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

Percentage Accurate: 97.8% → 98.8%

Time: 13.0s

Alternatives: 20

Speedup: 1.6×

• 33.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 97.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot x + \frac{z \cdot t}{16} \leq 5 \cdot 10^{+271}:\\ \;\;\;\;\mathsf{fma}\left(a, b \cdot -0.25, \mathsf{fma}\left(0.0625, z \cdot t, \mathsf{fma}\left(x, y, c\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, \mathsf{fma}\left(z, t \cdot 0.0625, \left(a \cdot b\right) \cdot -0.25\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (<= (+ (* y x) (/ (* z t) 16.0)) 5e+271)
   (fma a (* b -0.25) (fma 0.0625 (* z t) (fma x y c)))
   (fma y x (fma z (* t 0.0625) (* (* a b) -0.25)))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if (((y * x) + ((z * t) / 16.0)) <= 5e+271) {
		tmp = fma(a, (b * -0.25), fma(0.0625, (z * t), fma(x, y, c)));
	} else {
		tmp = fma(y, x, fma(z, (t * 0.0625), ((a * b) * -0.25)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if (Float64(Float64(y * x) + Float64(Float64(z * t) / 16.0)) <= 5e+271)
		tmp = fma(a, Float64(b * -0.25), fma(0.0625, Float64(z * t), fma(x, y, c)));
	else
		tmp = fma(y, x, fma(z, Float64(t * 0.0625), Float64(Float64(a * b) * -0.25)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[LessEqual[N[(N[(y * x), $MachinePrecision] + N[(N[(z * t), $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision], 5e+271], N[(a * N[(b * -0.25), $MachinePrecision] + N[(0.0625 * N[(z * t), $MachinePrecision] + N[(x * y + c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * x + N[(z * N[(t * 0.0625), $MachinePrecision] + N[(N[(a * b), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 x y) (/.f64 (*.f64 z t) #s(literal 16 binary64))) < 5.0000000000000003e271

   1. Initial program 99.0%

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

   3. Taylor expanded in x around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. associate-+r+ N/A

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

      11. +-commutative N/A

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

      12. associate-+l+ N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-fma.f64 99.5

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

   5. Applied rewrites 99.5%

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

   if 5.0000000000000003e271 < (+.f64 (*.f64 x y) (/.f64 (*.f64 z t) #s(literal 16 binary64)))

   1. Initial program 86.6%

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

   3. Taylor expanded in c around 0

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

      1. associate--l+ N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

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

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

      5. metadata-eval N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 86.6

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

   5. Applied rewrites 86.6%

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

   7. Applied rewrites 100.0%

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{x}, \mathsf{fma}\left(z, t \cdot 0.0625, \left(a \cdot b\right) \cdot -0.25\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot x + \frac{z \cdot t}{16} \leq 5 \cdot 10^{+271}:\\ \;\;\;\;\mathsf{fma}\left(a, b \cdot -0.25, \mathsf{fma}\left(0.0625, z \cdot t, \mathsf{fma}\left(x, y, c\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, \mathsf{fma}\left(z, t \cdot 0.0625, \left(a \cdot b\right) \cdot -0.25\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 76.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, x, 0.0625 \cdot \left(z \cdot t\right)\right)\\ t_2 := y \cdot x + \frac{z \cdot t}{16}\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+79}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{+151}:\\ \;\;\;\;c + a \cdot \left(b \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (fma y x (* 0.0625 (* z t)))) (t_2 (+ (* y x) (/ (* z t) 16.0))))
   (if (<= t_2 -2e+79) t_1 (if (<= t_2 1e+151) (+ c (* a (* b -0.25))) 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, (0.0625 * (z * t)));
	double t_2 = (y * x) + ((z * t) / 16.0);
	double tmp;
	if (t_2 <= -2e+79) {
		tmp = t_1;
	} else if (t_2 <= 1e+151) {
		tmp = c + (a * (b * -0.25));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = fma(y, x, Float64(0.0625 * Float64(z * t)))
	t_2 = Float64(Float64(y * x) + Float64(Float64(z * t) / 16.0))
	tmp = 0.0
	if (t_2 <= -2e+79)
		tmp = t_1;
	elseif (t_2 <= 1e+151)
		tmp = Float64(c + Float64(a * Float64(b * -0.25)));
	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[(0.0625 * N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y * x), $MachinePrecision] + N[(N[(z * t), $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+79], t$95$1, If[LessEqual[t$95$2, 1e+151], N[(c + N[(a * N[(b * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 x y) (/.f64 (*.f64 z t) #s(literal 16 binary64))) < -1.99999999999999993e79 or 1.00000000000000002e151 < (+.f64 (*.f64 x y) (/.f64 (*.f64 z t) #s(literal 16 binary64)))

   1. Initial program 95.1%

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

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

      4. associate--l+ N/A

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

      5. associate-+l+ N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-+.f64 N/A

