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

Percentage Accurate: 97.9% → 98.9%

Time: 8.3s

Alternatives: 12

Speedup: 0.5×

• 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: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.9% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (- (+ (/ (* t z) 16.0) (* y x)) (/ (* b a) 4.0))))
   (if (<= t_1 INFINITY) (+ c t_1) (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 t_1 = (((t * z) / 16.0) + (y * x)) - ((b * a) / 4.0);
	double tmp;
	if (t_1 <= ((double) INFINITY)) {
		tmp = c + t_1;
	} else {
		tmp = fma(y, x, (0.0625 * (t * z)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(Float64(Float64(t * z) / 16.0) + Float64(y * x)) - Float64(Float64(b * a) / 4.0))
	tmp = 0.0
	if (t_1 <= Inf)
		tmp = Float64(c + t_1);
	else
		tmp = fma(y, x, 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[(N[(t * z), $MachinePrecision] / 16.0), $MachinePrecision] + N[(y * x), $MachinePrecision]), $MachinePrecision] - N[(N[(b * a), $MachinePrecision] / 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, Infinity], N[(c + t$95$1), $MachinePrecision], N[(y * x + 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 40.0

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

   5. Applied rewrites 40.0%

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

   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 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 77.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, x, 0.0625 \cdot \left(t \cdot z\right)\right)\\ t_2 := \frac{t \cdot z}{16} + y \cdot x\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+146}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{+183}:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot a, b, c\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (fma y x (* 0.0625 (* t z)))) (t_2 (+ (/ (* t z) 16.0) (* y x))))
   (if (<= t_2 -1e+146) t_1 (if (<= t_2 4e+183) (fma (* -0.25 a) b c) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = fma(y, x, (0.0625 * (t * z)));
	double t_2 = ((t * z) / 16.0) + (y * x);
	double tmp;
	if (t_2 <= -1e+146) {
		tmp = t_1;
	} else if (t_2 <= 4e+183) {
		tmp = fma((-0.25 * a), b, c);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = fma(y, x, Float64(0.0625 * Float64(t * z)))
	t_2 = Float64(Float64(Float64(t * z) / 16.0) + Float64(y * x))
	tmp = 0.0
	if (t_2 <= -1e+146)
		tmp = t_1;
	elseif (t_2 <= 4e+183)
		tmp = fma(Float64(-0.25 * a), b, c);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(y * x + N[(0.0625 * N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(t * z), $MachinePrecision] / 16.0), $MachinePrecision] + N[(y * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+146], t$95$1, If[LessEqual[t$95$2, 4e+183], N[(N[(-0.25 * a), $MachinePrecision] * b + c), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 x y) (/.f64 (*.f64 z t) #s(literal 16 binary64))) < -9.99999999999999934e145 or 3.99999999999999979e183 < (+.f64 (*.f64 x y) (/.f64 (*.f64 z t) #s(literal 16 binary64)))

   1. Initial program 95.7%

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

   3. Taylor expanded in 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 89.8

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

   5. Applied rewrites 89.8%

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

   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 86.6%

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

   if -9.99999999999999934e145 < (+.f64 (*.f64 x y) (/.f64 (*.f64 z t) #s(literal 16 binary64))) < 3.99999999999999979e183

