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

Percentage Accurate: 97.5% → 98.4%

Time: 37.8s

Alternatives: 11

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: 97.5% 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: 98.4% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

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 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 98.8

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

5. Applied rewrites 98.8%

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

Alternative 2: 74.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(x, y, z \cdot \left(0.0625 \cdot t\right)\right)\\ t_2 := x \cdot y + \frac{t \cdot z}{16}\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{-10}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+122}:\\ \;\;\;\;c + a \cdot \left(b \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (fma x y (* z (* 0.0625 t)))) (t_2 (+ (* x y) (/ (* t z) 16.0))))
   (if (<= t_2 -5e-10) t_1 (if (<= t_2 2e+122) (+ 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(x, y, (z * (0.0625 * t)));
	double t_2 = (x * y) + ((t * z) / 16.0);
	double tmp;
	if (t_2 <= -5e-10) {
		tmp = t_1;
	} else if (t_2 <= 2e+122) {
		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(x, y, Float64(z * Float64(0.0625 * t)))
	t_2 = Float64(Float64(x * y) + Float64(Float64(t * z) / 16.0))
	tmp = 0.0
	if (t_2 <= -5e-10)
		tmp = t_1;
	elseif (t_2 <= 2e+122)
		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[(x * y + N[(z * N[(0.0625 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * y), $MachinePrecision] + N[(N[(t * z), $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e-10], t$95$1, If[LessEqual[t$95$2, 2e+122], 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))) < -5.00000000000000031e-10 or 2.00000000000000003e122 < (+.f64 (*.f64 x y) (/.f64 (*.f64 z t) #s(literal 16 binary64)))

   1. Initial program 97.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-+l- N/A

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

      8. lift-+.f64 N/A

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

      9. associate--l+ N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. div-inv N/A

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

      15. lower-*.f64 N/A

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

      16. metadata-eval N/A

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

      17. sub-neg N/A

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

   4. Applied rewrites 98.8%

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

   5. Taylor expanded in z around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 78.5

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

   7. Applied rewrites 78.5%

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

   if -5.00000000000000031e-10 < (+.f64 (*.f64 x y) (/.f64 (*.f64 z t) #s(literal 16 binary64))) < 2.00000000000000003e122

   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 77.8

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

   5. Applied rewrites 77.8%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y + \frac{t \cdot z}{16} \leq -5 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(x, y, z \cdot \left(0.0625 \cdot t\right)\right)\\ \mathbf{elif}\;x \cdot y + \frac{t \cdot z}{16} \leq 2 \cdot 10^{+122}:\\ \;\;\;\;c + a \cdot \left(b \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, y, z \cdot \left(0.0625 \cdot t\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 65.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := c + 0.0625 \cdot \left(t \cdot z\right)\\ \mathbf{if}\;t \cdot z \leq -2 \cdot 10^{+120}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \cdot z \leq -2 \cdot 10^{+66}:\\ \;\;\;\;c + a \cdot \left(b \cdot -0.25\right)\\ \mathbf{elif}\;t \cdot z \leq 10^{+85}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (+ c (* 0.0625 (* t z)))))
   (if (<= (* t z) -2e+120)
     t_1
     (if (<= (* t z) -2e+66)
       (+ c (* a (* b -0.25)))
       (if (<= (* t z) 1e+85) (fma y x c) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = c + (0.0625 * (t * z));
	double tmp;
	if ((t * z) <= -2e+120) {
		tmp = t_1;
	} else if ((t * z) <= -2e+66) {
		tmp = c + (a * (b * -0.25));
	} else if ((t * z) <= 1e+85) {
		tmp = fma(y, x, c);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(c + Float64(0.0625 * Float64(t * z)))
	tmp = 0.0
	if (Float64(t * z) <= -2e+120)
		tmp = t_1;
	elseif (Float64(t * z) <= -2e+66)
		tmp = Float64(c + Float64(a * Float64(b * -0.25)));
	elseif (Float64(t * z) <= 1e+85)
		tmp = fma(y, x, c);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(c + N[(0.0625 * N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t * z), $MachinePrecision], -2e+120], t$95$1, If[LessEqual[N[(t * z), $MachinePrecision], -2e+66], N[(c + N[(a * N[(b * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(t * z), $MachinePrecision], 1e+85], N[(y * x + c), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -2e120 or 1e85 < (*.f64 z t)

