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

Percentage Accurate: 97.9% → 98.9%

Time: 10.3s

Alternatives: 12

Speedup: 1.5×

• 33.4% 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: 97.9% 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: 98.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 97.3%

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

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. associate-+l+ N/A

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

   11. lift-/.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. associate-/l* N/A

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

   14. *-commutative N/A

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

   15. lower-fma.f64 N/A

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

   16. div-inv N/A

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

   17. lower-*.f64 N/A

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

   18. metadata-eval N/A

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

   19. lift-*.f64 N/A

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

   20. lower-fma.f64 N/A

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

   21. lift-/.f64 N/A

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

   22. div-inv N/A

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

4. Applied egg-rr 98.4%

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

Alternative 2: 77.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(t \cdot 0.0625, z, x \cdot y\right)\\ t_2 := x \cdot y + \frac{t \cdot z}{16}\\ \mathbf{if}\;t\_2 \leq -6 \cdot 10^{+158}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 8 \cdot 10^{+178}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot -0.25, a, c\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (fma (* t 0.0625) z (* x y))) (t_2 (+ (* x y) (/ (* t z) 16.0))))
   (if (<= t_2 -6e+158) t_1 (if (<= t_2 8e+178) (fma (* b -0.25) a c) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = fma((t * 0.0625), z, (x * y));
	double t_2 = (x * y) + ((t * z) / 16.0);
	double tmp;
	if (t_2 <= -6e+158) {
		tmp = t_1;
	} else if (t_2 <= 8e+178) {
		tmp = fma((b * -0.25), a, c);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = fma(Float64(t * 0.0625), z, Float64(x * y))
	t_2 = Float64(Float64(x * y) + Float64(Float64(t * z) / 16.0))
	tmp = 0.0
	if (t_2 <= -6e+158)
		tmp = t_1;
	elseif (t_2 <= 8e+178)
		tmp = fma(Float64(b * -0.25), a, c);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(t * 0.0625), $MachinePrecision] * z + N[(x * y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * y), $MachinePrecision] + N[(N[(t * z), $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -6e+158], t$95$1, If[LessEqual[t$95$2, 8e+178], N[(N[(b * -0.25), $MachinePrecision] * a + c), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 x y) (/.f64 (*.f64 z t) #s(literal 16 binary64))) < -6e158 or 8.0000000000000004e178 < (+.f64 (*.f64 x y) (/.f64 (*.f64 z t) #s(literal 16 binary64)))

   1. Initial program 93.9%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 89.1

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

   5. Simplified 89.1%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lower-fma.f64 90.0

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

   7. Applied egg-rr 90.0%

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

   8. Taylor expanded in x around inf

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

      1. lower-*.f64 86.7

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

   10. Simplified 86.7%

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

   if -6e158 < (+.f64 (*.f64 x y) (/.f64 (*.f64 z t) #s(literal 16 binary64))) < 8.0000000000000004e178

   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 73.9

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

   5. Simplified 73.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 73.9

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

   7. Applied egg-rr 73.9%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y + \frac{t \cdot z}{16} \leq -6 \cdot 10^{+158}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot 0.0625, z, x \cdot y\right)\\ \mathbf{elif}\;x \cdot y + \frac{t \cdot z}{16} \leq 8 \cdot 10^{+178}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot -0.25, a, c\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot 0.0625, z, x \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(0.0625, t \cdot z, x \cdot y\right)\\ t_2 := x \cdot y + \frac{t \cdot z}{16}\\ \mathbf{if}\;t\_2 \leq -6 \cdot 10^{+158}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 8 \cdot 10^{+178}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot -0.25, a, c\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (fma 0.0625 (* t z) (* x y))) (t_2 (+ (* x y) (/ (* t z) 16.0))))
   (if (<= t_2 -6e+158) t_1 (if (<= t_2 8e+178) (fma (* b -0.25) a c) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = fma(0.0625, (t * z), (x * y));
	double t_2 = (x * y) + ((t * z) / 16.0);
	double tmp;
	if (t_2 <= -6e+158) {
		tmp = t_1;
	} else if (t_2 <= 8e+178) {
		tmp = fma((b * -0.25), a, c);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = fma(0.0625, Float64(t * z), Float64(x * y))
	t_2 = Float64(Float64(x * y) + Float64(Float64(t * z) / 16.0))
	tmp = 0.0
	if (t_2 <= -6e+158)
		tmp = t_1;
	elseif (t_2 <= 8e+178)
		tmp = fma(Float64(b * -0.25), a, c);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(0.0625 * N[(t * z), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * y), $MachinePrecision] + N[(N[(t * z), $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -6e+158], t$95$1, If[LessEqual[t$95$2, 8e+178], N[(N[(b * -0.25), $MachinePrecision] * a + c), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 x y) (/.f64 (*.f64 z t) #s(literal 16 binary64))) < -6e158 or 8.0000000000000004e178 < (+.f64 (*.f64 x y) (/.f64 (*.f64 z t) #s(literal 16 binary64)))