   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 88.3

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

   5. Applied rewrites 88.3%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 74.3%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t \cdot z}{16} + y \cdot x \leq -1 \cdot 10^{+146}:\\ \;\;\;\;\mathsf{fma}\left(y, x, 0.0625 \cdot \left(t \cdot z\right)\right)\\ \mathbf{elif}\;\frac{t \cdot z}{16} + y \cdot x \leq 4 \cdot 10^{+183}:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot a, b, c\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, 0.0625 \cdot \left(t \cdot z\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \cdot z \leq -5 \cdot 10^{+57}:\\ \;\;\;\;\mathsf{fma}\left(y, x, 0.0625 \cdot \left(t \cdot z\right)\right) + c\\ \mathbf{elif}\;t \cdot z \leq 5 \cdot 10^{+109}:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot b, a, \mathsf{fma}\left(y, x, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot b, a, \mathsf{fma}\left(0.0625 \cdot t, z, c\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (<= (* t z) -5e+57)
   (+ (fma y x (* 0.0625 (* t z))) c)
   (if (<= (* t z) 5e+109)
     (fma (* -0.25 b) a (fma y x c))
     (fma (* -0.25 b) a (fma (* 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+57) {
		tmp = fma(y, x, (0.0625 * (t * z))) + c;
	} else if ((t * z) <= 5e+109) {
		tmp = fma((-0.25 * b), a, fma(y, x, c));
	} else {
		tmp = fma((-0.25 * b), a, fma((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+57)
		tmp = Float64(fma(y, x, Float64(0.0625 * Float64(t * z))) + c);
	elseif (Float64(t * z) <= 5e+109)
		tmp = fma(Float64(-0.25 * b), a, fma(y, x, c));
	else
		tmp = fma(Float64(-0.25 * b), a, fma(Float64(0.0625 * t), z, c));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[LessEqual[N[(t * z), $MachinePrecision], -5e+57], N[(N[(y * x + N[(0.0625 * N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision], If[LessEqual[N[(t * z), $MachinePrecision], 5e+109], N[(N[(-0.25 * b), $MachinePrecision] * a + N[(y * x + c), $MachinePrecision]), $MachinePrecision], N[(N[(-0.25 * b), $MachinePrecision] * a + N[(N[(0.0625 * t), $MachinePrecision] * z + c), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 95.7%

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

   3. Taylor expanded in 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 87.7

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

   5. Applied rewrites 87.7%

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

   if -4.99999999999999972e57 < (*.f64 z t) < 5.0000000000000001e109

   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 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 96.5

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

   5. Applied rewrites 96.5%

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

   if 5.0000000000000001e109 < (*.f64 z t)

   1. Initial program 95.9%

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

   3. Taylor expanded in 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 93.2

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

   5. Applied rewrites 93.2%

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

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot z \leq -5 \cdot 10^{+57}:\\ \;\;\;\;\mathsf{fma}\left(y, x, 0.0625 \cdot \left(t \cdot z\right)\right) + c\\ \mathbf{elif}\;t \cdot z \leq 5 \cdot 10^{+109}:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot b, a, \mathsf{fma}\left(y, x, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot b, a, \mathsf{fma}\left(0.0625 \cdot t, z, c\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 0.0625 \cdot \left(t \cdot z\right)\\ \mathbf{if}\;t \cdot z \leq -5 \cdot 10^{+57}:\\ \;\;\;\;\mathsf{fma}\left(y, x, t\_1\right) + c\\ \mathbf{elif}\;t \cdot z \leq 10^{+122}:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot b, a, \mathsf{fma}\left(y, x, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot b, a, t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (* 0.0625 (* t z))))
   (if (<= (* t z) -5e+57)
     (+ (fma y x t_1) c)
     (if (<= (* t z) 1e+122)
       (fma (* -0.25 b) a (fma y x c))
       (fma (* -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 = 0.0625 * (t * z);
	double tmp;
	if ((t * z) <= -5e+57) {
		tmp = fma(y, x, t_1) + c;
	} else if ((t * z) <= 1e+122) {
		tmp = fma((-0.25 * b), a, fma(y, x, c));
	} else {
		tmp = fma((-0.25 * b), a, 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) <= -5e+57)
		tmp = Float64(fma(y, x, t_1) + c);
	elseif (Float64(t * z) <= 1e+122)
		tmp = fma(Float64(-0.25 * b), a, fma(y, x, c));
	else
		tmp = fma(Float64(-0.25 * b), a, 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], -5e+57], N[(N[(y * x + t$95$1), $MachinePrecision] + c), $MachinePrecision], If[LessEqual[N[(t * z), $MachinePrecision], 1e+122], N[(N[(-0.25 * b), $MachinePrecision] * a + N[(y * x + c), $MachinePrecision]), $MachinePrecision], N[(N[(-0.25 * b), $MachinePrecision] * a + t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 95.7%