   1. Initial program 95.8%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 77.7

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

   5. Applied rewrites 77.7%

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

   if -2e120 < (*.f64 z t) < -1.99999999999999989e66

   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 88.0

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

   5. Applied rewrites 88.0%

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

   if -1.99999999999999989e66 < (*.f64 z t) < 1e85

   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 x around inf

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

      1. lower-*.f64 67.1

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

   5. Applied rewrites 67.1%

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

      1. *-commutative N/A

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

      2. lower-fma.f64 67.1

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

   7. Applied rewrites 67.1%

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

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot z \leq -2 \cdot 10^{+120}:\\ \;\;\;\;c + 0.0625 \cdot \left(t \cdot z\right)\\ \mathbf{elif}\;t \cdot z \leq -2 \cdot 10^{+66}:\\ \;\;\;\;c + a \cdot \left(b \cdot -0.25\right)\\ \mathbf{elif}\;t \cdot z \leq 10^{+85}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \mathbf{else}:\\ \;\;\;\;c + 0.0625 \cdot \left(t \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \cdot z \leq -2 \cdot 10^{+62}:\\ \;\;\;\;\mathsf{fma}\left(0.0625, t \cdot z, \mathsf{fma}\left(a, b \cdot -0.25, c\right)\right)\\ \mathbf{elif}\;t \cdot z \leq 5 \cdot 10^{+75}:\\ \;\;\;\;\mathsf{fma}\left(a, b \cdot -0.25, \mathsf{fma}\left(x, y, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.0625, t \cdot z, \mathsf{fma}\left(x, y, c\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (<= (* t z) -2e+62)
   (fma 0.0625 (* t z) (fma a (* b -0.25) c))
   (if (<= (* t z) 5e+75)
     (fma a (* b -0.25) (fma x y c))
     (fma 0.0625 (* t z) (fma x y c)))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if ((t * z) <= -2e+62) {
		tmp = fma(0.0625, (t * z), fma(a, (b * -0.25), c));
	} else if ((t * z) <= 5e+75) {
		tmp = fma(a, (b * -0.25), fma(x, y, c));
	} else {
		tmp = fma(0.0625, (t * z), fma(x, y, c));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if (Float64(t * z) <= -2e+62)
		tmp = fma(0.0625, Float64(t * z), fma(a, Float64(b * -0.25), c));
	elseif (Float64(t * z) <= 5e+75)
		tmp = fma(a, Float64(b * -0.25), fma(x, y, c));
	else
		tmp = fma(0.0625, Float64(t * z), fma(x, y, c));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[LessEqual[N[(t * z), $MachinePrecision], -2e+62], N[(0.0625 * N[(t * z), $MachinePrecision] + N[(a * N[(b * -0.25), $MachinePrecision] + c), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(t * z), $MachinePrecision], 5e+75], N[(a * N[(b * -0.25), $MachinePrecision] + N[(x * y + c), $MachinePrecision]), $MachinePrecision], N[(0.0625 * N[(t * z), $MachinePrecision] + N[(x * y + c), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -2.00000000000000007e62

   1. Initial program 94.6%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   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 92.1

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

   5. Applied rewrites 92.1%

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

   if -2.00000000000000007e62 < (*.f64 z t) < 5.0000000000000002e75

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-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 95.0

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

   5. Applied rewrites 95.0%

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

   if 5.0000000000000002e75 < (*.f64 z t)

