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

Percentage Accurate: 97.7% → 98.8%

Time: 7.9s

Alternatives: 12

Speedup: 1.5×

• 32.5% 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: 97.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.8% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. sub-neg N/A

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

   3. lift-+.f64 N/A

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

   4. +-commutative N/A

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

   5. associate-+l+ N/A

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

   6. lift-/.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. associate-/l* N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. div-inv N/A

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

   12. lower-*.f64 N/A

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

   13. metadata-eval N/A

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

   14. lift-*.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-fma.f64 N/A

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

   17. lift-/.f64 N/A

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

   18. div-inv N/A

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

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

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

   20. metadata-eval N/A

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

   21. metadata-eval N/A

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

   22. metadata-eval N/A

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

   23. metadata-eval N/A

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

   24. lower-*.f64 N/A

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

4. Applied rewrites 98.4%

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

Alternative 2: 76.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z \cdot t, 0.0625, x \cdot y\right)\\ t_2 := x \cdot y + \frac{z \cdot t}{16}\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+132}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq -5 \cdot 10^{+64}:\\ \;\;\;\;\mathsf{fma}\left(x, y, c\right)\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+119}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, a \cdot b, c\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (fma (* z t) 0.0625 (* x y))) (t_2 (+ (* x y) (/ (* z t) 16.0))))
   (if (<= t_2 -1e+132)
     t_1
     (if (<= t_2 -5e+64)
       (fma x y c)
       (if (<= t_2 5e+119) (fma -0.25 (* a b) c) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = fma((z * t), 0.0625, (x * y));
	double t_2 = (x * y) + ((z * t) / 16.0);
	double tmp;
	if (t_2 <= -1e+132) {
		tmp = t_1;
	} else if (t_2 <= -5e+64) {
		tmp = fma(x, y, c);
	} else if (t_2 <= 5e+119) {
		tmp = fma(-0.25, (a * b), c);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = fma(Float64(z * t), 0.0625, Float64(x * y))
	t_2 = Float64(Float64(x * y) + Float64(Float64(z * t) / 16.0))
	tmp = 0.0
	if (t_2 <= -1e+132)
		tmp = t_1;
	elseif (t_2 <= -5e+64)
		tmp = fma(x, y, c);
	elseif (t_2 <= 5e+119)
		tmp = fma(-0.25, Float64(a * b), c);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(z * t), $MachinePrecision] * 0.0625 + N[(x * y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * y), $MachinePrecision] + N[(N[(z * t), $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+132], t$95$1, If[LessEqual[t$95$2, -5e+64], N[(x * y + c), $MachinePrecision], If[LessEqual[t$95$2, 5e+119], N[(-0.25 * N[(a * b), $MachinePrecision] + c), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (*.f64 x y) (/.f64 (*.f64 z t) #s(literal 16 binary64))) < -9.99999999999999991e131 or 4.9999999999999999e119 < (+.f64 (*.f64 x y) (/.f64 (*.f64 z t) #s(literal 16 binary64)))

   1. Initial program 95.2%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 87.4

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

   5. Applied rewrites 87.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 53.6%

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

   10. Applied rewrites 54.2%

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

   11. Taylor expanded in c around 0

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

   13. Applied rewrites 80.6%

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

   if -9.99999999999999991e131 < (+.f64 (*.f64 x y) (/.f64 (*.f64 z t) #s(literal 16 binary64))) < -5e64

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 99.5

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 61.5%

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

   10. Applied rewrites 61.5%

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

   11. Taylor expanded in z around 0

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

   13. Applied rewrites 99.5%

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

   if -5e64 < (+.f64 (*.f64 x y) (/.f64 (*.f64 z t) #s(literal 16 binary64))) < 4.9999999999999999e119