   1. Initial program 93.9%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 89.1

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

   5. Simplified 89.1%

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

   6. Taylor expanded in c around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 85.8

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

   8. Simplified 85.8%

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

   if -6e158 < (+.f64 (*.f64 x y) (/.f64 (*.f64 z t) #s(literal 16 binary64))) < 8.0000000000000004e178

   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 73.9

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

   5. Simplified 73.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 73.9

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

   7. Applied egg-rr 73.9%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y + \frac{t \cdot z}{16} \leq -6 \cdot 10^{+158}:\\ \;\;\;\;\mathsf{fma}\left(0.0625, t \cdot z, x \cdot y\right)\\ \mathbf{elif}\;x \cdot y + \frac{t \cdot z}{16} \leq 8 \cdot 10^{+178}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot -0.25, a, c\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.0625, t \cdot z, x \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 65.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(b \cdot -0.25, a, c\right)\\ t_2 := \mathsf{fma}\left(t \cdot z, 0.0625, c\right)\\ \mathbf{if}\;t \cdot z \leq -5 \cdot 10^{+50}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t \cdot z \leq -5 \cdot 10^{-284}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \cdot z \leq 2 \cdot 10^{+36}:\\ \;\;\;\;\mathsf{fma}\left(x, y, c\right)\\ \mathbf{elif}\;t \cdot z \leq 2 \cdot 10^{+117}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (fma (* b -0.25) a c)) (t_2 (fma (* t z) 0.0625 c)))
   (if (<= (* t z) -5e+50)
     t_2
     (if (<= (* t z) -5e-284)
       t_1
       (if (<= (* t z) 2e+36) (fma x y c) (if (<= (* t z) 2e+117) 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((b * -0.25), a, c);
	double t_2 = fma((t * z), 0.0625, c);
	double tmp;
	if ((t * z) <= -5e+50) {
		tmp = t_2;
	} else if ((t * z) <= -5e-284) {
		tmp = t_1;
	} else if ((t * z) <= 2e+36) {
		tmp = fma(x, y, c);
	} else if ((t * z) <= 2e+117) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = fma(Float64(b * -0.25), a, c)
	t_2 = fma(Float64(t * z), 0.0625, c)
	tmp = 0.0
	if (Float64(t * z) <= -5e+50)
		tmp = t_2;
	elseif (Float64(t * z) <= -5e-284)
		tmp = t_1;
	elseif (Float64(t * z) <= 2e+36)
		tmp = fma(x, y, c);
	elseif (Float64(t * z) <= 2e+117)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(b * -0.25), $MachinePrecision] * a + c), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t * z), $MachinePrecision] * 0.0625 + c), $MachinePrecision]}, If[LessEqual[N[(t * z), $MachinePrecision], -5e+50], t$95$2, If[LessEqual[N[(t * z), $MachinePrecision], -5e-284], t$95$1, If[LessEqual[N[(t * z), $MachinePrecision], 2e+36], N[(x * y + c), $MachinePrecision], If[LessEqual[N[(t * z), $MachinePrecision], 2e+117], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -5e50 or 2.0000000000000001e117 < (*.f64 z t)

   1. Initial program 95.0%

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

   3. Taylor expanded in z around 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 77.4

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

   5. Simplified 77.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 77.4

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

   7. Applied egg-rr 77.4%

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

   if -5e50 < (*.f64 z t) < -4.99999999999999973e-284 or 2.00000000000000008e36 < (*.f64 z t) < 2.0000000000000001e117

