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

Percentage Accurate: 97.7% → 98.8%

Time: 8.1s

Alternatives: 13

Speedup: 0.5×

• 33.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 97.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.8% accurate, 0.5× speedup

Math

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

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 (* 0.0625 t) z (* y x)))))

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((0.0625 * t), z, (y * x));
	}
	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(Float64(0.0625 * t), z, Float64(y * x));
	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[(N[(0.0625 * t), $MachinePrecision] * z + N[(y * x), $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 99.6%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   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 83.3

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

   5. Applied rewrites 83.3%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 83.3%

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

4. Final simplification 99.3%

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

Alternative 2: 76.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(0.0625 \cdot t, z, y \cdot x\right)\\ t_2 := \frac{t \cdot z}{16} + y \cdot x\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+102}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+202}:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot a, b, c\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (fma (* 0.0625 t) z (* y x))) (t_2 (+ (/ (* t z) 16.0) (* y x))))
   (if (<= t_2 -2e+102) t_1 (if (<= t_2 2e+202) (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((0.0625 * t), z, (y * x));
	double t_2 = ((t * z) / 16.0) + (y * x);
	double tmp;
	if (t_2 <= -2e+102) {
		tmp = t_1;
	} else if (t_2 <= 2e+202) {
		tmp = fma((-0.25 * a), b, c);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = fma(Float64(0.0625 * t), z, Float64(y * x))
	t_2 = Float64(Float64(Float64(t * z) / 16.0) + Float64(y * x))
	tmp = 0.0
	if (t_2 <= -2e+102)
		tmp = t_1;
	elseif (t_2 <= 2e+202)
		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[(N[(0.0625 * t), $MachinePrecision] * z + N[(y * x), $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, -2e+102], t$95$1, If[LessEqual[t$95$2, 2e+202], 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))) < -1.99999999999999995e102 or 1.9999999999999998e202 < (+.f64 (*.f64 x y) (/.f64 (*.f64 z t) #s(literal 16 binary64)))

   1. Initial program 94.5%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   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 88.4

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

   5. Applied rewrites 88.4%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 85.3%

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

   if -1.99999999999999995e102 < (+.f64 (*.f64 x y) (/.f64 (*.f64 z t) #s(literal 16 binary64))) < 1.9999999999999998e202

   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 88.6

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

   5. Applied rewrites 88.6%

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

   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 79.3%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t \cdot z}{16} + y \cdot x \leq -2 \cdot 10^{+102}:\\ \;\;\;\;\mathsf{fma}\left(0.0625 \cdot t, z, y \cdot x\right)\\ \mathbf{elif}\;\frac{t \cdot z}{16} + y \cdot x \leq 2 \cdot 10^{+202}:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot a, b, c\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.0625 \cdot t, z, y \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 66.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-0.25 \cdot b, a, y \cdot x\right)\\ \mathbf{if}\;y \cdot x \leq -1 \cdot 10^{+98}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \cdot x \leq -1 \cdot 10^{-144}:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot a, b, c\right)\\ \mathbf{elif}\;y \cdot x \leq 2 \cdot 10^{-179}:\\ \;\;\;\;\mathsf{fma}\left(0.0625 \cdot z, t, c\right)\\ \mathbf{elif}\;y \cdot x \leq 2 \cdot 10^{-15}:\\ \;\;\;\;-0.25 \cdot \left(b \cdot a\right) + c\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (fma (* -0.25 b) a (* y x))))
   (if (<= (* y x) -1e+98)
     t_1
     (if (<= (* y x) -1e-144)
       (fma (* -0.25 a) b c)
       (if (<= (* y x) 2e-179)
         (fma (* 0.0625 z) t c)
         (if (<= (* y x) 2e-15) (+ (* -0.25 (* b a)) 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, (y * x));
	double tmp;
	if ((y * x) <= -1e+98) {
		tmp = t_1;
	} else if ((y * x) <= -1e-144) {
		tmp = fma((-0.25 * a), b, c);
	} else if ((y * x) <= 2e-179) {
		tmp = fma((0.0625 * z), t, c);
	} else if ((y * x) <= 2e-15) {
		tmp = (-0.25 * (b * a)) + 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, Float64(y * x))
	tmp = 0.0
	if (Float64(y * x) <= -1e+98)
		tmp = t_1;
	elseif (Float64(y * x) <= -1e-144)
		tmp = fma(Float64(-0.25 * a), b, c);
	elseif (Float64(y * x) <= 2e-179)
		tmp = fma(Float64(0.0625 * z), t, c);
	elseif (Float64(y * x) <= 2e-15)
		tmp = Float64(Float64(-0.25 * Float64(b * a)) + 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[(y * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(y * x), $MachinePrecision], -1e+98], t$95$1, If[LessEqual[N[(y * x), $MachinePrecision], -1e-144], N[(N[(-0.25 * a), $MachinePrecision] * b + c), $MachinePrecision], If[LessEqual[N[(y * x), $MachinePrecision], 2e-179], N[(N[(0.0625 * z), $MachinePrecision] * t + c), $MachinePrecision], If[LessEqual[N[(y * x), $MachinePrecision], 2e-15], N[(N[(-0.25 * N[(b * a), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 x y) < -9.99999999999999998e97 or 2.0000000000000002e-15 < (*.f64 x y)