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

   3. Taylor expanded in 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 87.7

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

   5. Applied rewrites 87.7%

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

   if -4.99999999999999972e57 < (*.f64 z t) < 1.00000000000000001e122

   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 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 96.5

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

   5. Applied rewrites 96.5%

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

   if 1.00000000000000001e122 < (*.f64 z t)

   1. Initial program 95.7%

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

   3. Taylor expanded in 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 92.8

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

   5. Applied rewrites 92.8%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 90.6%

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

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot z \leq -5 \cdot 10^{+57}:\\ \;\;\;\;\mathsf{fma}\left(y, x, 0.0625 \cdot \left(t \cdot z\right)\right) + c\\ \mathbf{elif}\;t \cdot z \leq 10^{+122}:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot b, a, \mathsf{fma}\left(y, x, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot b, a, 0.0625 \cdot \left(t \cdot z\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 90.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, x, 0.0625 \cdot \left(t \cdot z\right)\right) + c\\ \mathbf{if}\;t \cdot z \leq -5 \cdot 10^{+57}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \cdot z \leq 2 \cdot 10^{+167}:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot b, a, \mathsf{fma}\left(y, x, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (+ (fma y x (* 0.0625 (* t z))) c)))
   (if (<= (* t z) -5e+57)
     t_1
     (if (<= (* t z) 2e+167) (fma (* -0.25 b) a (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(y, x, (0.0625 * (t * z))) + c;
	double tmp;
	if ((t * z) <= -5e+57) {
		tmp = t_1;
	} else if ((t * z) <= 2e+167) {
		tmp = fma((-0.25 * b), a, fma(y, x, c));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(fma(y, x, Float64(0.0625 * Float64(t * z))) + c)
	tmp = 0.0
	if (Float64(t * z) <= -5e+57)
		tmp = t_1;
	elseif (Float64(t * z) <= 2e+167)
		tmp = fma(Float64(-0.25 * b), a, 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[(y * x + N[(0.0625 * N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision]}, If[LessEqual[N[(t * z), $MachinePrecision], -5e+57], t$95$1, If[LessEqual[N[(t * z), $MachinePrecision], 2e+167], N[(N[(-0.25 * b), $MachinePrecision] * a + N[(y * x + c), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -4.99999999999999972e57 or 2.0000000000000001e167 < (*.f64 z t)

   1. Initial program 95.4%

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

   3. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right)} + c \]
   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 88.7

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

   5. Applied rewrites 88.7%

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

   if -4.99999999999999972e57 < (*.f64 z t) < 2.0000000000000001e167

   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 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 95.5

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

   5. Applied rewrites 95.5%

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

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot z \leq -5 \cdot 10^{+57}:\\ \;\;\;\;\mathsf{fma}\left(y, x, 0.0625 \cdot \left(t \cdot z\right)\right) + c\\ \mathbf{elif}\;t \cdot z \leq 2 \cdot 10^{+167}:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot b, a, \mathsf{fma}\left(y, x, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, 0.0625 \cdot \left(t \cdot z\right)\right) + c\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 89.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \cdot z \leq -5 \cdot 10^{+57}:\\ \;\;\;\;\mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(y, x, c\right)\right)\\ \mathbf{elif}\;t \cdot z \leq 2 \cdot 10^{+167}:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot b, a, \mathsf{fma}\left(y, x, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, 0.0625 \cdot \left(t \cdot z\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (<= (* t z) -5e+57)
   (fma (* 0.0625 t) z (fma y x c))
   (if (<= (* t z) 2e+167)
     (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+57) {
		tmp = fma((0.0625 * t), z, fma(y, x, c));
	} else if ((t * z) <= 2e+167) {
		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+57)
		tmp = fma(Float64(0.0625 * t), z, fma(y, x, c));
	elseif (Float64(t * z) <= 2e+167)
		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+57], N[(N[(0.0625 * t), $MachinePrecision] * z + N[(y * x + c), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(t * z), $MachinePrecision], 2e+167], 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) < -4.99999999999999972e57