   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 71.3

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

   5. Simplified 71.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 71.3

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

   7. Applied egg-rr 71.3%

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

   if -4.99999999999999973e-284 < (*.f64 z t) < 2.00000000000000008e36

   1. Initial program 97.9%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 79.0

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

   5. Simplified 79.0%

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

   6. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 77.3

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

   8. Simplified 77.3%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot z \leq -5 \cdot 10^{+50}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot z, 0.0625, c\right)\\ \mathbf{elif}\;t \cdot z \leq -5 \cdot 10^{-284}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot -0.25, a, c\right)\\ \mathbf{elif}\;t \cdot z \leq 2 \cdot 10^{+36}:\\ \;\;\;\;\mathsf{fma}\left(x, y, c\right)\\ \mathbf{elif}\;t \cdot z \leq 2 \cdot 10^{+117}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot -0.25, a, c\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot z, 0.0625, c\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 63.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(b \cdot -0.25, a, c\right)\\ t_2 := 0.0625 \cdot \left(t \cdot z\right)\\ \mathbf{if}\;t \cdot z \leq -5 \cdot 10^{+96}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t \cdot z \leq -5 \cdot 10^{-284}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \cdot z \leq 2 \cdot 10^{+36}:\\ \;\;\;\;\mathsf{fma}\left(x, y, c\right)\\ \mathbf{elif}\;t \cdot z \leq 5 \cdot 10^{+136}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (fma (* b -0.25) a c)) (t_2 (* 0.0625 (* t z))))
   (if (<= (* t z) -5e+96)
     t_2
     (if (<= (* t z) -5e-284)
       t_1
       (if (<= (* t z) 2e+36) (fma x y c) (if (<= (* t z) 5e+136) 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((b * -0.25), a, c);
	double t_2 = 0.0625 * (t * z);
	double tmp;
	if ((t * z) <= -5e+96) {
		tmp = t_2;
	} else if ((t * z) <= -5e-284) {
		tmp = t_1;
	} else if ((t * z) <= 2e+36) {
		tmp = fma(x, y, c);
	} else if ((t * z) <= 5e+136) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = fma(Float64(b * -0.25), a, c)
	t_2 = Float64(0.0625 * Float64(t * z))
	tmp = 0.0
	if (Float64(t * z) <= -5e+96)
		tmp = t_2;
	elseif (Float64(t * z) <= -5e-284)
		tmp = t_1;
	elseif (Float64(t * z) <= 2e+36)
		tmp = fma(x, y, c);
	elseif (Float64(t * z) <= 5e+136)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(b * -0.25), $MachinePrecision] * a + c), $MachinePrecision]}, Block[{t$95$2 = N[(0.0625 * N[(t * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t * z), $MachinePrecision], -5e+96], t$95$2, If[LessEqual[N[(t * z), $MachinePrecision], -5e-284], t$95$1, If[LessEqual[N[(t * z), $MachinePrecision], 2e+36], N[(x * y + c), $MachinePrecision], If[LessEqual[N[(t * z), $MachinePrecision], 5e+136], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -5.0000000000000004e96 or 5.0000000000000002e136 < (*.f64 z t)

   1. Initial program 94.4%

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

   3. Taylor expanded in z around 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 72.8

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

   5. Simplified 72.8%

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

   if -5.0000000000000004e96 < (*.f64 z t) < -4.99999999999999973e-284 or 2.00000000000000008e36 < (*.f64 z t) < 5.0000000000000002e136