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 75.0

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

   7. Applied rewrites 75.0%

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

   if -1.99999999999999993e79 < (+.f64 (*.f64 x y) (/.f64 (*.f64 z t) #s(literal 16 binary64))) < 1.00000000000000002e151

   1. Initial program 100.0%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 81.0

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

   5. Applied rewrites 81.0%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot x + \frac{z \cdot t}{16} \leq -2 \cdot 10^{+79}:\\ \;\;\;\;\mathsf{fma}\left(y, x, 0.0625 \cdot \left(z \cdot t\right)\right)\\ \mathbf{elif}\;y \cdot x + \frac{z \cdot t}{16} \leq 10^{+151}:\\ \;\;\;\;c + a \cdot \left(b \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, 0.0625 \cdot \left(z \cdot t\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, x, \mathsf{fma}\left(z, t \cdot 0.0625, \left(a \cdot b\right) \cdot -0.25\right)\right)\\ \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+80}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \cdot t \leq 10^{+34}:\\ \;\;\;\;\mathsf{fma}\left(y, x, \mathsf{fma}\left(a, b \cdot -0.25, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (fma y x (fma z (* t 0.0625) (* (* a b) -0.25)))))
   (if (<= (* z t) -5e+80)
     t_1
     (if (<= (* z t) 1e+34) (fma y x (fma a (* b -0.25) c)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = fma(y, x, fma(z, (t * 0.0625), ((a * b) * -0.25)));
	double tmp;
	if ((z * t) <= -5e+80) {
		tmp = t_1;
	} else if ((z * t) <= 1e+34) {
		tmp = fma(y, x, fma(a, (b * -0.25), c));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = fma(y, x, fma(z, Float64(t * 0.0625), Float64(Float64(a * b) * -0.25)))
	tmp = 0.0
	if (Float64(z * t) <= -5e+80)
		tmp = t_1;
	elseif (Float64(z * t) <= 1e+34)
		tmp = fma(y, x, fma(a, Float64(b * -0.25), 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[(z * N[(t * 0.0625), $MachinePrecision] + N[(N[(a * b), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(z * t), $MachinePrecision], -5e+80], t$95$1, If[LessEqual[N[(z * t), $MachinePrecision], 1e+34], N[(y * x + N[(a * N[(b * -0.25), $MachinePrecision] + c), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -4.99999999999999961e80 or 9.99999999999999946e33 < (*.f64 z t)

   1. Initial program 94.2%

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

   3. Taylor expanded in c around 0

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

      1. associate--l+ N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

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

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

      5. metadata-eval N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 86.2

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

   5. Applied rewrites 86.2%

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

   7. Applied rewrites 90.9%

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

   if -4.99999999999999961e80 < (*.f64 z t) < 9.99999999999999946e33

   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. 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. lift-+.f64 N/A

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

      4. associate--l+ N/A

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

      5. associate-+l+ N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-+.f64 N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 62.4

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

   7. Applied rewrites 62.4%

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

   8. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 96.1

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

   10. Applied rewrites 96.1%

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

Alternative 4: 65.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot x \leq -1 \cdot 10^{+95}:\\ \;\;\;\;\mathsf{fma}\left(x, y, c\right)\\ \mathbf{elif}\;y \cdot x \leq 4 \cdot 10^{-131}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot 0.0625, t, c\right)\\ \mathbf{elif}\;y \cdot x \leq 10^{+151}:\\ \;\;\;\;c + a \cdot \left(b \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, y, c\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (<= (* y x) -1e+95)
   (fma x y c)
   (if (<= (* y x) 4e-131)
     (fma (* z 0.0625) t c)
     (if (<= (* y x) 1e+151) (+ c (* a (* b -0.25))) (fma x y c)))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if ((y * x) <= -1e+95) {
		tmp = fma(x, y, c);
	} else if ((y * x) <= 4e-131) {
		tmp = fma((z * 0.0625), t, c);
	} else if ((y * x) <= 1e+151) {
		tmp = c + (a * (b * -0.25));
	} else {
		tmp = fma(x, y, c);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if (Float64(y * x) <= -1e+95)
		tmp = fma(x, y, c);
	elseif (Float64(y * x) <= 4e-131)
		tmp = fma(Float64(z * 0.0625), t, c);
	elseif (Float64(y * x) <= 1e+151)
		tmp = Float64(c + Float64(a * Float64(b * -0.25)));
	else
		tmp = fma(x, y, c);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[LessEqual[N[(y * x), $MachinePrecision], -1e+95], N[(x * y + c), $MachinePrecision], If[LessEqual[N[(y * x), $MachinePrecision], 4e-131], N[(N[(z * 0.0625), $MachinePrecision] * t + c), $MachinePrecision], If[LessEqual[N[(y * x), $MachinePrecision], 1e+151], N[(c + N[(a * N[(b * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * y + c), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -1.00000000000000002e95 or 1.00000000000000002e151 < (*.f64 x y)

