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

Percentage Accurate: 97.8% → 99.0%

Time: 11.9s

Alternatives: 16

Speedup: 1.6×

• 32.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 97.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. associate-+l+ N/A

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

   4. associate-/l* N/A

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

   5. *-commutative N/A

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

   6. accelerator-lowering-fma.f64 N/A

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

   7. div-inv N/A

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

   8. *-lowering-*.f64 N/A

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

   9. metadata-eval N/A

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

   10. accelerator-lowering-fma.f64 N/A

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

   11. div-inv N/A

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

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

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. metadata-eval N/A

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

   16. metadata-eval N/A

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

   17. *-lowering-*.f64 N/A

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

   18. *-lowering-*.f64 N/A

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

   19. metadata-eval N/A

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

   20. metadata-eval 100.0

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

4. Applied egg-rr 100.0%

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

Alternative 2: 76.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(x, y, 0.0625 \cdot \left(t \cdot z\right)\right)\\ t_2 := x \cdot y + \frac{t \cdot z}{16}\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+74}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{+49}:\\ \;\;\;\;\mathsf{fma}\left(a, b \cdot -0.25, c\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (fma x y (* 0.0625 (* t z)))) (t_2 (+ (* x y) (/ (* t z) 16.0))))
   (if (<= t_2 -2e+74) t_1 (if (<= t_2 4e+49) (fma a (* b -0.25) c) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = fma(x, y, (0.0625 * (t * z)));
	double t_2 = (x * y) + ((t * z) / 16.0);
	double tmp;
	if (t_2 <= -2e+74) {
		tmp = t_1;
	} else if (t_2 <= 4e+49) {
		tmp = fma(a, (b * -0.25), c);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = fma(x, y, Float64(0.0625 * Float64(t * z)))
	t_2 = Float64(Float64(x * y) + Float64(Float64(t * z) / 16.0))
	tmp = 0.0
	if (t_2 <= -2e+74)
		tmp = t_1;
	elseif (t_2 <= 4e+49)
		tmp = fma(a, Float64(b * -0.25), c);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(x * y + N[(0.0625 * N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * y), $MachinePrecision] + N[(N[(t * z), $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+74], t$95$1, If[LessEqual[t$95$2, 4e+49], N[(a * N[(b * -0.25), $MachinePrecision] + c), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 x y) (/.f64 (*.f64 z t) #s(literal 16 binary64))) < -1.9999999999999999e74 or 3.99999999999999979e49 < (+.f64 (*.f64 x y) (/.f64 (*.f64 z t) #s(literal 16 binary64)))

   1. Initial program 98.9%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. div-inv N/A

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

      8. *-lowering-*.f64 N/A

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

      9. metadata-eval N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. div-inv N/A

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

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

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. *-lowering-*.f64 N/A

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

      18. *-lowering-*.f64 N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around inf

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

      1. *-lowering-*.f64 85.7

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

   7. Simplified 85.7%

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

   8. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-lowering-*.f64 72.6

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

   10. Simplified 72.6%

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

   if -1.9999999999999999e74 < (+.f64 (*.f64 x y) (/.f64 (*.f64 z t) #s(literal 16 binary64))) < 3.99999999999999979e49

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 N/A

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

      10. associate-+r+ N/A

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

      11. +-commutative N/A

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

      12. associate-+l+ N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. +-commutative N/A

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

      16. accelerator-lowering-fma.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in c around inf

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

   8. Simplified 84.8%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y + \frac{t \cdot z}{16} \leq -2 \cdot 10^{+74}:\\ \;\;\;\;\mathsf{fma}\left(x, y, 0.0625 \cdot \left(t \cdot z\right)\right)\\ \mathbf{elif}\;x \cdot y + \frac{t \cdot z}{16} \leq 4 \cdot 10^{+49}:\\ \;\;\;\;\mathsf{fma}\left(a, b \cdot -0.25, c\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, y, 0.0625 \cdot \left(t \cdot z\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 66.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(t \cdot z, 0.0625, c\right)\\ \mathbf{if}\;a \cdot b \leq -2 \cdot 10^{+19}:\\ \;\;\;\;\mathsf{fma}\left(a, b \cdot -0.25, c\right)\\ \mathbf{elif}\;a \cdot b \leq -1 \cdot 10^{-280}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \cdot b \leq 5 \cdot 10^{-169}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \mathbf{elif}\;a \cdot b \leq 10^{+96}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, b \cdot -0.25, x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (fma (* t z) 0.0625 c)))
   (if (<= (* a b) -2e+19)
     (fma a (* b -0.25) c)
     (if (<= (* a b) -1e-280)
       t_1
       (if (<= (* a b) 5e-169)
         (fma y x c)
         (if (<= (* a b) 1e+96) t_1 (fma a (* b -0.25) (* x y))))))))

