Diagrams.Solve.Tridiagonal:solveCyclicTriDiagonal from diagrams-solve-0.1, B

Percentage Accurate: 74.9% → 90.6%

Time: 11.2s

Alternatives: 19

Speedup: 0.5×

Specification

Math

\[\begin{array}{l} \\ \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t));
}

Fortran

real(8) function code(x, y, z, t, a, b)
    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
    code = (x + ((y * z) / t)) / ((a + 1.0d0) + ((y * b) / t))
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t));
}

Python

def code(x, y, z, t, a, b):
	return (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t))

Julia

function code(x, y, z, t, a, b)
	return Float64(Float64(x + Float64(Float64(y * z) / t)) / Float64(Float64(a + 1.0) + Float64(Float64(y * b) / t)))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t));
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[(N[(a + 1.0), $MachinePrecision] + N[(N[(y * b), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 74.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t));
}

Fortran

real(8) function code(x, y, z, t, a, b)
    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
    code = (x + ((y * z) / t)) / ((a + 1.0d0) + ((y * b) / t))
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t));
}

Python

def code(x, y, z, t, a, b):
	return (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t))

Julia

function code(x, y, z, t, a, b)
	return Float64(Float64(x + Float64(Float64(y * z) / t)) / Float64(Float64(a + 1.0) + Float64(Float64(y * b) / t)))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t));
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[(N[(a + 1.0), $MachinePrecision] + N[(N[(y * b), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 90.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(1 + a\right) + \frac{b \cdot y}{t}\\ t_2 := \frac{z \cdot y}{t}\\ t_3 := \frac{t\_2 + x}{t\_1}\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t} \cdot x, \frac{z}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{y}{t}, b, a\right), x, x\right)}, \frac{x}{\mathsf{fma}\left(\frac{y}{t}, b, 1 + a\right)}\right)\\ \mathbf{elif}\;t\_3 \leq -5 \cdot 10^{-324}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;\frac{z}{b} + \frac{\left(\frac{x}{b} - \frac{\frac{\mathsf{fma}\left(a, z, z\right)}{b}}{b}\right) \cdot t}{y}\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{+283}:\\ \;\;\;\;\frac{\frac{1}{\frac{t}{z \cdot y}} + x}{t\_1}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\left(\frac{x}{\mathsf{fma}\left(b, t\_2, z\right)} + \frac{y}{\mathsf{fma}\left(b, y, t\right)}\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (+ 1.0 a) (/ (* b y) t)))
        (t_2 (/ (* z y) t))
        (t_3 (/ (+ t_2 x) t_1)))
   (if (<= t_3 (- INFINITY))
     (fma
      (* (/ y t) x)
      (/ z (fma (fma (/ y t) b a) x x))
      (/ x (fma (/ y t) b (+ 1.0 a))))
     (if (<= t_3 -5e-324)
       t_3
       (if (<= t_3 0.0)
         (+ (/ z b) (/ (* (- (/ x b) (/ (/ (fma a z z) b) b)) t) y))
         (if (<= t_3 2e+283)
           (/ (+ (/ 1.0 (/ t (* z y))) x) t_1)
           (if (<= t_3 INFINITY)
             (* (+ (/ x (fma b t_2 z)) (/ y (fma b y t))) z)
             (/ z b))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (1.0 + a) + ((b * y) / t);
	double t_2 = (z * y) / t;
	double t_3 = (t_2 + x) / t_1;
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = fma(((y / t) * x), (z / fma(fma((y / t), b, a), x, x)), (x / fma((y / t), b, (1.0 + a))));
	} else if (t_3 <= -5e-324) {
		tmp = t_3;
	} else if (t_3 <= 0.0) {
		tmp = (z / b) + ((((x / b) - ((fma(a, z, z) / b) / b)) * t) / y);
	} else if (t_3 <= 2e+283) {
		tmp = ((1.0 / (t / (z * y))) + x) / t_1;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = ((x / fma(b, t_2, z)) + (y / fma(b, y, t))) * z;
	} else {
		tmp = z / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(1.0 + a) + Float64(Float64(b * y) / t))
	t_2 = Float64(Float64(z * y) / t)
	t_3 = Float64(Float64(t_2 + x) / t_1)
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = fma(Float64(Float64(y / t) * x), Float64(z / fma(fma(Float64(y / t), b, a), x, x)), Float64(x / fma(Float64(y / t), b, Float64(1.0 + a))));
	elseif (t_3 <= -5e-324)
		tmp = t_3;
	elseif (t_3 <= 0.0)
		tmp = Float64(Float64(z / b) + Float64(Float64(Float64(Float64(x / b) - Float64(Float64(fma(a, z, z) / b) / b)) * t) / y));
	elseif (t_3 <= 2e+283)
		tmp = Float64(Float64(Float64(1.0 / Float64(t / Float64(z * y))) + x) / t_1);
	elseif (t_3 <= Inf)
		tmp = Float64(Float64(Float64(x / fma(b, t_2, z)) + Float64(y / fma(b, y, t))) * z);
	else
		tmp = Float64(z / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(1.0 + a), $MachinePrecision] + N[(N[(b * y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(z * y), $MachinePrecision] / t), $MachinePrecision]}, Block[{t$95$3 = N[(N[(t$95$2 + x), $MachinePrecision] / t$95$1), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], N[(N[(N[(y / t), $MachinePrecision] * x), $MachinePrecision] * N[(z / N[(N[(N[(y / t), $MachinePrecision] * b + a), $MachinePrecision] * x + x), $MachinePrecision]), $MachinePrecision] + N[(x / N[(N[(y / t), $MachinePrecision] * b + N[(1.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -5e-324], t$95$3, If[LessEqual[t$95$3, 0.0], N[(N[(z / b), $MachinePrecision] + N[(N[(N[(N[(x / b), $MachinePrecision] - N[(N[(N[(a * z + z), $MachinePrecision] / b), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 2e+283], N[(N[(N[(1.0 / N[(t / N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[(N[(x / N[(b * t$95$2 + z), $MachinePrecision]), $MachinePrecision] + N[(y / N[(b * y + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], N[(z / b), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -inf.0

   1. Initial program 47.8%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. times-frac N/A

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

      4. associate-*r* N/A

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

      5. associate-*r/ N/A

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

      6. *-rgt-identity N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 72.0%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -4.94066e-324

   1. Initial program 99.8%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   if -4.94066e-324 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -0.0

   1. Initial program 46.1%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\left(\frac{z}{b} + \frac{t \cdot x}{b \cdot y}\right) - \frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2} \cdot y}} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{\left(\frac{z}{b} + \frac{t \cdot x}{b \cdot y}\right) + \left(\mathsf{neg}\left(\frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2} \cdot y}\right)\right)} \]

      2. associate-+l+ N/A

         \[\leadsto \color{blue}{\frac{z}{b} + \left(\frac{t \cdot x}{b \cdot y} + \left(\mathsf{neg}\left(\frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2} \cdot y}\right)\right)\right)} \]

      3. sub-neg N/A

         \[\leadsto \frac{z}{b} + \color{blue}{\left(\frac{t \cdot x}{b \cdot y} - \frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2} \cdot y}\right)} \]

      4. associate-/l* N/A

         \[\leadsto \frac{z}{b} + \left(\frac{t \cdot x}{b \cdot y} - \color{blue}{t \cdot \frac{z \cdot \left(1 + a\right)}{{b}^{2} \cdot y}}\right) \]

      5. associate-/l* N/A

         \[\leadsto \frac{z}{b} + \left(\color{blue}{t \cdot \frac{x}{b \cdot y}} - t \cdot \frac{z \cdot \left(1 + a\right)}{{b}^{2} \cdot y}\right) \]

      6. distribute-lft-out-- N/A

         \[\leadsto \frac{z}{b} + \color{blue}{t \cdot \left(\frac{x}{b \cdot y} - \frac{z \cdot \left(1 + a\right)}{{b}^{2} \cdot y}\right)} \]

      7. +-commutative N/A

         \[\leadsto \color{blue}{t \cdot \left(\frac{x}{b \cdot y} - \frac{z \cdot \left(1 + a\right)}{{b}^{2} \cdot y}\right) + \frac{z}{b}} \]

      8. lower-+.f64 N/A

         \[\leadsto \color{blue}{t \cdot \left(\frac{x}{b \cdot y} - \frac{z \cdot \left(1 + a\right)}{{b}^{2} \cdot y}\right) + \frac{z}{b}} \]

   5. Applied rewrites 77.7%

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

   if -0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 1.99999999999999991e283

   1. Initial program 98.6%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 98.7

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 98.7

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

   4. Applied rewrites 98.7%

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

   if 1.99999999999999991e283 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < +inf.0

   1. Initial program 43.2%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 68.6%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 72.9%

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

   if +inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t)))

