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

Percentage Accurate: 97.8% → 98.6%

Time: 12.4s

Alternatives: 12

Speedup: 0.5×

• 33.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 97.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = ((x * y) + ((z * t) / 16.0)) - ((a * b) / 4.0);
	double tmp;
	if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = t_1 + c;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	t_1 = ((x * y) + ((z * t) / 16.0)) - ((a * b) / 4.0)
	tmp = 0
	if t_1 <= math.inf:
		tmp = t_1 + c
	else:
		tmp = y * x
	return tmp

Julia

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

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c)
	t_1 = ((x * y) + ((z * t) / 16.0)) - ((a * b) / 4.0);
	tmp = 0.0;
	if (t_1 <= Inf)
		tmp = t_1 + c;
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

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

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

      1. *-commutative N/A

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

      2. associate--l+ N/A

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

      3. associate-*r/ N/A

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

      4. div-add-rev N/A

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

      5. +-commutative N/A

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

      6. associate-*r/ N/A

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

      7. div-sub N/A

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

      8. lower-*.f64 N/A

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

   5. Applied rewrites 0.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 100.0%

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

Alternative 2: 74.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, x, \left(t \cdot z\right) \cdot 0.0625\right)\\ t_2 := x \cdot y + \frac{z \cdot t}{16}\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+212}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq -1500000000:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot b, a, y \cdot x\right)\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+242}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, a \cdot b, c\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (fma y x (* (* t z) 0.0625))) (t_2 (+ (* x y) (/ (* z t) 16.0))))
   (if (<= t_2 -1e+212)
     t_1
     (if (<= t_2 -1500000000.0)
       (fma (* -0.25 b) a (* y x))
       (if (<= t_2 5e+242) (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, ((t * z) * 0.0625));
	double t_2 = (x * y) + ((z * t) / 16.0);
	double tmp;
	if (t_2 <= -1e+212) {
		tmp = t_1;
	} else if (t_2 <= -1500000000.0) {
		tmp = fma((-0.25 * b), a, (y * x));
	} else if (t_2 <= 5e+242) {
		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(Float64(t * z) * 0.0625))
	t_2 = Float64(Float64(x * y) + Float64(Float64(z * t) / 16.0))
	tmp = 0.0
	if (t_2 <= -1e+212)
		tmp = t_1;
	elseif (t_2 <= -1500000000.0)
		tmp = fma(Float64(-0.25 * b), a, Float64(y * x));
	elseif (t_2 <= 5e+242)
		tmp = fma(-0.25, Float64(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[(N[(t * z), $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * y), $MachinePrecision] + N[(N[(z * t), $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+212], t$95$1, If[LessEqual[t$95$2, -1500000000.0], N[(N[(-0.25 * b), $MachinePrecision] * a + N[(y * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e+242], N[(-0.25 * N[(a * b), $MachinePrecision] + c), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (*.f64 x y) (/.f64 (*.f64 z t) #s(literal 16 binary64))) < -9.9999999999999991e211 or 5.0000000000000004e242 < (+.f64 (*.f64 x y) (/.f64 (*.f64 z t) #s(literal 16 binary64)))

   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 a around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 95.4

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

   5. Applied rewrites 95.4%

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

   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 94.4%

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

   10. Applied rewrites 94.4%

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

   if -9.9999999999999991e211 < (+.f64 (*.f64 x y) (/.f64 (*.f64 z t) #s(literal 16 binary64))) < -1.5e9

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 93.3

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

   5. Applied rewrites 93.3%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 69.9%

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

   if -1.5e9 < (+.f64 (*.f64 x y) (/.f64 (*.f64 z t) #s(literal 16 binary64))) < 5.0000000000000004e242