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 ((a * b) <= -2e+19) {
		tmp = fma(a, (b * -0.25), c);
	} else if ((a * b) <= -1e-280) {
		tmp = t_1;
	} else if ((a * b) <= 5e-169) {
		tmp = fma(y, x, c);
	} else if ((a * b) <= 1e+96) {
		tmp = t_1;
	} else {
		tmp = fma(a, (b * -0.25), (x * y));
	}
	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(a * b) <= -2e+19)
		tmp = fma(a, Float64(b * -0.25), c);
	elseif (Float64(a * b) <= -1e-280)
		tmp = t_1;
	elseif (Float64(a * b) <= 5e-169)
		tmp = fma(y, x, c);
	elseif (Float64(a * b) <= 1e+96)
		tmp = t_1;
	else
		tmp = fma(a, Float64(b * -0.25), Float64(x * y));
	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[(a * b), $MachinePrecision], -2e+19], N[(a * N[(b * -0.25), $MachinePrecision] + c), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], -1e-280], t$95$1, If[LessEqual[N[(a * b), $MachinePrecision], 5e-169], N[(y * x + c), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], 1e+96], t$95$1, N[(a * N[(b * -0.25), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 a b) < -2e19

   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

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 N/A

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

      10. associate-+r+ N/A

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

      11. +-commutative N/A

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

      12. associate-+l+ N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. +-commutative N/A

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

      16. accelerator-lowering-fma.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in c around inf

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

   8. Simplified 79.7%

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

   if -2e19 < (*.f64 a b) < -9.9999999999999996e-281 or 5.0000000000000002e-169 < (*.f64 a b) < 1.00000000000000005e96

   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 inf

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

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{\left(\frac{1}{16} \cdot \left(t \cdot z\right) + 0\right)} + c \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{16}, t \cdot z, 0\right)} + c \]

      3. *-lowering-*.f64 73.1

         \[\leadsto \mathsf{fma}\left(0.0625, \color{blue}{t \cdot z}, 0\right) + c \]

   5. Simplified 73.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0625, t \cdot z, 0\right)} + c \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. *-lowering-*.f64 73.1

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

   7. Applied egg-rr 73.1%

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

   if -9.9999999999999996e-281 < (*.f64 a b) < 5.0000000000000002e-169

   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

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

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{\left(x \cdot y + 0\right)} + c \]

      2. accelerator-lowering-fma.f64 73.0

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, 0\right)} + c \]

   5. Simplified 73.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, 0\right)} + c \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 73.1

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

   7. Applied egg-rr 73.1%

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

   if 1.00000000000000005e96 < (*.f64 a b)

   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

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 N/A

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

      10. associate-+r+ N/A

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

      11. +-commutative N/A

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

      12. associate-+l+ N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. +-commutative N/A

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

      16. accelerator-lowering-fma.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf

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

      1. *-lowering-*.f64 75.0

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

   8. Simplified 75.0%

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

Alternative 4: 65.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(t \cdot z, 0.0625, c\right)\\ t_2 := \mathsf{fma}\left(a, b \cdot -0.25, c\right)\\ \mathbf{if}\;a \cdot b \leq -2 \cdot 10^{+19}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;a \cdot b \leq -1 \cdot 10^{-280}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \cdot b \leq 5 \cdot 10^{-169}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \mathbf{elif}\;a \cdot b \leq 10^{+96}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (fma (* t z) 0.0625 c)) (t_2 (fma a (* b -0.25) c)))
   (if (<= (* a b) -2e+19)
     t_2
     (if (<= (* a b) -1e-280)
       t_1
       (if (<= (* a b) 5e-169) (fma y x c) (if (<= (* a b) 1e+96) t_1 t_2))))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = fma((t * z), 0.0625, c);
	double t_2 = fma(a, (b * -0.25), c);
	double tmp;
	if ((a * b) <= -2e+19) {
		tmp = t_2;
	} else if ((a * b) <= -1e-280) {
		tmp = t_1;
	} else if ((a * b) <= 5e-169) {
		tmp = fma(y, x, c);
	} else if ((a * b) <= 1e+96) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = fma(Float64(t * z), 0.0625, c)
	t_2 = fma(a, Float64(b * -0.25), c)
	tmp = 0.0
	if (Float64(a * b) <= -2e+19)
		tmp = t_2;
	elseif (Float64(a * b) <= -1e-280)
		tmp = t_1;
	elseif (Float64(a * b) <= 5e-169)
		tmp = fma(y, x, c);
	elseif (Float64(a * b) <= 1e+96)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(t * z), $MachinePrecision] * 0.0625 + c), $MachinePrecision]}, Block[{t$95$2 = N[(a * N[(b * -0.25), $MachinePrecision] + c), $MachinePrecision]}, If[LessEqual[N[(a * b), $MachinePrecision], -2e+19], t$95$2, If[LessEqual[N[(a * b), $MachinePrecision], -1e-280], t$95$1, If[LessEqual[N[(a * b), $MachinePrecision], 5e-169], N[(y * x + c), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], 1e+96], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 a b) < -2e19 or 1.00000000000000005e96 < (*.f64 a b)