   1. Initial program 94.0%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 80.8

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

   5. Applied rewrites 80.8%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 73.1%

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

   if -1.00000000000000002e95 < (*.f64 x y) < 3.9999999999999999e-131

   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 a around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 71.9

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

   5. Applied rewrites 71.9%

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

   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 69.6%

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

   10. Applied rewrites 70.0%

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

   if 3.9999999999999999e-131 < (*.f64 x y) < 1.00000000000000002e151

   1. Initial program 100.0%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 70.4

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

   5. Applied rewrites 70.4%

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

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot x \leq -1 \cdot 10^{+95}:\\ \;\;\;\;\mathsf{fma}\left(x, y, c\right)\\ \mathbf{elif}\;y \cdot x \leq 4 \cdot 10^{-131}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot 0.0625, t, c\right)\\ \mathbf{elif}\;y \cdot x \leq 10^{+151}:\\ \;\;\;\;c + a \cdot \left(b \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, y, c\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 60.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot \left(b \cdot -0.25\right)\\ \mathbf{if}\;a \cdot b \leq -1 \cdot 10^{+250}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \cdot b \leq 2 \cdot 10^{-239}:\\ \;\;\;\;\mathsf{fma}\left(x, y, c\right)\\ \mathbf{elif}\;a \cdot b \leq 2 \cdot 10^{+59}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot 0.0625, t, c\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (* a (* b -0.25))))
   (if (<= (* a b) -1e+250)
     t_1
     (if (<= (* a b) 2e-239)
       (fma x y c)
       (if (<= (* a b) 2e+59) (fma (* z 0.0625) t c) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = a * (b * -0.25);
	double tmp;
	if ((a * b) <= -1e+250) {
		tmp = t_1;
	} else if ((a * b) <= 2e-239) {
		tmp = fma(x, y, c);
	} else if ((a * b) <= 2e+59) {
		tmp = fma((z * 0.0625), t, c);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(a * Float64(b * -0.25))
	tmp = 0.0
	if (Float64(a * b) <= -1e+250)
		tmp = t_1;
	elseif (Float64(a * b) <= 2e-239)
		tmp = fma(x, y, c);
	elseif (Float64(a * b) <= 2e+59)
		tmp = fma(Float64(z * 0.0625), t, c);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(a * N[(b * -0.25), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(a * b), $MachinePrecision], -1e+250], t$95$1, If[LessEqual[N[(a * b), $MachinePrecision], 2e-239], N[(x * y + c), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], 2e+59], N[(N[(z * 0.0625), $MachinePrecision] * t + c), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 a b) < -9.9999999999999992e249 or 1.99999999999999994e59 < (*.f64 a b)

   1. Initial program 92.2%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 75.3

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

   5. Applied rewrites 75.3%

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

   if -9.9999999999999992e249 < (*.f64 a b) < 2.0000000000000002e-239

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 90.9

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

   5. Applied rewrites 90.9%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 66.1%

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

   if 2.0000000000000002e-239 < (*.f64 a b) < 1.99999999999999994e59

   1. Initial program 98.0%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 87.4

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

   5. Applied rewrites 87.4%

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

   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 68.8%

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

   10. Applied rewrites 68.8%

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

Alternative 6: 60.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot \left(b \cdot -0.25\right)\\ \mathbf{if}\;a \cdot b \leq -1 \cdot 10^{+250}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \cdot b \leq 2 \cdot 10^{-239}:\\ \;\;\;\;\mathsf{fma}\left(x, y, c\right)\\ \mathbf{elif}\;a \cdot b \leq 2 \cdot 10^{+59}:\\ \;\;\;\;\mathsf{fma}\left(0.0625, z \cdot t, c\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (* a (* b -0.25))))
   (if (<= (* a b) -1e+250)
     t_1
     (if (<= (* a b) 2e-239)
       (fma x y c)
       (if (<= (* a b) 2e+59) (fma 0.0625 (* z t) c) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = a * (b * -0.25);
	double tmp;
	if ((a * b) <= -1e+250) {
		tmp = t_1;
	} else if ((a * b) <= 2e-239) {
		tmp = fma(x, y, c);
	} else if ((a * b) <= 2e+59) {
		tmp = fma(0.0625, (z * t), c);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(a * Float64(b * -0.25))
	tmp = 0.0
	if (Float64(a * b) <= -1e+250)
		tmp = t_1;
	elseif (Float64(a * b) <= 2e-239)
		tmp = fma(x, y, c);
	elseif (Float64(a * b) <= 2e+59)
		tmp = fma(0.0625, Float64(z * t), c);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(a * N[(b * -0.25), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(a * b), $MachinePrecision], -1e+250], t$95$1, If[LessEqual[N[(a * b), $MachinePrecision], 2e-239], N[(x * y + c), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], 2e+59], N[(0.0625 * N[(z * t), $MachinePrecision] + c), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 a b) < -9.9999999999999992e249 or 1.99999999999999994e59 < (*.f64 a b)