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 85.4

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

   5. Applied rewrites 85.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 73.1%

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

Alternative 3: 64.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(z \cdot t\right) \cdot 0.0625\\ \mathbf{if}\;z \cdot t \leq -2.85 \cdot 10^{+143}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \cdot t \leq -9.2 \cdot 10^{-237}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \mathbf{elif}\;z \cdot t \leq 1.95 \cdot 10^{+194}:\\ \;\;\;\;\mathsf{fma}\left(y, x, b \cdot \left(-0.25 \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (* (* z t) 0.0625)))
   (if (<= (* z t) -2.85e+143)
     t_1
     (if (<= (* z t) -9.2e-237)
       (fma y x c)
       (if (<= (* z t) 1.95e+194) (fma y x (* b (* -0.25 a))) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (z * t) * 0.0625;
	double tmp;
	if ((z * t) <= -2.85e+143) {
		tmp = t_1;
	} else if ((z * t) <= -9.2e-237) {
		tmp = fma(y, x, c);
	} else if ((z * t) <= 1.95e+194) {
		tmp = fma(y, x, (b * (-0.25 * a)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(z * t) * 0.0625)
	tmp = 0.0
	if (Float64(z * t) <= -2.85e+143)
		tmp = t_1;
	elseif (Float64(z * t) <= -9.2e-237)
		tmp = fma(y, x, c);
	elseif (Float64(z * t) <= 1.95e+194)
		tmp = fma(y, x, Float64(b * Float64(-0.25 * a)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(z * t), $MachinePrecision] * 0.0625), $MachinePrecision]}, If[LessEqual[N[(z * t), $MachinePrecision], -2.85e+143], t$95$1, If[LessEqual[N[(z * t), $MachinePrecision], -9.2e-237], N[(y * x + c), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 1.95e+194], N[(y * x + N[(b * N[(-0.25 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -2.85000000000000011e143 or 1.95000000000000008e194 < (*.f64 z t)

   1. Initial program 93.6%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. associate--l+ N/A

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

      3. associate-*r/ N/A

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

      4. div-add-rev N/A

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

      5. +-commutative N/A

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

      6. associate-*r/ N/A

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

      7. div-sub N/A

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

      8. lower-*.f64 N/A

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

   5. Applied rewrites 76.0%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 82.6%

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

   if -2.85000000000000011e143 < (*.f64 z t) < -9.20000000000000046e-237

   1. Initial program 98.5%

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

   3. Taylor expanded in z around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 95.7

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

   5. Applied rewrites 95.7%

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

   7. Applied rewrites 95.7%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 71.9%

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

   if -9.20000000000000046e-237 < (*.f64 z t) < 1.95000000000000008e194

   1. Initial program 99.2%

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

   3. Taylor expanded in z around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 88.9

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

   5. Applied rewrites 88.9%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 69.9%

      \[\leadsto \mathsf{fma}\left(-0.25 \cdot b, \color{blue}{a}, y \cdot x\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 69.1%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2.85 \cdot 10^{+143}:\\ \;\;\;\;\left(z \cdot t\right) \cdot 0.0625\\ \mathbf{elif}\;z \cdot t \leq -9.2 \cdot 10^{-237}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \mathbf{elif}\;z \cdot t \leq 1.95 \cdot 10^{+194}:\\ \;\;\;\;\mathsf{fma}\left(y, x, b \cdot \left(-0.25 \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot t\right) \cdot 0.0625\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-0.25, b \cdot a, \mathsf{fma}\left(t \cdot z, 0.0625, c\right)\right)\\ \mathbf{if}\;x \leq -2.5 \cdot 10^{-86}:\\ \;\;\;\;\left(\frac{t\_1}{x} + y\right) \cdot x\\ \mathbf{elif}\;x \leq 2.45 \cdot 10^{-64}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c + \left(a \cdot -0.25\right) \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (fma -0.25 (* b a) (fma (* t z) 0.0625 c))))
   (if (<= x -2.5e-86)
     (* (+ (/ t_1 x) y) x)
     (if (<= x 2.45e-64) t_1 (fma y x (+ c (* (* a -0.25) b)))))))

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((t * z), 0.0625, c));
	double tmp;
	if (x <= -2.5e-86) {
		tmp = ((t_1 / x) + y) * x;
	} else if (x <= 2.45e-64) {
		tmp = t_1;
	} else {
		tmp = fma(y, x, (c + ((a * -0.25) * b)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = fma(-0.25, Float64(b * a), fma(Float64(t * z), 0.0625, c))
	tmp = 0.0
	if (x <= -2.5e-86)
		tmp = Float64(Float64(Float64(t_1 / x) + y) * x);
	elseif (x <= 2.45e-64)
		tmp = t_1;
	else
		tmp = fma(y, x, Float64(c + Float64(Float64(a * -0.25) * b)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(-0.25 * N[(b * a), $MachinePrecision] + N[(N[(t * z), $MachinePrecision] * 0.0625 + c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.5e-86], N[(N[(N[(t$95$1 / x), $MachinePrecision] + y), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[x, 2.45e-64], t$95$1, N[(y * x + N[(c + N[(N[(a * -0.25), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.4999999999999999e-86