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 91.4

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

   5. Applied rewrites 91.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 83.8%

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

Alternative 3: 88.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -4 \cdot 10^{-23}:\\ \;\;\;\;\mathsf{fma}\left(y, x, \mathsf{fma}\left(t \cdot z, 0.0625, c\right)\right)\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{+42}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot 0.0625, z, -0.25 \cdot \left(a \cdot b\right)\right) + c\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot 0.0625, t, \mathsf{fma}\left(x, y, c\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (<= (* x y) -4e-23)
   (fma y x (fma (* t z) 0.0625 c))
   (if (<= (* x y) 2e+42)
     (+ (fma (* t 0.0625) z (* -0.25 (* a b))) c)
     (fma (* z 0.0625) t (fma x y c)))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if ((x * y) <= -4e-23) {
		tmp = fma(y, x, fma((t * z), 0.0625, c));
	} else if ((x * y) <= 2e+42) {
		tmp = fma((t * 0.0625), z, (-0.25 * (a * b))) + c;
	} else {
		tmp = fma((z * 0.0625), t, fma(x, y, c));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if (Float64(x * y) <= -4e-23)
		tmp = fma(y, x, fma(Float64(t * z), 0.0625, c));
	elseif (Float64(x * y) <= 2e+42)
		tmp = Float64(fma(Float64(t * 0.0625), z, Float64(-0.25 * Float64(a * b))) + c);
	else
		tmp = fma(Float64(z * 0.0625), t, fma(x, y, c));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[LessEqual[N[(x * y), $MachinePrecision], -4e-23], N[(y * x + N[(N[(t * z), $MachinePrecision] * 0.0625 + c), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 2e+42], N[(N[(N[(t * 0.0625), $MachinePrecision] * z + N[(-0.25 * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision], N[(N[(z * 0.0625), $MachinePrecision] * t + N[(x * y + c), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -3.99999999999999984e-23

   1. Initial program 93.1%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 91.5

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

   5. Applied rewrites 91.5%

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

   if -3.99999999999999984e-23 < (*.f64 x y) < 2.00000000000000009e42

   1. Initial program 100.0%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. div-inv N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f64 N/A

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

      17. lift-/.f64 N/A

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

      18. div-inv N/A

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

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

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

      20. metadata-eval N/A

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

      21. metadata-eval N/A

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

      22. metadata-eval N/A

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

      23. metadata-eval N/A

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

      24. lower-*.f64 N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 98.7

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

   7. Applied rewrites 98.7%

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

   if 2.00000000000000009e42 < (*.f64 x y)

   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 a around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 86.3

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

   5. Applied rewrites 86.3%

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

   7. Applied rewrites 88.0%

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

Alternative 4: 64.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z \cdot 0.0625, t, c\right)\\ \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+133}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \cdot t \leq -1 \cdot 10^{-239}:\\ \;\;\;\;\mathsf{fma}\left(x, y, c\right)\\ \mathbf{elif}\;z \cdot t \leq 10^{+117}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, a \cdot b, c\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (fma (* z 0.0625) t c)))
   (if (<= (* z t) -2e+133)
     t_1
     (if (<= (* z t) -1e-239)
       (fma x y c)
       (if (<= (* z t) 1e+117) (fma -0.25 (* a b) c) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = fma((z * 0.0625), t, c);
	double tmp;
	if ((z * t) <= -2e+133) {
		tmp = t_1;
	} else if ((z * t) <= -1e-239) {
		tmp = fma(x, y, c);
	} else if ((z * t) <= 1e+117) {
		tmp = fma(-0.25, (a * b), c);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = fma(Float64(z * 0.0625), t, c)
	tmp = 0.0
	if (Float64(z * t) <= -2e+133)
		tmp = t_1;
	elseif (Float64(z * t) <= -1e-239)
		tmp = fma(x, y, c);
	elseif (Float64(z * t) <= 1e+117)
		tmp = fma(-0.25, Float64(a * b), c);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(z * 0.0625), $MachinePrecision] * t + c), $MachinePrecision]}, If[LessEqual[N[(z * t), $MachinePrecision], -2e+133], t$95$1, If[LessEqual[N[(z * t), $MachinePrecision], -1e-239], N[(x * y + c), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 1e+117], N[(-0.25 * N[(a * b), $MachinePrecision] + c), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -2e133 or 1.00000000000000005e117 < (*.f64 z t)