   1. Initial program 95.7%

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

   3. Taylor expanded in 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 83.4

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

   5. Applied rewrites 83.4%

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

   if -4.99999999999999972e57 < (*.f64 z t) < 2.0000000000000001e167

   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 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 95.5

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

   5. Applied rewrites 95.5%

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

   if 2.0000000000000001e167 < (*.f64 z t)

   1. Initial program 95.0%

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

   3. Taylor expanded in 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 87.3

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

   5. Applied rewrites 87.3%

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

   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 87.3%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot z \leq -5 \cdot 10^{+57}:\\ \;\;\;\;\mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(y, x, c\right)\right)\\ \mathbf{elif}\;t \cdot z \leq 2 \cdot 10^{+167}:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot b, a, \mathsf{fma}\left(y, x, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, 0.0625 \cdot \left(t \cdot z\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 88.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, x, 0.0625 \cdot \left(t \cdot z\right)\right)\\ \mathbf{if}\;t \cdot z \leq -1.75 \cdot 10^{+128}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \cdot z \leq 3.8 \cdot 10^{+167}:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot b, a, \mathsf{fma}\left(y, x, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (fma y x (* 0.0625 (* t z)))))
   (if (<= (* t z) -1.75e+128)
     t_1
     (if (<= (* t z) 3.8e+167) (fma (* -0.25 b) a (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(y, x, (0.0625 * (t * z)));
	double tmp;
	if ((t * z) <= -1.75e+128) {
		tmp = t_1;
	} else if ((t * z) <= 3.8e+167) {
		tmp = fma((-0.25 * b), a, fma(y, x, c));
	} 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) <= -1.75e+128)
		tmp = t_1;
	elseif (Float64(t * z) <= 3.8e+167)
		tmp = fma(Float64(-0.25 * b), a, 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[(y * x + N[(0.0625 * N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t * z), $MachinePrecision], -1.75e+128], t$95$1, If[LessEqual[N[(t * z), $MachinePrecision], 3.8e+167], N[(N[(-0.25 * b), $MachinePrecision] * a + N[(y * x + c), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -1.74999999999999984e128 or 3.79999999999999994e167 < (*.f64 z t)

   1. Initial program 94.6%

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

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

   8. Applied rewrites 86.5%

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

   if -1.74999999999999984e128 < (*.f64 z t) < 3.79999999999999994e167

   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 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 93.7

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

   5. Applied rewrites 93.7%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot z \leq -1.75 \cdot 10^{+128}:\\ \;\;\;\;\mathsf{fma}\left(y, x, 0.0625 \cdot \left(t \cdot z\right)\right)\\ \mathbf{elif}\;t \cdot z \leq 3.8 \cdot 10^{+167}:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot b, a, \mathsf{fma}\left(y, x, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, 0.0625 \cdot \left(t \cdot z\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 66.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot x \leq -1 \cdot 10^{+62}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \mathbf{elif}\;y \cdot x \leq 10^{+31}:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot a, b, c\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (<= (* y x) -1e+62)
   (fma y x c)
   (if (<= (* y x) 1e+31) (fma (* -0.25 a) b c) (fma y x c))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if ((y * x) <= -1e+62) {
		tmp = fma(y, x, c);
	} else if ((y * x) <= 1e+31) {
		tmp = fma((-0.25 * a), b, c);
	} else {
		tmp = fma(y, x, c);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if (Float64(y * x) <= -1e+62)
		tmp = fma(y, x, c);
	elseif (Float64(y * x) <= 1e+31)
		tmp = fma(Float64(-0.25 * a), b, c);
	else
		tmp = fma(y, x, c);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[LessEqual[N[(y * x), $MachinePrecision], -1e+62], N[(y * x + c), $MachinePrecision], If[LessEqual[N[(y * x), $MachinePrecision], 1e+31], N[(N[(-0.25 * a), $MachinePrecision] * b + c), $MachinePrecision], N[(y * x + c), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -1.00000000000000004e62 or 9.9999999999999996e30 < (*.f64 x y)