   1. Initial program 97.1%

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

   3. Taylor expanded in 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 86.2

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

   5. Applied rewrites 86.2%

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

   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 76.7%

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

   if -9.99999999999999998e97 < (*.f64 x y) < -9.9999999999999995e-145

   1. Initial program 97.9%

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

   3. Taylor expanded in 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 83.4

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

   5. Applied rewrites 83.4%

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

   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 70.2%

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

   if -9.9999999999999995e-145 < (*.f64 x y) < 2e-179

   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 72.1

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

   5. Applied rewrites 72.1%

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

   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 72.1%

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

   if 2e-179 < (*.f64 x y) < 2.0000000000000002e-15

   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

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 84.7

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

   5. Applied rewrites 84.7%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot x \leq -1 \cdot 10^{+98}:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot b, a, y \cdot x\right)\\ \mathbf{elif}\;y \cdot x \leq -1 \cdot 10^{-144}:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot a, b, c\right)\\ \mathbf{elif}\;y \cdot x \leq 2 \cdot 10^{-179}:\\ \;\;\;\;\mathsf{fma}\left(0.0625 \cdot z, t, c\right)\\ \mathbf{elif}\;y \cdot x \leq 2 \cdot 10^{-15}:\\ \;\;\;\;-0.25 \cdot \left(b \cdot a\right) + c\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot b, a, y \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 63.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-0.25 \cdot a, b, c\right)\\ \mathbf{if}\;b \cdot a \leq -5 \cdot 10^{-13}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \cdot a \leq -5 \cdot 10^{-142}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \mathbf{elif}\;b \cdot a \leq 5 \cdot 10^{-234}:\\ \;\;\;\;\mathsf{fma}\left(0.0625 \cdot z, t, c\right)\\ \mathbf{elif}\;b \cdot a \leq 4 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (fma (* -0.25 a) b c)))
   (if (<= (* b a) -5e-13)
     t_1
     (if (<= (* b a) -5e-142)
       (fma y x c)
       (if (<= (* b a) 5e-234)
         (fma (* 0.0625 z) t c)
         (if (<= (* b a) 4e+15) (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 * a), b, c);
	double tmp;
	if ((b * a) <= -5e-13) {
		tmp = t_1;
	} else if ((b * a) <= -5e-142) {
		tmp = fma(y, x, c);
	} else if ((b * a) <= 5e-234) {
		tmp = fma((0.0625 * z), t, c);
	} else if ((b * a) <= 4e+15) {
		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(-0.25 * a), b, c)
	tmp = 0.0
	if (Float64(b * a) <= -5e-13)
		tmp = t_1;
	elseif (Float64(b * a) <= -5e-142)
		tmp = fma(y, x, c);
	elseif (Float64(b * a) <= 5e-234)
		tmp = fma(Float64(0.0625 * z), t, c);
	elseif (Float64(b * a) <= 4e+15)
		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[(-0.25 * a), $MachinePrecision] * b + c), $MachinePrecision]}, If[LessEqual[N[(b * a), $MachinePrecision], -5e-13], t$95$1, If[LessEqual[N[(b * a), $MachinePrecision], -5e-142], N[(y * x + c), $MachinePrecision], If[LessEqual[N[(b * a), $MachinePrecision], 5e-234], N[(N[(0.0625 * z), $MachinePrecision] * t + c), $MachinePrecision], If[LessEqual[N[(b * a), $MachinePrecision], 4e+15], N[(y * x + c), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 a b) < -4.9999999999999999e-13 or 4e15 < (*.f64 a b)