   1. Initial program 0.0%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{z}{b}} \]
   4. Step-by-step derivation

      1. lower-/.f64 100.0

         \[\leadsto \color{blue}{\frac{z}{b}} \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{z}{b}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{z \cdot y}{t} + x}{\left(1 + a\right) + \frac{b \cdot y}{t}} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t} \cdot x, \frac{z}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{y}{t}, b, a\right), x, x\right)}, \frac{x}{\mathsf{fma}\left(\frac{y}{t}, b, 1 + a\right)}\right)\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\left(1 + a\right) + \frac{b \cdot y}{t}} \leq -5 \cdot 10^{-324}:\\ \;\;\;\;\frac{\frac{z \cdot y}{t} + x}{\left(1 + a\right) + \frac{b \cdot y}{t}}\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\left(1 + a\right) + \frac{b \cdot y}{t}} \leq 0:\\ \;\;\;\;\frac{z}{b} + \frac{\left(\frac{x}{b} - \frac{\frac{\mathsf{fma}\left(a, z, z\right)}{b}}{b}\right) \cdot t}{y}\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\left(1 + a\right) + \frac{b \cdot y}{t}} \leq 2 \cdot 10^{+283}:\\ \;\;\;\;\frac{\frac{1}{\frac{t}{z \cdot y}} + x}{\left(1 + a\right) + \frac{b \cdot y}{t}}\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\left(1 + a\right) + \frac{b \cdot y}{t}} \leq \infty:\\ \;\;\;\;\left(\frac{x}{\mathsf{fma}\left(b, \frac{z \cdot y}{t}, z\right)} + \frac{y}{\mathsf{fma}\left(b, y, t\right)}\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 91.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(1 + a\right) + \frac{b \cdot y}{t}\\ t_2 := \frac{\frac{z \cdot y}{t} + x}{t\_1}\\ t_3 := \frac{1}{\mathsf{fma}\left(\left(\frac{\frac{1}{y}}{z} + \frac{\frac{a}{y}}{z}\right) - \frac{x}{z \cdot z} \cdot \frac{b}{y}, t, \frac{b}{z}\right)}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq -5 \cdot 10^{-324}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;\frac{z}{b} + \frac{\left(\frac{x}{b} - \frac{\frac{\mathsf{fma}\left(a, z, z\right)}{b}}{b}\right) \cdot t}{y}\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{+293}:\\ \;\;\;\;\frac{\frac{1}{\frac{t}{z \cdot y}} + x}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (+ 1.0 a) (/ (* b y) t)))
        (t_2 (/ (+ (/ (* z y) t) x) t_1))
        (t_3
         (/
          1.0
          (fma
           (- (+ (/ (/ 1.0 y) z) (/ (/ a y) z)) (* (/ x (* z z)) (/ b y)))
           t
           (/ b z)))))
   (if (<= t_2 (- INFINITY))
     t_3
     (if (<= t_2 -5e-324)
       t_2
       (if (<= t_2 0.0)
         (+ (/ z b) (/ (* (- (/ x b) (/ (/ (fma a z z) b) b)) t) y))
         (if (<= t_2 4e+293) (/ (+ (/ 1.0 (/ t (* z y))) x) t_1) t_3))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (1.0 + a) + ((b * y) / t);
	double t_2 = (((z * y) / t) + x) / t_1;
	double t_3 = 1.0 / fma(((((1.0 / y) / z) + ((a / y) / z)) - ((x / (z * z)) * (b / y))), t, (b / z));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_3;
	} else if (t_2 <= -5e-324) {
		tmp = t_2;
	} else if (t_2 <= 0.0) {
		tmp = (z / b) + ((((x / b) - ((fma(a, z, z) / b) / b)) * t) / y);
	} else if (t_2 <= 4e+293) {
		tmp = ((1.0 / (t / (z * y))) + x) / t_1;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(1.0 + a) + Float64(Float64(b * y) / t))
	t_2 = Float64(Float64(Float64(Float64(z * y) / t) + x) / t_1)
	t_3 = Float64(1.0 / fma(Float64(Float64(Float64(Float64(1.0 / y) / z) + Float64(Float64(a / y) / z)) - Float64(Float64(x / Float64(z * z)) * Float64(b / y))), t, Float64(b / z)))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_3;
	elseif (t_2 <= -5e-324)
		tmp = t_2;
	elseif (t_2 <= 0.0)
		tmp = Float64(Float64(z / b) + Float64(Float64(Float64(Float64(x / b) - Float64(Float64(fma(a, z, z) / b) / b)) * t) / y));
	elseif (t_2 <= 4e+293)
		tmp = Float64(Float64(Float64(1.0 / Float64(t / Float64(z * y))) + x) / t_1);
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(1.0 + a), $MachinePrecision] + N[(N[(b * y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(z * y), $MachinePrecision] / t), $MachinePrecision] + x), $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(1.0 / N[(N[(N[(N[(N[(1.0 / y), $MachinePrecision] / z), $MachinePrecision] + N[(N[(a / y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] - N[(N[(x / N[(z * z), $MachinePrecision]), $MachinePrecision] * N[(b / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t + N[(b / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$3, If[LessEqual[t$95$2, -5e-324], t$95$2, If[LessEqual[t$95$2, 0.0], N[(N[(z / b), $MachinePrecision] + N[(N[(N[(N[(x / b), $MachinePrecision] - N[(N[(N[(a * z + z), $MachinePrecision] / b), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 4e+293], N[(N[(N[(1.0 / N[(t / N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] / t$95$1), $MachinePrecision], t$95$3]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -inf.0 or 3.9999999999999997e293 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t)))

   1. Initial program 28.1%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 28.1

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 28.2

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 28.2

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lift-/.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. associate-/l* N/A

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

      21. *-commutative N/A

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

      22. lower-fma.f64 N/A

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

      23. lower-/.f64 44.9

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

   4. Applied rewrites 44.9%

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

   5. Taylor expanded in t around 0

      \[\leadsto \frac{1}{\color{blue}{t \cdot \left(\left(\frac{1}{y \cdot z} + \frac{a}{y \cdot z}\right) - \frac{b \cdot x}{y \cdot {z}^{2}}\right) + \frac{b}{z}}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{1}{\color{blue}{\left(\left(\frac{1}{y \cdot z} + \frac{a}{y \cdot z}\right) - \frac{b \cdot x}{y \cdot {z}^{2}}\right) \cdot t} + \frac{b}{z}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\left(\frac{1}{y \cdot z} + \frac{a}{y \cdot z}\right) - \frac{b \cdot x}{y \cdot {z}^{2}}, t, \frac{b}{z}\right)}} \]

   7. Applied rewrites 76.3%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -4.94066e-324

   1. Initial program 99.8%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   if -4.94066e-324 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -0.0

   1. Initial program 46.1%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\left(\frac{z}{b} + \frac{t \cdot x}{b \cdot y}\right) - \frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2} \cdot y}} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{\left(\frac{z}{b} + \frac{t \cdot x}{b \cdot y}\right) + \left(\mathsf{neg}\left(\frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2} \cdot y}\right)\right)} \]

      2. associate-+l+ N/A

         \[\leadsto \color{blue}{\frac{z}{b} + \left(\frac{t \cdot x}{b \cdot y} + \left(\mathsf{neg}\left(\frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2} \cdot y}\right)\right)\right)} \]

      3. sub-neg N/A

         \[\leadsto \frac{z}{b} + \color{blue}{\left(\frac{t \cdot x}{b \cdot y} - \frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2} \cdot y}\right)} \]

      4. associate-/l* N/A

         \[\leadsto \frac{z}{b} + \left(\frac{t \cdot x}{b \cdot y} - \color{blue}{t \cdot \frac{z \cdot \left(1 + a\right)}{{b}^{2} \cdot y}}\right) \]

      5. associate-/l* N/A

         \[\leadsto \frac{z}{b} + \left(\color{blue}{t \cdot \frac{x}{b \cdot y}} - t \cdot \frac{z \cdot \left(1 + a\right)}{{b}^{2} \cdot y}\right) \]

      6. distribute-lft-out-- N/A

         \[\leadsto \frac{z}{b} + \color{blue}{t \cdot \left(\frac{x}{b \cdot y} - \frac{z \cdot \left(1 + a\right)}{{b}^{2} \cdot y}\right)} \]

      7. +-commutative N/A

         \[\leadsto \color{blue}{t \cdot \left(\frac{x}{b \cdot y} - \frac{z \cdot \left(1 + a\right)}{{b}^{2} \cdot y}\right) + \frac{z}{b}} \]

      8. lower-+.f64 N/A

         \[\leadsto \color{blue}{t \cdot \left(\frac{x}{b \cdot y} - \frac{z \cdot \left(1 + a\right)}{{b}^{2} \cdot y}\right) + \frac{z}{b}} \]

   5. Applied rewrites 77.7%

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

   if -0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 3.9999999999999997e293

   1. Initial program 98.7%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 98.7

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 98.7

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

   4. Applied rewrites 98.7%

      \[\leadsto \frac{x + \color{blue}{\frac{1}{\frac{t}{z \cdot y}}}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{z \cdot y}{t} + x}{\left(1 + a\right) + \frac{b \cdot y}{t}} \leq -\infty:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\left(\frac{\frac{1}{y}}{z} + \frac{\frac{a}{y}}{z}\right) - \frac{x}{z \cdot z} \cdot \frac{b}{y}, t, \frac{b}{z}\right)}\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\left(1 + a\right) + \frac{b \cdot y}{t}} \leq -5 \cdot 10^{-324}:\\ \;\;\;\;\frac{\frac{z \cdot y}{t} + x}{\left(1 + a\right) + \frac{b \cdot y}{t}}\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\left(1 + a\right) + \frac{b \cdot y}{t}} \leq 0:\\ \;\;\;\;\frac{z}{b} + \frac{\left(\frac{x}{b} - \frac{\frac{\mathsf{fma}\left(a, z, z\right)}{b}}{b}\right) \cdot t}{y}\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\left(1 + a\right) + \frac{b \cdot y}{t}} \leq 4 \cdot 10^{+293}:\\ \;\;\;\;\frac{\frac{1}{\frac{t}{z \cdot y}} + x}{\left(1 + a\right) + \frac{b \cdot y}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\left(\frac{\frac{1}{y}}{z} + \frac{\frac{a}{y}}{z}\right) - \frac{x}{z \cdot z} \cdot \frac{b}{y}, t, \frac{b}{z}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(1 + a\right) + \frac{b \cdot y}{t}\\ t_2 := \frac{z \cdot y}{t}\\ t_3 := \frac{t\_2 + x}{t\_1}\\ \mathbf{if}\;t\_3 \leq -5 \cdot 10^{-324}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;\frac{z}{b} + \frac{\left(\frac{x}{b} - \frac{\frac{\mathsf{fma}\left(a, z, z\right)}{b}}{b}\right) \cdot t}{y}\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{+283}:\\ \;\;\;\;\frac{\frac{1}{\frac{t}{z \cdot y}} + x}{t\_1}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\left(\frac{x}{\mathsf{fma}\left(b, t\_2, z\right)} + \frac{y}{\mathsf{fma}\left(b, y, t\right)}\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (+ 1.0 a) (/ (* b y) t)))
        (t_2 (/ (* z y) t))
        (t_3 (/ (+ t_2 x) t_1)))
   (if (<= t_3 -5e-324)
     t_3
     (if (<= t_3 0.0)
       (+ (/ z b) (/ (* (- (/ x b) (/ (/ (fma a z z) b) b)) t) y))
       (if (<= t_3 2e+283)
         (/ (+ (/ 1.0 (/ t (* z y))) x) t_1)
         (if (<= t_3 INFINITY)
           (* (+ (/ x (fma b t_2 z)) (/ y (fma b y t))) z)
           (/ z b)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (1.0 + a) + ((b * y) / t);
	double t_2 = (z * y) / t;
	double t_3 = (t_2 + x) / t_1;
	double tmp;
	if (t_3 <= -5e-324) {
		tmp = t_3;
	} else if (t_3 <= 0.0) {
		tmp = (z / b) + ((((x / b) - ((fma(a, z, z) / b) / b)) * t) / y);
	} else if (t_3 <= 2e+283) {
		tmp = ((1.0 / (t / (z * y))) + x) / t_1;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = ((x / fma(b, t_2, z)) + (y / fma(b, y, t))) * z;
	} else {
		tmp = z / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(1.0 + a) + Float64(Float64(b * y) / t))
	t_2 = Float64(Float64(z * y) / t)
	t_3 = Float64(Float64(t_2 + x) / t_1)
	tmp = 0.0
	if (t_3 <= -5e-324)
		tmp = t_3;
	elseif (t_3 <= 0.0)
		tmp = Float64(Float64(z / b) + Float64(Float64(Float64(Float64(x / b) - Float64(Float64(fma(a, z, z) / b) / b)) * t) / y));
	elseif (t_3 <= 2e+283)
		tmp = Float64(Float64(Float64(1.0 / Float64(t / Float64(z * y))) + x) / t_1);
	elseif (t_3 <= Inf)
		tmp = Float64(Float64(Float64(x / fma(b, t_2, z)) + Float64(y / fma(b, y, t))) * z);
	else
		tmp = Float64(z / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(1.0 + a), $MachinePrecision] + N[(N[(b * y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(z * y), $MachinePrecision] / t), $MachinePrecision]}, Block[{t$95$3 = N[(N[(t$95$2 + x), $MachinePrecision] / t$95$1), $MachinePrecision]}, If[LessEqual[t$95$3, -5e-324], t$95$3, If[LessEqual[t$95$3, 0.0], N[(N[(z / b), $MachinePrecision] + N[(N[(N[(N[(x / b), $MachinePrecision] - N[(N[(N[(a * z + z), $MachinePrecision] / b), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 2e+283], N[(N[(N[(1.0 / N[(t / N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[(N[(x / N[(b * t$95$2 + z), $MachinePrecision]), $MachinePrecision] + N[(y / N[(b * y + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], N[(z / b), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -4.94066e-324