   1. Initial program 95.6%

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

   3. Taylor expanded in t around 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 82.4

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

   5. Applied rewrites 82.4%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 72.1%

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

   if -1.00000000000000004e62 < (*.f64 x y) < 9.9999999999999996e30

   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 68.8

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

   5. Applied rewrites 68.8%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 65.4%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot x \leq -1 \cdot 10^{+62}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \mathbf{elif}\;y \cdot x \leq 10^{+31}:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot a, b, c\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 62.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 0.0625 \cdot \left(t \cdot z\right)\\ \mathbf{if}\;t \cdot z \leq -1.3 \cdot 10^{+127}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \cdot z \leq 9.2 \cdot 10^{+127}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (* 0.0625 (* t z))))
   (if (<= (* t z) -1.3e+127) t_1 (if (<= (* t z) 9.2e+127) (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) <= -1.3e+127) {
		tmp = t_1;
	} else if ((t * z) <= 9.2e+127) {
		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) <= -1.3e+127)
		tmp = t_1;
	elseif (Float64(t * z) <= 9.2e+127)
		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], -1.3e+127], t$95$1, If[LessEqual[N[(t * z), $MachinePrecision], 9.2e+127], N[(y * x + c), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -1.3000000000000001e127 or 9.2000000000000007e127 < (*.f64 z t)

   1. Initial program 95.0%

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

   3. Taylor expanded in 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 73.8

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

   5. Applied rewrites 73.8%

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

   if -1.3000000000000001e127 < (*.f64 z t) < 9.2000000000000007e127

   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 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%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 61.0%

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

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot z \leq -1.3 \cdot 10^{+127}:\\ \;\;\;\;0.0625 \cdot \left(t \cdot z\right)\\ \mathbf{elif}\;t \cdot z \leq 9.2 \cdot 10^{+127}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \mathbf{else}:\\ \;\;\;\;0.0625 \cdot \left(t \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 64.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -0.25 \cdot \left(b \cdot a\right)\\ \mathbf{if}\;b \cdot a \leq -5 \cdot 10^{+150}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \cdot a \leq 10^{+159}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (* -0.25 (* b a))))
   (if (<= (* b a) -5e+150) t_1 (if (<= (* b a) 1e+159) (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+150) {
		tmp = t_1;
	} else if ((b * a) <= 1e+159) {
		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+150)
		tmp = t_1;
	elseif (Float64(b * a) <= 1e+159)
		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+150], t$95$1, If[LessEqual[N[(b * a), $MachinePrecision], 1e+159], N[(y * x + c), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a b) < -5.00000000000000009e150 or 9.9999999999999993e158 < (*.f64 a b)

   1. Initial program 93.1%

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

   3. Taylor expanded in 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 76.4

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

   5. Applied rewrites 76.4%

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

   if -5.00000000000000009e150 < (*.f64 a b) < 9.9999999999999993e158

   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 70.6

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

   5. Applied rewrites 70.6%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 60.2%

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

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot a \leq -5 \cdot 10^{+150}:\\ \;\;\;\;-0.25 \cdot \left(b \cdot a\right)\\ \mathbf{elif}\;b \cdot a \leq 10^{+159}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \left(b \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 49.2% 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 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 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 74.8

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

5. Applied rewrites 74.8%

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

6. Taylor expanded in b around 0

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

8. Applied rewrites 46.9%

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

Alternative 12: 28.7% 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 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 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 28.4

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

5. Applied rewrites 28.4%

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

Reproduce

herbie shell --seed 2024249 
(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))