   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 86.4

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

   5. Applied rewrites 86.4%

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

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot z \leq -2 \cdot 10^{+62}:\\ \;\;\;\;\mathsf{fma}\left(0.0625, t \cdot z, \mathsf{fma}\left(a, b \cdot -0.25, c\right)\right)\\ \mathbf{elif}\;t \cdot z \leq 5 \cdot 10^{+75}:\\ \;\;\;\;\mathsf{fma}\left(a, b \cdot -0.25, \mathsf{fma}\left(x, y, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.0625, t \cdot z, \mathsf{fma}\left(x, y, c\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 88.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \cdot z \leq -2 \cdot 10^{+121}:\\ \;\;\;\;\mathsf{fma}\left(a, b \cdot -0.25, 0.0625 \cdot \left(t \cdot z\right)\right)\\ \mathbf{elif}\;t \cdot z \leq 5 \cdot 10^{+75}:\\ \;\;\;\;\mathsf{fma}\left(a, b \cdot -0.25, \mathsf{fma}\left(x, y, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.0625, t \cdot z, \mathsf{fma}\left(x, y, c\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (<= (* t z) -2e+121)
   (fma a (* b -0.25) (* 0.0625 (* t z)))
   (if (<= (* t z) 5e+75)
     (fma a (* b -0.25) (fma x y c))
     (fma 0.0625 (* t z) (fma x y c)))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if ((t * z) <= -2e+121) {
		tmp = fma(a, (b * -0.25), (0.0625 * (t * z)));
	} else if ((t * z) <= 5e+75) {
		tmp = fma(a, (b * -0.25), fma(x, y, c));
	} else {
		tmp = fma(0.0625, (t * z), fma(x, y, c));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if (Float64(t * z) <= -2e+121)
		tmp = fma(a, Float64(b * -0.25), Float64(0.0625 * Float64(t * z)));
	elseif (Float64(t * z) <= 5e+75)
		tmp = fma(a, Float64(b * -0.25), fma(x, y, c));
	else
		tmp = fma(0.0625, Float64(t * z), fma(x, y, c));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[LessEqual[N[(t * z), $MachinePrecision], -2e+121], N[(a * N[(b * -0.25), $MachinePrecision] + N[(0.0625 * N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(t * z), $MachinePrecision], 5e+75], N[(a * N[(b * -0.25), $MachinePrecision] + N[(x * y + c), $MachinePrecision]), $MachinePrecision], N[(0.0625 * N[(t * z), $MachinePrecision] + N[(x * y + c), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -2.00000000000000007e121

   1. Initial program 93.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-+l- N/A

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

      8. lift-+.f64 N/A

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

      9. associate--l+ N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. div-inv N/A

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

      15. lower-*.f64 N/A

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

      16. metadata-eval N/A

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

      17. sub-neg N/A

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

   4. Applied rewrites 97.8%

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

   6. Applied rewrites 95.6%

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

   7. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 91.5

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

   9. Applied rewrites 91.5%

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

   if -2.00000000000000007e121 < (*.f64 z t) < 5.0000000000000002e75

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-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 94.1

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

   5. Applied rewrites 94.1%

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

   if 5.0000000000000002e75 < (*.f64 z t)

   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 86.4

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

   5. Applied rewrites 86.4%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot z \leq -2 \cdot 10^{+121}:\\ \;\;\;\;\mathsf{fma}\left(a, b \cdot -0.25, 0.0625 \cdot \left(t \cdot z\right)\right)\\ \mathbf{elif}\;t \cdot z \leq 5 \cdot 10^{+75}:\\ \;\;\;\;\mathsf{fma}\left(a, b \cdot -0.25, \mathsf{fma}\left(x, y, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.0625, t \cdot z, \mathsf{fma}\left(x, y, c\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 89.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(0.0625, t \cdot z, \mathsf{fma}\left(x, y, c\right)\right)\\ \mathbf{if}\;t \cdot z \leq -2 \cdot 10^{+121}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \cdot z \leq 5 \cdot 10^{+75}:\\ \;\;\;\;\mathsf{fma}\left(a, b \cdot -0.25, \mathsf{fma}\left(x, y, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (fma 0.0625 (* t z) (fma x y c))))
   (if (<= (* t z) -2e+121)
     t_1
     (if (<= (* t z) 5e+75) (fma a (* b -0.25) (fma x y c)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = fma(0.0625, (t * z), fma(x, y, c));
	double tmp;
	if ((t * z) <= -2e+121) {
		tmp = t_1;
	} else if ((t * z) <= 5e+75) {
		tmp = fma(a, (b * -0.25), fma(x, y, c));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = fma(0.0625, Float64(t * z), fma(x, y, c))
	tmp = 0.0
	if (Float64(t * z) <= -2e+121)
		tmp = t_1;
	elseif (Float64(t * z) <= 5e+75)
		tmp = fma(a, Float64(b * -0.25), fma(x, y, c));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(0.0625 * N[(t * z), $MachinePrecision] + N[(x * y + c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t * z), $MachinePrecision], -2e+121], t$95$1, If[LessEqual[N[(t * z), $MachinePrecision], 5e+75], N[(a * N[(b * -0.25), $MachinePrecision] + N[(x * y + c), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -2.00000000000000007e121 or 5.0000000000000002e75 < (*.f64 z t)

