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

Percentage Accurate: 74.5% → 92.7%

Time: 13.9s

Alternatives: 18

Speedup: 0.3×

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: 74.5% 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: 92.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)}\\ t_2 := \mathsf{fma}\left(y, \frac{b}{t}, a + 1\right)\\ t_3 := t \cdot \left(\frac{x}{b} - \frac{\mathsf{fma}\left(z, a, z\right)}{b \cdot b}\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+242}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t\_2}, \frac{y}{t}, \frac{x}{t\_2}\right)\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{-303}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{z}{b} + \frac{\mathsf{fma}\left(\left(-1 - a\right) \cdot t\_3, \frac{t}{y \cdot b}, t\_3\right)}{y}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+272}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(y, z \cdot \frac{1}{\mathsf{fma}\left(y \cdot b, 1, \mathsf{fma}\left(t, a, t\right)\right)}, \frac{x}{1 + \mathsf{fma}\left(y, \frac{b}{t}, a\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (* y z) t)) (+ (/ (* y b) t) (+ a 1.0))))
        (t_2 (fma y (/ b t) (+ a 1.0)))
        (t_3 (* t (- (/ x b) (/ (fma z a z) (* b b))))))
   (if (<= t_1 -2e+242)
     (fma (/ z t_2) (/ y t) (/ x t_2))
     (if (<= t_1 -1e-303)
       t_1
       (if (<= t_1 0.0)
         (+ (/ z b) (/ (fma (* (- -1.0 a) t_3) (/ t (* y b)) t_3) y))
         (if (<= t_1 2e+272)
           t_1
           (if (<= t_1 INFINITY)
             (fma
              y
              (* z (/ 1.0 (fma (* y b) 1.0 (fma t a t))))
              (/ x (+ 1.0 (fma y (/ b t) a))))
             (/ z b))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x + ((y * z) / t)) / (((y * b) / t) + (a + 1.0));
	double t_2 = fma(y, (b / t), (a + 1.0));
	double t_3 = t * ((x / b) - (fma(z, a, z) / (b * b)));
	double tmp;
	if (t_1 <= -2e+242) {
		tmp = fma((z / t_2), (y / t), (x / t_2));
	} else if (t_1 <= -1e-303) {
		tmp = t_1;
	} else if (t_1 <= 0.0) {
		tmp = (z / b) + (fma(((-1.0 - a) * t_3), (t / (y * b)), t_3) / y);
	} else if (t_1 <= 2e+272) {
		tmp = t_1;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = fma(y, (z * (1.0 / fma((y * b), 1.0, fma(t, a, t)))), (x / (1.0 + fma(y, (b / t), a))));
	} else {
		tmp = z / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x + Float64(Float64(y * z) / t)) / Float64(Float64(Float64(y * b) / t) + Float64(a + 1.0)))
	t_2 = fma(y, Float64(b / t), Float64(a + 1.0))
	t_3 = Float64(t * Float64(Float64(x / b) - Float64(fma(z, a, z) / Float64(b * b))))
	tmp = 0.0
	if (t_1 <= -2e+242)
		tmp = fma(Float64(z / t_2), Float64(y / t), Float64(x / t_2));
	elseif (t_1 <= -1e-303)
		tmp = t_1;
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(z / b) + Float64(fma(Float64(Float64(-1.0 - a) * t_3), Float64(t / Float64(y * b)), t_3) / y));
	elseif (t_1 <= 2e+272)
		tmp = t_1;
	elseif (t_1 <= Inf)
		tmp = fma(y, Float64(z * Float64(1.0 / fma(Float64(y * b), 1.0, fma(t, a, t)))), Float64(x / Float64(1.0 + fma(y, Float64(b / t), a))));
	else
		tmp = Float64(z / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(y * b), $MachinePrecision] / t), $MachinePrecision] + N[(a + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(y * N[(b / t), $MachinePrecision] + N[(a + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t * N[(N[(x / b), $MachinePrecision] - N[(N[(z * a + z), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+242], N[(N[(z / t$95$2), $MachinePrecision] * N[(y / t), $MachinePrecision] + N[(x / t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -1e-303], t$95$1, If[LessEqual[t$95$1, 0.0], N[(N[(z / b), $MachinePrecision] + N[(N[(N[(N[(-1.0 - a), $MachinePrecision] * t$95$3), $MachinePrecision] * N[(t / N[(y * b), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+272], t$95$1, If[LessEqual[t$95$1, Infinity], N[(y * N[(z * N[(1.0 / N[(N[(y * b), $MachinePrecision] * 1.0 + N[(t * a + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x / N[(1.0 + N[(y * N[(b / t), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]), $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))) < -2.0000000000000001e242

   1. Initial program 46.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-rgt-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-+.f64 N/A

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

   5. Applied rewrites 90.4%

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

   7. Applied rewrites 90.9%

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

   if -2.0000000000000001e242 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -9.99999999999999931e-304 or -0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 2.0000000000000001e272

   1. Initial program 98.5%

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

   if -9.99999999999999931e-304 < (/.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 55.5%

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

   3. Taylor expanded in y around -inf

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

   5. Applied rewrites 81.7%

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

   if 2.0000000000000001e272 < (/.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 46.5%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-rgt-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-+.f64 N/A

