		tmp = fma((z * 0.0625), t, c);
	} else {
		tmp = fma(y, x, ((a * -0.25) * b));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if (Float64(b * a) <= -5e+155)
		tmp = fma(-0.25, Float64(b * a), c);
	elseif (Float64(b * a) <= 2000.0)
		tmp = fma(Float64(z * 0.0625), t, c);
	else
		tmp = fma(y, x, Float64(Float64(a * -0.25) * b));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[LessEqual[N[(b * a), $MachinePrecision], -5e+155], N[(-0.25 * N[(b * a), $MachinePrecision] + c), $MachinePrecision], If[LessEqual[N[(b * a), $MachinePrecision], 2000.0], N[(N[(z * 0.0625), $MachinePrecision] * t + c), $MachinePrecision], N[(y * x + N[(N[(a * -0.25), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 a b) < -4.9999999999999999e155

   1. Initial program 95.3%

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

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

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 97.7

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

   5. Applied rewrites 97.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 95.4%

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

   if -4.9999999999999999e155 < (*.f64 a b) < 2e3

   1. Initial program 97.3%

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

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 93.8

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

   5. Applied rewrites 93.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 68.9%

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

   10. Applied rewrites 68.9%

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

   if 2e3 < (*.f64 a b)

   1. Initial program 98.4%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   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. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 84.6

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

   5. Applied rewrites 84.6%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 75.5%

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

   10. Applied rewrites 74.0%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot a \leq -5 \cdot 10^{+155}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, b \cdot a, c\right)\\ \mathbf{elif}\;b \cdot a \leq 2000:\\ \;\;\;\;\mathsf{fma}\left(z \cdot 0.0625, t, c\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, \left(a \cdot -0.25\right) \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 65.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-0.25, b \cdot a, c\right)\\ \mathbf{if}\;b \cdot a \leq -4 \cdot 10^{+155}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \cdot a \leq 2 \cdot 10^{+134}:\\ \;\;\;\;\mathsf{fma}\left(x, y, c\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (fma -0.25 (* b a) c)))
   (if (<= (* b a) -4e+155) t_1 (if (<= (* b a) 2e+134) (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(-0.25, (b * a), c);
	double tmp;
	if ((b * a) <= -4e+155) {
		tmp = t_1;
	} else if ((b * a) <= 2e+134) {
		tmp = fma(x, y, c);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = fma(-0.25, Float64(b * a), c)
	tmp = 0.0
	if (Float64(b * a) <= -4e+155)
		tmp = t_1;
	elseif (Float64(b * a) <= 2e+134)
		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[(-0.25 * N[(b * a), $MachinePrecision] + c), $MachinePrecision]}, If[LessEqual[N[(b * a), $MachinePrecision], -4e+155], t$95$1, If[LessEqual[N[(b * a), $MachinePrecision], 2e+134], N[(x * y + c), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a b) < -4.00000000000000003e155 or 1.99999999999999984e134 < (*.f64 a b)

   1. Initial program 96.4%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   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. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 90.8

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

   5. Applied rewrites 90.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 84.2%

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

   if -4.00000000000000003e155 < (*.f64 a b) < 1.99999999999999984e134

   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

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 92.7

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

   5. Applied rewrites 92.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 64.2%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 60.2%

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

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot a \leq -4 \cdot 10^{+155}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, b \cdot a, c\right)\\ \mathbf{elif}\;b \cdot a \leq 2 \cdot 10^{+134}:\\ \;\;\;\;\mathsf{fma}\left(x, y, c\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, b \cdot a, c\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 64.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(b \cdot a\right) \cdot -0.25\\ \mathbf{if}\;b \cdot a \leq -4 \cdot 10^{+155}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \cdot a \leq 10^{+154}:\\ \;\;\;\;\mathsf{fma}\left(x, y, c\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (* (* b a) -0.25)))
   (if (<= (* b a) -4e+155) t_1 (if (<= (* b a) 1e+154) (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 = (b * a) * -0.25;
	double tmp;
	if ((b * a) <= -4e+155) {
		tmp = t_1;
	} else if ((b * a) <= 1e+154) {
		tmp = fma(x, y, c);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(b * a) * -0.25)
	tmp = 0.0
	if (Float64(b * a) <= -4e+155)
		tmp = t_1;
	elseif (Float64(b * a) <= 1e+154)
		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 * a), $MachinePrecision] * -0.25), $MachinePrecision]}, If[LessEqual[N[(b * a), $MachinePrecision], -4e+155], t$95$1, If[LessEqual[N[(b * a), $MachinePrecision], 1e+154], N[(x * y + c), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a b) < -4.00000000000000003e155 or 1.00000000000000004e154 < (*.f64 a b)

   1. Initial program 96.3%

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

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 82.6

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

   5. Applied rewrites 82.6%

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

   if -4.00000000000000003e155 < (*.f64 a b) < 1.00000000000000004e154

   1. Initial program 97.7%

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

   3. Taylor expanded in 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

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 92.8

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

   5. Applied rewrites 92.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 64.6%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 60.1%