   1. Initial program 92.2%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 75.3

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

   5. Applied rewrites 75.3%

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

   if -9.9999999999999992e249 < (*.f64 a b) < 2.0000000000000002e-239

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 90.9

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

   5. Applied rewrites 90.9%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 66.1%

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

   if 2.0000000000000002e-239 < (*.f64 a b) < 1.99999999999999994e59

   1. Initial program 98.0%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 87.4

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

   5. Applied rewrites 87.4%

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

   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 68.8%

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

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b \leq -1 \cdot 10^{+250}:\\ \;\;\;\;a \cdot \left(b \cdot -0.25\right)\\ \mathbf{elif}\;a \cdot b \leq 2 \cdot 10^{-239}:\\ \;\;\;\;\mathsf{fma}\left(x, y, c\right)\\ \mathbf{elif}\;a \cdot b \leq 2 \cdot 10^{+59}:\\ \;\;\;\;\mathsf{fma}\left(0.0625, z \cdot t, c\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(b \cdot -0.25\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 88.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, x, \mathsf{fma}\left(a, b \cdot -0.25, c\right)\right)\\ \mathbf{if}\;y \cdot x \leq -1 \cdot 10^{+95}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \cdot x \leq 5 \cdot 10^{-58}:\\ \;\;\;\;\mathsf{fma}\left(a, b \cdot -0.25, \mathsf{fma}\left(0.0625, z \cdot t, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (fma y x (fma a (* b -0.25) c))))
   (if (<= (* y x) -1e+95)
     t_1
     (if (<= (* y x) 5e-58) (fma a (* b -0.25) (fma 0.0625 (* z t) c)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = fma(y, x, fma(a, (b * -0.25), c));
	double tmp;
	if ((y * x) <= -1e+95) {
		tmp = t_1;
	} else if ((y * x) <= 5e-58) {
		tmp = fma(a, (b * -0.25), fma(0.0625, (z * t), c));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = fma(y, x, fma(a, Float64(b * -0.25), c))
	tmp = 0.0
	if (Float64(y * x) <= -1e+95)
		tmp = t_1;
	elseif (Float64(y * x) <= 5e-58)
		tmp = fma(a, Float64(b * -0.25), fma(0.0625, Float64(z * t), c));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(y * x + N[(a * N[(b * -0.25), $MachinePrecision] + c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(y * x), $MachinePrecision], -1e+95], t$95$1, If[LessEqual[N[(y * x), $MachinePrecision], 5e-58], N[(a * N[(b * -0.25), $MachinePrecision] + N[(0.0625 * N[(z * t), $MachinePrecision] + c), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -1.00000000000000002e95 or 4.99999999999999977e-58 < (*.f64 x y)

   1. Initial program 95.8%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Add Preprocessing
   3. 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. lift-+.f64 N/A

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

      4. associate--l+ N/A

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

      5. associate-+l+ N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-+.f64 N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 78.6

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

   7. Applied rewrites 78.6%

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

   8. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 89.6

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

   10. Applied rewrites 89.6%

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

   if -1.00000000000000002e95 < (*.f64 x y) < 4.99999999999999977e-58

   1. Initial program 98.6%

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

   3. Taylor expanded in c around 0

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

      1. associate--l+ N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

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

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

      5. metadata-eval N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 64.8

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

   5. Applied rewrites 64.8%

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

   7. Applied rewrites 65.4%

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

   8. Taylor expanded in x around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 96.8

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

   10. Applied rewrites 96.8%

       \[\leadsto \color{blue}{\mathsf{fma}\left(a, b \cdot -0.25, \mathsf{fma}\left(0.0625, t \cdot z, c\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot x \leq -1 \cdot 10^{+95}:\\ \;\;\;\;\mathsf{fma}\left(y, x, \mathsf{fma}\left(a, b \cdot -0.25, c\right)\right)\\ \mathbf{elif}\;y \cdot x \leq 5 \cdot 10^{-58}:\\ \;\;\;\;\mathsf{fma}\left(a, b \cdot -0.25, \mathsf{fma}\left(0.0625, z \cdot t, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, \mathsf{fma}\left(a, b \cdot -0.25, c\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 88.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(a, b \cdot -0.25, c\right)\\ t_2 := \mathsf{fma}\left(y, x, t\_1\right)\\ \mathbf{if}\;y \cdot x \leq -1 \cdot 10^{+95}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \cdot x \leq 5 \cdot 10^{-58}:\\ \;\;\;\;\mathsf{fma}\left(0.0625, z \cdot t, t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (fma a (* b -0.25) c)) (t_2 (fma y x t_1)))
   (if (<= (* y x) -1e+95)
     t_2
     (if (<= (* y x) 5e-58) (fma 0.0625 (* z t) t_1) t_2))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = fma(a, (b * -0.25), c);
	double t_2 = fma(y, x, t_1);
	double tmp;
	if ((y * x) <= -1e+95) {
		tmp = t_2;
	} else if ((y * x) <= 5e-58) {
		tmp = fma(0.0625, (z * t), t_1);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = fma(a, Float64(b * -0.25), c)
	t_2 = fma(y, x, t_1)
	tmp = 0.0
	if (Float64(y * x) <= -1e+95)
		tmp = t_2;
	elseif (Float64(y * x) <= 5e-58)
		tmp = fma(0.0625, Float64(z * t), t_1);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(a * N[(b * -0.25), $MachinePrecision] + c), $MachinePrecision]}, Block[{t$95$2 = N[(y * x + t$95$1), $MachinePrecision]}, If[LessEqual[N[(y * x), $MachinePrecision], -1e+95], t$95$2, If[LessEqual[N[(y * x), $MachinePrecision], 5e-58], N[(0.0625 * N[(z * t), $MachinePrecision] + t$95$1), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -1.00000000000000002e95 or 4.99999999999999977e-58 < (*.f64 x y)

