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

Percentage Accurate: 97.4% → 98.8%

Time: 8.2s

Alternatives: 13

Speedup: 0.5×

• 32.9% 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: 97.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.8% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (- (+ (* y x) (/ (* t z) 16.0)) (/ (* b a) 4.0))))
   (if (<= t_1 INFINITY) (+ c t_1) (fma (* -0.25 b) a (* 0.0625 (* t z))))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = ((y * x) + ((t * z) / 16.0)) - ((b * a) / 4.0);
	double tmp;
	if (t_1 <= ((double) INFINITY)) {
		tmp = c + t_1;
	} else {
		tmp = fma((-0.25 * b), a, (0.0625 * (t * z)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(Float64(y * x) + Float64(Float64(t * z) / 16.0)) - Float64(Float64(b * a) / 4.0))
	tmp = 0.0
	if (t_1 <= Inf)
		tmp = Float64(c + t_1);
	else
		tmp = fma(Float64(-0.25 * b), a, Float64(0.0625 * Float64(t * z)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(N[(y * x), $MachinePrecision] + N[(N[(t * z), $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] - N[(N[(b * a), $MachinePrecision] / 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, Infinity], N[(c + t$95$1), $MachinePrecision], N[(N[(-0.25 * b), $MachinePrecision] * a + N[(0.0625 * N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 (*.f64 x y) (/.f64 (*.f64 z t) #s(literal 16 binary64))) (/.f64 (*.f64 a b) #s(literal 4 binary64))) < +inf.0

   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

   if +inf.0 < (-.f64 (+.f64 (*.f64 x y) (/.f64 (*.f64 z t) #s(literal 16 binary64))) (/.f64 (*.f64 a b) #s(literal 4 binary64)))

   1. Initial program 0.0%

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

   3. Taylor expanded in b 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. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. associate-*r* N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 42.9

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

   5. Applied rewrites 42.9%

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

   6. Taylor expanded in y around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 57.1

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

   8. Applied rewrites 57.1%

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

   9. Taylor expanded in c around 0

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

   11. Applied rewrites 57.1%

       \[\leadsto \mathsf{fma}\left(-0.25 \cdot b, a, \left(z \cdot t\right) \cdot 0.0625\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.8%

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

Alternative 2: 67.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \cdot z \leq -5 \cdot 10^{+57}:\\ \;\;\;\;\mathsf{fma}\left(0.0625 \cdot t, z, y \cdot x\right)\\ \mathbf{elif}\;t \cdot z \leq 10^{-234}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \mathbf{elif}\;t \cdot z \leq 2 \cdot 10^{+152}:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot b, a, y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, 0.0625 \cdot \left(t \cdot z\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (<= (* t z) -5e+57)
   (fma (* 0.0625 t) z (* y x))
   (if (<= (* t z) 1e-234)
     (fma y x c)
     (if (<= (* t z) 2e+152)
       (fma (* -0.25 b) a (* y x))
       (fma y x (* 0.0625 (* t z)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if ((t * z) <= -5e+57) {
		tmp = fma((0.0625 * t), z, (y * x));
	} else if ((t * z) <= 1e-234) {
		tmp = fma(y, x, c);
	} else if ((t * z) <= 2e+152) {
		tmp = fma((-0.25 * b), a, (y * x));
	} else {
		tmp = fma(y, x, (0.0625 * (t * z)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if (Float64(t * z) <= -5e+57)
		tmp = fma(Float64(0.0625 * t), z, Float64(y * x));
	elseif (Float64(t * z) <= 1e-234)
		tmp = fma(y, x, c);
	elseif (Float64(t * z) <= 2e+152)
		tmp = fma(Float64(-0.25 * b), a, Float64(y * x));
	else
		tmp = fma(y, x, Float64(0.0625 * Float64(t * z)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[LessEqual[N[(t * z), $MachinePrecision], -5e+57], N[(N[(0.0625 * t), $MachinePrecision] * z + N[(y * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(t * z), $MachinePrecision], 1e-234], N[(y * x + c), $MachinePrecision], If[LessEqual[N[(t * z), $MachinePrecision], 2e+152], N[(N[(-0.25 * b), $MachinePrecision] * a + N[(y * x), $MachinePrecision]), $MachinePrecision], N[(y * x + N[(0.0625 * N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 z t) < -4.99999999999999972e57