   1. Initial program 94.7%

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

   3. Taylor expanded in 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 84.2

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

   5. Applied rewrites 84.2%

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

   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 71.5%

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

   if -4.9999999999999999e-13 < (*.f64 a b) < -5.0000000000000002e-142 or 4.99999999999999979e-234 < (*.f64 a b) < 4e15

   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 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 96.9

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

   5. Applied rewrites 96.9%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 75.7%

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

   if -5.0000000000000002e-142 < (*.f64 a b) < 4.99999999999999979e-234

   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 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 98.6

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

   5. Applied rewrites 98.6%

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

   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 75.0%

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

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot a \leq -5 \cdot 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot a, b, c\right)\\ \mathbf{elif}\;b \cdot a \leq -5 \cdot 10^{-142}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \mathbf{elif}\;b \cdot a \leq 5 \cdot 10^{-234}:\\ \;\;\;\;\mathsf{fma}\left(0.0625 \cdot z, t, c\right)\\ \mathbf{elif}\;b \cdot a \leq 4 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot a, b, c\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 90.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-0.25 \cdot b, a, \mathsf{fma}\left(0.0625 \cdot t, z, c\right)\right)\\ \mathbf{if}\;t \cdot z \leq -2 \cdot 10^{+38}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \cdot z \leq 2 \cdot 10^{+120}:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot b, a, \mathsf{fma}\left(y, x, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (fma (* -0.25 b) a (fma (* 0.0625 t) z c))))
   (if (<= (* t z) -2e+38)
     t_1
     (if (<= (* t z) 2e+120) (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((-0.25 * b), a, fma((0.0625 * t), z, c));
	double tmp;
	if ((t * z) <= -2e+38) {
		tmp = t_1;
	} else if ((t * z) <= 2e+120) {
		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(Float64(-0.25 * b), a, fma(Float64(0.0625 * t), z, c))
	tmp = 0.0
	if (Float64(t * z) <= -2e+38)
		tmp = t_1;
	elseif (Float64(t * z) <= 2e+120)
		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[(-0.25 * b), $MachinePrecision] * a + N[(N[(0.0625 * t), $MachinePrecision] * z + c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t * z), $MachinePrecision], -2e+38], t$95$1, If[LessEqual[N[(t * z), $MachinePrecision], 2e+120], 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) < -1.99999999999999995e38 or 2e120 < (*.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 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 91.8