   1. Initial program 89.0%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   if -4.94066e-324 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -0.0

   1. Initial program 46.1%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\left(\frac{z}{b} + \frac{t \cdot x}{b \cdot y}\right) - \frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2} \cdot y}} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{\left(\frac{z}{b} + \frac{t \cdot x}{b \cdot y}\right) + \left(\mathsf{neg}\left(\frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2} \cdot y}\right)\right)} \]

      2. associate-+l+ N/A

         \[\leadsto \color{blue}{\frac{z}{b} + \left(\frac{t \cdot x}{b \cdot y} + \left(\mathsf{neg}\left(\frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2} \cdot y}\right)\right)\right)} \]

      3. sub-neg N/A

         \[\leadsto \frac{z}{b} + \color{blue}{\left(\frac{t \cdot x}{b \cdot y} - \frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2} \cdot y}\right)} \]

      4. associate-/l* N/A

         \[\leadsto \frac{z}{b} + \left(\frac{t \cdot x}{b \cdot y} - \color{blue}{t \cdot \frac{z \cdot \left(1 + a\right)}{{b}^{2} \cdot y}}\right) \]

      5. associate-/l* N/A

         \[\leadsto \frac{z}{b} + \left(\color{blue}{t \cdot \frac{x}{b \cdot y}} - t \cdot \frac{z \cdot \left(1 + a\right)}{{b}^{2} \cdot y}\right) \]

      6. distribute-lft-out-- N/A

         \[\leadsto \frac{z}{b} + \color{blue}{t \cdot \left(\frac{x}{b \cdot y} - \frac{z \cdot \left(1 + a\right)}{{b}^{2} \cdot y}\right)} \]

      7. +-commutative N/A

         \[\leadsto \color{blue}{t \cdot \left(\frac{x}{b \cdot y} - \frac{z \cdot \left(1 + a\right)}{{b}^{2} \cdot y}\right) + \frac{z}{b}} \]

      8. lower-+.f64 N/A

         \[\leadsto \color{blue}{t \cdot \left(\frac{x}{b \cdot y} - \frac{z \cdot \left(1 + a\right)}{{b}^{2} \cdot y}\right) + \frac{z}{b}} \]

   5. Applied rewrites 77.7%

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

   if -0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 1.99999999999999991e283

   1. Initial program 98.6%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 98.7

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 98.7

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

   4. Applied rewrites 98.7%

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

   if 1.99999999999999991e283 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < +inf.0

   1. Initial program 43.2%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 68.6%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 72.9%

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

   if +inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t)))

   1. Initial program 0.0%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{z}{b}} \]
   4. Step-by-step derivation

      1. lower-/.f64 100.0

         \[\leadsto \color{blue}{\frac{z}{b}} \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{z}{b}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{z \cdot y}{t} + x}{\left(1 + a\right) + \frac{b \cdot y}{t}} \leq -5 \cdot 10^{-324}:\\ \;\;\;\;\frac{\frac{z \cdot y}{t} + x}{\left(1 + a\right) + \frac{b \cdot y}{t}}\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\left(1 + a\right) + \frac{b \cdot y}{t}} \leq 0:\\ \;\;\;\;\frac{z}{b} + \frac{\left(\frac{x}{b} - \frac{\frac{\mathsf{fma}\left(a, z, z\right)}{b}}{b}\right) \cdot t}{y}\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\left(1 + a\right) + \frac{b \cdot y}{t}} \leq 2 \cdot 10^{+283}:\\ \;\;\;\;\frac{\frac{1}{\frac{t}{z \cdot y}} + x}{\left(1 + a\right) + \frac{b \cdot y}{t}}\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\left(1 + a\right) + \frac{b \cdot y}{t}} \leq \infty:\\ \;\;\;\;\left(\frac{x}{\mathsf{fma}\left(b, \frac{z \cdot y}{t}, z\right)} + \frac{y}{\mathsf{fma}\left(b, y, t\right)}\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(1 + a\right) + \frac{b \cdot y}{t}\\ t_2 := \frac{z \cdot y}{t}\\ t_3 := \frac{t\_2 + x}{t\_1}\\ \mathbf{if}\;t\_3 \leq -5 \cdot 10^{-324}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{b}, \frac{x}{y}, \frac{z}{b}\right)\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{+283}:\\ \;\;\;\;\frac{\frac{1}{\frac{t}{z \cdot y}} + x}{t\_1}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\left(\frac{x}{\mathsf{fma}\left(b, t\_2, z\right)} + \frac{y}{\mathsf{fma}\left(b, y, t\right)}\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (+ 1.0 a) (/ (* b y) t)))
        (t_2 (/ (* z y) t))
        (t_3 (/ (+ t_2 x) t_1)))
   (if (<= t_3 -5e-324)
     t_3
     (if (<= t_3 0.0)
       (fma (/ t b) (/ x y) (/ z b))
       (if (<= t_3 2e+283)
         (/ (+ (/ 1.0 (/ t (* z y))) x) t_1)
         (if (<= t_3 INFINITY)
           (* (+ (/ x (fma b t_2 z)) (/ y (fma b y t))) z)
           (/ z b)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (1.0 + a) + ((b * y) / t);
	double t_2 = (z * y) / t;
	double t_3 = (t_2 + x) / t_1;
	double tmp;
	if (t_3 <= -5e-324) {
		tmp = t_3;
	} else if (t_3 <= 0.0) {
		tmp = fma((t / b), (x / y), (z / b));
	} else if (t_3 <= 2e+283) {
		tmp = ((1.0 / (t / (z * y))) + x) / t_1;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = ((x / fma(b, t_2, z)) + (y / fma(b, y, t))) * z;
	} else {
		tmp = z / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(1.0 + a) + Float64(Float64(b * y) / t))
	t_2 = Float64(Float64(z * y) / t)
	t_3 = Float64(Float64(t_2 + x) / t_1)
	tmp = 0.0
	if (t_3 <= -5e-324)
		tmp = t_3;
	elseif (t_3 <= 0.0)
		tmp = fma(Float64(t / b), Float64(x / y), Float64(z / b));
	elseif (t_3 <= 2e+283)
		tmp = Float64(Float64(Float64(1.0 / Float64(t / Float64(z * y))) + x) / t_1);
	elseif (t_3 <= Inf)
		tmp = Float64(Float64(Float64(x / fma(b, t_2, z)) + Float64(y / fma(b, y, t))) * z);
	else
		tmp = Float64(z / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(1.0 + a), $MachinePrecision] + N[(N[(b * y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(z * y), $MachinePrecision] / t), $MachinePrecision]}, Block[{t$95$3 = N[(N[(t$95$2 + x), $MachinePrecision] / t$95$1), $MachinePrecision]}, If[LessEqual[t$95$3, -5e-324], t$95$3, If[LessEqual[t$95$3, 0.0], N[(N[(t / b), $MachinePrecision] * N[(x / y), $MachinePrecision] + N[(z / b), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 2e+283], N[(N[(N[(1.0 / N[(t / N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[(N[(x / N[(b * t$95$2 + z), $MachinePrecision]), $MachinePrecision] + N[(y / N[(b * y + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], N[(z / b), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -4.94066e-324

   1. Initial program 89.0%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   if -4.94066e-324 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -0.0

   1. Initial program 46.1%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 48.7

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 59.5

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 59.5

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

   4. Applied rewrites 59.5%

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

   5. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 54.3

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

   7. Applied rewrites 54.3%

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

   8. Taylor expanded in x around 0

      \[\leadsto \frac{z}{b} + \color{blue}{\frac{t \cdot x}{b \cdot y}} \]
   9. Step-by-step derivation

   10. Applied rewrites 72.4%

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

   if -0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 1.99999999999999991e283

   1. Initial program 98.6%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 98.7

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 98.7

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

   4. Applied rewrites 98.7%

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

   if 1.99999999999999991e283 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < +inf.0

   1. Initial program 43.2%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 68.6%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 72.9%

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

   if +inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t)))