   1. Initial program 96.6%

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

   3. Taylor expanded in b 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. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. associate-*r* N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 87.8

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

   5. Applied rewrites 87.8%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 84.7%

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

   if -4.99999999999999972e57 < (*.f64 z t) < 9.9999999999999996e-235

   1. Initial program 98.0%

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

   3. Taylor expanded in b 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. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. associate-*r* N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 75.4

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

   5. Applied rewrites 75.4%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 69.3%

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

   if 9.9999999999999996e-235 < (*.f64 z t) < 2.0000000000000001e152

   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 t around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 90.3

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

   5. Applied rewrites 90.3%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 74.7%

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

   if 2.0000000000000001e152 < (*.f64 z t)

   1. Initial program 92.3%

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

   3. Taylor expanded in b 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. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. associate-*r* N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 90.3

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

   5. Applied rewrites 90.3%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 85.6%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot z \leq -5 \cdot 10^{+57}:\\ \;\;\;\;\mathsf{fma}\left(0.0625 \cdot t, z, y \cdot x\right)\\ \mathbf{elif}\;t \cdot z \leq 10^{-234}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \mathbf{elif}\;t \cdot z \leq 2 \cdot 10^{+152}:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot b, a, y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, 0.0625 \cdot \left(t \cdot z\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 67.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, x, 0.0625 \cdot \left(t \cdot z\right)\right)\\ \mathbf{if}\;t \cdot z \leq -5 \cdot 10^{+57}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \cdot z \leq 10^{-234}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \mathbf{elif}\;t \cdot z \leq 2 \cdot 10^{+152}:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot b, a, y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (fma y x (* 0.0625 (* t z)))))
   (if (<= (* t z) -5e+57)
     t_1
     (if (<= (* t z) 1e-234)
       (fma y x c)
       (if (<= (* t z) 2e+152) (fma (* -0.25 b) a (* 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(y, x, (0.0625 * (t * z)));
	double tmp;
	if ((t * z) <= -5e+57) {
		tmp = t_1;
	} else if ((t * z) <= 1e-234) {
		tmp = fma(y, x, c);
	} else if ((t * z) <= 2e+152) {
		tmp = fma((-0.25 * b), a, (y * x));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = fma(y, x, Float64(0.0625 * Float64(t * z)))
	tmp = 0.0
	if (Float64(t * z) <= -5e+57)
		tmp = t_1;
	elseif (Float64(t * z) <= 1e-234)
		tmp = fma(y, x, c);
	elseif (Float64(t * z) <= 2e+152)
		tmp = fma(Float64(-0.25 * b), a, Float64(y * x));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(y * x + N[(0.0625 * N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t * z), $MachinePrecision], -5e+57], t$95$1, If[LessEqual[N[(t * z), $MachinePrecision], 1e-234], N[(y * x + c), $MachinePrecision], If[LessEqual[N[(t * z), $MachinePrecision], 2e+152], N[(N[(-0.25 * b), $MachinePrecision] * a + N[(y * x), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -4.99999999999999972e57 or 2.0000000000000001e152 < (*.f64 z t)

   1. Initial program 94.9%

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

   3. Taylor expanded in b 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. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. associate-*r* N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 88.8

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

   5. Applied rewrites 88.8%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 83.0%

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

   if -4.99999999999999972e57 < (*.f64 z t) < 9.9999999999999996e-235

   1. Initial program 98.0%

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

   3. Taylor expanded in b 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. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. associate-*r* N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 75.4