   1. Initial program 94.0%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 87.5

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

   5. Applied rewrites 87.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 78.5%

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

   10. Applied rewrites 79.4%

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

   if -2e133 < (*.f64 z t) < -1.0000000000000001e-239

   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

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 77.4

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

   5. Applied rewrites 77.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 34.1%

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

   10. Applied rewrites 34.1%

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

   11. Taylor expanded in z around 0

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

   13. Applied rewrites 74.0%

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

   if -1.0000000000000001e-239 < (*.f64 z t) < 1.00000000000000005e117

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 93.3

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

   5. Applied rewrites 93.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 67.3%

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

Alternative 5: 63.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+133}:\\ \;\;\;\;\left(z \cdot 0.0625\right) \cdot t\\ \mathbf{elif}\;z \cdot t \leq -1 \cdot 10^{-239}:\\ \;\;\;\;\mathsf{fma}\left(x, y, c\right)\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+130}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, a \cdot b, c\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot t\right) \cdot 0.0625\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (<= (* z t) -2e+133)
   (* (* z 0.0625) t)
   (if (<= (* z t) -1e-239)
     (fma x y c)
     (if (<= (* z t) 2e+130) (fma -0.25 (* a b) c) (* (* z t) 0.0625)))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if ((z * t) <= -2e+133) {
		tmp = (z * 0.0625) * t;
	} else if ((z * t) <= -1e-239) {
		tmp = fma(x, y, c);
	} else if ((z * t) <= 2e+130) {
		tmp = fma(-0.25, (a * b), c);
	} else {
		tmp = (z * t) * 0.0625;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if (Float64(z * t) <= -2e+133)
		tmp = Float64(Float64(z * 0.0625) * t);
	elseif (Float64(z * t) <= -1e-239)
		tmp = fma(x, y, c);
	elseif (Float64(z * t) <= 2e+130)
		tmp = fma(-0.25, Float64(a * b), c);
	else
		tmp = Float64(Float64(z * t) * 0.0625);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[LessEqual[N[(z * t), $MachinePrecision], -2e+133], N[(N[(z * 0.0625), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], -1e-239], N[(x * y + c), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 2e+130], N[(-0.25 * N[(a * b), $MachinePrecision] + c), $MachinePrecision], N[(N[(z * t), $MachinePrecision] * 0.0625), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 z t) < -2e133

   1. Initial program 93.1%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. div-inv N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f64 N/A

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

      17. lift-/.f64 N/A

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

      18. div-inv N/A

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

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

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

      20. metadata-eval N/A

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

      21. metadata-eval N/A

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

      22. metadata-eval N/A

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

      23. metadata-eval N/A

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

      24. lower-*.f64 N/A

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

   4. Applied rewrites 98.1%

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

   5. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 72.0

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

   7. Applied rewrites 72.0%

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

   9. Applied rewrites 73.5%

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

   if -2e133 < (*.f64 z t) < -1.0000000000000001e-239

   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

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 77.4

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

   5. Applied rewrites 77.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 34.1%

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

   10. Applied rewrites 34.1%

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

   11. Taylor expanded in z around 0

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

   13. Applied rewrites 74.0%

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

   if -1.0000000000000001e-239 < (*.f64 z t) < 2.0000000000000001e130

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 92.6

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

   5. Applied rewrites 92.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 67.3%

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

   if 2.0000000000000001e130 < (*.f64 z t)

   1. Initial program 94.9%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. div-inv N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f64 N/A