   1. Initial program 95.8%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Add Preprocessing
   3. 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. lift-+.f64 N/A

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

      4. associate--l+ N/A

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

      5. associate-+l+ N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-+.f64 N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 78.6

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

   7. Applied rewrites 78.6%

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

   8. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 89.6

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

   10. Applied rewrites 89.6%

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

   if -1.00000000000000002e95 < (*.f64 x y) < 4.99999999999999977e-58

   1. Initial program 98.6%

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

   3. Taylor expanded in x around 0

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

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

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. associate-+l+ N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 96.0

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

   5. Applied rewrites 96.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0625, t \cdot z, \mathsf{fma}\left(a, b \cdot -0.25, c\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot x \leq -1 \cdot 10^{+95}:\\ \;\;\;\;\mathsf{fma}\left(y, x, \mathsf{fma}\left(a, b \cdot -0.25, c\right)\right)\\ \mathbf{elif}\;y \cdot x \leq 5 \cdot 10^{-58}:\\ \;\;\;\;\mathsf{fma}\left(0.0625, z \cdot t, \mathsf{fma}\left(a, b \cdot -0.25, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, \mathsf{fma}\left(a, b \cdot -0.25, c\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 90.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, x, \mathsf{fma}\left(a, b \cdot -0.25, c\right)\right)\\ \mathbf{if}\;a \cdot b \leq -2 \cdot 10^{+114}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \cdot b \leq 2 \cdot 10^{+59}:\\ \;\;\;\;\mathsf{fma}\left(y, x, \mathsf{fma}\left(z, t \cdot 0.0625, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (fma y x (fma a (* b -0.25) c))))
   (if (<= (* a b) -2e+114)
     t_1
     (if (<= (* a b) 2e+59) (fma y x (fma z (* t 0.0625) c)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = fma(y, x, fma(a, (b * -0.25), c));
	double tmp;
	if ((a * b) <= -2e+114) {
		tmp = t_1;
	} else if ((a * b) <= 2e+59) {
		tmp = fma(y, x, fma(z, (t * 0.0625), c));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = fma(y, x, fma(a, Float64(b * -0.25), c))
	tmp = 0.0
	if (Float64(a * b) <= -2e+114)
		tmp = t_1;
	elseif (Float64(a * b) <= 2e+59)
		tmp = fma(y, x, fma(z, Float64(t * 0.0625), c));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(y * x + N[(a * N[(b * -0.25), $MachinePrecision] + c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(a * b), $MachinePrecision], -2e+114], t$95$1, If[LessEqual[N[(a * b), $MachinePrecision], 2e+59], N[(y * x + N[(z * N[(t * 0.0625), $MachinePrecision] + c), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a b) < -2e114 or 1.99999999999999994e59 < (*.f64 a b)

   1. Initial program 93.5%

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

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

      4. associate--l+ N/A

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

      5. associate-+l+ N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-+.f64 N/A

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

   4. Applied rewrites 98.9%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 83.8

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

   7. Applied rewrites 83.8%

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

   8. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 92.8

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

   10. Applied rewrites 92.8%

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

   if -2e114 < (*.f64 a b) < 1.99999999999999994e59

   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

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

      3. associate-+l+ N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 91.9

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

   5. Applied rewrites 91.9%

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

   7. Applied rewrites 92.5%

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

Alternative 10: 90.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(a, b \cdot -0.25, \mathsf{fma}\left(x, y, c\right)\right)\\ \mathbf{if}\;a \cdot b \leq -2 \cdot 10^{+114}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \cdot b \leq 2 \cdot 10^{+59}:\\ \;\;\;\;\mathsf{fma}\left(y, x, \mathsf{fma}\left(z, t \cdot 0.0625, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (fma a (* b -0.25) (fma x y c))))
   (if (<= (* a b) -2e+114)
     t_1
     (if (<= (* a b) 2e+59) (fma y x (fma z (* t 0.0625) c)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = fma(a, (b * -0.25), fma(x, y, c));
	double tmp;
	if ((a * b) <= -2e+114) {
		tmp = t_1;
	} else if ((a * b) <= 2e+59) {
		tmp = fma(y, x, fma(z, (t * 0.0625), c));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = fma(a, Float64(b * -0.25), fma(x, y, c))
	tmp = 0.0
	if (Float64(a * b) <= -2e+114)
		tmp = t_1;
	elseif (Float64(a * b) <= 2e+59)
		tmp = fma(y, x, fma(z, Float64(t * 0.0625), c));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(a * N[(b * -0.25), $MachinePrecision] + N[(x * y + c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(a * b), $MachinePrecision], -2e+114], t$95$1, If[LessEqual[N[(a * b), $MachinePrecision], 2e+59], N[(y * x + N[(z * N[(t * 0.0625), $MachinePrecision] + c), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a b) < -2e114 or 1.99999999999999994e59 < (*.f64 a b)