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

   5. Applied rewrites 75.4%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 69.3%

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

   if 9.9999999999999996e-235 < (*.f64 z t) < 2.0000000000000001e152

   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 t around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 90.3

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

   5. Applied rewrites 90.3%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 74.7%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot z \leq -5 \cdot 10^{+57}:\\ \;\;\;\;\mathsf{fma}\left(y, x, 0.0625 \cdot \left(t \cdot z\right)\right)\\ \mathbf{elif}\;t \cdot z \leq 10^{-234}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \mathbf{elif}\;t \cdot z \leq 2 \cdot 10^{+152}:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot b, a, y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, 0.0625 \cdot \left(t \cdot z\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 64.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, x, 0.0625 \cdot \left(t \cdot z\right)\right)\\ \mathbf{if}\;t \cdot z \leq -5 \cdot 10^{+57}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \cdot z \leq 5 \cdot 10^{-44}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \mathbf{elif}\;t \cdot z \leq 2 \cdot 10^{-9}:\\ \;\;\;\;-0.25 \cdot \left(b \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (fma y x (* 0.0625 (* t z)))))
   (if (<= (* t z) -5e+57)
     t_1
     (if (<= (* t z) 5e-44)
       (fma y x c)
       (if (<= (* t z) 2e-9) (* -0.25 (* b a)) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = fma(y, x, (0.0625 * (t * z)));
	double tmp;
	if ((t * z) <= -5e+57) {
		tmp = t_1;
	} else if ((t * z) <= 5e-44) {
		tmp = fma(y, x, c);
	} else if ((t * z) <= 2e-9) {
		tmp = -0.25 * (b * a);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = fma(y, x, Float64(0.0625 * Float64(t * z)))
	tmp = 0.0
	if (Float64(t * z) <= -5e+57)
		tmp = t_1;
	elseif (Float64(t * z) <= 5e-44)
		tmp = fma(y, x, c);
	elseif (Float64(t * z) <= 2e-9)
		tmp = Float64(-0.25 * Float64(b * a));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(y * x + N[(0.0625 * N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t * z), $MachinePrecision], -5e+57], t$95$1, If[LessEqual[N[(t * z), $MachinePrecision], 5e-44], N[(y * x + c), $MachinePrecision], If[LessEqual[N[(t * z), $MachinePrecision], 2e-9], N[(-0.25 * N[(b * a), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -4.99999999999999972e57 or 2.00000000000000012e-9 < (*.f64 z t)

   1. Initial program 95.8%

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

   3. Taylor expanded in b 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. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. associate-*r* N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 84.4

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

   5. Applied rewrites 84.4%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 78.8%

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

   if -4.99999999999999972e57 < (*.f64 z t) < 5.00000000000000039e-44

   1. Initial program 98.4%

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

   3. Taylor expanded in b 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. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. associate-*r* N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 74.5

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

   5. Applied rewrites 74.5%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 69.6%

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

   if 5.00000000000000039e-44 < (*.f64 z t) < 2.00000000000000012e-9

   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 b 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 65.4