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

   5. Applied rewrites 99.5%

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

   7. Applied rewrites 99.8%

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

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

   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 96.7

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

   5. Applied rewrites 96.7%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq -2 \cdot 10^{+242}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{\mathsf{fma}\left(y, \frac{b}{t}, a + 1\right)}, \frac{y}{t}, \frac{x}{\mathsf{fma}\left(y, \frac{b}{t}, a + 1\right)}\right)\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq -1 \cdot 10^{-303}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq 0:\\ \;\;\;\;\frac{z}{b} + \frac{\mathsf{fma}\left(\left(-1 - a\right) \cdot \left(t \cdot \left(\frac{x}{b} - \frac{\mathsf{fma}\left(z, a, z\right)}{b \cdot b}\right)\right), \frac{t}{y \cdot b}, t \cdot \left(\frac{x}{b} - \frac{\mathsf{fma}\left(z, a, z\right)}{b \cdot b}\right)\right)}{y}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq 2 \cdot 10^{+272}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(y, z \cdot \frac{1}{\mathsf{fma}\left(y \cdot b, 1, \mathsf{fma}\left(t, a, t\right)\right)}, \frac{x}{1 + \mathsf{fma}\left(y, \frac{b}{t}, a\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 77.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{y \cdot z}{t}\\ t_2 := \frac{t\_1}{\frac{y \cdot b}{t} + \left(a + 1\right)}\\ t_3 := \mathsf{fma}\left(y, b, \mathsf{fma}\left(t, a, t\right)\right)\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;z \cdot \frac{y}{t\_3}\\ \mathbf{elif}\;t\_2 \leq -1 \cdot 10^{-303}:\\ \;\;\;\;\frac{t\_1}{a + 1}\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{elif}\;t\_2 \leq 10^{+107}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{a + 1}\\ \mathbf{elif}\;t\_2 \leq 10^{+295}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}{\mathsf{fma}\left(y, \frac{b}{t}, 1\right)}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;y \cdot \frac{z}{t\_3}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ x (/ (* y z) t)))
        (t_2 (/ t_1 (+ (/ (* y b) t) (+ a 1.0))))
        (t_3 (fma y b (fma t a t))))
   (if (<= t_2 (- INFINITY))
     (* z (/ y t_3))
     (if (<= t_2 -1e-303)
       (/ t_1 (+ a 1.0))
       (if (<= t_2 0.0)
         (/ (fma t (/ x y) z) b)
         (if (<= t_2 1e+107)
           (/ (fma y (/ z t) x) (+ a 1.0))
           (if (<= t_2 1e+295)
             (/ (fma z (/ y t) x) (fma y (/ b t) 1.0))
             (if (<= t_2 INFINITY) (* y (/ z t_3)) (/ z b)))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x + ((y * z) / t);
	double t_2 = t_1 / (((y * b) / t) + (a + 1.0));
	double t_3 = fma(y, b, fma(t, a, t));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = z * (y / t_3);
	} else if (t_2 <= -1e-303) {
		tmp = t_1 / (a + 1.0);
	} else if (t_2 <= 0.0) {
		tmp = fma(t, (x / y), z) / b;
	} else if (t_2 <= 1e+107) {
		tmp = fma(y, (z / t), x) / (a + 1.0);
	} else if (t_2 <= 1e+295) {
		tmp = fma(z, (y / t), x) / fma(y, (b / t), 1.0);
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = y * (z / t_3);
	} else {
		tmp = z / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x + Float64(Float64(y * z) / t))
	t_2 = Float64(t_1 / Float64(Float64(Float64(y * b) / t) + Float64(a + 1.0)))
	t_3 = fma(y, b, fma(t, a, t))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(z * Float64(y / t_3));
	elseif (t_2 <= -1e-303)
		tmp = Float64(t_1 / Float64(a + 1.0));
	elseif (t_2 <= 0.0)
		tmp = Float64(fma(t, Float64(x / y), z) / b);
	elseif (t_2 <= 1e+107)
		tmp = Float64(fma(y, Float64(z / t), x) / Float64(a + 1.0));
	elseif (t_2 <= 1e+295)
		tmp = Float64(fma(z, Float64(y / t), x) / fma(y, Float64(b / t), 1.0));
	elseif (t_2 <= Inf)
		tmp = Float64(y * Float64(z / t_3));
	else
		tmp = Float64(z / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / N[(N[(N[(y * b), $MachinePrecision] / t), $MachinePrecision] + N[(a + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(y * b + N[(t * a + t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(z * N[(y / t$95$3), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -1e-303], N[(t$95$1 / N[(a + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.0], N[(N[(t * N[(x / y), $MachinePrecision] + z), $MachinePrecision] / b), $MachinePrecision], If[LessEqual[t$95$2, 1e+107], N[(N[(y * N[(z / t), $MachinePrecision] + x), $MachinePrecision] / N[(a + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e+295], N[(N[(z * N[(y / t), $MachinePrecision] + x), $MachinePrecision] / N[(y * N[(b / t), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(y * N[(z / t$95$3), $MachinePrecision]), $MachinePrecision], N[(z / b), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 7 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 24.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-rgt-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-+.f64 N/A

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

   5. Applied rewrites 86.6%

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

   7. Applied rewrites 86.6%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 61.5%

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

   12. Applied rewrites 86.5%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -9.99999999999999931e-304

   1. Initial program 98.5%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   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 76.2

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

   5. Applied rewrites 76.2%

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

   if -9.99999999999999931e-304 < (/.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 55.5%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-rgt-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-+.f64 N/A

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

   5. Applied rewrites 63.1%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 79.6%

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

   if -0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 9.9999999999999997e106

   1. Initial program 97.9%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   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. clear-num N/A

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

      5. inv-pow N/A

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

      6. div-inv N/A

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

      7. unpow-prod-down N/A

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

      8. inv-pow N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-/.f64 97.9

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

   4. Applied rewrites 97.9%

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

   5. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{x + \frac{y \cdot z}{t}}{1 + a}} \]
   6. 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

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 77.3

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

   7. Applied rewrites 77.3%

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

   if 9.9999999999999997e106 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 9.9999999999999998e294

   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 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. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 82.5

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

   5. Applied rewrites 82.5%

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

   if 9.9999999999999998e294 < (/.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 37.5%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-rgt-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-+.f64 N/A

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

   5. Applied rewrites 99.4%

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

   7. Applied rewrites 99.7%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 53.1%

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

   12. Applied rewrites 99.5%

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

   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 96.7

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

   5. Applied rewrites 96.7%

      \[\leadsto \color{blue}{\frac{z}{b}} \]
3. Recombined 7 regimes into one program.