   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 67.7

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

   5. Simplified 67.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 67.7

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

   7. Applied egg-rr 67.7%

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

   if -4.99999999999999973e-284 < (*.f64 z t) < 2.00000000000000008e36

   1. Initial program 97.9%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 79.0

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

   5. Simplified 79.0%

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

   6. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 77.3

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

   8. Simplified 77.3%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot z \leq -5 \cdot 10^{+96}:\\ \;\;\;\;0.0625 \cdot \left(t \cdot z\right)\\ \mathbf{elif}\;t \cdot z \leq -5 \cdot 10^{-284}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot -0.25, a, c\right)\\ \mathbf{elif}\;t \cdot z \leq 2 \cdot 10^{+36}:\\ \;\;\;\;\mathsf{fma}\left(x, y, c\right)\\ \mathbf{elif}\;t \cdot z \leq 5 \cdot 10^{+136}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot -0.25, a, c\right)\\ \mathbf{else}:\\ \;\;\;\;0.0625 \cdot \left(t \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 89.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \cdot z \leq -5 \cdot 10^{+50}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot 0.0625, z, \mathsf{fma}\left(x, y, c\right)\right)\\ \mathbf{elif}\;t \cdot z \leq 10000000000000:\\ \;\;\;\;\mathsf{fma}\left(a, b \cdot -0.25, \mathsf{fma}\left(x, y, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.0625, t \cdot z, \mathsf{fma}\left(a, b \cdot -0.25, c\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (<= (* t z) -5e+50)
   (fma (* t 0.0625) z (fma x y c))
   (if (<= (* t z) 10000000000000.0)
     (fma a (* b -0.25) (fma x y c))
     (fma 0.0625 (* t z) (fma a (* b -0.25) c)))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if ((t * z) <= -5e+50) {
		tmp = fma((t * 0.0625), z, fma(x, y, c));
	} else if ((t * z) <= 10000000000000.0) {
		tmp = fma(a, (b * -0.25), fma(x, y, c));
	} else {
		tmp = fma(0.0625, (t * z), fma(a, (b * -0.25), c));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if (Float64(t * z) <= -5e+50)
		tmp = fma(Float64(t * 0.0625), z, fma(x, y, c));
	elseif (Float64(t * z) <= 10000000000000.0)
		tmp = fma(a, Float64(b * -0.25), fma(x, y, c));
	else
		tmp = fma(0.0625, Float64(t * z), fma(a, Float64(b * -0.25), c));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[LessEqual[N[(t * z), $MachinePrecision], -5e+50], N[(N[(t * 0.0625), $MachinePrecision] * z + N[(x * y + c), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(t * z), $MachinePrecision], 10000000000000.0], N[(a * N[(b * -0.25), $MachinePrecision] + N[(x * y + c), $MachinePrecision]), $MachinePrecision], N[(0.0625 * N[(t * z), $MachinePrecision] + N[(a * N[(b * -0.25), $MachinePrecision] + c), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -5e50

   1. Initial program 98.2%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 89.5

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

   5. Simplified 89.5%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lower-fma.f64 89.5

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

   7. Applied egg-rr 89.5%

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

   if -5e50 < (*.f64 z t) < 1e13

   1. Initial program 98.5%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   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 96.9

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

   5. Simplified 96.9%

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

   if 1e13 < (*.f64 z t)

   1. Initial program 93.8%

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

   3. Taylor expanded in 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 89.5

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

   5. Simplified 89.5%

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

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot z \leq -5 \cdot 10^{+50}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot 0.0625, z, \mathsf{fma}\left(x, y, c\right)\right)\\ \mathbf{elif}\;t \cdot z \leq 10000000000000:\\ \;\;\;\;\mathsf{fma}\left(a, b \cdot -0.25, \mathsf{fma}\left(x, y, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.0625, t \cdot z, \mathsf{fma}\left(a, b \cdot -0.25, c\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 88.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \cdot z \leq -5 \cdot 10^{+50}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot 0.0625, z, \mathsf{fma}\left(x, y, c\right)\right)\\ \mathbf{elif}\;t \cdot z \leq 5 \cdot 10^{+156}:\\ \;\;\;\;\mathsf{fma}\left(a, b \cdot -0.25, \mathsf{fma}\left(x, y, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot z, 0.0625, c\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (<= (* t z) -5e+50)
   (fma (* t 0.0625) z (fma x y c))
   (if (<= (* t z) 5e+156)
     (fma a (* b -0.25) (fma x y c))
     (fma (* t z) 0.0625 c))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if ((t * z) <= -5e+50) {
		tmp = fma((t * 0.0625), z, fma(x, y, c));
	} else if ((t * z) <= 5e+156) {
		tmp = fma(a, (b * -0.25), fma(x, y, c));
	} else {
		tmp = fma((t * z), 0.0625, c);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if (Float64(t * z) <= -5e+50)
		tmp = fma(Float64(t * 0.0625), z, fma(x, y, c));
	elseif (Float64(t * z) <= 5e+156)
		tmp = fma(a, Float64(b * -0.25), fma(x, y, c));
	else
		tmp = fma(Float64(t * z), 0.0625, c);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[LessEqual[N[(t * z), $MachinePrecision], -5e+50], N[(N[(t * 0.0625), $MachinePrecision] * z + N[(x * y + c), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(t * z), $MachinePrecision], 5e+156], N[(a * N[(b * -0.25), $MachinePrecision] + N[(x * y + c), $MachinePrecision]), $MachinePrecision], N[(N[(t * z), $MachinePrecision] * 0.0625 + c), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -5e50