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

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot a \leq -4 \cdot 10^{+155}:\\ \;\;\;\;\left(b \cdot a\right) \cdot -0.25\\ \mathbf{elif}\;b \cdot a \leq 10^{+154}:\\ \;\;\;\;\mathsf{fma}\left(x, y, c\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot a\right) \cdot -0.25\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 62.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(z \cdot t\right) \cdot 0.0625\\ \mathbf{if}\;z \cdot t \leq -2.55 \cdot 10^{+175}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \cdot t \leq 8.8 \cdot 10^{+88}:\\ \;\;\;\;\mathsf{fma}\left(x, y, c\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (* (* z t) 0.0625)))
   (if (<= (* z t) -2.55e+175) t_1 (if (<= (* z t) 8.8e+88) (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 = (z * t) * 0.0625;
	double tmp;
	if ((z * t) <= -2.55e+175) {
		tmp = t_1;
	} else if ((z * t) <= 8.8e+88) {
		tmp = fma(x, y, c);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(z * t) * 0.0625)
	tmp = 0.0
	if (Float64(z * t) <= -2.55e+175)
		tmp = t_1;
	elseif (Float64(z * t) <= 8.8e+88)
		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[(z * t), $MachinePrecision] * 0.0625), $MachinePrecision]}, If[LessEqual[N[(z * t), $MachinePrecision], -2.55e+175], t$95$1, If[LessEqual[N[(z * t), $MachinePrecision], 8.8e+88], N[(x * y + c), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -2.55000000000000003e175 or 8.80000000000000035e88 < (*.f64 z t)

   1. Initial program 92.5%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. associate--l+ N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. div-inv N/A

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

      13. lower-*.f64 N/A

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

      14. metadata-eval N/A

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

      15. lower--.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. lower--.f64 98.8

          \[\leadsto \mathsf{fma}\left(t \cdot 0.0625, z, y \cdot x - \color{blue}{\left(\frac{a \cdot b}{4} - c\right)}\right) \]

   4. Applied rewrites 98.8%

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

   5. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 66.8

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

   7. Applied rewrites 66.8%

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

   if -2.55000000000000003e175 < (*.f64 z t) < 8.80000000000000035e88

   1. Initial program 99.4%

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

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 65.9

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

   5. Applied rewrites 65.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 37.0%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 56.9%

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

Alternative 13: 99.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 97.2%

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

   1. lift-+.f64 N/A

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

   2. lift--.f64 N/A

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

   3. associate-+l- N/A

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

   4. lift-+.f64 N/A

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

   5. +-commutative N/A

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

   6. associate--l+ N/A

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

   7. lift-/.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. associate-/l* N/A

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

   10. *-commutative N/A

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

   11. lower-fma.f64 N/A

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

   12. div-inv N/A

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

   13. lower-*.f64 N/A

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

   14. metadata-eval N/A

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

   15. lower--.f64 N/A

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

   16. lift-*.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 N/A

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

   19. lower--.f64 99.2

       \[\leadsto \mathsf{fma}\left(t \cdot 0.0625, z, y \cdot x - \color{blue}{\left(\frac{a \cdot b}{4} - c\right)}\right) \]

4. Applied rewrites 99.2%

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

   1. lift--.f64 N/A

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

   2. lift--.f64 N/A

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

   3. associate--r- N/A

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

   4. +-commutative N/A

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

   5. lift-*.f64 N/A

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

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

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

   7. metadata-eval N/A

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

   8. lift-*.f64 N/A

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

   9. associate-+l+ N/A

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

   10. +-commutative N/A

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

   11. associate-+l+ N/A

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

   12. lift-*.f64 N/A

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

   13. lower-fma.f64 N/A

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

   14. lower-+.f64 99.2

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 99.2

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

6. Applied rewrites 99.2%

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

7. Final simplification 99.2%

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

Alternative 14: 99.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 97.2%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. lift-+.f64 N/A

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

   4. +-commutative N/A

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

   5. associate-+l+ N/A

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

   6. lift-/.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. associate-/l* N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. div-inv N/A

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

   12. lower-*.f64 N/A

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

   13. metadata-eval N/A

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

   14. lift-*.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-fma.f64 N/A

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

   17. lift-/.f64 N/A

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

   18. div-inv N/A

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

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

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

   20. metadata-eval N/A

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

   21. metadata-eval N/A

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

   22. metadata-eval N/A

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

   23. metadata-eval N/A

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

   24. lower-*.f64 N/A

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

4. Applied rewrites 99.2%

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

5. Final simplification 99.2%

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

Alternative 15: 49.6% 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 97.2%

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

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

   3. *-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. +-commutative N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-*.f64 71.5

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

5. Applied rewrites 71.5%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 48.9%

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

9. Taylor expanded in z around 0

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

11. Applied rewrites 45.9%

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

Alternative 16: 29.3% 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 97.2%

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

3. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 25.7

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

5. Applied rewrites 25.7%

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

6. Final simplification 25.7%

   \[\leadsto x \cdot y \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024332 
(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))