   1. Initial program 97.0%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. associate--l+ N/A

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

      3. associate-*r/ N/A

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

      4. div-add-rev N/A

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

      5. +-commutative N/A

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

      6. associate-*r/ N/A

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

      7. div-sub N/A

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

      8. lower-*.f64 N/A

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

   5. Applied rewrites 94.2%

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

   if -2.4999999999999999e-86 < x < 2.4500000000000001e-64

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 87.7

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

   5. Applied rewrites 87.7%

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

   if 2.4500000000000001e-64 < x

   1. Initial program 95.5%

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

   3. Taylor expanded in z around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 80.7

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

   5. Applied rewrites 80.7%

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

   7. Applied rewrites 80.7%

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

Alternative 5: 88.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+207} \lor \neg \left(z \cdot t \leq 5000000\right):\\ \;\;\;\;\mathsf{fma}\left(y, x, \mathsf{fma}\left(t \cdot z, 0.0625, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c + \left(a \cdot -0.25\right) \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (or (<= (* z t) -2e+207) (not (<= (* z t) 5000000.0)))
   (fma y x (fma (* t z) 0.0625 c))
   (fma y x (+ c (* (* a -0.25) b)))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if (((z * t) <= -2e+207) || !((z * t) <= 5000000.0)) {
		tmp = fma(y, x, fma((t * z), 0.0625, c));
	} else {
		tmp = fma(y, x, (c + ((a * -0.25) * b)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if ((Float64(z * t) <= -2e+207) || !(Float64(z * t) <= 5000000.0))
		tmp = fma(y, x, fma(Float64(t * z), 0.0625, c));
	else
		tmp = fma(y, x, Float64(c + Float64(Float64(a * -0.25) * b)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[Or[LessEqual[N[(z * t), $MachinePrecision], -2e+207], N[Not[LessEqual[N[(z * t), $MachinePrecision], 5000000.0]], $MachinePrecision]], N[(y * x + N[(N[(t * z), $MachinePrecision] * 0.0625 + c), $MachinePrecision]), $MachinePrecision], N[(y * x + N[(c + N[(N[(a * -0.25), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -2.0000000000000001e207 or 5e6 < (*.f64 z t)

   1. Initial program 94.8%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 87.1

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

   5. Applied rewrites 87.1%

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

   if -2.0000000000000001e207 < (*.f64 z t) < 5e6

   1. Initial program 99.4%

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

   3. Taylor expanded in z around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 96.4

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

   5. Applied rewrites 96.4%

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

   7. Applied rewrites 96.4%

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

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+207} \lor \neg \left(z \cdot t \leq 5000000\right):\\ \;\;\;\;\mathsf{fma}\left(y, x, \mathsf{fma}\left(t \cdot z, 0.0625, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c + \left(a \cdot -0.25\right) \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 88.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+207} \lor \neg \left(z \cdot t \leq 5000000\right):\\ \;\;\;\;\mathsf{fma}\left(y, x, \mathsf{fma}\left(t \cdot z, 0.0625, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, b \cdot a, \mathsf{fma}\left(y, x, c\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (or (<= (* z t) -2e+207) (not (<= (* z t) 5000000.0)))
   (fma y x (fma (* t z) 0.0625 c))
   (fma -0.25 (* b a) (fma y x c))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if (((z * t) <= -2e+207) || !((z * t) <= 5000000.0)) {
		tmp = fma(y, x, fma((t * z), 0.0625, c));
	} else {
		tmp = fma(-0.25, (b * a), fma(y, x, c));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if ((Float64(z * t) <= -2e+207) || !(Float64(z * t) <= 5000000.0))
		tmp = fma(y, x, fma(Float64(t * z), 0.0625, c));
	else
		tmp = fma(-0.25, Float64(b * a), fma(y, x, c));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[Or[LessEqual[N[(z * t), $MachinePrecision], -2e+207], N[Not[LessEqual[N[(z * t), $MachinePrecision], 5000000.0]], $MachinePrecision]], N[(y * x + N[(N[(t * z), $MachinePrecision] * 0.0625 + c), $MachinePrecision]), $MachinePrecision], N[(-0.25 * N[(b * a), $MachinePrecision] + N[(y * x + c), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -2.0000000000000001e207 or 5e6 < (*.f64 z t)