   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

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 N/A

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

      10. associate-+r+ N/A

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

      11. +-commutative N/A

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

      12. associate-+l+ N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. +-commutative N/A

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

      16. accelerator-lowering-fma.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in c around inf

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

   8. Simplified 74.8%

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

   if -2e19 < (*.f64 a b) < -9.9999999999999996e-281 or 5.0000000000000002e-169 < (*.f64 a b) < 1.00000000000000005e96

   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 inf

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

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{\left(\frac{1}{16} \cdot \left(t \cdot z\right) + 0\right)} + c \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{16}, t \cdot z, 0\right)} + c \]

      3. *-lowering-*.f64 73.1

         \[\leadsto \mathsf{fma}\left(0.0625, \color{blue}{t \cdot z}, 0\right) + c \]

   5. Simplified 73.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0625, t \cdot z, 0\right)} + c \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. *-lowering-*.f64 73.1

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

   7. Applied egg-rr 73.1%

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

   if -9.9999999999999996e-281 < (*.f64 a b) < 5.0000000000000002e-169

   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

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

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{\left(x \cdot y + 0\right)} + c \]

      2. accelerator-lowering-fma.f64 73.0

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, 0\right)} + c \]

   5. Simplified 73.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, 0\right)} + c \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 73.1

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

   7. Applied egg-rr 73.1%

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

Alternative 5: 60.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(a, b \cdot -0.25, c\right)\\ t_2 := \left(t \cdot 0.0625\right) \cdot z\\ \mathbf{if}\;t \cdot z \leq -5 \cdot 10^{+248}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t \cdot z \leq -2 \cdot 10^{-40}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \cdot z \leq -3.5 \cdot 10^{-299}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \mathbf{elif}\;t \cdot z \leq 5 \cdot 10^{+36}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (fma a (* b -0.25) c)) (t_2 (* (* t 0.0625) z)))
   (if (<= (* t z) -5e+248)
     t_2
     (if (<= (* t z) -2e-40)
       t_1
       (if (<= (* t z) -3.5e-299)
         (fma y x c)
         (if (<= (* t z) 5e+36) t_1 t_2))))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = fma(a, (b * -0.25), c);
	double t_2 = (t * 0.0625) * z;
	double tmp;
	if ((t * z) <= -5e+248) {
		tmp = t_2;
	} else if ((t * z) <= -2e-40) {
		tmp = t_1;
	} else if ((t * z) <= -3.5e-299) {
		tmp = fma(y, x, c);
	} else if ((t * z) <= 5e+36) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = fma(a, Float64(b * -0.25), c)
	t_2 = Float64(Float64(t * 0.0625) * z)
	tmp = 0.0
	if (Float64(t * z) <= -5e+248)
		tmp = t_2;
	elseif (Float64(t * z) <= -2e-40)
		tmp = t_1;
	elseif (Float64(t * z) <= -3.5e-299)
		tmp = fma(y, x, c);
	elseif (Float64(t * z) <= 5e+36)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(a * N[(b * -0.25), $MachinePrecision] + c), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t * 0.0625), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[N[(t * z), $MachinePrecision], -5e+248], t$95$2, If[LessEqual[N[(t * z), $MachinePrecision], -2e-40], t$95$1, If[LessEqual[N[(t * z), $MachinePrecision], -3.5e-299], N[(y * x + c), $MachinePrecision], If[LessEqual[N[(t * z), $MachinePrecision], 5e+36], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -4.9999999999999996e248 or 4.99999999999999977e36 < (*.f64 z t)

   1. Initial program 97.8%

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

   3. Taylor expanded in z around inf

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

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{\frac{1}{16} \cdot \left(t \cdot z\right) + 0} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{16}, t \cdot z, 0\right)} \]

      3. *-lowering-*.f64 75.7

         \[\leadsto \mathsf{fma}\left(0.0625, \color{blue}{t \cdot z}, 0\right) \]

   5. Simplified 75.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0625, t \cdot z, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 77.9

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

   7. Applied egg-rr 77.9%

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

   if -4.9999999999999996e248 < (*.f64 z t) < -1.9999999999999999e-40 or -3.49999999999999991e-299 < (*.f64 z t) < 4.99999999999999977e36