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

   5. Applied rewrites 91.8%

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

   if -1.99999999999999995e38 < (*.f64 z t) < 2e120

   1. Initial program 98.9%

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

   3. 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 95.7

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

   5. Applied rewrites 95.7%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot z \leq -2 \cdot 10^{+38}:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot b, a, \mathsf{fma}\left(0.0625 \cdot t, z, c\right)\right)\\ \mathbf{elif}\;t \cdot z \leq 2 \cdot 10^{+120}:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot b, a, \mathsf{fma}\left(y, x, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot b, a, \mathsf{fma}\left(0.0625 \cdot t, z, c\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 89.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \cdot z \leq -1 \cdot 10^{+72}:\\ \;\;\;\;\mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(y, x, c\right)\right)\\ \mathbf{elif}\;t \cdot z \leq 10^{+132}:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot b, a, \mathsf{fma}\left(y, x, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot b, a, 0.0625 \cdot \left(t \cdot z\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (<= (* t z) -1e+72)
   (fma (* 0.0625 t) z (fma y x c))
   (if (<= (* t z) 1e+132)
     (fma (* -0.25 b) a (fma y x c))
     (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 tmp;
	if ((t * z) <= -1e+72) {
		tmp = fma((0.0625 * t), z, fma(y, x, c));
	} else if ((t * z) <= 1e+132) {
		tmp = fma((-0.25 * b), a, fma(y, x, c));
	} else {
		tmp = fma((-0.25 * b), a, (0.0625 * (t * z)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if (Float64(t * z) <= -1e+72)
		tmp = fma(Float64(0.0625 * t), z, fma(y, x, c));
	elseif (Float64(t * z) <= 1e+132)
		tmp = fma(Float64(-0.25 * b), a, fma(y, x, c));
	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_] := If[LessEqual[N[(t * z), $MachinePrecision], -1e+72], N[(N[(0.0625 * t), $MachinePrecision] * z + N[(y * x + c), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(t * z), $MachinePrecision], 1e+132], N[(N[(-0.25 * b), $MachinePrecision] * a + N[(y * x + c), $MachinePrecision]), $MachinePrecision], N[(N[(-0.25 * b), $MachinePrecision] * a + N[(0.0625 * N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -9.99999999999999944e71

   1. Initial program 92.9%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   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 85.7

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

   5. Applied rewrites 85.7%

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

   if -9.99999999999999944e71 < (*.f64 z t) < 9.99999999999999991e131

   1. Initial program 99.0%

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

   3. Taylor expanded in 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 94.8

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

   5. Applied rewrites 94.8%

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

   if 9.99999999999999991e131 < (*.f64 z t)

   1. Initial program 93.5%

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

   3. 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 93.7

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

   5. Applied rewrites 93.7%

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

   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 87.5%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot z \leq -1 \cdot 10^{+72}:\\ \;\;\;\;\mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(y, x, c\right)\right)\\ \mathbf{elif}\;t \cdot z \leq 10^{+132}:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot b, a, \mathsf{fma}\left(y, x, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot b, a, 0.0625 \cdot \left(t \cdot z\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 89.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(y, x, c\right)\right)\\ \mathbf{if}\;t \cdot z \leq -1 \cdot 10^{+72}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \cdot z \leq 2 \cdot 10^{+116}:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot b, a, \mathsf{fma}\left(y, x, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (fma (* 0.0625 t) z (fma y x c))))
   (if (<= (* t z) -1e+72)
     t_1
     (if (<= (* t z) 2e+116) (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((0.0625 * t), z, fma(y, x, c));
	double tmp;
	if ((t * z) <= -1e+72) {
		tmp = t_1;
	} else if ((t * z) <= 2e+116) {
		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(Float64(0.0625 * t), z, fma(y, x, c))
	tmp = 0.0
	if (Float64(t * z) <= -1e+72)
		tmp = t_1;
	elseif (Float64(t * z) <= 2e+116)
		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[(0.0625 * t), $MachinePrecision] * z + N[(y * x + c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t * z), $MachinePrecision], -1e+72], t$95$1, If[LessEqual[N[(t * z), $MachinePrecision], 2e+116], 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) < -9.99999999999999944e71 or 2.00000000000000003e116 < (*.f64 z t)

   1. Initial program 93.4%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   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 85.4