   1. Initial program 93.5%

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

   3. Taylor expanded in z around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 89.4

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

   5. Applied rewrites 89.4%

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

   if -2e114 < (*.f64 a b) < 1.99999999999999994e59

   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

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

      3. associate-+l+ N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 91.9

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

   5. Applied rewrites 91.9%

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

   7. Applied rewrites 92.5%

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

Alternative 11: 90.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(a, b \cdot -0.25, \mathsf{fma}\left(x, y, c\right)\right)\\ \mathbf{if}\;a \cdot b \leq -2 \cdot 10^{+114}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \cdot b \leq 2 \cdot 10^{+59}:\\ \;\;\;\;\mathsf{fma}\left(0.0625, z \cdot t, \mathsf{fma}\left(x, y, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (fma a (* b -0.25) (fma x y c))))
   (if (<= (* a b) -2e+114)
     t_1
     (if (<= (* a b) 2e+59) (fma 0.0625 (* z t) (fma x y c)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = fma(a, (b * -0.25), fma(x, y, c));
	double tmp;
	if ((a * b) <= -2e+114) {
		tmp = t_1;
	} else if ((a * b) <= 2e+59) {
		tmp = fma(0.0625, (z * t), fma(x, y, c));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = fma(a, Float64(b * -0.25), fma(x, y, c))
	tmp = 0.0
	if (Float64(a * b) <= -2e+114)
		tmp = t_1;
	elseif (Float64(a * b) <= 2e+59)
		tmp = fma(0.0625, Float64(z * t), fma(x, y, c));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(a * N[(b * -0.25), $MachinePrecision] + N[(x * y + c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(a * b), $MachinePrecision], -2e+114], t$95$1, If[LessEqual[N[(a * b), $MachinePrecision], 2e+59], N[(0.0625 * N[(z * t), $MachinePrecision] + N[(x * y + c), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a b) < -2e114 or 1.99999999999999994e59 < (*.f64 a b)

   1. Initial program 93.5%

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

   3. Taylor expanded in z around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 89.4

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

   5. Applied rewrites 89.4%

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

   if -2e114 < (*.f64 a b) < 1.99999999999999994e59

   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

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

      3. associate-+l+ N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 91.9

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

   5. Applied rewrites 91.9%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b \leq -2 \cdot 10^{+114}:\\ \;\;\;\;\mathsf{fma}\left(a, b \cdot -0.25, \mathsf{fma}\left(x, y, c\right)\right)\\ \mathbf{elif}\;a \cdot b \leq 2 \cdot 10^{+59}:\\ \;\;\;\;\mathsf{fma}\left(0.0625, z \cdot t, \mathsf{fma}\left(x, y, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, b \cdot -0.25, \mathsf{fma}\left(x, y, c\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 86.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, x, \left(a \cdot b\right) \cdot -0.25\right)\\ \mathbf{if}\;a \cdot b \leq -5 \cdot 10^{+197}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \cdot b \leq 2 \cdot 10^{+59}:\\ \;\;\;\;\mathsf{fma}\left(0.0625, z \cdot t, \mathsf{fma}\left(x, y, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (fma y x (* (* a b) -0.25))))
   (if (<= (* a b) -5e+197)
     t_1
     (if (<= (* a b) 2e+59) (fma 0.0625 (* z t) (fma x y c)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = fma(y, x, ((a * b) * -0.25));
	double tmp;
	if ((a * b) <= -5e+197) {
		tmp = t_1;
	} else if ((a * b) <= 2e+59) {
		tmp = fma(0.0625, (z * t), fma(x, y, c));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = fma(y, x, Float64(Float64(a * b) * -0.25))
	tmp = 0.0
	if (Float64(a * b) <= -5e+197)
		tmp = t_1;
	elseif (Float64(a * b) <= 2e+59)
		tmp = fma(0.0625, Float64(z * t), fma(x, y, c));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(y * x + N[(N[(a * b), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(a * b), $MachinePrecision], -5e+197], t$95$1, If[LessEqual[N[(a * b), $MachinePrecision], 2e+59], N[(0.0625 * N[(z * t), $MachinePrecision] + N[(x * y + c), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a b) < -5.00000000000000009e197 or 1.99999999999999994e59 < (*.f64 a b)