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 N/A

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

      10. associate-+r+ N/A

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

      11. +-commutative N/A

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

      12. associate-+l+ N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. +-commutative N/A

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

      16. accelerator-lowering-fma.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in c around inf

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

   8. Simplified 67.9%

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

   if -1.9999999999999999e-40 < (*.f64 z t) < -3.49999999999999991e-299

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{\left(x \cdot y + 0\right)} + c \]

      2. accelerator-lowering-fma.f64 74.5

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, 0\right)} + c \]

   5. Simplified 74.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, 0\right)} + c \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 74.5

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

   7. Applied egg-rr 74.5%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot z \leq -5 \cdot 10^{+248}:\\ \;\;\;\;\left(t \cdot 0.0625\right) \cdot z\\ \mathbf{elif}\;t \cdot z \leq -2 \cdot 10^{-40}:\\ \;\;\;\;\mathsf{fma}\left(a, b \cdot -0.25, c\right)\\ \mathbf{elif}\;t \cdot z \leq -3.5 \cdot 10^{-299}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \mathbf{elif}\;t \cdot z \leq 5 \cdot 10^{+36}:\\ \;\;\;\;\mathsf{fma}\left(a, b \cdot -0.25, c\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t \cdot 0.0625\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 88.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(0.0625, t \cdot z, \mathsf{fma}\left(a, b \cdot -0.25, c\right)\right)\\ \mathbf{if}\;t \cdot z \leq -1:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \cdot z \leq 5 \cdot 10^{+36}:\\ \;\;\;\;\mathsf{fma}\left(a, b \cdot -0.25, \mathsf{fma}\left(x, y, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (fma 0.0625 (* t z) (fma a (* b -0.25) c))))
   (if (<= (* t z) -1.0)
     t_1
     (if (<= (* t z) 5e+36) (fma a (* b -0.25) (fma x y c)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = fma(0.0625, (t * z), fma(a, (b * -0.25), c));
	double tmp;
	if ((t * z) <= -1.0) {
		tmp = t_1;
	} else if ((t * z) <= 5e+36) {
		tmp = fma(a, (b * -0.25), fma(x, y, c));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = fma(0.0625, Float64(t * z), fma(a, Float64(b * -0.25), c))
	tmp = 0.0
	if (Float64(t * z) <= -1.0)
		tmp = t_1;
	elseif (Float64(t * z) <= 5e+36)
		tmp = fma(a, Float64(b * -0.25), fma(x, y, c));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(0.0625 * N[(t * z), $MachinePrecision] + N[(a * N[(b * -0.25), $MachinePrecision] + c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t * z), $MachinePrecision], -1.0], t$95$1, If[LessEqual[N[(t * z), $MachinePrecision], 5e+36], N[(a * N[(b * -0.25), $MachinePrecision] + N[(x * y + c), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -1 or 4.99999999999999977e36 < (*.f64 z t)

   1. Initial program 98.5%

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

   3. Taylor expanded in x around 0

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

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

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. associate-+l+ N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. accelerator-lowering-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 90.8

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

   5. Simplified 90.8%

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

   if -1 < (*.f64 z t) < 4.99999999999999977e36

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 N/A

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

      10. +-commutative N/A

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

      11. accelerator-lowering-fma.f64 96.2

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

   5. Simplified 96.2%

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

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot z \leq -1:\\ \;\;\;\;\mathsf{fma}\left(0.0625, t \cdot z, \mathsf{fma}\left(a, b \cdot -0.25, c\right)\right)\\ \mathbf{elif}\;t \cdot z \leq 5 \cdot 10^{+36}:\\ \;\;\;\;\mathsf{fma}\left(a, b \cdot -0.25, \mathsf{fma}\left(x, y, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.0625, t \cdot z, \mathsf{fma}\left(a, b \cdot -0.25, c\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 57.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t \cdot 0.0625\right) \cdot z\\ \mathbf{if}\;t \cdot z \leq -3 \cdot 10^{+49}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \cdot z \leq 2 \cdot 10^{-128}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \mathbf{elif}\;t \cdot z \leq 5 \cdot 10^{+36}:\\ \;\;\;\;a \cdot \left(b \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (* (* t 0.0625) z)))
   (if (<= (* t z) -3e+49)
     t_1
     (if (<= (* t z) 2e-128)
       (fma y x c)
       (if (<= (* t z) 5e+36) (* a (* b -0.25)) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (t * 0.0625) * z;
	double tmp;
	if ((t * z) <= -3e+49) {
		tmp = t_1;
	} else if ((t * z) <= 2e-128) {
		tmp = fma(y, x, c);
	} else if ((t * z) <= 5e+36) {
		tmp = a * (b * -0.25);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(t * 0.0625) * z)
	tmp = 0.0
	if (Float64(t * z) <= -3e+49)
		tmp = t_1;
	elseif (Float64(t * z) <= 2e-128)
		tmp = fma(y, x, c);
	elseif (Float64(t * z) <= 5e+36)
		tmp = Float64(a * Float64(b * -0.25));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(t * 0.0625), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[N[(t * z), $MachinePrecision], -3e+49], t$95$1, If[LessEqual[N[(t * z), $MachinePrecision], 2e-128], N[(y * x + c), $MachinePrecision], If[LessEqual[N[(t * z), $MachinePrecision], 5e+36], N[(a * N[(b * -0.25), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -3.0000000000000002e49 or 4.99999999999999977e36 < (*.f64 z t)