   1. Initial program 0.0%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{z}{b}} \]
   4. Step-by-step derivation

      1. lower-/.f64 100.0

         \[\leadsto \color{blue}{\frac{z}{b}} \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{z}{b}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{z \cdot y}{t} + x}{\left(1 + a\right) + \frac{b \cdot y}{t}} \leq -5 \cdot 10^{-324}:\\ \;\;\;\;\frac{\frac{z \cdot y}{t} + x}{\left(1 + a\right) + \frac{b \cdot y}{t}}\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\left(1 + a\right) + \frac{b \cdot y}{t}} \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{b}, \frac{x}{y}, \frac{z}{b}\right)\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\left(1 + a\right) + \frac{b \cdot y}{t}} \leq 2 \cdot 10^{+283}:\\ \;\;\;\;\frac{\frac{1}{\frac{t}{z \cdot y}} + x}{\left(1 + a\right) + \frac{b \cdot y}{t}}\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\left(1 + a\right) + \frac{b \cdot y}{t}} \leq \infty:\\ \;\;\;\;\left(\frac{x}{\mathsf{fma}\left(b, \frac{z \cdot y}{t}, z\right)} + \frac{y}{\mathsf{fma}\left(b, y, t\right)}\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot y}{t}\\ t_2 := \frac{t\_1 + x}{\left(1 + a\right) + \frac{b \cdot y}{t}}\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{-324}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{b}, \frac{x}{y}, \frac{z}{b}\right)\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+283}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\left(\frac{x}{\mathsf{fma}\left(b, t\_1, z\right)} + \frac{y}{\mathsf{fma}\left(b, y, t\right)}\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (* z y) t)) (t_2 (/ (+ t_1 x) (+ (+ 1.0 a) (/ (* b y) t)))))
   (if (<= t_2 -5e-324)
     t_2
     (if (<= t_2 0.0)
       (fma (/ t b) (/ x y) (/ z b))
       (if (<= t_2 2e+283)
         t_2
         (if (<= t_2 INFINITY)
           (* (+ (/ x (fma b t_1 z)) (/ y (fma b y t))) z)
           (/ z b)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (z * y) / t;
	double t_2 = (t_1 + x) / ((1.0 + a) + ((b * y) / t));
	double tmp;
	if (t_2 <= -5e-324) {
		tmp = t_2;
	} else if (t_2 <= 0.0) {
		tmp = fma((t / b), (x / y), (z / b));
	} else if (t_2 <= 2e+283) {
		tmp = t_2;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = ((x / fma(b, t_1, z)) + (y / fma(b, y, t))) * z;
	} else {
		tmp = z / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(z * y) / t)
	t_2 = Float64(Float64(t_1 + x) / Float64(Float64(1.0 + a) + Float64(Float64(b * y) / t)))
	tmp = 0.0
	if (t_2 <= -5e-324)
		tmp = t_2;
	elseif (t_2 <= 0.0)
		tmp = fma(Float64(t / b), Float64(x / y), Float64(z / b));
	elseif (t_2 <= 2e+283)
		tmp = t_2;
	elseif (t_2 <= Inf)
		tmp = Float64(Float64(Float64(x / fma(b, t_1, z)) + Float64(y / fma(b, y, t))) * z);
	else
		tmp = Float64(z / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(z * y), $MachinePrecision] / t), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 + x), $MachinePrecision] / N[(N[(1.0 + a), $MachinePrecision] + N[(N[(b * y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e-324], t$95$2, If[LessEqual[t$95$2, 0.0], N[(N[(t / b), $MachinePrecision] * N[(x / y), $MachinePrecision] + N[(z / b), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e+283], t$95$2, If[LessEqual[t$95$2, Infinity], N[(N[(N[(x / N[(b * t$95$1 + z), $MachinePrecision]), $MachinePrecision] + N[(y / N[(b * y + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], N[(z / b), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -4.94066e-324 or -0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 1.99999999999999991e283

   1. Initial program 93.9%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   if -4.94066e-324 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -0.0

   1. Initial program 46.1%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 48.7

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 59.5

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 59.5

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

   4. Applied rewrites 59.5%

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

   5. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 54.3

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

   7. Applied rewrites 54.3%

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

   8. Taylor expanded in x around 0

      \[\leadsto \frac{z}{b} + \color{blue}{\frac{t \cdot x}{b \cdot y}} \]
   9. Step-by-step derivation

   10. Applied rewrites 72.4%

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

   if 1.99999999999999991e283 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < +inf.0

   1. Initial program 43.2%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 68.6%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 72.9%

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

   if +inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t)))

   1. Initial program 0.0%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{z}{b}} \]
   4. Step-by-step derivation

      1. lower-/.f64 100.0

         \[\leadsto \color{blue}{\frac{z}{b}} \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{z}{b}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{z \cdot y}{t} + x}{\left(1 + a\right) + \frac{b \cdot y}{t}} \leq -5 \cdot 10^{-324}:\\ \;\;\;\;\frac{\frac{z \cdot y}{t} + x}{\left(1 + a\right) + \frac{b \cdot y}{t}}\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\left(1 + a\right) + \frac{b \cdot y}{t}} \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{b}, \frac{x}{y}, \frac{z}{b}\right)\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\left(1 + a\right) + \frac{b \cdot y}{t}} \leq 2 \cdot 10^{+283}:\\ \;\;\;\;\frac{\frac{z \cdot y}{t} + x}{\left(1 + a\right) + \frac{b \cdot y}{t}}\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\left(1 + a\right) + \frac{b \cdot y}{t}} \leq \infty:\\ \;\;\;\;\left(\frac{x}{\mathsf{fma}\left(b, \frac{z \cdot y}{t}, z\right)} + \frac{y}{\mathsf{fma}\left(b, y, t\right)}\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot y}{t} + x\\ t_2 := \frac{t\_1}{\left(1 + a\right) + \frac{b \cdot y}{t}}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{y}{t}, b, a\right), t, t\right)} \cdot z\\ \mathbf{elif}\;t\_2 \leq -5 \cdot 10^{-324}:\\ \;\;\;\;\frac{t\_1}{1 + a}\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(y, z, t \cdot x\right)}{y}}{b}\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{+293}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{1 + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (/ (* z y) t) x)) (t_2 (/ t_1 (+ (+ 1.0 a) (/ (* b y) t)))))
   (if (<= t_2 (- INFINITY))
     (* (/ y (fma (fma (/ y t) b a) t t)) z)
     (if (<= t_2 -5e-324)
       (/ t_1 (+ 1.0 a))
       (if (<= t_2 0.0)
         (/ (/ (fma y z (* t x)) y) b)
         (if (<= t_2 4e+293) (/ (fma (/ y t) z x) (+ 1.0 a)) (/ z b)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((z * y) / t) + x;
	double t_2 = t_1 / ((1.0 + a) + ((b * y) / t));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = (y / fma(fma((y / t), b, a), t, t)) * z;
	} else if (t_2 <= -5e-324) {
		tmp = t_1 / (1.0 + a);
	} else if (t_2 <= 0.0) {
		tmp = (fma(y, z, (t * x)) / y) / b;
	} else if (t_2 <= 4e+293) {
		tmp = fma((y / t), z, x) / (1.0 + a);
	} else {
		tmp = z / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(z * y) / t) + x)
	t_2 = Float64(t_1 / Float64(Float64(1.0 + a) + Float64(Float64(b * y) / t)))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(Float64(y / fma(fma(Float64(y / t), b, a), t, t)) * z);
	elseif (t_2 <= -5e-324)
		tmp = Float64(t_1 / Float64(1.0 + a));
	elseif (t_2 <= 0.0)
		tmp = Float64(Float64(fma(y, z, Float64(t * x)) / y) / b);
	elseif (t_2 <= 4e+293)
		tmp = Float64(fma(Float64(y / t), z, x) / Float64(1.0 + a));
	else
		tmp = Float64(z / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(z * y), $MachinePrecision] / t), $MachinePrecision] + x), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / N[(N[(1.0 + a), $MachinePrecision] + N[(N[(b * y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(N[(y / N[(N[(N[(y / t), $MachinePrecision] * b + a), $MachinePrecision] * t + t), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[t$95$2, -5e-324], N[(t$95$1 / N[(1.0 + a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.0], N[(N[(N[(y * z + N[(t * x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] / b), $MachinePrecision], If[LessEqual[t$95$2, 4e+293], N[(N[(N[(y / t), $MachinePrecision] * z + x), $MachinePrecision] / N[(1.0 + a), $MachinePrecision]), $MachinePrecision], N[(z / b), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -inf.0

   1. Initial program 47.8%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. *-lft-identity N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

          \[\leadsto \frac{y}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{y}{t}, b, a\right)}, t, t\right)} \cdot z \]

      12. lower-/.f64 67.0

          \[\leadsto \frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{y}{t}}, b, a\right), t, t\right)} \cdot z \]

   5. Applied rewrites 67.0%

      \[\leadsto \color{blue}{\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{y}{t}, b, a\right), t, t\right)} \cdot z} \]

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -4.94066e-324

   1. Initial program 99.8%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{x + \frac{y \cdot z}{t}}{\color{blue}{1 + a}} \]
   4. Step-by-step derivation

      1. lower-+.f64 77.8

         \[\leadsto \frac{x + \frac{y \cdot z}{t}}{\color{blue}{1 + a}} \]

   5. Applied rewrites 77.8%

      \[\leadsto \frac{x + \frac{y \cdot z}{t}}{\color{blue}{1 + a}} \]

   if -4.94066e-324 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -0.0

   1. Initial program 46.1%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 48.7

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 59.5

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 59.5

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

   4. Applied rewrites 59.5%

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

   5. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 54.3

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

   7. Applied rewrites 54.3%

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

   8. Taylor expanded in x around 0

      \[\leadsto \frac{t \cdot x + y \cdot z}{\color{blue}{b} \cdot y} \]
   9. Step-by-step derivation

   10. Applied rewrites 54.4%

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

   12. Applied rewrites 72.4%

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

   if -0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 3.9999999999999997e293

   1. Initial program 98.7%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{x + \frac{y \cdot z}{t}}{1 + a}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x + \frac{y \cdot z}{t}}{1 + a}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z}{t} + x}}{1 + a} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{t} \cdot z} + x}{1 + a} \]

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 74.7

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

   5. Applied rewrites 74.7%

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

   if 3.9999999999999997e293 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t)))

   1. Initial program 20.0%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{z}{b}} \]
   4. Step-by-step derivation

      1. lower-/.f64 73.7

         \[\leadsto \color{blue}{\frac{z}{b}} \]