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq -\infty:\\ \;\;\;\;z \cdot \frac{y}{\mathsf{fma}\left(y, b, \mathsf{fma}\left(t, a, t\right)\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq -1 \cdot 10^{-303}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{a + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq 10^{+107}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{a + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq 10^{+295}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}{\mathsf{fma}\left(y, \frac{b}{t}, 1\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq \infty:\\ \;\;\;\;y \cdot \frac{z}{\mathsf{fma}\left(y, b, \mathsf{fma}\left(t, a, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)}\\ t_2 := \mathsf{fma}\left(y, \frac{b}{t}, a + 1\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+242}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t\_2}, \frac{y}{t}, \frac{x}{t\_2}\right)\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{-303}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+272}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(y, z \cdot \frac{1}{\mathsf{fma}\left(y \cdot b, 1, \mathsf{fma}\left(t, a, t\right)\right)}, \frac{x}{1 + \mathsf{fma}\left(y, \frac{b}{t}, a\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (* y z) t)) (+ (/ (* y b) t) (+ a 1.0))))
        (t_2 (fma y (/ b t) (+ a 1.0))))
   (if (<= t_1 -2e+242)
     (fma (/ z t_2) (/ y t) (/ x t_2))
     (if (<= t_1 -1e-303)
       t_1
       (if (<= t_1 0.0)
         (/ (fma t (/ x y) z) b)
         (if (<= t_1 2e+272)
           t_1
           (if (<= t_1 INFINITY)
             (fma
              y
              (* z (/ 1.0 (fma (* y b) 1.0 (fma t a t))))
              (/ x (+ 1.0 (fma y (/ b t) a))))
             (/ z b))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x + ((y * z) / t)) / (((y * b) / t) + (a + 1.0));
	double t_2 = fma(y, (b / t), (a + 1.0));
	double tmp;
	if (t_1 <= -2e+242) {
		tmp = fma((z / t_2), (y / t), (x / t_2));
	} else if (t_1 <= -1e-303) {
		tmp = t_1;
	} else if (t_1 <= 0.0) {
		tmp = fma(t, (x / y), z) / b;
	} else if (t_1 <= 2e+272) {
		tmp = t_1;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = fma(y, (z * (1.0 / fma((y * b), 1.0, fma(t, a, t)))), (x / (1.0 + fma(y, (b / t), a))));
	} else {
		tmp = z / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x + Float64(Float64(y * z) / t)) / Float64(Float64(Float64(y * b) / t) + Float64(a + 1.0)))
	t_2 = fma(y, Float64(b / t), Float64(a + 1.0))
	tmp = 0.0
	if (t_1 <= -2e+242)
		tmp = fma(Float64(z / t_2), Float64(y / t), Float64(x / t_2));
	elseif (t_1 <= -1e-303)
		tmp = t_1;
	elseif (t_1 <= 0.0)
		tmp = Float64(fma(t, Float64(x / y), z) / b);
	elseif (t_1 <= 2e+272)
		tmp = t_1;
	elseif (t_1 <= Inf)
		tmp = fma(y, Float64(z * Float64(1.0 / fma(Float64(y * b), 1.0, fma(t, a, t)))), Float64(x / Float64(1.0 + fma(y, Float64(b / t), a))));
	else
		tmp = Float64(z / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(y * b), $MachinePrecision] / t), $MachinePrecision] + N[(a + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(y * N[(b / t), $MachinePrecision] + N[(a + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+242], N[(N[(z / t$95$2), $MachinePrecision] * N[(y / t), $MachinePrecision] + N[(x / t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -1e-303], t$95$1, If[LessEqual[t$95$1, 0.0], N[(N[(t * N[(x / y), $MachinePrecision] + z), $MachinePrecision] / b), $MachinePrecision], If[LessEqual[t$95$1, 2e+272], t$95$1, If[LessEqual[t$95$1, Infinity], N[(y * N[(z * N[(1.0 / N[(N[(y * b), $MachinePrecision] * 1.0 + N[(t * a + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x / N[(1.0 + N[(y * N[(b / t), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]), $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))) < -2.0000000000000001e242

   1. Initial program 46.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-rgt-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-+.f64 N/A

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

   5. Applied rewrites 90.4%

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

   7. Applied rewrites 90.9%

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

   if -2.0000000000000001e242 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -9.99999999999999931e-304 or -0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 2.0000000000000001e272

   1. Initial program 98.5%

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

   if -9.99999999999999931e-304 < (/.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 55.5%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-rgt-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-+.f64 N/A

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

   5. Applied rewrites 63.1%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 79.6%

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

   if 2.0000000000000001e272 < (/.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 46.5%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-rgt-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-+.f64 N/A

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

   5. Applied rewrites 99.5%

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

   7. Applied rewrites 99.8%

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

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

   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 96.7

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

   5. Applied rewrites 96.7%

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

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq -2 \cdot 10^{+242}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{\mathsf{fma}\left(y, \frac{b}{t}, a + 1\right)}, \frac{y}{t}, \frac{x}{\mathsf{fma}\left(y, \frac{b}{t}, a + 1\right)}\right)\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq -1 \cdot 10^{-303}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq 2 \cdot 10^{+272}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(y, z \cdot \frac{1}{\mathsf{fma}\left(y \cdot b, 1, \mathsf{fma}\left(t, a, t\right)\right)}, \frac{x}{1 + \mathsf{fma}\left(y, \frac{b}{t}, a\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)}\\ t_2 := \mathsf{fma}\left(y, \frac{b}{t}, a\right)\\ t_3 := \frac{x}{1 + t\_2}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+242}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{\mathsf{fma}\left(t, t\_2, t\right)}, t\_3\right)\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{-303}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+272}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(y, z \cdot \frac{1}{\mathsf{fma}\left(y \cdot b, 1, \mathsf{fma}\left(t, a, t\right)\right)}, t\_3\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (* y z) t)) (+ (/ (* y b) t) (+ a 1.0))))
        (t_2 (fma y (/ b t) a))
        (t_3 (/ x (+ 1.0 t_2))))
   (if (<= t_1 -2e+242)
     (fma y (/ z (fma t t_2 t)) t_3)
     (if (<= t_1 -1e-303)
       t_1
       (if (<= t_1 0.0)
         (/ (fma t (/ x y) z) b)
         (if (<= t_1 2e+272)
           t_1
           (if (<= t_1 INFINITY)
             (fma y (* z (/ 1.0 (fma (* y b) 1.0 (fma t a t)))) t_3)
             (/ z b))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x + ((y * z) / t)) / (((y * b) / t) + (a + 1.0));
	double t_2 = fma(y, (b / t), a);
	double t_3 = x / (1.0 + t_2);
	double tmp;
	if (t_1 <= -2e+242) {
		tmp = fma(y, (z / fma(t, t_2, t)), t_3);
	} else if (t_1 <= -1e-303) {
		tmp = t_1;
	} else if (t_1 <= 0.0) {
		tmp = fma(t, (x / y), z) / b;
	} else if (t_1 <= 2e+272) {
		tmp = t_1;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = fma(y, (z * (1.0 / fma((y * b), 1.0, fma(t, a, t)))), t_3);
	} else {
		tmp = z / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x + Float64(Float64(y * z) / t)) / Float64(Float64(Float64(y * b) / t) + Float64(a + 1.0)))
	t_2 = fma(y, Float64(b / t), a)
	t_3 = Float64(x / Float64(1.0 + t_2))
	tmp = 0.0
	if (t_1 <= -2e+242)
		tmp = fma(y, Float64(z / fma(t, t_2, t)), t_3);
	elseif (t_1 <= -1e-303)
		tmp = t_1;
	elseif (t_1 <= 0.0)
		tmp = Float64(fma(t, Float64(x / y), z) / b);
	elseif (t_1 <= 2e+272)
		tmp = t_1;
	elseif (t_1 <= Inf)
		tmp = fma(y, Float64(z * Float64(1.0 / fma(Float64(y * b), 1.0, fma(t, a, t)))), t_3);
	else
		tmp = Float64(z / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(y * b), $MachinePrecision] / t), $MachinePrecision] + N[(a + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(y * N[(b / t), $MachinePrecision] + a), $MachinePrecision]}, Block[{t$95$3 = N[(x / N[(1.0 + t$95$2), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+242], N[(y * N[(z / N[(t * t$95$2 + t), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision], If[LessEqual[t$95$1, -1e-303], t$95$1, If[LessEqual[t$95$1, 0.0], N[(N[(t * N[(x / y), $MachinePrecision] + z), $MachinePrecision] / b), $MachinePrecision], If[LessEqual[t$95$1, 2e+272], t$95$1, If[LessEqual[t$95$1, Infinity], N[(y * N[(z * N[(1.0 / N[(N[(y * b), $MachinePrecision] * 1.0 + N[(t * a + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$3), $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))) < -2.0000000000000001e242