   1. Initial program 94.8%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 87.1

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

   5. Applied rewrites 87.1%

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

   if -2.0000000000000001e207 < (*.f64 z t) < 5e6

   1. Initial program 99.4%

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

   3. Taylor expanded in z around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 96.4

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

   5. Applied rewrites 96.4%

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

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+207} \lor \neg \left(z \cdot t \leq 5000000\right):\\ \;\;\;\;\mathsf{fma}\left(y, x, \mathsf{fma}\left(t \cdot z, 0.0625, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, b \cdot a, \mathsf{fma}\left(y, x, c\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+213} \lor \neg \left(z \cdot t \leq 5 \cdot 10^{+32}\right):\\ \;\;\;\;\mathsf{fma}\left(y, x, \left(t \cdot z\right) \cdot 0.0625\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, b \cdot a, \mathsf{fma}\left(y, x, c\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (or (<= (* z t) -2e+213) (not (<= (* z t) 5e+32)))
   (fma y x (* (* t z) 0.0625))
   (fma -0.25 (* b a) (fma y x c))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if (((z * t) <= -2e+213) || !((z * t) <= 5e+32)) {
		tmp = fma(y, x, ((t * z) * 0.0625));
	} else {
		tmp = fma(-0.25, (b * a), fma(y, x, c));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if ((Float64(z * t) <= -2e+213) || !(Float64(z * t) <= 5e+32))
		tmp = fma(y, x, Float64(Float64(t * z) * 0.0625));
	else
		tmp = fma(-0.25, Float64(b * a), fma(y, x, c));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[Or[LessEqual[N[(z * t), $MachinePrecision], -2e+213], N[Not[LessEqual[N[(z * t), $MachinePrecision], 5e+32]], $MachinePrecision]], N[(y * x + N[(N[(t * z), $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision], N[(-0.25 * N[(b * a), $MachinePrecision] + N[(y * x + c), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -1.99999999999999997e213 or 4.9999999999999997e32 < (*.f64 z t)

   1. Initial program 94.4%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 88.1

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

   5. Applied rewrites 88.1%

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

   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 84.8%

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

   10. Applied rewrites 84.8%

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

   if -1.99999999999999997e213 < (*.f64 z t) < 4.9999999999999997e32

   1. Initial program 99.4%

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

   3. Taylor expanded in z around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 95.4

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

   5. Applied rewrites 95.4%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+213} \lor \neg \left(z \cdot t \leq 5 \cdot 10^{+32}\right):\\ \;\;\;\;\mathsf{fma}\left(y, x, \left(t \cdot z\right) \cdot 0.0625\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, b \cdot a, \mathsf{fma}\left(y, x, c\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 66.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot b \leq -5 \cdot 10^{+113} \lor \neg \left(a \cdot b \leq 5 \cdot 10^{+26}\right):\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot b, a, y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (or (<= (* a b) -5e+113) (not (<= (* a b) 5e+26)))
   (fma (* -0.25 b) a (* y x))
   (fma y x c)))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if (((a * b) <= -5e+113) || !((a * b) <= 5e+26)) {
		tmp = fma((-0.25 * b), a, (y * x));
	} else {
		tmp = fma(y, x, c);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if ((Float64(a * b) <= -5e+113) || !(Float64(a * b) <= 5e+26))
		tmp = fma(Float64(-0.25 * b), a, Float64(y * x));
	else
		tmp = fma(y, x, c);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[Or[LessEqual[N[(a * b), $MachinePrecision], -5e+113], N[Not[LessEqual[N[(a * b), $MachinePrecision], 5e+26]], $MachinePrecision]], N[(N[(-0.25 * b), $MachinePrecision] * a + N[(y * x), $MachinePrecision]), $MachinePrecision], N[(y * x + c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a b) < -5e113 or 5.0000000000000001e26 < (*.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 z around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 84.8