   5. Applied rewrites 73.7%

      \[\leadsto \color{blue}{\frac{z}{b}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{z \cdot y}{t} + x}{\left(1 + a\right) + \frac{b \cdot y}{t}} \leq -\infty:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{y}{t}, b, a\right), t, t\right)} \cdot z\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\left(1 + a\right) + \frac{b \cdot y}{t}} \leq -5 \cdot 10^{-324}:\\ \;\;\;\;\frac{\frac{z \cdot y}{t} + x}{1 + a}\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\left(1 + a\right) + \frac{b \cdot y}{t}} \leq 0:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(y, z, t \cdot x\right)}{y}}{b}\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\left(1 + a\right) + \frac{b \cdot y}{t}} \leq 4 \cdot 10^{+293}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{1 + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot y}{t} + x\\ t_2 := \frac{t\_1}{\left(1 + a\right) + \frac{b \cdot y}{t}}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;t\_2 \leq -5 \cdot 10^{-324}:\\ \;\;\;\;\frac{t\_1}{1 + a}\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(y, z, t \cdot x\right)}{y}}{b}\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{+293}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{1 + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (/ (* z y) t) x)) (t_2 (/ t_1 (+ (+ 1.0 a) (/ (* b y) t)))))
   (if (<= t_2 (- INFINITY))
     (/ z b)
     (if (<= t_2 -5e-324)
       (/ t_1 (+ 1.0 a))
       (if (<= t_2 0.0)
         (/ (/ (fma y z (* t x)) y) b)
         (if (<= t_2 4e+293) (/ (fma (/ y t) z x) (+ 1.0 a)) (/ z b)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((z * y) / t) + x;
	double t_2 = t_1 / ((1.0 + a) + ((b * y) / t));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = z / b;
	} else if (t_2 <= -5e-324) {
		tmp = t_1 / (1.0 + a);
	} else if (t_2 <= 0.0) {
		tmp = (fma(y, z, (t * x)) / y) / b;
	} else if (t_2 <= 4e+293) {
		tmp = fma((y / t), z, x) / (1.0 + a);
	} else {
		tmp = z / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(z * y) / t) + x)
	t_2 = Float64(t_1 / Float64(Float64(1.0 + a) + Float64(Float64(b * y) / t)))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(z / b);
	elseif (t_2 <= -5e-324)
		tmp = Float64(t_1 / Float64(1.0 + a));
	elseif (t_2 <= 0.0)
		tmp = Float64(Float64(fma(y, z, Float64(t * x)) / y) / b);
	elseif (t_2 <= 4e+293)
		tmp = Float64(fma(Float64(y / t), z, x) / Float64(1.0 + a));
	else
		tmp = Float64(z / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(z * y), $MachinePrecision] / t), $MachinePrecision] + x), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / N[(N[(1.0 + a), $MachinePrecision] + N[(N[(b * y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(z / b), $MachinePrecision], If[LessEqual[t$95$2, -5e-324], N[(t$95$1 / N[(1.0 + a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.0], N[(N[(N[(y * z + N[(t * x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] / b), $MachinePrecision], If[LessEqual[t$95$2, 4e+293], N[(N[(N[(y / t), $MachinePrecision] * z + x), $MachinePrecision] / N[(1.0 + a), $MachinePrecision]), $MachinePrecision], N[(z / b), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -inf.0 or 3.9999999999999997e293 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t)))

   1. Initial program 28.1%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{z}{b}} \]
   4. Step-by-step derivation

      1. lower-/.f64 70.2

         \[\leadsto \color{blue}{\frac{z}{b}} \]

   5. Applied rewrites 70.2%

      \[\leadsto \color{blue}{\frac{z}{b}} \]

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -4.94066e-324

   1. Initial program 99.8%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{x + \frac{y \cdot z}{t}}{\color{blue}{1 + a}} \]
   4. Step-by-step derivation

      1. lower-+.f64 77.8

         \[\leadsto \frac{x + \frac{y \cdot z}{t}}{\color{blue}{1 + a}} \]

   5. Applied rewrites 77.8%

      \[\leadsto \frac{x + \frac{y \cdot z}{t}}{\color{blue}{1 + a}} \]

   if -4.94066e-324 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -0.0

   1. Initial program 46.1%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 48.7

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 59.5

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 59.5

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

   4. Applied rewrites 59.5%

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

   5. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 54.3

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

   7. Applied rewrites 54.3%

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

   8. Taylor expanded in x around 0

      \[\leadsto \frac{t \cdot x + y \cdot z}{\color{blue}{b} \cdot y} \]
   9. Step-by-step derivation

   10. Applied rewrites 54.4%

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

   12. Applied rewrites 72.4%

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

   if -0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 3.9999999999999997e293

   1. Initial program 98.7%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{x + \frac{y \cdot z}{t}}{1 + a}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x + \frac{y \cdot z}{t}}{1 + a}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z}{t} + x}}{1 + a} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{t} \cdot z} + x}{1 + a} \]

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 74.7

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

   5. Applied rewrites 74.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{1 + a}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{z \cdot y}{t} + x}{\left(1 + a\right) + \frac{b \cdot y}{t}} \leq -\infty:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\left(1 + a\right) + \frac{b \cdot y}{t}} \leq -5 \cdot 10^{-324}:\\ \;\;\;\;\frac{\frac{z \cdot y}{t} + x}{1 + a}\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\left(1 + a\right) + \frac{b \cdot y}{t}} \leq 0:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(y, z, t \cdot x\right)}{y}}{b}\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\left(1 + a\right) + \frac{b \cdot y}{t}} \leq 4 \cdot 10^{+293}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{1 + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{z \cdot y}{t} + x}{\left(1 + a\right) + \frac{b \cdot y}{t}}\\ t_2 := \frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{1 + a}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-324}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(y, z, t \cdot x\right)}{y}}{b}\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+293}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (+ (/ (* z y) t) x) (+ (+ 1.0 a) (/ (* b y) t))))
        (t_2 (/ (fma (/ y t) z x) (+ 1.0 a))))
   (if (<= t_1 (- INFINITY))
     (/ z b)
     (if (<= t_1 -5e-324)
       t_2
       (if (<= t_1 0.0)
         (/ (/ (fma y z (* t x)) y) b)
         (if (<= t_1 4e+293) t_2 (/ z b)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (((z * y) / t) + x) / ((1.0 + a) + ((b * y) / t));
	double t_2 = fma((y / t), z, x) / (1.0 + a);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = z / b;
	} else if (t_1 <= -5e-324) {
		tmp = t_2;
	} else if (t_1 <= 0.0) {
		tmp = (fma(y, z, (t * x)) / y) / b;
	} else if (t_1 <= 4e+293) {
		tmp = t_2;
	} else {
		tmp = z / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(Float64(z * y) / t) + x) / Float64(Float64(1.0 + a) + Float64(Float64(b * y) / t)))
	t_2 = Float64(fma(Float64(y / t), z, x) / Float64(1.0 + a))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(z / b);
	elseif (t_1 <= -5e-324)
		tmp = t_2;
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(fma(y, z, Float64(t * x)) / y) / b);
	elseif (t_1 <= 4e+293)
		tmp = t_2;
	else
		tmp = Float64(z / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(N[(z * y), $MachinePrecision] / t), $MachinePrecision] + x), $MachinePrecision] / N[(N[(1.0 + a), $MachinePrecision] + N[(N[(b * y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(y / t), $MachinePrecision] * z + x), $MachinePrecision] / N[(1.0 + a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(z / b), $MachinePrecision], If[LessEqual[t$95$1, -5e-324], t$95$2, If[LessEqual[t$95$1, 0.0], N[(N[(N[(y * z + N[(t * x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] / b), $MachinePrecision], If[LessEqual[t$95$1, 4e+293], t$95$2, N[(z / b), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -inf.0 or 3.9999999999999997e293 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t)))

   1. Initial program 28.1%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{z}{b}} \]
   4. Step-by-step derivation

      1. lower-/.f64 70.2

         \[\leadsto \color{blue}{\frac{z}{b}} \]

   5. Applied rewrites 70.2%

      \[\leadsto \color{blue}{\frac{z}{b}} \]

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -4.94066e-324 or -0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 3.9999999999999997e293

   1. Initial program 99.1%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{x + \frac{y \cdot z}{t}}{1 + a}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x + \frac{y \cdot z}{t}}{1 + a}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z}{t} + x}}{1 + a} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{t} \cdot z} + x}{1 + a} \]

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 75.5

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

   5. Applied rewrites 75.5%

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

   if -4.94066e-324 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -0.0

   1. Initial program 46.1%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 48.7

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 59.5

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 59.5

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

   4. Applied rewrites 59.5%

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

   5. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 54.3

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

   7. Applied rewrites 54.3%

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

   8. Taylor expanded in x around 0

      \[\leadsto \frac{t \cdot x + y \cdot z}{\color{blue}{b} \cdot y} \]
   9. Step-by-step derivation

   10. Applied rewrites 54.4%

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

   12. Applied rewrites 72.4%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{z \cdot y}{t} + x}{\left(1 + a\right) + \frac{b \cdot y}{t}} \leq -\infty:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\left(1 + a\right) + \frac{b \cdot y}{t}} \leq -5 \cdot 10^{-324}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{1 + a}\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\left(1 + a\right) + \frac{b \cdot y}{t}} \leq 0:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(y, z, t \cdot x\right)}{y}}{b}\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\left(1 + a\right) + \frac{b \cdot y}{t}} \leq 4 \cdot 10^{+293}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{1 + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 72.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{z \cdot y}{t} + x}{\left(1 + a\right) + \frac{b \cdot y}{t}}\\ t_2 := \frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{1 + a}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-324}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+293}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (+ (/ (* z y) t) x) (+ (+ 1.0 a) (/ (* b y) t))))
        (t_2 (/ (fma (/ y t) z x) (+ 1.0 a))))
   (if (<= t_1 (- INFINITY))
     (/ z b)
     (if (<= t_1 -5e-324)
       t_2
       (if (<= t_1 0.0) (/ z b) (if (<= t_1 4e+293) t_2 (/ z b)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (((z * y) / t) + x) / ((1.0 + a) + ((b * y) / t));
	double t_2 = fma((y / t), z, x) / (1.0 + a);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = z / b;
	} else if (t_1 <= -5e-324) {
		tmp = t_2;
	} else if (t_1 <= 0.0) {
		tmp = z / b;
	} else if (t_1 <= 4e+293) {
		tmp = t_2;
	} else {
		tmp = z / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(Float64(z * y) / t) + x) / Float64(Float64(1.0 + a) + Float64(Float64(b * y) / t)))
	t_2 = Float64(fma(Float64(y / t), z, x) / Float64(1.0 + a))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(z / b);
	elseif (t_1 <= -5e-324)
		tmp = t_2;
	elseif (t_1 <= 0.0)
		tmp = Float64(z / b);
	elseif (t_1 <= 4e+293)
		tmp = t_2;
	else
		tmp = Float64(z / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(N[(z * y), $MachinePrecision] / t), $MachinePrecision] + x), $MachinePrecision] / N[(N[(1.0 + a), $MachinePrecision] + N[(N[(b * y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(y / t), $MachinePrecision] * z + x), $MachinePrecision] / N[(1.0 + a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(z / b), $MachinePrecision], If[LessEqual[t$95$1, -5e-324], t$95$2, If[LessEqual[t$95$1, 0.0], N[(z / b), $MachinePrecision], If[LessEqual[t$95$1, 4e+293], t$95$2, N[(z / b), $MachinePrecision]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -inf.0 or -4.94066e-324 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -0.0 or 3.9999999999999997e293 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t)))

