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

Percentage Accurate: 97.6% → 98.4%

Time: 8.9s

Alternatives: 13

Speedup: 1.4×

• 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: 97.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.4% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.8%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-/.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-/l* N/A

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

   7. distribute-lft-neg-in N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-neg.f64 N/A

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

   10. div-inv N/A

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

   11. lower-*.f64 N/A

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

   12. metadata-eval 99.2

       \[\leadsto \mathsf{fma}\left(-a, b \cdot \color{blue}{0.25}, x \cdot y + \frac{z \cdot t}{16}\right) + c \]

   13. lift-+.f64 N/A

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

   14. +-commutative N/A

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

   15. lift-/.f64 N/A

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

   16. clear-num N/A

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

   17. associate-/r/ N/A

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

   18. lower-fma.f64 N/A

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

   19. metadata-eval 99.2

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

   20. lift-*.f64 N/A

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

   21. *-commutative N/A

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

   22. lower-*.f64 99.2

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

   23. lift-*.f64 N/A

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

   24. *-commutative N/A

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

   25. lower-*.f64 99.2

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

4. Applied rewrites 99.2%

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

5. Final simplification 99.2%

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

Alternative 2: 63.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-0.25, b \cdot a, c\right)\\ \mathbf{if}\;z \cdot t \leq -1.7 \cdot 10^{+95}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot t, 0.0625, c\right)\\ \mathbf{elif}\;z \cdot t \leq -9.6 \cdot 10^{-272}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \cdot t \leq 1.3 \cdot 10^{-134}:\\ \;\;\;\;\mathsf{fma}\left(x, y, c\right)\\ \mathbf{elif}\;z \cdot t \leq 2.5 \cdot 10^{+237}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot t\right) \cdot 0.0625\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (fma -0.25 (* b a) c)))
   (if (<= (* z t) -1.7e+95)
     (fma (* z t) 0.0625 c)
     (if (<= (* z t) -9.6e-272)
       t_1
       (if (<= (* z t) 1.3e-134)
         (fma x y c)
         (if (<= (* z t) 2.5e+237) t_1 (* (* z t) 0.0625)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = fma(-0.25, (b * a), c);
	double tmp;
	if ((z * t) <= -1.7e+95) {
		tmp = fma((z * t), 0.0625, c);
	} else if ((z * t) <= -9.6e-272) {
		tmp = t_1;
	} else if ((z * t) <= 1.3e-134) {
		tmp = fma(x, y, c);
	} else if ((z * t) <= 2.5e+237) {
		tmp = t_1;
	} else {
		tmp = (z * t) * 0.0625;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = fma(-0.25, Float64(b * a), c)
	tmp = 0.0
	if (Float64(z * t) <= -1.7e+95)
		tmp = fma(Float64(z * t), 0.0625, c);
	elseif (Float64(z * t) <= -9.6e-272)
		tmp = t_1;
	elseif (Float64(z * t) <= 1.3e-134)
		tmp = fma(x, y, c);
	elseif (Float64(z * t) <= 2.5e+237)
		tmp = t_1;
	else
		tmp = Float64(Float64(z * t) * 0.0625);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(-0.25 * N[(b * a), $MachinePrecision] + c), $MachinePrecision]}, If[LessEqual[N[(z * t), $MachinePrecision], -1.7e+95], N[(N[(z * t), $MachinePrecision] * 0.0625 + c), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], -9.6e-272], t$95$1, If[LessEqual[N[(z * t), $MachinePrecision], 1.3e-134], N[(x * y + c), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 2.5e+237], t$95$1, N[(N[(z * t), $MachinePrecision] * 0.0625), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 z t) < -1.70000000000000011e95

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 88.5

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

   5. Applied rewrites 88.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 76.6%

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

   if -1.70000000000000011e95 < (*.f64 z t) < -9.59999999999999959e-272 or 1.30000000000000011e-134 < (*.f64 z t) < 2.5000000000000001e237

   1. Initial program 99.2%

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

   3. Taylor expanded in z around 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-sub-sign-inv N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 88.0

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

   5. Applied rewrites 88.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 66.9%

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

   if -9.59999999999999959e-272 < (*.f64 z t) < 1.30000000000000011e-134

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 74.8

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

   5. Applied rewrites 74.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 30.3%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 74.8%

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

   if 2.5000000000000001e237 < (*.f64 z t)