   1. Initial program 98.3%

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

   3. Taylor expanded in z around inf

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

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{\frac{1}{16} \cdot \left(t \cdot z\right) + 0} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{16}, t \cdot z, 0\right)} \]

      3. *-lowering-*.f64 68.6

         \[\leadsto \mathsf{fma}\left(0.0625, \color{blue}{t \cdot z}, 0\right) \]

   5. Simplified 68.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0625, t \cdot z, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 70.3

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

   7. Applied egg-rr 70.3%

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

   if -3.0000000000000002e49 < (*.f64 z t) < 2.00000000000000011e-128

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{\left(x \cdot y + 0\right)} + c \]

      2. accelerator-lowering-fma.f64 65.3

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, 0\right)} + c \]

   5. Simplified 65.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, 0\right)} + c \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 65.3

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

   7. Applied egg-rr 65.3%

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

   if 2.00000000000000011e-128 < (*.f64 z t) < 4.99999999999999977e36

   1. Initial program 99.9%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 51.7

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

   5. Simplified 51.7%

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

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot z \leq -3 \cdot 10^{+49}:\\ \;\;\;\;\left(t \cdot 0.0625\right) \cdot z\\ \mathbf{elif}\;t \cdot z \leq 2 \cdot 10^{-128}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \mathbf{elif}\;t \cdot z \leq 5 \cdot 10^{+36}:\\ \;\;\;\;a \cdot \left(b \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t \cdot 0.0625\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 87.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \cdot z \leq -3 \cdot 10^{+49}:\\ \;\;\;\;\mathsf{fma}\left(a, b \cdot -0.25, 0.0625 \cdot \left(t \cdot z\right)\right)\\ \mathbf{elif}\;t \cdot z \leq 5 \cdot 10^{+25}:\\ \;\;\;\;\mathsf{fma}\left(a, b \cdot -0.25, \mathsf{fma}\left(x, y, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot 0.0625, z, \mathsf{fma}\left(x, y, c\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (<= (* t z) -3e+49)
   (fma a (* b -0.25) (* 0.0625 (* t z)))
   (if (<= (* t z) 5e+25)
     (fma a (* b -0.25) (fma x y c))
     (fma (* t 0.0625) z (fma x y c)))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if ((t * z) <= -3e+49) {
		tmp = fma(a, (b * -0.25), (0.0625 * (t * z)));
	} else if ((t * z) <= 5e+25) {
		tmp = fma(a, (b * -0.25), fma(x, y, c));
	} else {
		tmp = fma((t * 0.0625), z, fma(x, y, c));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if (Float64(t * z) <= -3e+49)
		tmp = fma(a, Float64(b * -0.25), Float64(0.0625 * Float64(t * z)));
	elseif (Float64(t * z) <= 5e+25)
		tmp = fma(a, Float64(b * -0.25), fma(x, y, c));
	else
		tmp = fma(Float64(t * 0.0625), z, fma(x, y, c));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[LessEqual[N[(t * z), $MachinePrecision], -3e+49], N[(a * N[(b * -0.25), $MachinePrecision] + N[(0.0625 * N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(t * z), $MachinePrecision], 5e+25], N[(a * N[(b * -0.25), $MachinePrecision] + N[(x * y + c), $MachinePrecision]), $MachinePrecision], N[(N[(t * 0.0625), $MachinePrecision] * z + N[(x * y + c), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -3.0000000000000002e49

   1. Initial program 98.1%

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

   3. Taylor expanded in x around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 N/A

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

      10. associate-+r+ N/A

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

      11. +-commutative N/A

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

      12. associate-+l+ N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. +-commutative N/A

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

      16. accelerator-lowering-fma.f64 98.2

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

   5. Simplified 98.2%

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

   6. Taylor expanded in t around inf

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

      1. *-lowering-*.f64 N/A

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

      2. *-lowering-*.f64 82.9

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

   8. Simplified 82.9%

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

   if -3.0000000000000002e49 < (*.f64 z t) < 5.00000000000000024e25

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 N/A

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

      10. +-commutative N/A

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

      11. accelerator-lowering-fma.f64 95.4

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

   5. Simplified 95.4%

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

   if 5.00000000000000024e25 < (*.f64 z t)