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

   5. Applied rewrites 65.4%

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

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot z \leq -5 \cdot 10^{+57}:\\ \;\;\;\;\mathsf{fma}\left(y, x, 0.0625 \cdot \left(t \cdot z\right)\right)\\ \mathbf{elif}\;t \cdot z \leq 5 \cdot 10^{-44}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \mathbf{elif}\;t \cdot z \leq 2 \cdot 10^{-9}:\\ \;\;\;\;-0.25 \cdot \left(b \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, 0.0625 \cdot \left(t \cdot z\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 88.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-0.25 \cdot b, a, \mathsf{fma}\left(0.0625 \cdot t, z, c\right)\right)\\ \mathbf{if}\;b \cdot a \leq -5 \cdot 10^{+161}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \cdot a \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(y, x, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (fma (* -0.25 b) a (fma (* 0.0625 t) z c))))
   (if (<= (* b a) -5e+161)
     t_1
     (if (<= (* b a) 5e-7) (fma (* 0.0625 t) z (fma y x c)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = fma((-0.25 * b), a, fma((0.0625 * t), z, c));
	double tmp;
	if ((b * a) <= -5e+161) {
		tmp = t_1;
	} else if ((b * a) <= 5e-7) {
		tmp = fma((0.0625 * t), z, fma(y, x, c));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = fma(Float64(-0.25 * b), a, fma(Float64(0.0625 * t), z, c))
	tmp = 0.0
	if (Float64(b * a) <= -5e+161)
		tmp = t_1;
	elseif (Float64(b * a) <= 5e-7)
		tmp = fma(Float64(0.0625 * t), z, fma(y, x, c));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(-0.25 * b), $MachinePrecision] * a + N[(N[(0.0625 * t), $MachinePrecision] * z + c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(b * a), $MachinePrecision], -5e+161], t$95$1, If[LessEqual[N[(b * a), $MachinePrecision], 5e-7], N[(N[(0.0625 * t), $MachinePrecision] * z + N[(y * x + c), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a b) < -4.9999999999999997e161 or 4.99999999999999977e-7 < (*.f64 a b)

   1. Initial program 96.6%

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

   3. Taylor expanded in y around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 93.4

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

   5. Applied rewrites 93.4%

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

   if -4.9999999999999997e161 < (*.f64 a b) < 4.99999999999999977e-7

   1. Initial program 97.6%

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

   3. Taylor expanded in b 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. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. associate-*r* N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 96.0

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

   5. Applied rewrites 96.0%

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

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot a \leq -5 \cdot 10^{+161}:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot b, a, \mathsf{fma}\left(0.0625 \cdot t, z, c\right)\right)\\ \mathbf{elif}\;b \cdot a \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(y, x, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot b, a, \mathsf{fma}\left(0.0625 \cdot t, z, c\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 88.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \cdot z \leq -5 \cdot 10^{-28}:\\ \;\;\;\;\mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(y, x, c\right)\right)\\ \mathbf{elif}\;t \cdot z \leq 2 \cdot 10^{+152}:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot b, a, \mathsf{fma}\left(y, x, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, 0.0625 \cdot \left(t \cdot z\right)\right) + c\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (<= (* t z) -5e-28)
   (fma (* 0.0625 t) z (fma y x c))
   (if (<= (* t z) 2e+152)
     (fma (* -0.25 b) a (fma y x c))
     (+ (fma y x (* 0.0625 (* t z))) c))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if ((t * z) <= -5e-28) {
		tmp = fma((0.0625 * t), z, fma(y, x, c));
	} else if ((t * z) <= 2e+152) {
		tmp = fma((-0.25 * b), a, fma(y, x, c));
	} else {
		tmp = fma(y, x, (0.0625 * (t * z))) + c;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if (Float64(t * z) <= -5e-28)
		tmp = fma(Float64(0.0625 * t), z, fma(y, x, c));
	elseif (Float64(t * z) <= 2e+152)
		tmp = fma(Float64(-0.25 * b), a, fma(y, x, c));
	else
		tmp = Float64(fma(y, x, Float64(0.0625 * Float64(t * z))) + c);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[LessEqual[N[(t * z), $MachinePrecision], -5e-28], N[(N[(0.0625 * t), $MachinePrecision] * z + N[(y * x + c), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(t * z), $MachinePrecision], 2e+152], N[(N[(-0.25 * b), $MachinePrecision] * a + N[(y * x + c), $MachinePrecision]), $MachinePrecision], N[(N[(y * x + N[(0.0625 * N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -5.0000000000000002e-28

   1. Initial program 97.5%

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

   3. Taylor expanded in b 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. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. associate-*r* N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 87.5

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

   5. Applied rewrites 87.5%

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

   if -5.0000000000000002e-28 < (*.f64 z t) < 2.0000000000000001e152

   1. Initial program 98.5%

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

   3. Taylor expanded in t around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 94.6

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

   5. Applied rewrites 94.6%

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

   if 2.0000000000000001e152 < (*.f64 z t)