   1. Initial program 90.9%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. distribute-lft-neg-in N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. div-inv N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval 95.5

          \[\leadsto \mathsf{fma}\left(-a, b \cdot \color{blue}{0.25}, x \cdot y + \frac{z \cdot t}{16}\right) + c \]

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-/.f64 N/A

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

      16. clear-num N/A

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

      17. associate-/r/ N/A

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

      18. lower-fma.f64 N/A

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

      19. metadata-eval 95.5

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 95.5

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

      23. lift-*.f64 N/A

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

      24. *-commutative N/A

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

      25. lower-*.f64 95.5

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

   4. Applied rewrites 95.5%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 11.3

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

   7. Applied rewrites 11.3%

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

   8. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 86.7

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

   10. Applied rewrites 86.7%

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

Alternative 3: 89.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z \cdot t, 0.0625, c\right)\\ \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{+37}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, b \cdot a, \mathsf{fma}\left(y, x, c\right)\right)\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+79}:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot b, a, t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (fma (* z t) 0.0625 c)))
   (if (<= (* x y) -1e+37)
     (fma -0.25 (* b a) (fma y x c))
     (if (<= (* x y) 5e+79) (fma (* -0.25 b) a t_1) (fma y x t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = fma((z * t), 0.0625, c);
	double tmp;
	if ((x * y) <= -1e+37) {
		tmp = fma(-0.25, (b * a), fma(y, x, c));
	} else if ((x * y) <= 5e+79) {
		tmp = fma((-0.25 * b), a, t_1);
	} else {
		tmp = fma(y, x, t_1);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = fma(Float64(z * t), 0.0625, c)
	tmp = 0.0
	if (Float64(x * y) <= -1e+37)
		tmp = fma(-0.25, Float64(b * a), fma(y, x, c));
	elseif (Float64(x * y) <= 5e+79)
		tmp = fma(Float64(-0.25 * b), a, t_1);
	else
		tmp = fma(y, x, t_1);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(z * t), $MachinePrecision] * 0.0625 + c), $MachinePrecision]}, If[LessEqual[N[(x * y), $MachinePrecision], -1e+37], N[(-0.25 * N[(b * a), $MachinePrecision] + N[(y * x + c), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 5e+79], N[(N[(-0.25 * b), $MachinePrecision] * a + t$95$1), $MachinePrecision], N[(y * x + t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -9.99999999999999954e36

   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 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-sub-sign-inv N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 91.6

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

   5. Applied rewrites 91.6%

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

   if -9.99999999999999954e36 < (*.f64 x y) < 5e79

   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

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 96.2

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

   5. Applied rewrites 96.2%

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

   7. Applied rewrites 96.8%

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

   if 5e79 < (*.f64 x y)

   1. Initial program 97.9%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 86.6

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

   5. Applied rewrites 86.6%

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

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{+37}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, b \cdot a, \mathsf{fma}\left(y, x, c\right)\right)\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+79}:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot b, a, \mathsf{fma}\left(z \cdot t, 0.0625, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, \mathsf{fma}\left(z \cdot t, 0.0625, c\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 88.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z \cdot t, 0.0625, c\right)\\ \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+30}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, b \cdot a, \mathsf{fma}\left(y, x, c\right)\right)\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+79}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, b \cdot a, t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (fma (* z t) 0.0625 c)))
   (if (<= (* x y) -2e+30)
     (fma -0.25 (* b a) (fma y x c))
     (if (<= (* x y) 5e+79) (fma -0.25 (* b a) t_1) (fma y x t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = fma((z * t), 0.0625, c);
	double tmp;
	if ((x * y) <= -2e+30) {
		tmp = fma(-0.25, (b * a), fma(y, x, c));
	} else if ((x * y) <= 5e+79) {
		tmp = fma(-0.25, (b * a), t_1);
	} else {
		tmp = fma(y, x, t_1);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = fma(Float64(z * t), 0.0625, c)
	tmp = 0.0
	if (Float64(x * y) <= -2e+30)
		tmp = fma(-0.25, Float64(b * a), fma(y, x, c));
	elseif (Float64(x * y) <= 5e+79)
		tmp = fma(-0.25, Float64(b * a), t_1);
	else
		tmp = fma(y, x, t_1);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(z * t), $MachinePrecision] * 0.0625 + c), $MachinePrecision]}, If[LessEqual[N[(x * y), $MachinePrecision], -2e+30], N[(-0.25 * N[(b * a), $MachinePrecision] + N[(y * x + c), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 5e+79], N[(-0.25 * N[(b * a), $MachinePrecision] + t$95$1), $MachinePrecision], N[(y * x + t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -2e30