   1. Initial program 34.6%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{z}{b}} \]
   4. Step-by-step derivation

      1. lower-/.f64 69.6

         \[\leadsto \color{blue}{\frac{z}{b}} \]

   5. Applied rewrites 69.6%

      \[\leadsto \color{blue}{\frac{z}{b}} \]

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -4.94066e-324 or -0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 3.9999999999999997e293

   1. Initial program 99.1%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{x + \frac{y \cdot z}{t}}{1 + a}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x + \frac{y \cdot z}{t}}{1 + a}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z}{t} + x}}{1 + a} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{t} \cdot z} + x}{1 + a} \]

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 75.5

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

   5. Applied rewrites 75.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{1 + a}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{z \cdot y}{t} + x}{\left(1 + a\right) + \frac{b \cdot y}{t}} \leq -\infty:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\left(1 + a\right) + \frac{b \cdot y}{t}} \leq -5 \cdot 10^{-324}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{1 + a}\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\left(1 + a\right) + \frac{b \cdot y}{t}} \leq 0:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\left(1 + a\right) + \frac{b \cdot y}{t}} \leq 4 \cdot 10^{+293}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{1 + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 88.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{z \cdot y}{t} + x}{\left(1 + a\right) + \frac{b \cdot y}{t}}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-324}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{b}, \frac{x}{y}, \frac{z}{b}\right)\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+293}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (+ (/ (* z y) t) x) (+ (+ 1.0 a) (/ (* b y) t)))))
   (if (<= t_1 -5e-324)
     t_1
     (if (<= t_1 0.0)
       (fma (/ t b) (/ x y) (/ z b))
       (if (<= t_1 4e+293) t_1 (/ z b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (((z * y) / t) + x) / ((1.0 + a) + ((b * y) / t));
	double tmp;
	if (t_1 <= -5e-324) {
		tmp = t_1;
	} else if (t_1 <= 0.0) {
		tmp = fma((t / b), (x / y), (z / b));
	} else if (t_1 <= 4e+293) {
		tmp = t_1;
	} else {
		tmp = z / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(Float64(z * y) / t) + x) / Float64(Float64(1.0 + a) + Float64(Float64(b * y) / t)))
	tmp = 0.0
	if (t_1 <= -5e-324)
		tmp = t_1;
	elseif (t_1 <= 0.0)
		tmp = fma(Float64(t / b), Float64(x / y), Float64(z / b));
	elseif (t_1 <= 4e+293)
		tmp = t_1;
	else
		tmp = Float64(z / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(N[(z * y), $MachinePrecision] / t), $MachinePrecision] + x), $MachinePrecision] / N[(N[(1.0 + a), $MachinePrecision] + N[(N[(b * y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-324], t$95$1, If[LessEqual[t$95$1, 0.0], N[(N[(t / b), $MachinePrecision] * N[(x / y), $MachinePrecision] + N[(z / b), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 4e+293], t$95$1, N[(z / b), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -4.94066e-324 or -0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 3.9999999999999997e293

   1. Initial program 93.9%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   if -4.94066e-324 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -0.0

   1. Initial program 46.1%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 48.7

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 59.5

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 59.5

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

   4. Applied rewrites 59.5%

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

   5. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 54.3

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

   7. Applied rewrites 54.3%

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

   8. Taylor expanded in x around 0

      \[\leadsto \frac{z}{b} + \color{blue}{\frac{t \cdot x}{b \cdot y}} \]
   9. Step-by-step derivation

   10. Applied rewrites 72.4%

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

   if 3.9999999999999997e293 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t)))

   1. Initial program 20.0%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{z}{b}} \]
   4. Step-by-step derivation

      1. lower-/.f64 73.7

         \[\leadsto \color{blue}{\frac{z}{b}} \]

   5. Applied rewrites 73.7%

      \[\leadsto \color{blue}{\frac{z}{b}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{z \cdot y}{t} + x}{\left(1 + a\right) + \frac{b \cdot y}{t}} \leq -5 \cdot 10^{-324}:\\ \;\;\;\;\frac{\frac{z \cdot y}{t} + x}{\left(1 + a\right) + \frac{b \cdot y}{t}}\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\left(1 + a\right) + \frac{b \cdot y}{t}} \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{b}, \frac{x}{y}, \frac{z}{b}\right)\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\left(1 + a\right) + \frac{b \cdot y}{t}} \leq 4 \cdot 10^{+293}:\\ \;\;\;\;\frac{\frac{z \cdot y}{t} + x}{\left(1 + a\right) + \frac{b \cdot y}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 67.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{z \cdot y}{t} + x}{\left(1 + a\right) + \frac{b \cdot y}{t}}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+293}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(\frac{y}{t}, b, 1 + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (+ (/ (* z y) t) x) (+ (+ 1.0 a) (/ (* b y) t)))))
   (if (<= t_1 (- INFINITY))
     (/ z b)
     (if (<= t_1 4e+293) (/ x (fma (/ y t) b (+ 1.0 a))) (/ z b)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (((z * y) / t) + x) / ((1.0 + a) + ((b * y) / t));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = z / b;
	} else if (t_1 <= 4e+293) {
		tmp = x / fma((y / t), b, (1.0 + a));
	} else {
		tmp = z / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(Float64(z * y) / t) + x) / Float64(Float64(1.0 + a) + Float64(Float64(b * y) / t)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(z / b);
	elseif (t_1 <= 4e+293)
		tmp = Float64(x / fma(Float64(y / t), b, Float64(1.0 + a)));
	else
		tmp = Float64(z / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(N[(z * y), $MachinePrecision] / t), $MachinePrecision] + x), $MachinePrecision] / N[(N[(1.0 + a), $MachinePrecision] + N[(N[(b * y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(z / b), $MachinePrecision], If[LessEqual[t$95$1, 4e+293], N[(x / N[(N[(y / t), $MachinePrecision] * b + N[(1.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z / b), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -inf.0 or 3.9999999999999997e293 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t)))

   1. Initial program 28.1%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{z}{b}} \]
   4. Step-by-step derivation

      1. lower-/.f64 70.2

         \[\leadsto \color{blue}{\frac{z}{b}} \]

   5. Applied rewrites 70.2%

      \[\leadsto \color{blue}{\frac{z}{b}} \]

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 3.9999999999999997e293

   1. Initial program 89.6%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-+.f64 65.3

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

   5. Applied rewrites 65.3%

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(\frac{y}{t}, b, 1 + a\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{z \cdot y}{t} + x}{\left(1 + a\right) + \frac{b \cdot y}{t}} \leq -\infty:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\left(1 + a\right) + \frac{b \cdot y}{t}} \leq 4 \cdot 10^{+293}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(\frac{y}{t}, b, 1 + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 83.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\frac{z \cdot y}{t} + x}{\left(1 + a\right) + \frac{b \cdot y}{t}} \leq \infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{z}{t}, y, x\right)}{\mathsf{fma}\left(\frac{b}{t}, y, 1 + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (/ (+ (/ (* z y) t) x) (+ (+ 1.0 a) (/ (* b y) t))) INFINITY)
   (/ (fma (/ z t) y x) (fma (/ b t) y (+ 1.0 a)))
   (/ z b)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((((z * y) / t) + x) / ((1.0 + a) + ((b * y) / t))) <= ((double) INFINITY)) {
		tmp = fma((z / t), y, x) / fma((b / t), y, (1.0 + a));
	} else {
		tmp = z / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(z * y) / t) + x) / Float64(Float64(1.0 + a) + Float64(Float64(b * y) / t))) <= Inf)
		tmp = Float64(fma(Float64(z / t), y, x) / fma(Float64(b / t), y, Float64(1.0 + a)));
	else
		tmp = Float64(z / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(N[(N[(N[(z * y), $MachinePrecision] / t), $MachinePrecision] + x), $MachinePrecision] / N[(N[(1.0 + a), $MachinePrecision] + N[(N[(b * y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[(z / t), $MachinePrecision] * y + x), $MachinePrecision] / N[(N[(b / t), $MachinePrecision] * y + N[(1.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < +inf.0

   1. Initial program 81.4%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 83.2

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 82.7

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 82.7

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

   4. Applied rewrites 82.7%

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

   if +inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t)))

   1. Initial program 0.0%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{z}{b}} \]
   4. Step-by-step derivation

      1. lower-/.f64 100.0

         \[\leadsto \color{blue}{\frac{z}{b}} \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{z}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{z \cdot y}{t} + x}{\left(1 + a\right) + \frac{b \cdot y}{t}} \leq \infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{z}{t}, y, x\right)}{\mathsf{fma}\left(\frac{b}{t}, y, 1 + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 70.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{y}{t}, z, x\right)\\ \mathbf{if}\;a \leq -0.00038:\\ \;\;\;\;\frac{t\_1}{1 + a}\\ \mathbf{elif}\;a \leq 62:\\ \;\;\;\;\frac{t\_1}{\mathsf{fma}\left(\frac{y}{t}, b, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{z \cdot y}{t} + x}{1 + a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma (/ y t) z x)))
   (if (<= a -0.00038)
     (/ t_1 (+ 1.0 a))
     (if (<= a 62.0)
       (/ t_1 (fma (/ y t) b 1.0))
       (/ (+ (/ (* z y) t) x) (+ 1.0 a))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma((y / t), z, x);
	double tmp;
	if (a <= -0.00038) {
		tmp = t_1 / (1.0 + a);
	} else if (a <= 62.0) {
		tmp = t_1 / fma((y / t), b, 1.0);
	} else {
		tmp = (((z * y) / t) + x) / (1.0 + a);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(Float64(y / t), z, x)
	tmp = 0.0
	if (a <= -0.00038)
		tmp = Float64(t_1 / Float64(1.0 + a));
	elseif (a <= 62.0)
		tmp = Float64(t_1 / fma(Float64(y / t), b, 1.0));
	else
		tmp = Float64(Float64(Float64(Float64(z * y) / t) + x) / Float64(1.0 + a));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(y / t), $MachinePrecision] * z + x), $MachinePrecision]}, If[LessEqual[a, -0.00038], N[(t$95$1 / N[(1.0 + a), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 62.0], N[(t$95$1 / N[(N[(y / t), $MachinePrecision] * b + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(z * y), $MachinePrecision] / t), $MachinePrecision] + x), $MachinePrecision] / N[(1.0 + a), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < -3.8000000000000002e-4