   1. Initial program 95.8%

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

   3. Taylor expanded in a around 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 86.4

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

   5. Applied rewrites 86.4%

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

   if -2.00000000000000007e121 < (*.f64 z t) < 5.0000000000000002e75

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-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 94.1

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

   5. Applied rewrites 94.1%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot z \leq -2 \cdot 10^{+121}:\\ \;\;\;\;\mathsf{fma}\left(0.0625, t \cdot z, \mathsf{fma}\left(x, y, c\right)\right)\\ \mathbf{elif}\;t \cdot z \leq 5 \cdot 10^{+75}:\\ \;\;\;\;\mathsf{fma}\left(a, b \cdot -0.25, \mathsf{fma}\left(x, y, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.0625, t \cdot z, \mathsf{fma}\left(x, y, c\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(x, y, -0.25 \cdot \left(a \cdot b\right)\right)\\ \mathbf{if}\;a \cdot b \leq -1 \cdot 10^{+177}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \cdot b \leq 10^{+25}:\\ \;\;\;\;\mathsf{fma}\left(0.0625, t \cdot z, \mathsf{fma}\left(x, y, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (fma x y (* -0.25 (* a b)))))
   (if (<= (* a b) -1e+177)
     t_1
     (if (<= (* a b) 1e+25) (fma 0.0625 (* t z) (fma x y c)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = fma(x, y, (-0.25 * (a * b)));
	double tmp;
	if ((a * b) <= -1e+177) {
		tmp = t_1;
	} else if ((a * b) <= 1e+25) {
		tmp = fma(0.0625, (t * z), fma(x, y, c));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = fma(x, y, Float64(-0.25 * Float64(a * b)))
	tmp = 0.0
	if (Float64(a * b) <= -1e+177)
		tmp = t_1;
	elseif (Float64(a * b) <= 1e+25)
		tmp = fma(0.0625, Float64(t * z), fma(x, y, c));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(x * y + N[(-0.25 * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(a * b), $MachinePrecision], -1e+177], t$95$1, If[LessEqual[N[(a * b), $MachinePrecision], 1e+25], N[(0.0625 * N[(t * z), $MachinePrecision] + N[(x * y + c), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a b) < -1e177 or 1.00000000000000009e25 < (*.f64 a b)

   1. Initial program 97.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-+l- N/A

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

      8. lift-+.f64 N/A

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

      9. associate--l+ N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. div-inv N/A