   1. Initial program 96.7%

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

   3. Taylor expanded in z around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 90.2

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

   5. Applied rewrites 90.2%

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

   if -2e30 < (*.f64 x y) < 5e79

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 96.8

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

   5. Applied rewrites 96.8%

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

   if 5e79 < (*.f64 x y)

   1. Initial program 97.9%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 86.6

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

   5. Applied rewrites 86.6%

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

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+30}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, b \cdot a, \mathsf{fma}\left(y, x, c\right)\right)\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+79}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, b \cdot a, \mathsf{fma}\left(z \cdot t, 0.0625, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, \mathsf{fma}\left(z \cdot t, 0.0625, c\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 88.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -4 \cdot 10^{+95}:\\ \;\;\;\;\mathsf{fma}\left(y, x, \mathsf{fma}\left(z \cdot t, 0.0625, c\right)\right)\\ \mathbf{elif}\;z \cdot t \leq 10^{+188}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, b \cdot a, \mathsf{fma}\left(y, x, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot a, b, \left(t \cdot 0.0625\right) \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (<= (* z t) -4e+95)
   (fma y x (fma (* z t) 0.0625 c))
   (if (<= (* z t) 1e+188)
     (fma -0.25 (* b a) (fma y x c))
     (fma (* -0.25 a) b (* (* t 0.0625) z)))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if ((z * t) <= -4e+95) {
		tmp = fma(y, x, fma((z * t), 0.0625, c));
	} else if ((z * t) <= 1e+188) {
		tmp = fma(-0.25, (b * a), fma(y, x, c));
	} else {
		tmp = fma((-0.25 * a), b, ((t * 0.0625) * z));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if (Float64(z * t) <= -4e+95)
		tmp = fma(y, x, fma(Float64(z * t), 0.0625, c));
	elseif (Float64(z * t) <= 1e+188)
		tmp = fma(-0.25, Float64(b * a), fma(y, x, c));
	else
		tmp = fma(Float64(-0.25 * a), b, Float64(Float64(t * 0.0625) * z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[LessEqual[N[(z * t), $MachinePrecision], -4e+95], N[(y * x + N[(N[(z * t), $MachinePrecision] * 0.0625 + c), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 1e+188], N[(-0.25 * N[(b * a), $MachinePrecision] + N[(y * x + c), $MachinePrecision]), $MachinePrecision], N[(N[(-0.25 * a), $MachinePrecision] * b + N[(N[(t * 0.0625), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -4.00000000000000008e95

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 88.5

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

   5. Applied rewrites 88.5%

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

   if -4.00000000000000008e95 < (*.f64 z t) < 1e188

   1. Initial program 99.4%

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

   3. Taylor expanded in z around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 92.9

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

   5. Applied rewrites 92.9%

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

   if 1e188 < (*.f64 z t)

   1. Initial program 92.8%

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

   3. Taylor expanded in 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-sub-sign-inv N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 89.2