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

      17. lift-/.f64 N/A

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

      18. div-inv N/A

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

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

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

      20. metadata-eval N/A

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

      21. metadata-eval N/A

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

      22. metadata-eval N/A

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

      23. metadata-eval N/A

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

      24. lower-*.f64 N/A

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

   4. Applied rewrites 94.9%

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

   5. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 72.5

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

   7. Applied rewrites 72.5%

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

Alternative 6: 88.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(t \cdot z, 0.0625, c\right)\\ \mathbf{if}\;x \cdot y \leq -4 \cdot 10^{-23}:\\ \;\;\;\;\mathsf{fma}\left(y, x, t\_1\right)\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{+42}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, b \cdot a, t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot 0.0625, t, \mathsf{fma}\left(x, y, c\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (fma (* t z) 0.0625 c)))
   (if (<= (* x y) -4e-23)
     (fma y x t_1)
     (if (<= (* x y) 2e+42)
       (fma -0.25 (* b a) t_1)
       (fma (* z 0.0625) t (fma x y c))))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = fma((t * z), 0.0625, c);
	double tmp;
	if ((x * y) <= -4e-23) {
		tmp = fma(y, x, t_1);
	} else if ((x * y) <= 2e+42) {
		tmp = fma(-0.25, (b * a), t_1);
	} else {
		tmp = fma((z * 0.0625), t, fma(x, y, c));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = fma(Float64(t * z), 0.0625, c)
	tmp = 0.0
	if (Float64(x * y) <= -4e-23)
		tmp = fma(y, x, t_1);
	elseif (Float64(x * y) <= 2e+42)
		tmp = fma(-0.25, Float64(b * a), t_1);
	else
		tmp = fma(Float64(z * 0.0625), t, fma(x, y, c));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(t * z), $MachinePrecision] * 0.0625 + c), $MachinePrecision]}, If[LessEqual[N[(x * y), $MachinePrecision], -4e-23], N[(y * x + t$95$1), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 2e+42], N[(-0.25 * N[(b * a), $MachinePrecision] + t$95$1), $MachinePrecision], N[(N[(z * 0.0625), $MachinePrecision] * t + N[(x * y + c), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -3.99999999999999984e-23

   1. Initial program 93.1%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 91.5

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

   5. Applied rewrites 91.5%

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

   if -3.99999999999999984e-23 < (*.f64 x y) < 2.00000000000000009e42

   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 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. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 98.7

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

   5. Applied rewrites 98.7%

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

   if 2.00000000000000009e42 < (*.f64 x y)

   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 a around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 86.3

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

   5. Applied rewrites 86.3%

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

   7. Applied rewrites 88.0%

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

Alternative 7: 88.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot b \leq -2 \cdot 10^{+177} \lor \neg \left(a \cdot b \leq 2 \cdot 10^{+167}\right):\\ \;\;\;\;\mathsf{fma}\left(-0.25, b \cdot a, \mathsf{fma}\left(y, x, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, \mathsf{fma}\left(t \cdot z, 0.0625, c\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (or (<= (* a b) -2e+177) (not (<= (* a b) 2e+167)))
   (fma -0.25 (* b a) (fma y x c))
   (fma y x (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 (((a * b) <= -2e+177) || !((a * b) <= 2e+167)) {
		tmp = fma(-0.25, (b * a), fma(y, x, c));
	} else {
		tmp = fma(y, x, fma((t * z), 0.0625, c));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if ((Float64(a * b) <= -2e+177) || !(Float64(a * b) <= 2e+167))
		tmp = fma(-0.25, Float64(b * a), fma(y, x, c));
	else
		tmp = fma(y, x, fma(Float64(t * z), 0.0625, c));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[Or[LessEqual[N[(a * b), $MachinePrecision], -2e+177], N[Not[LessEqual[N[(a * b), $MachinePrecision], 2e+167]], $MachinePrecision]], N[(-0.25 * N[(b * a), $MachinePrecision] + N[(y * x + c), $MachinePrecision]), $MachinePrecision], N[(y * x + N[(N[(t * z), $MachinePrecision] * 0.0625 + c), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a b) < -2e177 or 2.0000000000000001e167 < (*.f64 a b)