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

   5. Applied rewrites 85.4%

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

   if -9.99999999999999944e71 < (*.f64 z t) < 2.00000000000000003e116

   1. Initial program 99.0%

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

   3. Taylor expanded in 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%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot z \leq -1 \cdot 10^{+72}:\\ \;\;\;\;\mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(y, x, c\right)\right)\\ \mathbf{elif}\;t \cdot z \leq 2 \cdot 10^{+116}:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot b, a, \mathsf{fma}\left(y, x, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(y, x, c\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 86.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \cdot z \leq -2 \cdot 10^{+80}:\\ \;\;\;\;\mathsf{fma}\left(0.0625 \cdot t, z, y \cdot x\right)\\ \mathbf{elif}\;t \cdot z \leq 5 \cdot 10^{+243}:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot b, a, \mathsf{fma}\left(y, x, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.0625 \cdot \left(t \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (<= (* t z) -2e+80)
   (fma (* 0.0625 t) z (* y x))
   (if (<= (* t z) 5e+243)
     (fma (* -0.25 b) a (fma y x c))
     (* 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) <= -2e+80) {
		tmp = fma((0.0625 * t), z, (y * x));
	} else if ((t * z) <= 5e+243) {
		tmp = fma((-0.25 * b), a, fma(y, x, c));
	} else {
		tmp = 0.0625 * (t * z);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if (Float64(t * z) <= -2e+80)
		tmp = fma(Float64(0.0625 * t), z, Float64(y * x));
	elseif (Float64(t * z) <= 5e+243)
		tmp = fma(Float64(-0.25 * b), a, fma(y, x, c));
	else
		tmp = 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], -2e+80], N[(N[(0.0625 * t), $MachinePrecision] * z + N[(y * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(t * z), $MachinePrecision], 5e+243], N[(N[(-0.25 * b), $MachinePrecision] * a + N[(y * x + c), $MachinePrecision]), $MachinePrecision], N[(0.0625 * N[(t * z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -2e80

   1. Initial program 92.0%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   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 86.5

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

   5. Applied rewrites 86.5%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 81.5%

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

   if -2e80 < (*.f64 z t) < 5.00000000000000037e243

   1. Initial program 98.6%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   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 91.3

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

   5. Applied rewrites 91.3%

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

   if 5.00000000000000037e243 < (*.f64 z t)

   1. Initial program 94.1%

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

   3. 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 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot z \leq -2 \cdot 10^{+80}:\\ \;\;\;\;\mathsf{fma}\left(0.0625 \cdot t, z, y \cdot x\right)\\ \mathbf{elif}\;t \cdot z \leq 5 \cdot 10^{+243}:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot b, a, \mathsf{fma}\left(y, x, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.0625 \cdot \left(t \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 62.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-0.25 \cdot a, b, c\right)\\ \mathbf{if}\;b \cdot a \leq -5 \cdot 10^{-13}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \cdot a \leq 4 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (fma (* -0.25 a) b c)))
   (if (<= (* b a) -5e-13) t_1 (if (<= (* b a) 4e+15) (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 * a), b, c);
	double tmp;
	if ((b * a) <= -5e-13) {
		tmp = t_1;
	} else if ((b * a) <= 4e+15) {
		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(-0.25 * a), b, c)
	tmp = 0.0
	if (Float64(b * a) <= -5e-13)
		tmp = t_1;
	elseif (Float64(b * a) <= 4e+15)
		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[(-0.25 * a), $MachinePrecision] * b + c), $MachinePrecision]}, If[LessEqual[N[(b * a), $MachinePrecision], -5e-13], t$95$1, If[LessEqual[N[(b * a), $MachinePrecision], 4e+15], N[(y * x + c), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a b) < -4.9999999999999999e-13 or 4e15 < (*.f64 a b)

   1. Initial program 94.7%

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

   3. Taylor expanded in 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 84.2

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

   5. Applied rewrites 84.2%

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

   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 71.5%

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

   if -4.9999999999999999e-13 < (*.f64 a b) < 4e15

   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 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 97.8