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

   5. Applied rewrites 89.2%

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

   7. Applied rewrites 92.9%

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

   8. Taylor expanded in c around 0

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

   10. Applied rewrites 86.7%

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

   12. Applied rewrites 90.3%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -4 \cdot 10^{+95}:\\ \;\;\;\;\mathsf{fma}\left(y, x, \mathsf{fma}\left(z \cdot t, 0.0625, c\right)\right)\\ \mathbf{elif}\;z \cdot t \leq 10^{+188}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, b \cdot a, \mathsf{fma}\left(y, x, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot a, b, \left(t \cdot 0.0625\right) \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 89.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-0.25, b \cdot a, \mathsf{fma}\left(y, x, c\right)\right)\\ \mathbf{if}\;b \cdot a \leq -1 \cdot 10^{-8}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \cdot a \leq 10^{+163}:\\ \;\;\;\;\mathsf{fma}\left(y, x, \mathsf{fma}\left(z \cdot t, 0.0625, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (fma -0.25 (* b a) (fma y x c))))
   (if (<= (* b a) -1e-8)
     t_1
     (if (<= (* b a) 1e+163) (fma y x (fma (* z t) 0.0625 c)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = fma(-0.25, (b * a), fma(y, x, c));
	double tmp;
	if ((b * a) <= -1e-8) {
		tmp = t_1;
	} else if ((b * a) <= 1e+163) {
		tmp = fma(y, x, fma((z * t), 0.0625, c));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = fma(-0.25, Float64(b * a), fma(y, x, c))
	tmp = 0.0
	if (Float64(b * a) <= -1e-8)
		tmp = t_1;
	elseif (Float64(b * a) <= 1e+163)
		tmp = fma(y, x, fma(Float64(z * t), 0.0625, c));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(-0.25 * N[(b * a), $MachinePrecision] + N[(y * x + c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(b * a), $MachinePrecision], -1e-8], t$95$1, If[LessEqual[N[(b * a), $MachinePrecision], 1e+163], N[(y * x + N[(N[(z * t), $MachinePrecision] * 0.0625 + c), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a b) < -1e-8 or 9.9999999999999994e162 < (*.f64 a b)

   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 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-sub-sign-inv N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 86.8

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

   5. Applied rewrites 86.8%

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

   if -1e-8 < (*.f64 a b) < 9.9999999999999994e162

   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 a around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 95.6

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

   5. Applied rewrites 95.6%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot a \leq -1 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, b \cdot a, \mathsf{fma}\left(y, x, c\right)\right)\\ \mathbf{elif}\;b \cdot a \leq 10^{+163}:\\ \;\;\;\;\mathsf{fma}\left(y, x, \mathsf{fma}\left(z \cdot t, 0.0625, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, b \cdot a, \mathsf{fma}\left(y, x, c\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1.3 \cdot 10^{+115}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot t, 0.0625, c\right)\\ \mathbf{elif}\;z \cdot t \leq 4.3 \cdot 10^{+241}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, b \cdot a, \mathsf{fma}\left(y, x, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot t\right) \cdot 0.0625\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (<= (* z t) -1.3e+115)
   (fma (* z t) 0.0625 c)
   (if (<= (* z t) 4.3e+241)
     (fma -0.25 (* b a) (fma y x c))
     (* (* z t) 0.0625))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if ((z * t) <= -1.3e+115) {
		tmp = fma((z * t), 0.0625, c);
	} else if ((z * t) <= 4.3e+241) {
		tmp = fma(-0.25, (b * a), fma(y, x, c));
	} else {
		tmp = (z * t) * 0.0625;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if (Float64(z * t) <= -1.3e+115)
		tmp = fma(Float64(z * t), 0.0625, c);
	elseif (Float64(z * t) <= 4.3e+241)
		tmp = fma(-0.25, Float64(b * a), fma(y, x, c));
	else
		tmp = Float64(Float64(z * t) * 0.0625);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[LessEqual[N[(z * t), $MachinePrecision], -1.3e+115], N[(N[(z * t), $MachinePrecision] * 0.0625 + c), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 4.3e+241], N[(-0.25 * N[(b * a), $MachinePrecision] + N[(y * x + c), $MachinePrecision]), $MachinePrecision], N[(N[(z * t), $MachinePrecision] * 0.0625), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -1.3e115

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 89.0

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

   5. Applied rewrites 89.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 79.9%

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

   if -1.3e115 < (*.f64 z t) < 4.30000000000000004e241

   1. Initial program 99.5%

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

   3. Taylor expanded in z around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 91.2

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

   5. Applied rewrites 91.2%

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

   if 4.30000000000000004e241 < (*.f64 z t)

   1. Initial program 90.9%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. distribute-lft-neg-in N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. div-inv N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval 95.5

          \[\leadsto \mathsf{fma}\left(-a, b \cdot \color{blue}{0.25}, x \cdot y + \frac{z \cdot t}{16}\right) + c \]