   1. Initial program 46.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-rgt-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-+.f64 N/A

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

   5. Applied rewrites 90.4%

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

   if -2.0000000000000001e242 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -9.99999999999999931e-304 or -0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 2.0000000000000001e272

   1. Initial program 98.5%

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

   if -9.99999999999999931e-304 < (/.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 55.5%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-rgt-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-+.f64 N/A

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

   5. Applied rewrites 63.1%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 79.6%

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

   if 2.0000000000000001e272 < (/.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 46.5%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-rgt-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-+.f64 N/A

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

   5. Applied rewrites 99.5%

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

   7. Applied rewrites 99.8%

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

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

   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 96.7

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

   5. Applied rewrites 96.7%

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

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq -2 \cdot 10^{+242}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{\mathsf{fma}\left(t, \mathsf{fma}\left(y, \frac{b}{t}, a\right), t\right)}, \frac{x}{1 + \mathsf{fma}\left(y, \frac{b}{t}, a\right)}\right)\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq -1 \cdot 10^{-303}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq 2 \cdot 10^{+272}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(y, z \cdot \frac{1}{\mathsf{fma}\left(y \cdot b, 1, \mathsf{fma}\left(t, a, t\right)\right)}, \frac{x}{1 + \mathsf{fma}\left(y, \frac{b}{t}, a\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)}\\ t_2 := \mathsf{fma}\left(y, z \cdot \frac{1}{\mathsf{fma}\left(y \cdot b, 1, \mathsf{fma}\left(t, a, t\right)\right)}, \frac{x}{1 + \mathsf{fma}\left(y, \frac{b}{t}, a\right)}\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+242}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{-303}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+272}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (* y z) t)) (+ (/ (* y b) t) (+ a 1.0))))
        (t_2
         (fma
          y
          (* z (/ 1.0 (fma (* y b) 1.0 (fma t a t))))
          (/ x (+ 1.0 (fma y (/ b t) a))))))
   (if (<= t_1 -2e+242)
     t_2
     (if (<= t_1 -1e-303)
       t_1
       (if (<= t_1 0.0)
         (/ (fma t (/ x y) z) b)
         (if (<= t_1 2e+272) t_1 (if (<= t_1 INFINITY) t_2 (/ z b))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x + ((y * z) / t)) / (((y * b) / t) + (a + 1.0));
	double t_2 = fma(y, (z * (1.0 / fma((y * b), 1.0, fma(t, a, t)))), (x / (1.0 + fma(y, (b / t), a))));
	double tmp;
	if (t_1 <= -2e+242) {
		tmp = t_2;
	} else if (t_1 <= -1e-303) {
		tmp = t_1;
	} else if (t_1 <= 0.0) {
		tmp = fma(t, (x / y), z) / b;
	} else if (t_1 <= 2e+272) {
		tmp = t_1;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_2;
	} else {
		tmp = z / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x + Float64(Float64(y * z) / t)) / Float64(Float64(Float64(y * b) / t) + Float64(a + 1.0)))
	t_2 = fma(y, Float64(z * Float64(1.0 / fma(Float64(y * b), 1.0, fma(t, a, t)))), Float64(x / Float64(1.0 + fma(y, Float64(b / t), a))))
	tmp = 0.0
	if (t_1 <= -2e+242)
		tmp = t_2;
	elseif (t_1 <= -1e-303)
		tmp = t_1;
	elseif (t_1 <= 0.0)
		tmp = Float64(fma(t, Float64(x / y), z) / b);
	elseif (t_1 <= 2e+272)
		tmp = t_1;
	elseif (t_1 <= Inf)
		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[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(y * b), $MachinePrecision] / t), $MachinePrecision] + N[(a + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(y * N[(z * N[(1.0 / N[(N[(y * b), $MachinePrecision] * 1.0 + N[(t * a + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x / N[(1.0 + N[(y * N[(b / t), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+242], t$95$2, If[LessEqual[t$95$1, -1e-303], t$95$1, If[LessEqual[t$95$1, 0.0], N[(N[(t * N[(x / y), $MachinePrecision] + z), $MachinePrecision] / b), $MachinePrecision], If[LessEqual[t$95$1, 2e+272], t$95$1, If[LessEqual[t$95$1, Infinity], t$95$2, 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))) < -2.0000000000000001e242 or 2.0000000000000001e272 < (/.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 46.2%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-rgt-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-+.f64 N/A

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

   5. Applied rewrites 94.1%

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

   7. Applied rewrites 94.2%

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

   if -2.0000000000000001e242 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -9.99999999999999931e-304 or -0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 2.0000000000000001e272

   1. Initial program 98.5%

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

   if -9.99999999999999931e-304 < (/.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 55.5%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-rgt-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-+.f64 N/A