   1. Initial program 95.7%

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

   3. Taylor expanded in z around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 91.3

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

   5. Applied rewrites 91.3%

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

   if -2e177 < (*.f64 a b) < 2.0000000000000001e167

   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

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 93.0

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

   5. Applied rewrites 93.0%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b \leq -2 \cdot 10^{+177} \lor \neg \left(a \cdot b \leq 2 \cdot 10^{+167}\right):\\ \;\;\;\;\mathsf{fma}\left(-0.25, b \cdot a, \mathsf{fma}\left(y, x, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, \mathsf{fma}\left(t \cdot z, 0.0625, c\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 86.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+133}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot 0.0625, t, c\right)\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+183}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, b \cdot a, \mathsf{fma}\left(y, x, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot t, 0.0625, x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (<= (* z t) -2e+133)
   (fma (* z 0.0625) t c)
   (if (<= (* z t) 5e+183)
     (fma -0.25 (* b a) (fma y x c))
     (fma (* z t) 0.0625 (* x y)))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if ((z * t) <= -2e+133) {
		tmp = fma((z * 0.0625), t, c);
	} else if ((z * t) <= 5e+183) {
		tmp = fma(-0.25, (b * a), fma(y, x, c));
	} else {
		tmp = fma((z * t), 0.0625, (x * y));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if (Float64(z * t) <= -2e+133)
		tmp = fma(Float64(z * 0.0625), t, c);
	elseif (Float64(z * t) <= 5e+183)
		tmp = fma(-0.25, Float64(b * a), fma(y, x, c));
	else
		tmp = fma(Float64(z * t), 0.0625, Float64(x * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[LessEqual[N[(z * t), $MachinePrecision], -2e+133], N[(N[(z * 0.0625), $MachinePrecision] * t + c), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 5e+183], N[(-0.25 * N[(b * a), $MachinePrecision] + N[(y * x + c), $MachinePrecision]), $MachinePrecision], N[(N[(z * t), $MachinePrecision] * 0.0625 + N[(x * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -2e133

   1. Initial program 93.1%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 83.3

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

   5. Applied rewrites 83.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 78.2%

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

   10. Applied rewrites 79.7%

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

   if -2e133 < (*.f64 z t) < 5.00000000000000009e183

   1. Initial program 99.4%

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

   3. Taylor expanded in z around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 92.4

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

   5. Applied rewrites 92.4%

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

   if 5.00000000000000009e183 < (*.f64 z t)

   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

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 94.2

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

   5. Applied rewrites 94.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 82.2%

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

   10. Applied rewrites 82.2%

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

   11. Taylor expanded in c around 0

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

   13. Applied rewrites 85.0%

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

Alternative 9: 62.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+133} \lor \neg \left(z \cdot t \leq 2 \cdot 10^{+177}\right):\\ \;\;\;\;\left(z \cdot 0.0625\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, y, c\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (or (<= (* z t) -2e+133) (not (<= (* z t) 2e+177)))
   (* (* z 0.0625) t)
   (fma x y c)))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if (((z * t) <= -2e+133) || !((z * t) <= 2e+177)) {
		tmp = (z * 0.0625) * t;
	} else {
		tmp = fma(x, y, c);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if ((Float64(z * t) <= -2e+133) || !(Float64(z * t) <= 2e+177))
		tmp = Float64(Float64(z * 0.0625) * t);
	else
		tmp = fma(x, y, c);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[Or[LessEqual[N[(z * t), $MachinePrecision], -2e+133], N[Not[LessEqual[N[(z * t), $MachinePrecision], 2e+177]], $MachinePrecision]], N[(N[(z * 0.0625), $MachinePrecision] * t), $MachinePrecision], N[(x * y + c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -2e133 or 2e177 < (*.f64 z t)

   1. Initial program 93.5%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. div-inv N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f64 N/A