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-/.f64 N/A

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

      16. clear-num N/A

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

      17. associate-/r/ N/A

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

      18. lower-fma.f64 N/A

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

      19. metadata-eval 95.5

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 95.5

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

      23. lift-*.f64 N/A

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

      24. *-commutative N/A

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

      25. lower-*.f64 95.5

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

   4. Applied rewrites 95.5%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 11.3

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

   7. Applied rewrites 11.3%

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

   8. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 86.7

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

   10. Applied rewrites 86.7%

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

Alternative 8: 64.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z \cdot t, 0.0625, c\right)\\ \mathbf{if}\;z \cdot t \leq -255000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \cdot t \leq 520000:\\ \;\;\;\;\mathsf{fma}\left(-0.25, b \cdot a, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (fma (* z t) 0.0625 c)))
   (if (<= (* z t) -255000.0)
     t_1
     (if (<= (* z t) 520000.0) (fma -0.25 (* b a) (* x y)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = fma((z * t), 0.0625, c);
	double tmp;
	if ((z * t) <= -255000.0) {
		tmp = t_1;
	} else if ((z * t) <= 520000.0) {
		tmp = fma(-0.25, (b * a), (x * y));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = fma(Float64(z * t), 0.0625, c)
	tmp = 0.0
	if (Float64(z * t) <= -255000.0)
		tmp = t_1;
	elseif (Float64(z * t) <= 520000.0)
		tmp = fma(-0.25, Float64(b * a), Float64(x * y));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(z * t), $MachinePrecision] * 0.0625 + c), $MachinePrecision]}, If[LessEqual[N[(z * t), $MachinePrecision], -255000.0], t$95$1, If[LessEqual[N[(z * t), $MachinePrecision], 520000.0], N[(-0.25 * N[(b * a), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -255000 or 5.2e5 < (*.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

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 80.6

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

   5. Applied rewrites 80.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 69.1%

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

   if -255000 < (*.f64 z t) < 5.2e5

   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 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-sub-sign-inv N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 97.3

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

   5. Applied rewrites 97.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 64.0%

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

   9. Taylor expanded in c around 0

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

   11. Applied rewrites 73.3%

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

Alternative 9: 65.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+120}:\\ \;\;\;\;\mathsf{fma}\left(x, y, c\right)\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+79}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, b \cdot a, c\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, y, c\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (<= (* x y) -2e+120)
   (fma x y c)
   (if (<= (* x y) 5e+79) (fma -0.25 (* b a) c) (fma x y c))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if ((x * y) <= -2e+120) {
		tmp = fma(x, y, c);
	} else if ((x * y) <= 5e+79) {
		tmp = fma(-0.25, (b * a), c);
	} else {
		tmp = fma(x, y, c);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if (Float64(x * y) <= -2e+120)
		tmp = fma(x, y, c);
	elseif (Float64(x * y) <= 5e+79)
		tmp = fma(-0.25, Float64(b * a), c);
	else
		tmp = fma(x, y, c);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[LessEqual[N[(x * y), $MachinePrecision], -2e+120], N[(x * y + c), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 5e+79], N[(-0.25 * N[(b * a), $MachinePrecision] + c), $MachinePrecision], N[(x * y + c), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -2e120 or 5e79 < (*.f64 x y)

   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 a around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 85.6

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

   5. Applied rewrites 85.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 26.0%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 75.2%

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

   if -2e120 < (*.f64 x y) < 5e79

   1. Initial program 99.4%

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

   3. Taylor expanded in z around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 69.2

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

   5. Applied rewrites 69.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 64.5%

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

Alternative 10: 63.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(b \cdot a\right) \cdot -0.25\\ \mathbf{if}\;b \cdot a \leq -1 \cdot 10^{+174}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \cdot a \leq 4 \cdot 10^{+143}:\\ \;\;\;\;\mathsf{fma}\left(x, y, c\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (* (* b a) -0.25)))
   (if (<= (* b a) -1e+174) t_1 (if (<= (* b a) 4e+143) (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 = (b * a) * -0.25;
	double tmp;
	if ((b * a) <= -1e+174) {
		tmp = t_1;
	} else if ((b * a) <= 4e+143) {
		tmp = fma(x, y, c);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(b * a) * -0.25)
	tmp = 0.0
	if (Float64(b * a) <= -1e+174)
		tmp = t_1;
	elseif (Float64(b * a) <= 4e+143)
		tmp = fma(x, y, c);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(b * a), $MachinePrecision] * -0.25), $MachinePrecision]}, If[LessEqual[N[(b * a), $MachinePrecision], -1e+174], t$95$1, If[LessEqual[N[(b * a), $MachinePrecision], 4e+143], N[(x * y + c), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a b) < -1.00000000000000007e174 or 4.0000000000000001e143 < (*.f64 a b)