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

   5. Applied rewrites 63.1%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 79.6%

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

   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 96.7

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

   5. Applied rewrites 96.7%

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

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq -2 \cdot 10^{+242}:\\ \;\;\;\;\mathsf{fma}\left(y, z \cdot \frac{1}{\mathsf{fma}\left(y \cdot b, 1, \mathsf{fma}\left(t, a, t\right)\right)}, \frac{x}{1 + \mathsf{fma}\left(y, \frac{b}{t}, a\right)}\right)\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq -1 \cdot 10^{-303}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq 2 \cdot 10^{+272}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(y, z \cdot \frac{1}{\mathsf{fma}\left(y \cdot b, 1, \mathsf{fma}\left(t, a, t\right)\right)}, \frac{x}{1 + \mathsf{fma}\left(y, \frac{b}{t}, a\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 92.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)}\\ t_2 := \mathsf{fma}\left(y, b, \mathsf{fma}\left(t, a, t\right)\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;z \cdot \frac{y}{t\_2}\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{-303}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{elif}\;t\_1 \leq 10^{+295}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;y \cdot \frac{z}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (* y z) t)) (+ (/ (* y b) t) (+ a 1.0))))
        (t_2 (fma y b (fma t a t))))
   (if (<= t_1 (- INFINITY))
     (* z (/ y t_2))
     (if (<= t_1 -1e-303)
       t_1
       (if (<= t_1 0.0)
         (/ (fma t (/ x y) z) b)
         (if (<= t_1 1e+295)
           t_1
           (if (<= t_1 INFINITY) (* y (/ z t_2)) (/ z b))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x + ((y * z) / t)) / (((y * b) / t) + (a + 1.0));
	double t_2 = fma(y, b, fma(t, a, t));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = z * (y / t_2);
	} else if (t_1 <= -1e-303) {
		tmp = t_1;
	} else if (t_1 <= 0.0) {
		tmp = fma(t, (x / y), z) / b;
	} else if (t_1 <= 1e+295) {
		tmp = t_1;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = y * (z / t_2);
	} else {
		tmp = z / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x + Float64(Float64(y * z) / t)) / Float64(Float64(Float64(y * b) / t) + Float64(a + 1.0)))
	t_2 = fma(y, b, fma(t, a, t))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(z * Float64(y / t_2));
	elseif (t_1 <= -1e-303)
		tmp = t_1;
	elseif (t_1 <= 0.0)
		tmp = Float64(fma(t, Float64(x / y), z) / b);
	elseif (t_1 <= 1e+295)
		tmp = t_1;
	elseif (t_1 <= Inf)
		tmp = Float64(y * Float64(z / 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[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(y * b), $MachinePrecision] / t), $MachinePrecision] + N[(a + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(y * b + N[(t * a + t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(z * N[(y / t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -1e-303], t$95$1, If[LessEqual[t$95$1, 0.0], N[(N[(t * N[(x / y), $MachinePrecision] + z), $MachinePrecision] / b), $MachinePrecision], If[LessEqual[t$95$1, 1e+295], t$95$1, If[LessEqual[t$95$1, Infinity], N[(y * N[(z / t$95$2), $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 24.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-rgt-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-+.f64 N/A

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

   5. Applied rewrites 86.6%

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

   7. Applied rewrites 86.6%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 61.5%

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

   12. Applied rewrites 86.5%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -9.99999999999999931e-304 or -0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 9.9999999999999998e294

   1. Initial program 98.6%

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

   if -9.99999999999999931e-304 < (/.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 55.5%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-rgt-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-+.f64 N/A

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

   5. Applied rewrites 63.1%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 79.6%

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

   if 9.9999999999999998e294 < (/.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 37.5%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-rgt-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-+.f64 N/A

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

   5. Applied rewrites 99.4%

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

   7. Applied rewrites 99.7%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 53.1%

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

   12. Applied rewrites 99.5%

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

   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 96.7

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

   5. Applied rewrites 96.7%

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

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq -\infty:\\ \;\;\;\;z \cdot \frac{y}{\mathsf{fma}\left(y, b, \mathsf{fma}\left(t, a, t\right)\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq -1 \cdot 10^{-303}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq 10^{+295}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq \infty:\\ \;\;\;\;y \cdot \frac{z}{\mathsf{fma}\left(y, b, \mathsf{fma}\left(t, a, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 78.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{y \cdot z}{t}\\ t_2 := \frac{t\_1}{a + 1}\\ t_3 := \frac{t\_1}{\frac{y \cdot b}{t} + \left(a + 1\right)}\\ t_4 := \mathsf{fma}\left(y, b, \mathsf{fma}\left(t, a, t\right)\right)\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;z \cdot \frac{y}{t\_4}\\ \mathbf{elif}\;t\_3 \leq -1 \cdot 10^{-303}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{elif}\;t\_3 \leq 10^{+295}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;y \cdot \frac{z}{t\_4}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ x (/ (* y z) t)))
        (t_2 (/ t_1 (+ a 1.0)))
        (t_3 (/ t_1 (+ (/ (* y b) t) (+ a 1.0))))
        (t_4 (fma y b (fma t a t))))
   (if (<= t_3 (- INFINITY))
     (* z (/ y t_4))
     (if (<= t_3 -1e-303)
       t_2
       (if (<= t_3 0.0)
         (/ (fma t (/ x y) z) b)
         (if (<= t_3 1e+295)
           t_2
           (if (<= t_3 INFINITY) (* y (/ z t_4)) (/ z b))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x + ((y * z) / t);
	double t_2 = t_1 / (a + 1.0);
	double t_3 = t_1 / (((y * b) / t) + (a + 1.0));
	double t_4 = fma(y, b, fma(t, a, t));
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = z * (y / t_4);
	} else if (t_3 <= -1e-303) {
		tmp = t_2;
	} else if (t_3 <= 0.0) {
		tmp = fma(t, (x / y), z) / b;
	} else if (t_3 <= 1e+295) {
		tmp = t_2;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = y * (z / t_4);
	} else {
		tmp = z / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x + Float64(Float64(y * z) / t))
	t_2 = Float64(t_1 / Float64(a + 1.0))
	t_3 = Float64(t_1 / Float64(Float64(Float64(y * b) / t) + Float64(a + 1.0)))
	t_4 = fma(y, b, fma(t, a, t))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(z * Float64(y / t_4));
	elseif (t_3 <= -1e-303)
		tmp = t_2;
	elseif (t_3 <= 0.0)
		tmp = Float64(fma(t, Float64(x / y), z) / b);
	elseif (t_3 <= 1e+295)
		tmp = t_2;
	elseif (t_3 <= Inf)
		tmp = Float64(y * Float64(z / t_4));
	else
		tmp = Float64(z / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / N[(a + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 / N[(N[(N[(y * b), $MachinePrecision] / t), $MachinePrecision] + N[(a + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(y * b + N[(t * a + t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], N[(z * N[(y / t$95$4), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -1e-303], t$95$2, If[LessEqual[t$95$3, 0.0], N[(N[(t * N[(x / y), $MachinePrecision] + z), $MachinePrecision] / b), $MachinePrecision], If[LessEqual[t$95$3, 1e+295], t$95$2, If[LessEqual[t$95$3, Infinity], N[(y * N[(z / t$95$4), $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 24.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-rgt-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-+.f64 N/A