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

   5. Applied rewrites 84.8%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 83.9%

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

   if -5e113 < (*.f64 a b) < 5.0000000000000001e26

   1. Initial program 98.1%

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

   3. Taylor expanded in z around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 69.1

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

   5. Applied rewrites 69.1%

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

   7. Applied rewrites 69.1%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 65.8%

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

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b \leq -5 \cdot 10^{+113} \lor \neg \left(a \cdot b \leq 5 \cdot 10^{+26}\right):\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot b, a, y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 65.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot b \leq -5 \cdot 10^{+113} \lor \neg \left(a \cdot b \leq 5 \cdot 10^{+152}\right):\\ \;\;\;\;\mathsf{fma}\left(-0.25, a \cdot b, c\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (or (<= (* a b) -5e+113) (not (<= (* a b) 5e+152)))
   (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 (((a * b) <= -5e+113) || !((a * b) <= 5e+152)) {
		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(a * b) <= -5e+113) || !(Float64(a * b) <= 5e+152))
		tmp = fma(-0.25, Float64(a * b), c);
	else
		tmp = fma(y, x, c);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[Or[LessEqual[N[(a * b), $MachinePrecision], -5e+113], N[Not[LessEqual[N[(a * b), $MachinePrecision], 5e+152]], $MachinePrecision]], N[(-0.25 * N[(a * b), $MachinePrecision] + c), $MachinePrecision], N[(y * x + c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a b) < -5e113 or 5e152 < (*.f64 a b)

   1. Initial program 96.0%

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

   3. Taylor expanded in z around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 85.8

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

   5. Applied rewrites 85.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 76.7%

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

   if -5e113 < (*.f64 a b) < 5e152

   1. Initial program 98.3%

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

   3. Taylor expanded in z around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 70.5

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

   5. Applied rewrites 70.5%

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

   7. Applied rewrites 70.5%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 63.8%

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

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b \leq -5 \cdot 10^{+113} \lor \neg \left(a \cdot b \leq 5 \cdot 10^{+152}\right):\\ \;\;\;\;\mathsf{fma}\left(-0.25, a \cdot b, c\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 64.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2.85 \cdot 10^{+143} \lor \neg \left(z \cdot t \leq 3.55 \cdot 10^{+152}\right):\\ \;\;\;\;\left(z \cdot t\right) \cdot 0.0625\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (or (<= (* z t) -2.85e+143) (not (<= (* z t) 3.55e+152)))
   (* (* z t) 0.0625)
   (fma y x c)))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if (((z * t) <= -2.85e+143) || !((z * t) <= 3.55e+152)) {
		tmp = (z * t) * 0.0625;
	} else {
		tmp = fma(y, x, c);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if ((Float64(z * t) <= -2.85e+143) || !(Float64(z * t) <= 3.55e+152))
		tmp = Float64(Float64(z * t) * 0.0625);
	else
		tmp = fma(y, x, c);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[Or[LessEqual[N[(z * t), $MachinePrecision], -2.85e+143], N[Not[LessEqual[N[(z * t), $MachinePrecision], 3.55e+152]], $MachinePrecision]], N[(N[(z * t), $MachinePrecision] * 0.0625), $MachinePrecision], N[(y * x + c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -2.85000000000000011e143 or 3.55000000000000008e152 < (*.f64 z t)

   1. Initial program 94.3%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. associate--l+ N/A

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

      3. associate-*r/ N/A

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

      4. div-add-rev N/A

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

      5. +-commutative N/A

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

      6. associate-*r/ N/A

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

      7. div-sub N/A

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

      8. lower-*.f64 N/A

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

   5. Applied rewrites 75.7%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 78.7%

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

   if -2.85000000000000011e143 < (*.f64 z t) < 3.55000000000000008e152

   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 z around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 92.5

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

   5. Applied rewrites 92.5%

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

   7. Applied rewrites 92.5%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 63.1%

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

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2.85 \cdot 10^{+143} \lor \neg \left(z \cdot t \leq 3.55 \cdot 10^{+152}\right):\\ \;\;\;\;\left(z \cdot t\right) \cdot 0.0625\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 49.5% 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 97.6%

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

3. Taylor expanded in z around 0

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

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

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

   2. metadata-eval N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. +-commutative N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 75.0

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

5. Applied rewrites 75.0%

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

7. Applied rewrites 75.0%

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

8. Taylor expanded in a around 0

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

10. Applied rewrites 51.3%

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

11. Final simplification 51.3%

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

Alternative 12: 28.1% 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 97.6%

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

3. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. associate--l+ N/A

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

   3. associate-*r/ N/A

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

   4. div-add-rev N/A

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

   5. +-commutative N/A

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

   6. associate-*r/ N/A

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

   7. div-sub N/A

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

   8. lower-*.f64 N/A

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

5. Applied rewrites 78.3%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 31.6%

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

Reproduce

herbie shell --seed 2024329 
(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))