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

      17. lift-/.f64 N/A

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

      18. div-inv N/A

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

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

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

      20. metadata-eval N/A

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

      21. metadata-eval N/A

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

      22. metadata-eval N/A

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

      23. metadata-eval N/A

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

      24. lower-*.f64 N/A

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

   4. Applied rewrites 96.6%

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

   5. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 73.9

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

   7. Applied rewrites 73.9%

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

   9. Applied rewrites 74.8%

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

   if -2e133 < (*.f64 z t) < 2e177

   1. Initial program 99.4%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 69.9

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

   5. Applied rewrites 69.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 37.5%

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

   10. Applied rewrites 37.5%

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

   11. Taylor expanded in z around 0

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

   13. Applied rewrites 63.1%

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

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+133} \lor \neg \left(z \cdot t \leq 2 \cdot 10^{+177}\right):\\ \;\;\;\;\left(z \cdot 0.0625\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, y, c\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 62.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+133}:\\ \;\;\;\;\left(z \cdot 0.0625\right) \cdot t\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+177}:\\ \;\;\;\;\mathsf{fma}\left(x, y, c\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot t\right) \cdot 0.0625\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (<= (* z t) -2e+133)
   (* (* z 0.0625) t)
   (if (<= (* z t) 2e+177) (fma x y c) (* (* z t) 0.0625))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if ((z * t) <= -2e+133) {
		tmp = (z * 0.0625) * t;
	} else if ((z * t) <= 2e+177) {
		tmp = fma(x, y, c);
	} else {
		tmp = (z * t) * 0.0625;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if (Float64(z * t) <= -2e+133)
		tmp = Float64(Float64(z * 0.0625) * t);
	elseif (Float64(z * t) <= 2e+177)
		tmp = fma(x, y, c);
	else
		tmp = Float64(Float64(z * t) * 0.0625);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[LessEqual[N[(z * t), $MachinePrecision], -2e+133], N[(N[(z * 0.0625), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 2e+177], N[(x * y + c), $MachinePrecision], N[(N[(z * t), $MachinePrecision] * 0.0625), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -2e133

   1. Initial program 93.1%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. div-inv N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f64 N/A

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

      17. lift-/.f64 N/A

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

      18. div-inv N/A

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

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

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

      20. metadata-eval N/A

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

      21. metadata-eval N/A

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

      22. metadata-eval N/A

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

      23. metadata-eval N/A

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

      24. lower-*.f64 N/A

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

   4. Applied rewrites 98.1%

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

   5. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 72.0

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

   7. Applied rewrites 72.0%

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

   9. Applied rewrites 73.5%

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

   if -2e133 < (*.f64 z t) < 2e177

   1. Initial program 99.4%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 69.9

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

   5. Applied rewrites 69.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 37.5%

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

   10. Applied rewrites 37.5%

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

   11. Taylor expanded in z around 0

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

   13. Applied rewrites 63.1%

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

   if 2e177 < (*.f64 z t)

   1. Initial program 94.1%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. div-inv N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f64 N/A

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

      17. lift-/.f64 N/A

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

      18. div-inv N/A

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

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

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

      20. metadata-eval N/A

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

      21. metadata-eval N/A

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

      22. metadata-eval N/A

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

      23. metadata-eval N/A

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

      24. lower-*.f64 N/A

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

   4. Applied rewrites 94.1%

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

   5. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 77.0

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

   7. Applied rewrites 77.0%

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

Alternative 11: 48.6% accurate, 6.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 97.3%

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

3. Taylor expanded in a around 0

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

   1. associate-+r+ N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. +-commutative N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-*.f64 75.7

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

5. Applied rewrites 75.7%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 52.0%

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

10. Applied rewrites 52.4%

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

11. Taylor expanded in z around 0

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

13. Applied rewrites 46.0%

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

Alternative 12: 28.5% accurate, 7.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.3%

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

3. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 26.0

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

5. Applied rewrites 26.0%

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

Reproduce

herbie shell --seed 2024299 
(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))