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

   5. Applied rewrites 86.6%

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

   7. Applied rewrites 86.6%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 61.5%

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

   12. Applied rewrites 86.5%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -9.99999999999999931e-304 or -0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 9.9999999999999998e294

   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. 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 74.8

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

   5. Applied rewrites 74.8%

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

   if -9.99999999999999931e-304 < (/.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 55.5%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-rgt-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-+.f64 N/A

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

   5. Applied rewrites 63.1%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 79.6%

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

   if 9.9999999999999998e294 < (/.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 37.5%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-rgt-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-+.f64 N/A

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

   5. Applied rewrites 99.4%

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

   7. Applied rewrites 99.7%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 53.1%

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

   12. Applied rewrites 99.5%

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

   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 96.7

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

   5. Applied rewrites 96.7%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq -\infty:\\ \;\;\;\;z \cdot \frac{y}{\mathsf{fma}\left(y, b, \mathsf{fma}\left(t, a, t\right)\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq -1 \cdot 10^{-303}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{a + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq 10^{+295}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{a + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq \infty:\\ \;\;\;\;y \cdot \frac{z}{\mathsf{fma}\left(y, b, \mathsf{fma}\left(t, a, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 91.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{y \cdot z}{t}\\ t_2 := \frac{t\_1}{\frac{y \cdot b}{t} + \left(a + 1\right)}\\ t_3 := \mathsf{fma}\left(y, b, \mathsf{fma}\left(t, a, t\right)\right)\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;z \cdot \frac{y}{t\_3}\\ \mathbf{elif}\;t\_2 \leq 10^{+295}:\\ \;\;\;\;\frac{t\_1}{\mathsf{fma}\left(b, \frac{y}{t}, a + 1\right)}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;y \cdot \frac{z}{t\_3}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ x (/ (* y z) t)))
        (t_2 (/ t_1 (+ (/ (* y b) t) (+ a 1.0))))
        (t_3 (fma y b (fma t a t))))
   (if (<= t_2 (- INFINITY))
     (* z (/ y t_3))
     (if (<= t_2 1e+295)
       (/ t_1 (fma b (/ y t) (+ a 1.0)))
       (if (<= t_2 INFINITY) (* y (/ z t_3)) (/ z b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x + ((y * z) / t);
	double t_2 = t_1 / (((y * b) / t) + (a + 1.0));
	double t_3 = fma(y, b, fma(t, a, t));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = z * (y / t_3);
	} else if (t_2 <= 1e+295) {
		tmp = t_1 / fma(b, (y / t), (a + 1.0));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = y * (z / t_3);
	} else {
		tmp = z / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x + Float64(Float64(y * z) / t))
	t_2 = Float64(t_1 / Float64(Float64(Float64(y * b) / t) + Float64(a + 1.0)))
	t_3 = fma(y, b, fma(t, a, t))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(z * Float64(y / t_3));
	elseif (t_2 <= 1e+295)
		tmp = Float64(t_1 / fma(b, Float64(y / t), Float64(a + 1.0)));
	elseif (t_2 <= Inf)
		tmp = Float64(y * Float64(z / t_3));
	else
		tmp = Float64(z / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / N[(N[(N[(y * b), $MachinePrecision] / t), $MachinePrecision] + N[(a + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(y * b + N[(t * a + t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(z * N[(y / t$95$3), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e+295], N[(t$95$1 / N[(b * N[(y / t), $MachinePrecision] + N[(a + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(y * N[(z / t$95$3), $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

   1. Initial program 24.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-rgt-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-+.f64 N/A

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

   5. Applied rewrites 86.6%

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

   7. Applied rewrites 86.6%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 61.5%

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

   12. Applied rewrites 86.5%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 9.9999999999999998e294

   1. Initial program 90.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 90.4

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

   4. Applied rewrites 90.4%

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

   if 9.9999999999999998e294 < (/.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 37.5%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-rgt-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-+.f64 N/A

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

   5. Applied rewrites 99.4%

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

   7. Applied rewrites 99.7%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 53.1%

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

   12. Applied rewrites 99.5%

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

   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 96.7

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

   5. Applied rewrites 96.7%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq -\infty:\\ \;\;\;\;z \cdot \frac{y}{\mathsf{fma}\left(y, b, \mathsf{fma}\left(t, a, t\right)\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq 10^{+295}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\mathsf{fma}\left(b, \frac{y}{t}, a + 1\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq \infty:\\ \;\;\;\;y \cdot \frac{z}{\mathsf{fma}\left(y, b, \mathsf{fma}\left(t, a, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 85.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)}\\ t_2 := z \cdot \frac{y}{\mathsf{fma}\left(y, b, \mathsf{fma}\left(t, a, t\right)\right)}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+219}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\mathsf{fma}\left(y, \frac{b}{t}, a + 1\right)}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (* y z) t)) (+ (/ (* y b) t) (+ a 1.0))))
        (t_2 (* z (/ y (fma y b (fma t a t))))))
   (if (<= t_1 (- INFINITY))
     t_2
     (if (<= t_1 4e+219)
       (/ (fma y (/ z t) x) (fma y (/ b t) (+ a 1.0)))
       (if (<= t_1 INFINITY) t_2 (/ z b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x + ((y * z) / t)) / (((y * b) / t) + (a + 1.0));
	double t_2 = z * (y / fma(y, b, fma(t, a, t)));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_2;
	} else if (t_1 <= 4e+219) {
		tmp = fma(y, (z / t), x) / fma(y, (b / t), (a + 1.0));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_2;
	} else {
		tmp = z / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x + Float64(Float64(y * z) / t)) / Float64(Float64(Float64(y * b) / t) + Float64(a + 1.0)))
	t_2 = Float64(z * Float64(y / fma(y, b, fma(t, a, t))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = t_2;
	elseif (t_1 <= 4e+219)
		tmp = Float64(fma(y, Float64(z / t), x) / fma(y, Float64(b / t), Float64(a + 1.0)));
	elseif (t_1 <= Inf)
		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[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(y * b), $MachinePrecision] / t), $MachinePrecision] + N[(a + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(z * N[(y / N[(y * b + N[(t * a + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], t$95$2, If[LessEqual[t$95$1, 4e+219], N[(N[(y * N[(z / t), $MachinePrecision] + x), $MachinePrecision] / N[(y * N[(b / t), $MachinePrecision] + N[(a + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], 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 3.99999999999999986e219 < (/.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 43.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-rgt-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-+.f64 N/A