   1. Initial program 98.5%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. div-inv N/A

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

      8. *-lowering-*.f64 N/A

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

      9. metadata-eval N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. div-inv N/A

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

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

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. *-lowering-*.f64 N/A

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

      18. *-lowering-*.f64 N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around inf

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

      1. *-lowering-*.f64 87.8

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

   7. Simplified 87.8%

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

      1. associate-+l+ N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. accelerator-lowering-fma.f64 87.8

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

   9. Applied egg-rr 87.8%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot z \leq -3 \cdot 10^{+49}:\\ \;\;\;\;\mathsf{fma}\left(a, b \cdot -0.25, 0.0625 \cdot \left(t \cdot z\right)\right)\\ \mathbf{elif}\;t \cdot z \leq 5 \cdot 10^{+25}:\\ \;\;\;\;\mathsf{fma}\left(a, b \cdot -0.25, \mathsf{fma}\left(x, y, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot 0.0625, z, \mathsf{fma}\left(x, y, c\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 90.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(t \cdot 0.0625, z, \mathsf{fma}\left(x, y, c\right)\right)\\ \mathbf{if}\;t \cdot z \leq -3 \cdot 10^{+49}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \cdot z \leq 5 \cdot 10^{+25}:\\ \;\;\;\;\mathsf{fma}\left(a, b \cdot -0.25, \mathsf{fma}\left(x, y, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (fma (* t 0.0625) z (fma x y c))))
   (if (<= (* t z) -3e+49)
     t_1
     (if (<= (* t z) 5e+25) (fma a (* b -0.25) (fma x y c)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = fma((t * 0.0625), z, fma(x, y, c));
	double tmp;
	if ((t * z) <= -3e+49) {
		tmp = t_1;
	} else if ((t * z) <= 5e+25) {
		tmp = fma(a, (b * -0.25), fma(x, y, c));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = fma(Float64(t * 0.0625), z, fma(x, y, c))
	tmp = 0.0
	if (Float64(t * z) <= -3e+49)
		tmp = t_1;
	elseif (Float64(t * z) <= 5e+25)
		tmp = fma(a, Float64(b * -0.25), fma(x, y, c));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(t * 0.0625), $MachinePrecision] * z + N[(x * y + c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t * z), $MachinePrecision], -3e+49], t$95$1, If[LessEqual[N[(t * z), $MachinePrecision], 5e+25], N[(a * N[(b * -0.25), $MachinePrecision] + N[(x * y + c), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -3.0000000000000002e49 or 5.00000000000000024e25 < (*.f64 z t)

   1. Initial program 98.3%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. div-inv N/A

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

      8. *-lowering-*.f64 N/A

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

      9. metadata-eval N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. div-inv N/A

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

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

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. *-lowering-*.f64 N/A

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

      18. *-lowering-*.f64 N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around inf

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

      1. *-lowering-*.f64 85.2

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

   7. Simplified 85.2%

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

      1. associate-+l+ N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. accelerator-lowering-fma.f64 85.2

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

   9. Applied egg-rr 85.2%

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

   if -3.0000000000000002e49 < (*.f64 z t) < 5.00000000000000024e25

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 N/A

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

      10. +-commutative N/A

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

      11. accelerator-lowering-fma.f64 95.4

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

   5. Simplified 95.4%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot z \leq -3 \cdot 10^{+49}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot 0.0625, z, \mathsf{fma}\left(x, y, c\right)\right)\\ \mathbf{elif}\;t \cdot z \leq 5 \cdot 10^{+25}:\\ \;\;\;\;\mathsf{fma}\left(a, b \cdot -0.25, \mathsf{fma}\left(x, y, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot 0.0625, z, \mathsf{fma}\left(x, y, c\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 89.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(0.0625, t \cdot z, \mathsf{fma}\left(x, y, c\right)\right)\\ \mathbf{if}\;t \cdot z \leq -3 \cdot 10^{+49}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \cdot z \leq 5 \cdot 10^{+25}:\\ \;\;\;\;\mathsf{fma}\left(a, b \cdot -0.25, \mathsf{fma}\left(x, y, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (fma 0.0625 (* t z) (fma x y c))))
   (if (<= (* t z) -3e+49)
     t_1
     (if (<= (* t z) 5e+25) (fma a (* b -0.25) (fma x y c)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = fma(0.0625, (t * z), fma(x, y, c));
	double tmp;
	if ((t * z) <= -3e+49) {
		tmp = t_1;
	} else if ((t * z) <= 5e+25) {
		tmp = fma(a, (b * -0.25), fma(x, y, c));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = fma(0.0625, Float64(t * z), fma(x, y, c))
	tmp = 0.0
	if (Float64(t * z) <= -3e+49)
		tmp = t_1;
	elseif (Float64(t * z) <= 5e+25)
		tmp = fma(a, Float64(b * -0.25), fma(x, y, c));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(0.0625 * N[(t * z), $MachinePrecision] + N[(x * y + c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t * z), $MachinePrecision], -3e+49], t$95$1, If[LessEqual[N[(t * z), $MachinePrecision], 5e+25], N[(a * N[(b * -0.25), $MachinePrecision] + N[(x * y + c), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -3.0000000000000002e49 or 5.00000000000000024e25 < (*.f64 z t)