   1. Initial program 92.3%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 92.8

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

   5. Applied rewrites 92.8%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot z \leq -5 \cdot 10^{-28}:\\ \;\;\;\;\mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(y, x, c\right)\right)\\ \mathbf{elif}\;t \cdot z \leq 2 \cdot 10^{+152}:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot b, a, \mathsf{fma}\left(y, x, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, 0.0625 \cdot \left(t \cdot z\right)\right) + c\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 87.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \cdot z \leq -5 \cdot 10^{-28}:\\ \;\;\;\;\mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(y, x, c\right)\right)\\ \mathbf{elif}\;t \cdot z \leq 5 \cdot 10^{+189}:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot b, a, \mathsf{fma}\left(y, x, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, 0.0625 \cdot \left(t \cdot z\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (<= (* t z) -5e-28)
   (fma (* 0.0625 t) z (fma y x c))
   (if (<= (* t z) 5e+189)
     (fma (* -0.25 b) a (fma y x c))
     (fma y x (* 0.0625 (* t z))))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if ((t * z) <= -5e-28) {
		tmp = fma((0.0625 * t), z, fma(y, x, c));
	} else if ((t * z) <= 5e+189) {
		tmp = fma((-0.25 * b), a, fma(y, x, c));
	} else {
		tmp = fma(y, x, (0.0625 * (t * z)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if (Float64(t * z) <= -5e-28)
		tmp = fma(Float64(0.0625 * t), z, fma(y, x, c));
	elseif (Float64(t * z) <= 5e+189)
		tmp = fma(Float64(-0.25 * b), a, fma(y, x, c));
	else
		tmp = fma(y, x, Float64(0.0625 * Float64(t * z)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[LessEqual[N[(t * z), $MachinePrecision], -5e-28], N[(N[(0.0625 * t), $MachinePrecision] * z + N[(y * x + c), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(t * z), $MachinePrecision], 5e+189], N[(N[(-0.25 * b), $MachinePrecision] * a + N[(y * x + c), $MachinePrecision]), $MachinePrecision], N[(y * x + N[(0.0625 * N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -5.0000000000000002e-28

   1. Initial program 97.5%

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

   3. Taylor expanded in b 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. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. associate-*r* N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 87.5

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

   5. Applied rewrites 87.5%

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

   if -5.0000000000000002e-28 < (*.f64 z t) < 5.0000000000000004e189

   1. Initial program 98.5%

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

   3. Taylor expanded in t around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 94.7

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

   5. Applied rewrites 94.7%

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

   if 5.0000000000000004e189 < (*.f64 z t)

   1. Initial program 91.7%

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

   3. Taylor expanded in b 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. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. associate-*r* N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 89.5

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

   5. Applied rewrites 89.5%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 89.8%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot z \leq -5 \cdot 10^{-28}:\\ \;\;\;\;\mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(y, x, c\right)\right)\\ \mathbf{elif}\;t \cdot z \leq 5 \cdot 10^{+189}:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot b, a, \mathsf{fma}\left(y, x, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, 0.0625 \cdot \left(t \cdot z\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 87.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \cdot z \leq -1 \cdot 10^{+78}:\\ \;\;\;\;\mathsf{fma}\left(0.0625 \cdot t, z, y \cdot x\right)\\ \mathbf{elif}\;t \cdot z \leq 5 \cdot 10^{+189}:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot b, a, \mathsf{fma}\left(y, x, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, 0.0625 \cdot \left(t \cdot z\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (<= (* t z) -1e+78)
   (fma (* 0.0625 t) z (* y x))
   (if (<= (* t z) 5e+189)
     (fma (* -0.25 b) a (fma y x c))
     (fma y x (* 0.0625 (* t z))))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if ((t * z) <= -1e+78) {
		tmp = fma((0.0625 * t), z, (y * x));
	} else if ((t * z) <= 5e+189) {
		tmp = fma((-0.25 * b), a, fma(y, x, c));
	} else {
		tmp = fma(y, x, (0.0625 * (t * z)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if (Float64(t * z) <= -1e+78)
		tmp = fma(Float64(0.0625 * t), z, Float64(y * x));
	elseif (Float64(t * z) <= 5e+189)
		tmp = fma(Float64(-0.25 * b), a, fma(y, x, c));
	else
		tmp = fma(y, x, Float64(0.0625 * Float64(t * z)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[LessEqual[N[(t * z), $MachinePrecision], -1e+78], N[(N[(0.0625 * t), $MachinePrecision] * z + N[(y * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(t * z), $MachinePrecision], 5e+189], N[(N[(-0.25 * b), $MachinePrecision] * a + N[(y * x + c), $MachinePrecision]), $MachinePrecision], N[(y * x + N[(0.0625 * N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -1.00000000000000001e78