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

   5. Applied rewrites 85.2%

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

   7. Applied rewrites 85.3%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 59.5%

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

   12. Applied rewrites 87.7%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 3.99999999999999986e219

   1. Initial program 90.1%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   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. lower-fma.f64 N/A

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

      7. lower-/.f64 86.2

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/l* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 84.6

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

   4. Applied rewrites 84.6%

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

   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 96.7

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

   5. Applied rewrites 96.7%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq -\infty:\\ \;\;\;\;z \cdot \frac{y}{\mathsf{fma}\left(y, b, \mathsf{fma}\left(t, a, t\right)\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq 4 \cdot 10^{+219}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\mathsf{fma}\left(y, \frac{b}{t}, a + 1\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq \infty:\\ \;\;\;\;z \cdot \frac{y}{\mathsf{fma}\left(y, b, \mathsf{fma}\left(t, a, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 68.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)}\\ t_2 := z \cdot \frac{y}{\mathsf{fma}\left(y, b, \mathsf{fma}\left(t, a, t\right)\right)}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+216}:\\ \;\;\;\;\frac{x}{1 + \mathsf{fma}\left(y, \frac{b}{t}, a\right)}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (* y z) t)) (+ (/ (* y b) t) (+ a 1.0))))
        (t_2 (* z (/ y (fma y b (fma t a t))))))
   (if (<= t_1 (- INFINITY))
     t_2
     (if (<= t_1 2e+216)
       (/ x (+ 1.0 (fma y (/ b t) a)))
       (if (<= t_1 INFINITY) t_2 (/ z b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x + ((y * z) / t)) / (((y * b) / t) + (a + 1.0));
	double t_2 = z * (y / fma(y, b, fma(t, a, t)));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_2;
	} else if (t_1 <= 2e+216) {
		tmp = x / (1.0 + fma(y, (b / t), a));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_2;
	} else {
		tmp = z / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x + Float64(Float64(y * z) / t)) / Float64(Float64(Float64(y * b) / t) + Float64(a + 1.0)))
	t_2 = Float64(z * Float64(y / fma(y, b, fma(t, a, t))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = t_2;
	elseif (t_1 <= 2e+216)
		tmp = Float64(x / Float64(1.0 + fma(y, Float64(b / t), a)));
	elseif (t_1 <= Inf)
		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[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(y * b), $MachinePrecision] / t), $MachinePrecision] + N[(a + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(z * N[(y / N[(y * b + N[(t * a + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], t$95$2, If[LessEqual[t$95$1, 2e+216], N[(x / N[(1.0 + N[(y * N[(b / t), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], 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 2e216 < (/.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 46.2%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-rgt-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-+.f64 N/A

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

   5. Applied rewrites 85.9%

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

   7. Applied rewrites 86.1%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 59.0%

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

   12. Applied rewrites 85.6%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 2e216

   1. Initial program 90.0%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   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. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 65.4

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

   5. Applied rewrites 65.4%

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

   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 96.7

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

   5. Applied rewrites 96.7%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq -\infty:\\ \;\;\;\;z \cdot \frac{y}{\mathsf{fma}\left(y, b, \mathsf{fma}\left(t, a, t\right)\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq 2 \cdot 10^{+216}:\\ \;\;\;\;\frac{x}{1 + \mathsf{fma}\left(y, \frac{b}{t}, a\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq \infty:\\ \;\;\;\;z \cdot \frac{y}{\mathsf{fma}\left(y, b, \mathsf{fma}\left(t, a, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 69.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{if}\;y \leq -3.6 \cdot 10^{+140}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 10^{+61}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}{a + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (fma t (/ x y) z) b)))
   (if (<= y -3.6e+140)
     t_1
     (if (<= y 1e+61) (/ (fma z (/ y t) x) (+ a 1.0)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(t, (x / y), z) / b;
	double tmp;
	if (y <= -3.6e+140) {
		tmp = t_1;
	} else if (y <= 1e+61) {
		tmp = fma(z, (y / t), x) / (a + 1.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(fma(t, Float64(x / y), z) / b)
	tmp = 0.0
	if (y <= -3.6e+140)
		tmp = t_1;
	elseif (y <= 1e+61)
		tmp = Float64(fma(z, Float64(y / t), x) / Float64(a + 1.0));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t * N[(x / y), $MachinePrecision] + z), $MachinePrecision] / b), $MachinePrecision]}, If[LessEqual[y, -3.6e+140], t$95$1, If[LessEqual[y, 1e+61], N[(N[(z * N[(y / t), $MachinePrecision] + x), $MachinePrecision] / N[(a + 1.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.6e140 or 9.99999999999999949e60 < y

   1. Initial program 46.7%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-rgt-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-+.f64 N/A

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

   5. Applied rewrites 58.9%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 80.3%

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

   if -3.6e140 < y < 9.99999999999999949e60

   1. Initial program 87.6%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   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. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-+.f64 71.4

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

   5. Applied rewrites 71.4%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.6 \cdot 10^{+140}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{elif}\;y \leq 10^{+61}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}{a + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 61.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{a + 1}\\ \mathbf{if}\;t \leq -3.2 \cdot 10^{+34}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 3.3 \cdot 10^{+57}:\\ \;\;\;\;z \cdot \frac{y}{\mathsf{fma}\left(y, b, \mathsf{fma}\left(t, a, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ x (+ a 1.0))))
   (if (<= t -3.2e+34)
     t_1
     (if (<= t 3.3e+57) (* z (/ y (fma y b (fma t a t)))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x / (a + 1.0);
	double tmp;
	if (t <= -3.2e+34) {
		tmp = t_1;
	} else if (t <= 3.3e+57) {
		tmp = z * (y / fma(y, b, fma(t, a, t)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x / Float64(a + 1.0))
	tmp = 0.0
	if (t <= -3.2e+34)
		tmp = t_1;
	elseif (t <= 3.3e+57)
		tmp = Float64(z * Float64(y / fma(y, b, fma(t, a, t))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x / N[(a + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -3.2e+34], t$95$1, If[LessEqual[t, 3.3e+57], N[(z * N[(y / N[(y * b + N[(t * a + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -3.1999999999999998e34 or 3.3000000000000001e57 < t

   1. Initial program 81.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 72.0

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

   5. Applied rewrites 72.0%

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

   if -3.1999999999999998e34 < t < 3.3000000000000001e57

   1. Initial program 70.1%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-rgt-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-+.f64 N/A