   1. Initial program 98.3%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. +-commutative N/A

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

      7. accelerator-lowering-fma.f64 83.5

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

   5. Simplified 83.5%

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

   if -3.0000000000000002e49 < (*.f64 z t) < 5.00000000000000024e25

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 N/A

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

      10. +-commutative N/A

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

      11. accelerator-lowering-fma.f64 95.4

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

   5. Simplified 95.4%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot z \leq -3 \cdot 10^{+49}:\\ \;\;\;\;\mathsf{fma}\left(0.0625, t \cdot z, \mathsf{fma}\left(x, y, c\right)\right)\\ \mathbf{elif}\;t \cdot z \leq 5 \cdot 10^{+25}:\\ \;\;\;\;\mathsf{fma}\left(a, b \cdot -0.25, \mathsf{fma}\left(x, y, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.0625, t \cdot z, \mathsf{fma}\left(x, y, c\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 85.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot b \leq -2 \cdot 10^{+19}:\\ \;\;\;\;\mathsf{fma}\left(a, b \cdot -0.25, c\right)\\ \mathbf{elif}\;a \cdot b \leq 2 \cdot 10^{+146}:\\ \;\;\;\;\mathsf{fma}\left(0.0625, t \cdot z, \mathsf{fma}\left(x, y, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, b \cdot -0.25, x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (<= (* a b) -2e+19)
   (fma a (* b -0.25) c)
   (if (<= (* a b) 2e+146)
     (fma 0.0625 (* t z) (fma x y c))
     (fma a (* b -0.25) (* x y)))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if ((a * b) <= -2e+19) {
		tmp = fma(a, (b * -0.25), c);
	} else if ((a * b) <= 2e+146) {
		tmp = fma(0.0625, (t * z), fma(x, y, c));
	} else {
		tmp = fma(a, (b * -0.25), (x * y));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if (Float64(a * b) <= -2e+19)
		tmp = fma(a, Float64(b * -0.25), c);
	elseif (Float64(a * b) <= 2e+146)
		tmp = fma(0.0625, Float64(t * z), fma(x, y, c));
	else
		tmp = fma(a, Float64(b * -0.25), Float64(x * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[LessEqual[N[(a * b), $MachinePrecision], -2e+19], N[(a * N[(b * -0.25), $MachinePrecision] + c), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], 2e+146], N[(0.0625 * N[(t * z), $MachinePrecision] + N[(x * y + c), $MachinePrecision]), $MachinePrecision], N[(a * N[(b * -0.25), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 a b) < -2e19

   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

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 N/A

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

      10. associate-+r+ N/A

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

      11. +-commutative N/A

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

      12. associate-+l+ N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. +-commutative N/A

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

      16. accelerator-lowering-fma.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in c around inf

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

   8. Simplified 79.7%

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

   if -2e19 < (*.f64 a b) < 1.99999999999999987e146

   1. Initial program 99.0%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. +-commutative N/A

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

      7. accelerator-lowering-fma.f64 91.7

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

   5. Simplified 91.7%

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

   if 1.99999999999999987e146 < (*.f64 a b)

   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

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 N/A

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

      10. associate-+r+ N/A

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

      11. +-commutative N/A

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

      12. associate-+l+ N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. +-commutative N/A

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

      16. accelerator-lowering-fma.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf