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

      15. lower-*.f64 N/A

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

      16. metadata-eval N/A

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

      17. sub-neg N/A

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

   4. Applied rewrites 98.8%

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

   5. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 80.4

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

   7. Applied rewrites 80.4%

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

   if -1e177 < (*.f64 a b) < 1.00000000000000009e25

   1. Initial program 98.8%

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

   3. Taylor expanded in a around 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 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b \leq -1 \cdot 10^{+177}:\\ \;\;\;\;\mathsf{fma}\left(x, y, -0.25 \cdot \left(a \cdot b\right)\right)\\ \mathbf{elif}\;a \cdot b \leq 10^{+25}:\\ \;\;\;\;\mathsf{fma}\left(0.0625, t \cdot z, \mathsf{fma}\left(x, y, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, y, -0.25 \cdot \left(a \cdot b\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 65.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := c + 0.0625 \cdot \left(t \cdot z\right)\\ \mathbf{if}\;t \cdot z \leq -2 \cdot 10^{+62}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \cdot z \leq 10^{+85}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (+ c (* 0.0625 (* t z)))))
   (if (<= (* t z) -2e+62) t_1 (if (<= (* t z) 1e+85) (fma y x c) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = c + (0.0625 * (t * z));
	double tmp;
	if ((t * z) <= -2e+62) {
		tmp = t_1;
	} else if ((t * z) <= 1e+85) {
		tmp = fma(y, x, c);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(c + Float64(0.0625 * Float64(t * z)))
	tmp = 0.0
	if (Float64(t * z) <= -2e+62)
		tmp = t_1;
	elseif (Float64(t * z) <= 1e+85)
		tmp = fma(y, x, c);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(c + N[(0.0625 * N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t * z), $MachinePrecision], -2e+62], t$95$1, If[LessEqual[N[(t * z), $MachinePrecision], 1e+85], N[(y * x + c), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -2.00000000000000007e62 or 1e85 < (*.f64 z t)

   1. Initial program 96.2%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 74.0

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

   5. Applied rewrites 74.0%

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

   if -2.00000000000000007e62 < (*.f64 z t) < 1e85

   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 x around inf

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

      1. lower-*.f64 67.5

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

   5. Applied rewrites 67.5%

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

      1. *-commutative N/A

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

      2. lower-fma.f64 67.5

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

   7. Applied rewrites 67.5%

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

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot z \leq -2 \cdot 10^{+62}:\\ \;\;\;\;c + 0.0625 \cdot \left(t \cdot z\right)\\ \mathbf{elif}\;t \cdot z \leq 10^{+85}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \mathbf{else}:\\ \;\;\;\;c + 0.0625 \cdot \left(t \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 63.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 0.0625 \cdot \left(t \cdot z\right)\\ \mathbf{if}\;t \cdot z \leq -2 \cdot 10^{+121}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \cdot z \leq 10^{+209}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (* 0.0625 (* t z))))
   (if (<= (* t z) -2e+121) t_1 (if (<= (* t z) 1e+209) (fma y x c) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = 0.0625 * (t * z);
	double tmp;
	if ((t * z) <= -2e+121) {
		tmp = t_1;
	} else if ((t * z) <= 1e+209) {
		tmp = fma(y, x, c);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(0.0625 * Float64(t * z))
	tmp = 0.0
	if (Float64(t * z) <= -2e+121)
		tmp = t_1;
	elseif (Float64(t * z) <= 1e+209)
		tmp = fma(y, x, c);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(0.0625 * N[(t * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t * z), $MachinePrecision], -2e+121], t$95$1, If[LessEqual[N[(t * z), $MachinePrecision], 1e+209], N[(y * x + c), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -2.00000000000000007e121 or 1.0000000000000001e209 < (*.f64 z t)

   1. Initial program 94.3%

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

   3. Taylor expanded in z around 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 83.0

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

   5. Applied rewrites 83.0%

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

   if -2.00000000000000007e121 < (*.f64 z t) < 1.0000000000000001e209

   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 x around inf

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

      1. lower-*.f64 61.6

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

   5. Applied rewrites 61.6%

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

      1. *-commutative N/A

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

      2. lower-fma.f64 61.6

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

   7. Applied rewrites 61.6%

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

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot z \leq -2 \cdot 10^{+121}:\\ \;\;\;\;0.0625 \cdot \left(t \cdot z\right)\\ \mathbf{elif}\;t \cdot z \leq 10^{+209}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \mathbf{else}:\\ \;\;\;\;0.0625 \cdot \left(t \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 48.6% accurate, 6.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

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 x around inf

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

   1. lower-*.f64 48.2

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

5. Applied rewrites 48.2%

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

   1. *-commutative N/A

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

   2. lower-fma.f64 48.2

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

7. Applied rewrites 48.2%

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

Alternative 11: 28.3% accurate, 7.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 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 x around inf

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

   1. lower-*.f64 31.1

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

5. Applied rewrites 31.1%

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

Reproduce

herbie shell --seed 2024220 
(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))