   1. Initial program 98.2%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 89.5

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

   5. Simplified 89.5%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lower-fma.f64 89.5

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

   7. Applied egg-rr 89.5%

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

   if -5e50 < (*.f64 z t) < 4.99999999999999992e156

   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 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 93.8

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

   5. Simplified 93.8%

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

   if 4.99999999999999992e156 < (*.f64 z t)

   1. Initial program 90.0%

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

   3. Taylor expanded in z around 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 90.5

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

   5. Simplified 90.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 90.5

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

   7. Applied egg-rr 90.5%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot z \leq -5 \cdot 10^{+50}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot 0.0625, z, \mathsf{fma}\left(x, y, c\right)\right)\\ \mathbf{elif}\;t \cdot z \leq 5 \cdot 10^{+156}:\\ \;\;\;\;\mathsf{fma}\left(a, b \cdot -0.25, \mathsf{fma}\left(x, y, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot z, 0.0625, c\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 88.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \cdot z \leq -5 \cdot 10^{+50}:\\ \;\;\;\;\mathsf{fma}\left(0.0625, t \cdot z, \mathsf{fma}\left(x, y, c\right)\right)\\ \mathbf{elif}\;t \cdot z \leq 5 \cdot 10^{+156}:\\ \;\;\;\;\mathsf{fma}\left(a, b \cdot -0.25, \mathsf{fma}\left(x, y, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot z, 0.0625, c\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (<= (* t z) -5e+50)
   (fma 0.0625 (* t z) (fma x y c))
   (if (<= (* t z) 5e+156)
     (fma a (* b -0.25) (fma x y c))
     (fma (* t z) 0.0625 c))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if ((t * z) <= -5e+50) {
		tmp = fma(0.0625, (t * z), fma(x, y, c));
	} else if ((t * z) <= 5e+156) {
		tmp = fma(a, (b * -0.25), fma(x, y, c));
	} else {
		tmp = fma((t * z), 0.0625, c);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if (Float64(t * z) <= -5e+50)
		tmp = fma(0.0625, Float64(t * z), fma(x, y, c));
	elseif (Float64(t * z) <= 5e+156)
		tmp = fma(a, Float64(b * -0.25), fma(x, y, c));
	else
		tmp = fma(Float64(t * z), 0.0625, c);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[LessEqual[N[(t * z), $MachinePrecision], -5e+50], N[(0.0625 * N[(t * z), $MachinePrecision] + N[(x * y + c), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(t * z), $MachinePrecision], 5e+156], N[(a * N[(b * -0.25), $MachinePrecision] + N[(x * y + c), $MachinePrecision]), $MachinePrecision], N[(N[(t * z), $MachinePrecision] * 0.0625 + c), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -5e50

   1. Initial program 98.2%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 89.5

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

   5. Simplified 89.5%

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

   if -5e50 < (*.f64 z t) < 4.99999999999999992e156

   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 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 93.8

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

   5. Simplified 93.8%

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

   if 4.99999999999999992e156 < (*.f64 z t)