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

   5. Applied rewrites 97.8%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 68.4%

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

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot a \leq -5 \cdot 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot a, b, c\right)\\ \mathbf{elif}\;b \cdot a \leq 4 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot a, b, c\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 60.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot a \leq -5 \cdot 10^{+193}:\\ \;\;\;\;-0.25 \cdot \left(b \cdot a\right)\\ \mathbf{elif}\;b \cdot a \leq 10^{+16}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-0.25 \cdot a\right) \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (<= (* b a) -5e+193)
   (* -0.25 (* b a))
   (if (<= (* b a) 1e+16) (fma y x c) (* (* -0.25 a) b))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if ((b * a) <= -5e+193) {
		tmp = -0.25 * (b * a);
	} else if ((b * a) <= 1e+16) {
		tmp = fma(y, x, c);
	} else {
		tmp = (-0.25 * a) * b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if (Float64(b * a) <= -5e+193)
		tmp = Float64(-0.25 * Float64(b * a));
	elseif (Float64(b * a) <= 1e+16)
		tmp = fma(y, x, c);
	else
		tmp = Float64(Float64(-0.25 * a) * b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[LessEqual[N[(b * a), $MachinePrecision], -5e+193], N[(-0.25 * N[(b * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(b * a), $MachinePrecision], 1e+16], N[(y * x + c), $MachinePrecision], N[(N[(-0.25 * a), $MachinePrecision] * b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 a b) < -4.99999999999999972e193

   1. Initial program 86.4%

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

   3. Taylor expanded in b around 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 72.0

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

   5. Applied rewrites 72.0%

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

   if -4.99999999999999972e193 < (*.f64 a b) < 1e16

   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 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 91.6

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

   5. Applied rewrites 91.6%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 64.6%

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

   if 1e16 < (*.f64 a b)

   1. Initial program 96.9%

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

   3. Taylor expanded in b around 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 64.4

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

   5. Applied rewrites 64.4%

      \[\leadsto \color{blue}{-0.25 \cdot \left(b \cdot a\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 64.4%

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

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot a \leq -5 \cdot 10^{+193}:\\ \;\;\;\;-0.25 \cdot \left(b \cdot a\right)\\ \mathbf{elif}\;b \cdot a \leq 10^{+16}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-0.25 \cdot a\right) \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 60.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -0.25 \cdot \left(b \cdot a\right)\\ \mathbf{if}\;b \cdot a \leq -5 \cdot 10^{+193}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \cdot a \leq 10^{+16}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (* -0.25 (* b a))))
   (if (<= (* b a) -5e+193) t_1 (if (<= (* b a) 1e+16) (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+193) {
		tmp = t_1;
	} else if ((b * a) <= 1e+16) {
		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+193)
		tmp = t_1;
	elseif (Float64(b * a) <= 1e+16)
		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+193], t$95$1, If[LessEqual[N[(b * a), $MachinePrecision], 1e+16], N[(y * x + c), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a b) < -4.99999999999999972e193 or 1e16 < (*.f64 a b)

   1. Initial program 92.8%

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

   3. Taylor expanded in 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 67.4

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

   5. Applied rewrites 67.4%

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

   if -4.99999999999999972e193 < (*.f64 a b) < 1e16

   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 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 91.6

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

   5. Applied rewrites 91.6%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 64.6%

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

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot a \leq -5 \cdot 10^{+193}:\\ \;\;\;\;-0.25 \cdot \left(b \cdot a\right)\\ \mathbf{elif}\;b \cdot a \leq 10^{+16}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \left(b \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 48.8% accurate, 6.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 97.3%

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

3. Taylor expanded in 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 71.4

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

5. Applied rewrites 71.4%

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

6. Taylor expanded in t around 0

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

8. Applied rewrites 49.0%

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

Alternative 13: 28.8% accurate, 7.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.3%

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

3. Taylor expanded in 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 27.0

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

5. Applied rewrites 27.0%

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

Reproduce

herbie shell --seed 2024243 
(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))