   1. Initial program 96.4%

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

   3. Taylor expanded in b 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. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. associate-*r* N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 88.9

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

   5. Applied rewrites 88.9%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 85.6%

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

   if -1.00000000000000001e78 < (*.f64 z t) < 5.0000000000000004e189

   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 t around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 91.1

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

   5. Applied rewrites 91.1%

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

   if 5.0000000000000004e189 < (*.f64 z t)

   1. Initial program 91.7%

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

   3. Taylor expanded in b 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. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. associate-*r* N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 89.5

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

   5. Applied rewrites 89.5%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 89.8%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot z \leq -1 \cdot 10^{+78}:\\ \;\;\;\;\mathsf{fma}\left(0.0625 \cdot t, z, y \cdot x\right)\\ \mathbf{elif}\;t \cdot z \leq 5 \cdot 10^{+189}:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot b, a, \mathsf{fma}\left(y, x, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, 0.0625 \cdot \left(t \cdot z\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 63.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(0.0625 \cdot z, t, c\right)\\ \mathbf{if}\;t \cdot z \leq -5 \cdot 10^{+109}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \cdot z \leq 10^{+230}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (fma (* 0.0625 z) t c)))
   (if (<= (* t z) -5e+109) t_1 (if (<= (* t z) 1e+230) (fma y x c) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = fma((0.0625 * z), t, c);
	double tmp;
	if ((t * z) <= -5e+109) {
		tmp = t_1;
	} else if ((t * z) <= 1e+230) {
		tmp = fma(y, x, c);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = fma(Float64(0.0625 * z), t, c)
	tmp = 0.0
	if (Float64(t * z) <= -5e+109)
		tmp = t_1;
	elseif (Float64(t * z) <= 1e+230)
		tmp = fma(y, x, c);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(0.0625 * z), $MachinePrecision] * t + c), $MachinePrecision]}, If[LessEqual[N[(t * z), $MachinePrecision], -5e+109], t$95$1, If[LessEqual[N[(t * z), $MachinePrecision], 1e+230], N[(y * x + c), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -5.0000000000000001e109 or 1.0000000000000001e230 < (*.f64 z t)

   1. Initial program 93.8%

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

   3. Taylor expanded in b 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. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. associate-*r* N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 90.0

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

   5. Applied rewrites 90.0%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 84.6%

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

   if -5.0000000000000001e109 < (*.f64 z t) < 1.0000000000000001e230

   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 b 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. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. associate-*r* N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 71.8