   1. Initial program 92.7%

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

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

      4. associate--l+ N/A

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

      5. associate-+l+ N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-+.f64 N/A

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

   4. Applied rewrites 98.7%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 87.6

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

   7. Applied rewrites 87.6%

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

   if -5.00000000000000009e197 < (*.f64 a b) < 1.99999999999999994e59

   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

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

      3. associate-+l+ N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 91.0

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

   5. Applied rewrites 91.0%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b \leq -5 \cdot 10^{+197}:\\ \;\;\;\;\mathsf{fma}\left(y, x, \left(a \cdot b\right) \cdot -0.25\right)\\ \mathbf{elif}\;a \cdot b \leq 2 \cdot 10^{+59}:\\ \;\;\;\;\mathsf{fma}\left(0.0625, z \cdot t, \mathsf{fma}\left(x, y, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, \left(a \cdot b\right) \cdot -0.25\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 66.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, x, \left(a \cdot b\right) \cdot -0.25\right)\\ \mathbf{if}\;y \cdot x \leq -2 \cdot 10^{+64}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \cdot x \leq 2 \cdot 10^{-27}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot 0.0625, t, c\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (fma y x (* (* a b) -0.25))))
   (if (<= (* y x) -2e+64)
     t_1
     (if (<= (* y x) 2e-27) (fma (* z 0.0625) t c) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = fma(y, x, ((a * b) * -0.25));
	double tmp;
	if ((y * x) <= -2e+64) {
		tmp = t_1;
	} else if ((y * x) <= 2e-27) {
		tmp = fma((z * 0.0625), t, c);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = fma(y, x, Float64(Float64(a * b) * -0.25))
	tmp = 0.0
	if (Float64(y * x) <= -2e+64)
		tmp = t_1;
	elseif (Float64(y * x) <= 2e-27)
		tmp = fma(Float64(z * 0.0625), t, 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[(a * b), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(y * x), $MachinePrecision], -2e+64], t$95$1, If[LessEqual[N[(y * x), $MachinePrecision], 2e-27], N[(N[(z * 0.0625), $MachinePrecision] * t + c), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -2.00000000000000004e64 or 2.0000000000000001e-27 < (*.f64 x y)

   1. Initial program 95.7%

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

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

      4. associate--l+ N/A

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

      5. associate-+l+ N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-+.f64 N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 80.4

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

   7. Applied rewrites 80.4%

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

   if -2.00000000000000004e64 < (*.f64 x y) < 2.0000000000000001e-27

   1. Initial program 98.6%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 71.7

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

   5. Applied rewrites 71.7%

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

   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 69.0%

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

   10. Applied rewrites 69.4%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot x \leq -2 \cdot 10^{+64}:\\ \;\;\;\;\mathsf{fma}\left(y, x, \left(a \cdot b\right) \cdot -0.25\right)\\ \mathbf{elif}\;y \cdot x \leq 2 \cdot 10^{-27}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot 0.0625, t, c\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, \left(a \cdot b\right) \cdot -0.25\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 66.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(a, b \cdot -0.25, y \cdot x\right)\\ \mathbf{if}\;y \cdot x \leq -2 \cdot 10^{+64}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \cdot x \leq 2 \cdot 10^{-27}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot 0.0625, t, c\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (fma a (* b -0.25) (* y x))))
   (if (<= (* y x) -2e+64)
     t_1
     (if (<= (* y x) 2e-27) (fma (* z 0.0625) t c) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = fma(a, (b * -0.25), (y * x));
	double tmp;
	if ((y * x) <= -2e+64) {
		tmp = t_1;
	} else if ((y * x) <= 2e-27) {
		tmp = fma((z * 0.0625), t, c);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = fma(a, Float64(b * -0.25), Float64(y * x))
	tmp = 0.0
	if (Float64(y * x) <= -2e+64)
		tmp = t_1;
	elseif (Float64(y * x) <= 2e-27)
		tmp = fma(Float64(z * 0.0625), t, c);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(a * N[(b * -0.25), $MachinePrecision] + N[(y * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(y * x), $MachinePrecision], -2e+64], t$95$1, If[LessEqual[N[(y * x), $MachinePrecision], 2e-27], N[(N[(z * 0.0625), $MachinePrecision] * t + c), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -2.00000000000000004e64 or 2.0000000000000001e-27 < (*.f64 x y)