   1. Initial program 72.6%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{x + \frac{y \cdot z}{t}}{1 + a}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x + \frac{y \cdot z}{t}}{1 + a}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z}{t} + x}}{1 + a} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{t} \cdot z} + x}{1 + a} \]

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 68.2

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

   5. Applied rewrites 68.2%

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

   if -3.8000000000000002e-4 < a < 62

   1. Initial program 75.7%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{x + \frac{y \cdot z}{t}}{1 + \frac{b \cdot y}{t}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x + \frac{y \cdot z}{t}}{1 + \frac{b \cdot y}{t}}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z}{t} + x}}{1 + \frac{b \cdot y}{t}} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{t} \cdot z} + x}{1 + \frac{b \cdot y}{t}} \]

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 79.7

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

   5. Applied rewrites 79.7%

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

   if 62 < a

   1. Initial program 75.0%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{x + \frac{y \cdot z}{t}}{\color{blue}{1 + a}} \]
   4. Step-by-step derivation

      1. lower-+.f64 64.2

         \[\leadsto \frac{x + \frac{y \cdot z}{t}}{\color{blue}{1 + a}} \]

   5. Applied rewrites 64.2%

      \[\leadsto \frac{x + \frac{y \cdot z}{t}}{\color{blue}{1 + a}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.00038:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{1 + a}\\ \mathbf{elif}\;a \leq 62:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{\mathsf{fma}\left(\frac{y}{t}, b, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{z \cdot y}{t} + x}{1 + a}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 70.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{t}{b}, \frac{x}{y}, \frac{z}{b}\right)\\ \mathbf{if}\;y \leq -3 \cdot 10^{+38}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{+27}:\\ \;\;\;\;\frac{\frac{z \cdot y}{t} + x}{1 + a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma (/ t b) (/ x y) (/ z b))))
   (if (<= y -3e+38)
     t_1
     (if (<= y 3.1e+27) (/ (+ (/ (* z y) t) x) (+ 1.0 a)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma((t / b), (x / y), (z / b));
	double tmp;
	if (y <= -3e+38) {
		tmp = t_1;
	} else if (y <= 3.1e+27) {
		tmp = (((z * y) / t) + x) / (1.0 + a);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(Float64(t / b), Float64(x / y), Float64(z / b))
	tmp = 0.0
	if (y <= -3e+38)
		tmp = t_1;
	elseif (y <= 3.1e+27)
		tmp = Float64(Float64(Float64(Float64(z * y) / t) + x) / Float64(1.0 + a));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t / b), $MachinePrecision] * N[(x / y), $MachinePrecision] + N[(z / b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -3e+38], t$95$1, If[LessEqual[y, 3.1e+27], N[(N[(N[(N[(z * y), $MachinePrecision] / t), $MachinePrecision] + x), $MachinePrecision] / N[(1.0 + a), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.0000000000000001e38 or 3.09999999999999996e27 < y

   1. Initial program 55.7%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 62.9

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 67.4

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 67.4

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

   4. Applied rewrites 67.4%

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

   5. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 39.0

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

   7. Applied rewrites 39.0%

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

   8. Taylor expanded in x around 0

      \[\leadsto \frac{z}{b} + \color{blue}{\frac{t \cdot x}{b \cdot y}} \]
   9. Step-by-step derivation

   10. Applied rewrites 64.8%

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

   if -3.0000000000000001e38 < y < 3.09999999999999996e27

   1. Initial program 93.7%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{x + \frac{y \cdot z}{t}}{\color{blue}{1 + a}} \]
   4. Step-by-step derivation

      1. lower-+.f64 79.8

         \[\leadsto \frac{x + \frac{y \cdot z}{t}}{\color{blue}{1 + a}} \]

   5. Applied rewrites 79.8%

      \[\leadsto \frac{x + \frac{y \cdot z}{t}}{\color{blue}{1 + a}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3 \cdot 10^{+38}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{b}, \frac{x}{y}, \frac{z}{b}\right)\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{+27}:\\ \;\;\;\;\frac{\frac{z \cdot y}{t} + x}{1 + a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{b}, \frac{x}{y}, \frac{z}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 56.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.5 \cdot 10^{+47}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{+27}:\\ \;\;\;\;\frac{1}{\frac{1 + a}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -3.5e+47)
   (/ z b)
   (if (<= y 1.4e+27) (/ 1.0 (/ (+ 1.0 a) x)) (/ z b))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -3.5e+47) {
		tmp = z / b;
	} else if (y <= 1.4e+27) {
		tmp = 1.0 / ((1.0 + a) / x);
	} else {
		tmp = z / b;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    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) :: tmp
    if (y <= (-3.5d+47)) then
        tmp = z / b
    else if (y <= 1.4d+27) then
        tmp = 1.0d0 / ((1.0d0 + a) / x)
    else
        tmp = z / b
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -3.5e+47) {
		tmp = z / b;
	} else if (y <= 1.4e+27) {
		tmp = 1.0 / ((1.0 + a) / x);
	} else {
		tmp = z / b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -3.5e+47:
		tmp = z / b
	elif y <= 1.4e+27:
		tmp = 1.0 / ((1.0 + a) / x)
	else:
		tmp = z / b
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -3.5e+47)
		tmp = Float64(z / b);
	elseif (y <= 1.4e+27)
		tmp = Float64(1.0 / Float64(Float64(1.0 + a) / x));
	else
		tmp = Float64(z / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -3.5e+47)
		tmp = z / b;
	elseif (y <= 1.4e+27)
		tmp = 1.0 / ((1.0 + a) / x);
	else
		tmp = z / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -3.5e+47], N[(z / b), $MachinePrecision], If[LessEqual[y, 1.4e+27], N[(1.0 / N[(N[(1.0 + a), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(z / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.50000000000000015e47 or 1.4e27 < y

   1. Initial program 54.7%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{z}{b}} \]
   4. Step-by-step derivation

      1. lower-/.f64 59.1

         \[\leadsto \color{blue}{\frac{z}{b}} \]

   5. Applied rewrites 59.1%

      \[\leadsto \color{blue}{\frac{z}{b}} \]

   if -3.50000000000000015e47 < y < 1.4e27

   1. Initial program 92.6%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 92.4

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 87.5

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 87.5

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lift-/.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. associate-/l* N/A

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

      21. *-commutative N/A

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

      22. lower-fma.f64 N/A

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

      23. lower-/.f64 84.6

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

   4. Applied rewrites 84.6%

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

   5. Taylor expanded in y around 0

      \[\leadsto \frac{1}{\color{blue}{\frac{1 + a}{x}}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1 + a}{x}}} \]

      2. lower-+.f64 62.7

         \[\leadsto \frac{1}{\frac{\color{blue}{1 + a}}{x}} \]

   7. Applied rewrites 62.7%

      \[\leadsto \frac{1}{\color{blue}{\frac{1 + a}{x}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 41.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.5 \cdot 10^{+47}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;y \leq -3.3 \cdot 10^{-252}:\\ \;\;\;\;\frac{x}{a}\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-111}:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -3.5e+47)
   (/ z b)
   (if (<= y -3.3e-252) (/ x a) (if (<= y 5e-111) (* 1.0 x) (/ z b)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -3.5e+47) {
		tmp = z / b;
	} else if (y <= -3.3e-252) {
		tmp = x / a;
	} else if (y <= 5e-111) {
		tmp = 1.0 * x;
	} else {
		tmp = z / b;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    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) :: tmp
    if (y <= (-3.5d+47)) then
        tmp = z / b
    else if (y <= (-3.3d-252)) then
        tmp = x / a
    else if (y <= 5d-111) then
        tmp = 1.0d0 * x
    else
        tmp = z / b
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -3.5e+47) {
		tmp = z / b;
	} else if (y <= -3.3e-252) {
		tmp = x / a;
	} else if (y <= 5e-111) {
		tmp = 1.0 * x;
	} else {
		tmp = z / b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -3.5e+47:
		tmp = z / b
	elif y <= -3.3e-252:
		tmp = x / a
	elif y <= 5e-111:
		tmp = 1.0 * x
	else:
		tmp = z / b
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -3.5e+47)
		tmp = Float64(z / b);
	elseif (y <= -3.3e-252)
		tmp = Float64(x / a);
	elseif (y <= 5e-111)
		tmp = Float64(1.0 * x);
	else
		tmp = Float64(z / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -3.5e+47)
		tmp = z / b;
	elseif (y <= -3.3e-252)
		tmp = x / a;
	elseif (y <= 5e-111)
		tmp = 1.0 * x;
	else
		tmp = z / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -3.5e+47], N[(z / b), $MachinePrecision], If[LessEqual[y, -3.3e-252], N[(x / a), $MachinePrecision], If[LessEqual[y, 5e-111], N[(1.0 * x), $MachinePrecision], N[(z / b), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.50000000000000015e47 or 5.0000000000000003e-111 < y

   1. Initial program 60.6%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{z}{b}} \]
   4. Step-by-step derivation

      1. lower-/.f64 53.5

         \[\leadsto \color{blue}{\frac{z}{b}} \]

   5. Applied rewrites 53.5%

      \[\leadsto \color{blue}{\frac{z}{b}} \]

   if -3.50000000000000015e47 < y < -3.30000000000000009e-252

   1. Initial program 89.6%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 88.2

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 84.0

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 84.0

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

   4. Applied rewrites 84.0%

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

   5. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\frac{x + \frac{y \cdot z}{t}}{a}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x + \frac{y \cdot z}{t}}{a}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z}{t} + x}}{a} \]

      3. associate-/l* N/A

         \[\leadsto \frac{\color{blue}{y \cdot \frac{z}{t}} + x}{a} \]

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 44.1

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

   7. Applied rewrites 44.1%

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

   8. Taylor expanded in x around inf

      \[\leadsto \frac{x}{\color{blue}{a}} \]
   9. Step-by-step derivation

   10. Applied rewrites 38.1%

       \[\leadsto \frac{x}{\color{blue}{a}} \]

   if -3.30000000000000009e-252 < y < 5.0000000000000003e-111

   1. Initial program 97.7%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 84.8%

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

   6. Taylor expanded in a around inf

      \[\leadsto \frac{\frac{x}{z} + \frac{y}{t}}{a} \cdot z \]
   7. Step-by-step derivation

   8. Applied rewrites 39.9%

      \[\leadsto \frac{\frac{y}{t} + \frac{x}{z}}{a} \cdot z \]

   9. Taylor expanded in y around 0

      \[\leadsto \frac{x \cdot z}{\color{blue}{z + a \cdot z}} \]
   10. Step-by-step derivation

   11. Applied rewrites 71.8%

       \[\leadsto x \cdot \color{blue}{\frac{z}{\mathsf{fma}\left(z, a, z\right)}} \]

   12. Taylor expanded in a around 0

       \[\leadsto x \cdot 1 \]
   13. Step-by-step derivation

   14. Applied rewrites 49.0%

       \[\leadsto x \cdot 1 \]
3. Recombined 3 regimes into one program.