   1. Initial program 97.3%

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

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 78.0

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

   5. Applied rewrites 78.0%

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

   if -1.00000000000000007e174 < (*.f64 a b) < 4.0000000000000001e143

   1. Initial program 99.4%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 88.9

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

   5. Applied rewrites 88.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 59.1%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 59.5%

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

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot a \leq -1 \cdot 10^{+174}:\\ \;\;\;\;\left(b \cdot a\right) \cdot -0.25\\ \mathbf{elif}\;b \cdot a \leq 4 \cdot 10^{+143}:\\ \;\;\;\;\mathsf{fma}\left(x, y, c\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot a\right) \cdot -0.25\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 62.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(z \cdot t\right) \cdot 0.0625\\ \mathbf{if}\;z \cdot t \leq -3.15 \cdot 10^{+111}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \cdot t \leq 4.2 \cdot 10^{+159}:\\ \;\;\;\;\mathsf{fma}\left(x, y, c\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (* (* z t) 0.0625)))
   (if (<= (* z t) -3.15e+111)
     t_1
     (if (<= (* z t) 4.2e+159) (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 = (z * t) * 0.0625;
	double tmp;
	if ((z * t) <= -3.15e+111) {
		tmp = t_1;
	} else if ((z * t) <= 4.2e+159) {
		tmp = fma(x, y, c);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(z * t) * 0.0625)
	tmp = 0.0
	if (Float64(z * t) <= -3.15e+111)
		tmp = t_1;
	elseif (Float64(z * t) <= 4.2e+159)
		tmp = fma(x, y, c);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(z * t), $MachinePrecision] * 0.0625), $MachinePrecision]}, If[LessEqual[N[(z * t), $MachinePrecision], -3.15e+111], t$95$1, If[LessEqual[N[(z * t), $MachinePrecision], 4.2e+159], N[(x * y + c), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -3.1500000000000001e111 or 4.19999999999999978e159 < (*.f64 z t)

   1. Initial program 97.4%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. distribute-lft-neg-in N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. div-inv N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval 98.7

          \[\leadsto \mathsf{fma}\left(-a, b \cdot \color{blue}{0.25}, x \cdot y + \frac{z \cdot t}{16}\right) + c \]

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-/.f64 N/A

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

      16. clear-num N/A

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

      17. associate-/r/ N/A

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

      18. lower-fma.f64 N/A

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

      19. metadata-eval 98.7

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 98.7

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

      23. lift-*.f64 N/A

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

      24. *-commutative N/A

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

      25. lower-*.f64 98.7

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

   4. Applied rewrites 98.7%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 13.7

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

   7. Applied rewrites 13.7%

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

   8. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 68.9

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

   10. Applied rewrites 68.9%

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

   if -3.1500000000000001e111 < (*.f64 z t) < 4.19999999999999978e159

   1. Initial program 99.4%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 67.0

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

   5. Applied rewrites 67.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 36.0%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 60.2%

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

Alternative 12: 48.9% accurate, 6.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.8%

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

3. Taylor expanded in a around 0

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

   1. associate-+r+ N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. +-commutative N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-*.f64 71.7

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

5. Applied rewrites 71.7%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 47.7%

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

9. Taylor expanded in z around 0

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

11. Applied rewrites 47.5%

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

Alternative 13: 28.6% accurate, 7.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.8%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-/.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-/l* N/A

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

   7. distribute-lft-neg-in N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-neg.f64 N/A

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

   10. div-inv N/A

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

   11. lower-*.f64 N/A

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

   12. metadata-eval 99.2

       \[\leadsto \mathsf{fma}\left(-a, b \cdot \color{blue}{0.25}, x \cdot y + \frac{z \cdot t}{16}\right) + c \]

   13. lift-+.f64 N/A

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

   14. +-commutative N/A

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

   15. lift-/.f64 N/A

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

   16. clear-num N/A

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

   17. associate-/r/ N/A

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

   18. lower-fma.f64 N/A

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

   19. metadata-eval 99.2

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

   20. lift-*.f64 N/A

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

   21. *-commutative N/A

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

   22. lower-*.f64 99.2

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

   23. lift-*.f64 N/A

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

   24. *-commutative N/A

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

   25. lower-*.f64 99.2

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

4. Applied rewrites 99.2%

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

5. Taylor expanded in x around inf

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

   1. lower-*.f64 27.7

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

7. Applied rewrites 27.7%

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

Reproduce

herbie shell --seed 2024298 
(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))