   1. Initial program 90.0%

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

   3. Taylor expanded in z around 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 90.5

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

   5. Simplified 90.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 90.5

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

   7. Applied egg-rr 90.5%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot z \leq -5 \cdot 10^{+50}:\\ \;\;\;\;\mathsf{fma}\left(0.0625, t \cdot z, \mathsf{fma}\left(x, y, c\right)\right)\\ \mathbf{elif}\;t \cdot z \leq 5 \cdot 10^{+156}:\\ \;\;\;\;\mathsf{fma}\left(a, b \cdot -0.25, \mathsf{fma}\left(x, y, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot z, 0.0625, c\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 87.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(x, y, \left(a \cdot b\right) \cdot -0.25\right)\\ \mathbf{if}\;a \cdot b \leq -5 \cdot 10^{+201}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \cdot b \leq 2 \cdot 10^{+166}:\\ \;\;\;\;\mathsf{fma}\left(0.0625, t \cdot z, \mathsf{fma}\left(x, y, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (fma x y (* (* a b) -0.25))))
   (if (<= (* a b) -5e+201)
     t_1
     (if (<= (* a b) 2e+166) (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, ((a * b) * -0.25));
	double tmp;
	if ((a * b) <= -5e+201) {
		tmp = t_1;
	} else if ((a * b) <= 2e+166) {
		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(Float64(a * b) * -0.25))
	tmp = 0.0
	if (Float64(a * b) <= -5e+201)
		tmp = t_1;
	elseif (Float64(a * b) <= 2e+166)
		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[(N[(a * b), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(a * b), $MachinePrecision], -5e+201], t$95$1, If[LessEqual[N[(a * b), $MachinePrecision], 2e+166], 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) < -4.9999999999999995e201 or 1.99999999999999988e166 < (*.f64 a b)

   1. Initial program 94.5%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   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 89.9

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

   5. Simplified 89.9%

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

   6. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 84.6

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

   8. Simplified 84.6%

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

   if -4.9999999999999995e201 < (*.f64 a b) < 1.99999999999999988e166

   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 88.8

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

   5. Simplified 88.8%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b \leq -5 \cdot 10^{+201}:\\ \;\;\;\;\mathsf{fma}\left(x, y, \left(a \cdot b\right) \cdot -0.25\right)\\ \mathbf{elif}\;a \cdot b \leq 2 \cdot 10^{+166}:\\ \;\;\;\;\mathsf{fma}\left(0.0625, t \cdot z, \mathsf{fma}\left(x, y, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, y, \left(a \cdot b\right) \cdot -0.25\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 63.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 0.0625 \cdot \left(t \cdot z\right)\\ \mathbf{if}\;t \cdot z \leq -2 \cdot 10^{+182}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \cdot z \leq 2 \cdot 10^{+162}:\\ \;\;\;\;\mathsf{fma}\left(x, y, c\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (* 0.0625 (* t z))))
   (if (<= (* t z) -2e+182) t_1 (if (<= (* t z) 2e+162) (fma x y c) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = 0.0625 * (t * z);
	double tmp;
	if ((t * z) <= -2e+182) {
		tmp = t_1;
	} else if ((t * z) <= 2e+162) {
		tmp = fma(x, y, c);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(0.0625 * Float64(t * z))
	tmp = 0.0
	if (Float64(t * z) <= -2e+182)
		tmp = t_1;
	elseif (Float64(t * z) <= 2e+162)
		tmp = fma(x, y, c);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(0.0625 * N[(t * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t * z), $MachinePrecision], -2e+182], t$95$1, If[LessEqual[N[(t * z), $MachinePrecision], 2e+162], N[(x * y + c), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -2.0000000000000001e182 or 1.9999999999999999e162 < (*.f64 z t)

   1. Initial program 93.0%

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

   3. Taylor expanded in z around 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 83.3

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

   5. Simplified 83.3%

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

   if -2.0000000000000001e182 < (*.f64 z t) < 1.9999999999999999e162

   1. Initial program 98.9%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 72.0

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

   5. Simplified 72.0%

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

   6. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 63.1

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

   8. Simplified 63.1%

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

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot z \leq -2 \cdot 10^{+182}:\\ \;\;\;\;0.0625 \cdot \left(t \cdot z\right)\\ \mathbf{elif}\;t \cdot z \leq 2 \cdot 10^{+162}:\\ \;\;\;\;\mathsf{fma}\left(x, y, c\right)\\ \mathbf{else}:\\ \;\;\;\;0.0625 \cdot \left(t \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 48.1% accurate, 6.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 97.3%

   \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
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 76.8

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

5. Simplified 76.8%

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

6. Taylor expanded in t around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 50.7

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

8. Simplified 50.7%

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

Alternative 12: 28.6% accurate, 7.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.3%

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

3. Taylor expanded in x around inf

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

   1. lower-*.f64 27.5

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

5. Simplified 27.5%

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

Reproduce

herbie shell --seed 2024207 
(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))