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

      1. *-lowering-*.f64 81.6

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

   8. Simplified 81.6%

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

Alternative 12: 63.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot \left(b \cdot -0.25\right)\\ \mathbf{if}\;a \cdot b \leq -2 \cdot 10^{+66}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \cdot b \leq 2 \cdot 10^{+146}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (* a (* b -0.25))))
   (if (<= (* a b) -2e+66) t_1 (if (<= (* a b) 2e+146) (fma y x c) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = a * (b * -0.25);
	double tmp;
	if ((a * b) <= -2e+66) {
		tmp = t_1;
	} else if ((a * b) <= 2e+146) {
		tmp = fma(y, x, c);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(a * Float64(b * -0.25))
	tmp = 0.0
	if (Float64(a * b) <= -2e+66)
		tmp = t_1;
	elseif (Float64(a * b) <= 2e+146)
		tmp = fma(y, x, c);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(a * N[(b * -0.25), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(a * b), $MachinePrecision], -2e+66], t$95$1, If[LessEqual[N[(a * b), $MachinePrecision], 2e+146], N[(y * x + c), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a b) < -1.99999999999999989e66 or 1.99999999999999987e146 < (*.f64 a b)

   1. Initial program 100.0%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 71.9

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

   5. Simplified 71.9%

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

   if -1.99999999999999989e66 < (*.f64 a b) < 1.99999999999999987e146

   1. Initial program 99.1%

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

   3. Taylor expanded in x around inf

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

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{\left(x \cdot y + 0\right)} + c \]

      2. accelerator-lowering-fma.f64 55.8

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, 0\right)} + c \]

   5. Simplified 55.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, 0\right)} + c \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 55.8

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

   7. Applied egg-rr 55.8%

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

Alternative 13: 98.5% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

3. Taylor expanded in x around 0

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

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

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

   2. metadata-eval N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

   7. accelerator-lowering-fma.f64 N/A

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

   8. *-commutative N/A

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

   9. *-lowering-*.f64 N/A

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

   10. associate-+r+ N/A

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

   11. +-commutative N/A

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

   12. associate-+l+ N/A

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

   13. accelerator-lowering-fma.f64 N/A

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

   14. *-lowering-*.f64 N/A

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

   15. +-commutative N/A

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

   16. accelerator-lowering-fma.f64 99.4

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

5. Simplified 99.4%

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

Alternative 14: 42.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{+69}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+111}:\\ \;\;\;\;c\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (<= (* x y) -1e+69) (* x y) (if (<= (* x y) 5e+111) c (* x y))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if ((x * y) <= -1e+69) {
		tmp = x * y;
	} else if ((x * y) <= 5e+111) {
		tmp = c;
	} else {
		tmp = x * y;
	}
	return tmp;
}

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
    real(8) :: tmp
    if ((x * y) <= (-1d+69)) then
        tmp = x * y
    else if ((x * y) <= 5d+111) then
        tmp = c
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if ((x * y) <= -1e+69) {
		tmp = x * y;
	} else if ((x * y) <= 5e+111) {
		tmp = c;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	tmp = 0
	if (x * y) <= -1e+69:
		tmp = x * y
	elif (x * y) <= 5e+111:
		tmp = c
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if (Float64(x * y) <= -1e+69)
		tmp = Float64(x * y);
	elseif (Float64(x * y) <= 5e+111)
		tmp = c;
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c)
	tmp = 0.0;
	if ((x * y) <= -1e+69)
		tmp = x * y;
	elseif ((x * y) <= 5e+111)
		tmp = c;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[LessEqual[N[(x * y), $MachinePrecision], -1e+69], N[(x * y), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 5e+111], c, N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -1.0000000000000001e69 or 4.9999999999999997e111 < (*.f64 x y)

   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. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. div-inv N/A

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

      8. *-lowering-*.f64 N/A

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

      9. metadata-eval N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. div-inv N/A

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

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

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. *-lowering-*.f64 N/A

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

      18. *-lowering-*.f64 N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around inf

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

      1. *-lowering-*.f64 57.4

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

   7. Simplified 57.4%

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

   if -1.0000000000000001e69 < (*.f64 x y) < 4.9999999999999997e111

   1. Initial program 99.1%

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

   3. Taylor expanded in c around inf

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

   5. Simplified 28.9%

      \[\leadsto \color{blue}{c} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 48.8% accurate, 6.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

3. Taylor expanded in x around inf

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

   1. +-rgt-identity N/A

      \[\leadsto \color{blue}{\left(x \cdot y + 0\right)} + c \]

   2. accelerator-lowering-fma.f64 44.4

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, 0\right)} + c \]

5. Simplified 44.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, 0\right)} + c \]
6. Step-by-step derivation

   1. +-rgt-identity N/A

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

   2. *-commutative N/A

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

   3. accelerator-lowering-fma.f64 44.4

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

7. Applied egg-rr 44.4%

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

Alternative 16: 22.6% accurate, 47.0× speedup

Math

\[\begin{array}{l} \\ c \end{array} \]

FPCore

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

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	return 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 = c
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := c

Derivation

1. Initial program 99.3%

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

3. Taylor expanded in c around inf

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

5. Simplified 24.0%

   \[\leadsto \color{blue}{c} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024196 
(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))