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

   5. Applied rewrites 71.8%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 62.1%

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

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot z \leq -5 \cdot 10^{+109}:\\ \;\;\;\;\mathsf{fma}\left(0.0625 \cdot z, t, c\right)\\ \mathbf{elif}\;t \cdot z \leq 10^{+230}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.0625 \cdot z, t, c\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 61.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 0.0625 \cdot \left(t \cdot z\right)\\ \mathbf{if}\;t \cdot z \leq -3.25 \cdot 10^{+98}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \cdot z \leq 10^{+277}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (* 0.0625 (* t z))))
   (if (<= (* t z) -3.25e+98) t_1 (if (<= (* t z) 1e+277) (fma y x c) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = 0.0625 * (t * z);
	double tmp;
	if ((t * z) <= -3.25e+98) {
		tmp = t_1;
	} else if ((t * z) <= 1e+277) {
		tmp = fma(y, x, c);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(0.0625 * Float64(t * z))
	tmp = 0.0
	if (Float64(t * z) <= -3.25e+98)
		tmp = t_1;
	elseif (Float64(t * z) <= 1e+277)
		tmp = fma(y, x, c);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(0.0625 * N[(t * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t * z), $MachinePrecision], -3.25e+98], t$95$1, If[LessEqual[N[(t * z), $MachinePrecision], 1e+277], N[(y * x + c), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -3.25e98 or 1e277 < (*.f64 z t)

   1. Initial program 93.6%

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

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\frac{1}{16} \cdot \left(t \cdot z\right)} \]
   4. 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 82.9

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

   5. Applied rewrites 82.9%

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

   if -3.25e98 < (*.f64 z t) < 1e277

   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 b 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. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. associate-*r* N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 72.2

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

   5. Applied rewrites 72.2%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 61.9%

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

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot z \leq -3.25 \cdot 10^{+98}:\\ \;\;\;\;0.0625 \cdot \left(t \cdot z\right)\\ \mathbf{elif}\;t \cdot z \leq 10^{+277}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \mathbf{else}:\\ \;\;\;\;0.0625 \cdot \left(t \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 63.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -0.25 \cdot \left(b \cdot a\right)\\ \mathbf{if}\;b \cdot a \leq -5 \cdot 10^{+161}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \cdot a \leq 10^{+179}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (* -0.25 (* b a))))
   (if (<= (* b a) -5e+161) t_1 (if (<= (* b a) 1e+179) (fma y x c) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = -0.25 * (b * a);
	double tmp;
	if ((b * a) <= -5e+161) {
		tmp = t_1;
	} else if ((b * a) <= 1e+179) {
		tmp = fma(y, x, c);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(-0.25 * Float64(b * a))
	tmp = 0.0
	if (Float64(b * a) <= -5e+161)
		tmp = t_1;
	elseif (Float64(b * a) <= 1e+179)
		tmp = fma(y, x, c);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(-0.25 * N[(b * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(b * a), $MachinePrecision], -5e+161], t$95$1, If[LessEqual[N[(b * a), $MachinePrecision], 1e+179], N[(y * x + c), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a b) < -4.9999999999999997e161 or 9.9999999999999998e178 < (*.f64 a b)

   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 b 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 70.5

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

   5. Applied rewrites 70.5%

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

   if -4.9999999999999997e161 < (*.f64 a b) < 9.9999999999999998e178

   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. Taylor expanded in b 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. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. associate-*r* N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 92.0

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

   5. Applied rewrites 92.0%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 58.0%

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

4. Final simplification 61.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot a \leq -5 \cdot 10^{+161}:\\ \;\;\;\;-0.25 \cdot \left(b \cdot a\right)\\ \mathbf{elif}\;b \cdot a \leq 10^{+179}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \left(b \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 48.3% accurate, 6.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 97.2%

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

3. Taylor expanded in b 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. +-commutative N/A

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

   2. associate-+l+ N/A

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

   3. associate-*r* N/A

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

   4. +-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. +-commutative N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 77.5

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

5. Applied rewrites 77.5%

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

6. Taylor expanded in t around 0

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

8. Applied rewrites 47.3%

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

Alternative 13: 28.2% accurate, 7.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.2%

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

3. Taylor expanded in y around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 26.0

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

5. Applied rewrites 26.0%

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

Reproduce

herbie shell --seed 2024273 
(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))