4. Final simplification 48.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.5 \cdot 10^{+47}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;y \leq -3.3 \cdot 10^{-252}:\\ \;\;\;\;\frac{x}{a}\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-111}:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 56.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.5 \cdot 10^{+47}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{+27}:\\ \;\;\;\;\frac{x}{1 + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -3.5e+47) (/ z b) (if (<= y 1.4e+27) (/ x (+ 1.0 a)) (/ z b))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -3.5e+47) {
		tmp = z / b;
	} else if (y <= 1.4e+27) {
		tmp = x / (1.0 + a);
	} else {
		tmp = z / b;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    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) :: tmp
    if (y <= (-3.5d+47)) then
        tmp = z / b
    else if (y <= 1.4d+27) then
        tmp = x / (1.0d0 + a)
    else
        tmp = z / b
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -3.5e+47) {
		tmp = z / b;
	} else if (y <= 1.4e+27) {
		tmp = x / (1.0 + a);
	} else {
		tmp = z / b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -3.5e+47:
		tmp = z / b
	elif y <= 1.4e+27:
		tmp = x / (1.0 + a)
	else:
		tmp = z / b
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -3.5e+47)
		tmp = Float64(z / b);
	elseif (y <= 1.4e+27)
		tmp = Float64(x / Float64(1.0 + a));
	else
		tmp = Float64(z / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -3.5e+47)
		tmp = z / b;
	elseif (y <= 1.4e+27)
		tmp = x / (1.0 + a);
	else
		tmp = z / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -3.5e+47], N[(z / b), $MachinePrecision], If[LessEqual[y, 1.4e+27], N[(x / N[(1.0 + a), $MachinePrecision]), $MachinePrecision], N[(z / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.50000000000000015e47 or 1.4e27 < y

   1. Initial program 54.7%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{z}{b}} \]
   4. Step-by-step derivation

      1. lower-/.f64 59.1

         \[\leadsto \color{blue}{\frac{z}{b}} \]

   5. Applied rewrites 59.1%

      \[\leadsto \color{blue}{\frac{z}{b}} \]

   if -3.50000000000000015e47 < y < 1.4e27

   1. Initial program 92.6%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x}{1 + a}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{1 + a}} \]

      2. lower-+.f64 62.6

         \[\leadsto \frac{x}{\color{blue}{1 + a}} \]

   5. Applied rewrites 62.6%

      \[\leadsto \color{blue}{\frac{x}{1 + a}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 40.6% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9 \cdot 10^{+39}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-111}:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -9e+39) (/ z b) (if (<= y 5e-111) (* 1.0 x) (/ z b))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -9e+39) {
		tmp = z / b;
	} else if (y <= 5e-111) {
		tmp = 1.0 * x;
	} else {
		tmp = z / b;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    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) :: tmp
    if (y <= (-9d+39)) then
        tmp = z / b
    else if (y <= 5d-111) then
        tmp = 1.0d0 * x
    else
        tmp = z / b
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -9e+39) {
		tmp = z / b;
	} else if (y <= 5e-111) {
		tmp = 1.0 * x;
	} else {
		tmp = z / b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -9e+39:
		tmp = z / b
	elif y <= 5e-111:
		tmp = 1.0 * x
	else:
		tmp = z / b
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -9e+39)
		tmp = Float64(z / b);
	elseif (y <= 5e-111)
		tmp = Float64(1.0 * x);
	else
		tmp = Float64(z / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -9e+39)
		tmp = z / b;
	elseif (y <= 5e-111)
		tmp = 1.0 * x;
	else
		tmp = z / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -9e+39], N[(z / b), $MachinePrecision], If[LessEqual[y, 5e-111], N[(1.0 * x), $MachinePrecision], N[(z / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -8.99999999999999991e39 or 5.0000000000000003e-111 < y

   1. Initial program 60.6%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{z}{b}} \]
   4. Step-by-step derivation

      1. lower-/.f64 53.1

         \[\leadsto \color{blue}{\frac{z}{b}} \]

   5. Applied rewrites 53.1%

      \[\leadsto \color{blue}{\frac{z}{b}} \]

   if -8.99999999999999991e39 < y < 5.0000000000000003e-111

   1. Initial program 94.2%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 85.2%

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

   6. Taylor expanded in a around inf

      \[\leadsto \frac{\frac{x}{z} + \frac{y}{t}}{a} \cdot z \]
   7. Step-by-step derivation

   8. Applied rewrites 37.9%

      \[\leadsto \frac{\frac{y}{t} + \frac{x}{z}}{a} \cdot z \]

   9. Taylor expanded in y around 0

      \[\leadsto \frac{x \cdot z}{\color{blue}{z + a \cdot z}} \]
   10. Step-by-step derivation

   11. Applied rewrites 64.5%

       \[\leadsto x \cdot \color{blue}{\frac{z}{\mathsf{fma}\left(z, a, z\right)}} \]

   12. Taylor expanded in a around 0

       \[\leadsto x \cdot 1 \]
   13. Step-by-step derivation

   14. Applied rewrites 35.8%

       \[\leadsto x \cdot 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 45.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9 \cdot 10^{+39}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-111}:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 19.6% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ 1 \cdot x \end{array} \]

FPCore

(FPCore (x y z t a b) :precision binary64 (* 1.0 x))

C

double code(double x, double y, double z, double t, double a, double b) {
	return 1.0 * x;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    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
    code = 1.0d0 * x
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return 1.0 * x;
}

Python

def code(x, y, z, t, a, b):
	return 1.0 * x

Julia

function code(x, y, z, t, a, b)
	return Float64(1.0 * x)
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = 1.0 * x;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(1.0 * x), $MachinePrecision]

Derivation

1. Initial program 74.7%

   \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
2. Add Preprocessing

3. Taylor expanded in z around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 70.9%

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

6. Taylor expanded in a around inf

   \[\leadsto \frac{\frac{x}{z} + \frac{y}{t}}{a} \cdot z \]
7. Step-by-step derivation

8. Applied rewrites 29.8%

   \[\leadsto \frac{\frac{y}{t} + \frac{x}{z}}{a} \cdot z \]

9. Taylor expanded in y around 0

   \[\leadsto \frac{x \cdot z}{\color{blue}{z + a \cdot z}} \]
10. Step-by-step derivation

11. Applied rewrites 38.5%

    \[\leadsto x \cdot \color{blue}{\frac{z}{\mathsf{fma}\left(z, a, z\right)}} \]

12. Taylor expanded in a around 0

    \[\leadsto x \cdot 1 \]
13. Step-by-step derivation

14. Applied rewrites 19.6%

    \[\leadsto x \cdot 1 \]

15. Final simplification 19.6%

    \[\leadsto 1 \cdot x \]
16. Add Preprocessing

Developer Target 1: 79.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \mathbf{if}\;t < -1.3659085366310088 \cdot 10^{-271}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t < 3.036967103737246 \cdot 10^{-130}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1
         (* 1.0 (* (+ x (* (/ y t) z)) (/ 1.0 (+ (+ a 1.0) (* (/ y t) b)))))))
   (if (< t -1.3659085366310088e-271)
     t_1
     (if (< t 3.036967103737246e-130) (/ z b) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = 1.0 * ((x + ((y / t) * z)) * (1.0 / ((a + 1.0) + ((y / t) * b))));
	double tmp;
	if (t < -1.3659085366310088e-271) {
		tmp = t_1;
	} else if (t < 3.036967103737246e-130) {
		tmp = z / b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    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) :: t_1
    real(8) :: tmp
    t_1 = 1.0d0 * ((x + ((y / t) * z)) * (1.0d0 / ((a + 1.0d0) + ((y / t) * b))))
    if (t < (-1.3659085366310088d-271)) then
        tmp = t_1
    else if (t < 3.036967103737246d-130) then
        tmp = z / b
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = 1.0 * ((x + ((y / t) * z)) * (1.0 / ((a + 1.0) + ((y / t) * b))));
	double tmp;
	if (t < -1.3659085366310088e-271) {
		tmp = t_1;
	} else if (t < 3.036967103737246e-130) {
		tmp = z / b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = 1.0 * ((x + ((y / t) * z)) * (1.0 / ((a + 1.0) + ((y / t) * b))))
	tmp = 0
	if t < -1.3659085366310088e-271:
		tmp = t_1
	elif t < 3.036967103737246e-130:
		tmp = z / b
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(1.0 * Float64(Float64(x + Float64(Float64(y / t) * z)) * Float64(1.0 / Float64(Float64(a + 1.0) + Float64(Float64(y / t) * b)))))
	tmp = 0.0
	if (t < -1.3659085366310088e-271)
		tmp = t_1;
	elseif (t < 3.036967103737246e-130)
		tmp = Float64(z / b);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = 1.0 * ((x + ((y / t) * z)) * (1.0 / ((a + 1.0) + ((y / t) * b))));
	tmp = 0.0;
	if (t < -1.3659085366310088e-271)
		tmp = t_1;
	elseif (t < 3.036967103737246e-130)
		tmp = z / b;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(1.0 * N[(N[(x + N[(N[(y / t), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(N[(a + 1.0), $MachinePrecision] + N[(N[(y / t), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Less[t, -1.3659085366310088e-271], t$95$1, If[Less[t, 3.036967103737246e-130], N[(z / b), $MachinePrecision], t$95$1]]]

Reproduce

herbie shell --seed 2024276 
(FPCore (x y z t a b)
  :name "Diagrams.Solve.Tridiagonal:solveCyclicTriDiagonal from diagrams-solve-0.1, B"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< t -1707385670788761/12500000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (* 1 (* (+ x (* (/ y t) z)) (/ 1 (+ (+ a 1) (* (/ y t) b))))) (if (< t 1518483551868623/5000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (/ z b) (* 1 (* (+ x (* (/ y t) z)) (/ 1 (+ (+ a 1) (* (/ y t) b))))))))

  (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t))))