   1. Initial program 95.7%

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

   3. Taylor expanded in c around 0

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

      1. associate--l+ N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

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

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

      5. metadata-eval N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 85.6

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

   5. Applied rewrites 85.6%

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

   7. Applied rewrites 89.9%

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

   8. Taylor expanded in z around 0

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

   10. Applied rewrites 77.8%

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

   if -2.00000000000000004e64 < (*.f64 x y) < 2.0000000000000001e-27

   1. Initial program 98.6%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 71.7

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

   5. Applied rewrites 71.7%

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

   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 69.0%

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

   10. Applied rewrites 69.4%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot x \leq -2 \cdot 10^{+64}:\\ \;\;\;\;\mathsf{fma}\left(a, b \cdot -0.25, y \cdot x\right)\\ \mathbf{elif}\;y \cdot x \leq 2 \cdot 10^{-27}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot 0.0625, t, c\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, b \cdot -0.25, y \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 61.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot \left(b \cdot -0.25\right)\\ \mathbf{if}\;a \cdot b \leq -1 \cdot 10^{+250}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \cdot b \leq 2 \cdot 10^{+160}:\\ \;\;\;\;\mathsf{fma}\left(x, y, c\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (* a (* b -0.25))))
   (if (<= (* a b) -1e+250) t_1 (if (<= (* a b) 2e+160) (fma x y c) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = a * (b * -0.25);
	double tmp;
	if ((a * b) <= -1e+250) {
		tmp = t_1;
	} else if ((a * b) <= 2e+160) {
		tmp = fma(x, y, c);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(a * Float64(b * -0.25))
	tmp = 0.0
	if (Float64(a * b) <= -1e+250)
		tmp = t_1;
	elseif (Float64(a * b) <= 2e+160)
		tmp = fma(x, y, c);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(a * N[(b * -0.25), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(a * b), $MachinePrecision], -1e+250], t$95$1, If[LessEqual[N[(a * b), $MachinePrecision], 2e+160], N[(x * y + c), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a b) < -9.9999999999999992e249 or 2.00000000000000001e160 < (*.f64 a b)

   1. Initial program 90.7%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 81.5

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

   5. Applied rewrites 81.5%

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

   if -9.9999999999999992e249 < (*.f64 a b) < 2.00000000000000001e160

   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

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

      3. associate-+l+ N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 88.0

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

   5. Applied rewrites 88.0%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 60.8%

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

Alternative 16: 61.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 0.0625 \cdot \left(z \cdot t\right)\\ \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+199}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \cdot t \leq 10^{+34}:\\ \;\;\;\;\mathsf{fma}\left(x, y, c\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (* 0.0625 (* z t))))
   (if (<= (* z t) -1e+199) t_1 (if (<= (* z t) 1e+34) (fma x y c) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = 0.0625 * (z * t);
	double tmp;
	if ((z * t) <= -1e+199) {
		tmp = t_1;
	} else if ((z * t) <= 1e+34) {
		tmp = fma(x, y, c);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(0.0625 * Float64(z * t))
	tmp = 0.0
	if (Float64(z * t) <= -1e+199)
		tmp = t_1;
	elseif (Float64(z * t) <= 1e+34)
		tmp = fma(x, y, c);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(0.0625 * N[(z * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(z * t), $MachinePrecision], -1e+199], t$95$1, If[LessEqual[N[(z * t), $MachinePrecision], 1e+34], N[(x * y + c), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -1.0000000000000001e199 or 9.99999999999999946e33 < (*.f64 z t)

   1. Initial program 92.8%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 60.8

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

   5. Applied rewrites 60.8%

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

   if -1.0000000000000001e199 < (*.f64 z t) < 9.99999999999999946e33

   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

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

      3. associate-+l+ N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 70.3

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

   5. Applied rewrites 70.3%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 63.8%

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

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+199}:\\ \;\;\;\;0.0625 \cdot \left(z \cdot t\right)\\ \mathbf{elif}\;z \cdot t \leq 10^{+34}:\\ \;\;\;\;\mathsf{fma}\left(x, y, c\right)\\ \mathbf{else}:\\ \;\;\;\;0.0625 \cdot \left(z \cdot t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 98.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

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. 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. lift-+.f64 N/A

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

   4. associate--l+ N/A

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

   5. associate-+l+ N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-+.f64 N/A

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

4. Applied rewrites 99.6%

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

Alternative 18: 98.5% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

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. 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. lift-+.f64 N/A

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

   4. associate--l+ N/A

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

   5. associate-+l+ N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-+.f64 N/A

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

4. Applied rewrites 99.6%

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

   1. lift-+.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. associate-+l+ N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*r* N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. associate-*r* N/A

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

   11. lift-*.f64 N/A

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

   12. lower-fma.f64 98.9

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

6. Applied rewrites 98.9%

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

Alternative 19: 48.5% 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 97.3%

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

3. Taylor expanded in a around 0

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

   1. associate-+r+ N/A

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

   2. +-commutative N/A

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

   3. associate-+l+ N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. +-commutative N/A

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

   7. lower-fma.f64 72.4

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

5. Applied rewrites 72.4%

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

6. Taylor expanded in t around 0

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

8. Applied rewrites 49.9%

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

Alternative 20: 28.9% accurate, 7.8× speedup

Math

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

FPCore

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

C

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

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 = y * x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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 x around inf

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

   1. lower-*.f64 27.3

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

5. Applied rewrites 27.3%

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

6. Final simplification 27.3%

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

Reproduce

herbie shell --seed 2024221 
(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))