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

   5. Applied rewrites 67.0%

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

   7. Applied rewrites 79.6%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 54.9%

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

   12. Applied rewrites 62.2%

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

4. Final simplification 65.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.2 \cdot 10^{+34}:\\ \;\;\;\;\frac{x}{a + 1}\\ \mathbf{elif}\;t \leq 3.3 \cdot 10^{+57}:\\ \;\;\;\;z \cdot \frac{y}{\mathsf{fma}\left(y, b, \mathsf{fma}\left(t, a, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 42.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.5 \cdot 10^{-17}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;y \leq 9.8 \cdot 10^{-83}:\\ \;\;\;\;\frac{x}{a}\\ \mathbf{elif}\;y \leq 9.4 \cdot 10^{+49}:\\ \;\;\;\;\frac{x}{1}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -6.5e-17)
   (/ z b)
   (if (<= y 9.8e-83) (/ x a) (if (<= y 9.4e+49) (/ x 1.0) (/ z b)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -6.5e-17) {
		tmp = z / b;
	} else if (y <= 9.8e-83) {
		tmp = x / a;
	} else if (y <= 9.4e+49) {
		tmp = x / 1.0;
	} 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 <= (-6.5d-17)) then
        tmp = z / b
    else if (y <= 9.8d-83) then
        tmp = x / a
    else if (y <= 9.4d+49) then
        tmp = x / 1.0d0
    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 <= -6.5e-17) {
		tmp = z / b;
	} else if (y <= 9.8e-83) {
		tmp = x / a;
	} else if (y <= 9.4e+49) {
		tmp = x / 1.0;
	} else {
		tmp = z / b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -6.5e-17:
		tmp = z / b
	elif y <= 9.8e-83:
		tmp = x / a
	elif y <= 9.4e+49:
		tmp = x / 1.0
	else:
		tmp = z / b
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -6.5e-17)
		tmp = Float64(z / b);
	elseif (y <= 9.8e-83)
		tmp = Float64(x / a);
	elseif (y <= 9.4e+49)
		tmp = Float64(x / 1.0);
	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 <= -6.5e-17)
		tmp = z / b;
	elseif (y <= 9.8e-83)
		tmp = x / a;
	elseif (y <= 9.4e+49)
		tmp = x / 1.0;
	else
		tmp = z / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -6.5e-17], N[(z / b), $MachinePrecision], If[LessEqual[y, 9.8e-83], N[(x / a), $MachinePrecision], If[LessEqual[y, 9.4e+49], N[(x / 1.0), $MachinePrecision], N[(z / b), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.4999999999999996e-17 or 9.3999999999999995e49 < y

   1. Initial program 53.6%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 59.2

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

   5. Applied rewrites 59.2%

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

   if -6.4999999999999996e-17 < y < 9.8e-83

   1. Initial program 97.1%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   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. clear-num N/A

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

      5. inv-pow N/A

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

      6. div-inv N/A

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

      7. unpow-prod-down N/A

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

      8. inv-pow N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-/.f64 97.1

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

   4. Applied rewrites 97.1%

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

   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 42.3

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

   7. Applied rewrites 42.3%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 36.6%

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

   if 9.8e-83 < y < 9.3999999999999995e49

   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. 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. clear-num N/A

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

      5. inv-pow N/A

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

      6. div-inv N/A

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

      7. unpow-prod-down N/A

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

      8. inv-pow N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-/.f64 72.6

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

   4. Applied rewrites 72.6%

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

   5. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 51.9

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

   7. Applied rewrites 51.9%

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

   8. Taylor expanded in a around 0

      \[\leadsto \frac{x}{1} \]
   9. Step-by-step derivation

   10. Applied rewrites 44.3%

       \[\leadsto \frac{x}{1} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 14: 42.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.5 \cdot 10^{-17}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;y \leq 9.8 \cdot 10^{-83}:\\ \;\;\;\;\frac{x}{a}\\ \mathbf{elif}\;y \leq 9.4 \cdot 10^{+49}:\\ \;\;\;\;x - x \cdot a\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -6.5e-17)
   (/ z b)
   (if (<= y 9.8e-83) (/ x a) (if (<= y 9.4e+49) (- x (* x a)) (/ z b)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -6.5e-17) {
		tmp = z / b;
	} else if (y <= 9.8e-83) {
		tmp = x / a;
	} else if (y <= 9.4e+49) {
		tmp = x - (x * 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 <= (-6.5d-17)) then
        tmp = z / b
    else if (y <= 9.8d-83) then
        tmp = x / a
    else if (y <= 9.4d+49) then
        tmp = x - (x * 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 <= -6.5e-17) {
		tmp = z / b;
	} else if (y <= 9.8e-83) {
		tmp = x / a;
	} else if (y <= 9.4e+49) {
		tmp = x - (x * a);
	} else {
		tmp = z / b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -6.5e-17:
		tmp = z / b
	elif y <= 9.8e-83:
		tmp = x / a
	elif y <= 9.4e+49:
		tmp = x - (x * a)
	else:
		tmp = z / b
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -6.5e-17)
		tmp = Float64(z / b);
	elseif (y <= 9.8e-83)
		tmp = Float64(x / a);
	elseif (y <= 9.4e+49)
		tmp = Float64(x - Float64(x * 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 <= -6.5e-17)
		tmp = z / b;
	elseif (y <= 9.8e-83)
		tmp = x / a;
	elseif (y <= 9.4e+49)
		tmp = x - (x * a);
	else
		tmp = z / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -6.5e-17], N[(z / b), $MachinePrecision], If[LessEqual[y, 9.8e-83], N[(x / a), $MachinePrecision], If[LessEqual[y, 9.4e+49], N[(x - N[(x * a), $MachinePrecision]), $MachinePrecision], N[(z / b), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.4999999999999996e-17 or 9.3999999999999995e49 < y

   1. Initial program 53.6%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 59.2

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

   5. Applied rewrites 59.2%

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

   if -6.4999999999999996e-17 < y < 9.8e-83

   1. Initial program 97.1%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   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. clear-num N/A

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

      5. inv-pow N/A

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

      6. div-inv N/A

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

      7. unpow-prod-down N/A

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

      8. inv-pow N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-/.f64 97.1

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

   4. Applied rewrites 97.1%

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

   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 42.3

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

   7. Applied rewrites 42.3%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 36.6%

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

   if 9.8e-83 < y < 9.3999999999999995e49

   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. 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. clear-num N/A

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

      5. inv-pow N/A

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

      6. div-inv N/A

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

      7. unpow-prod-down N/A

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

      8. inv-pow N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-/.f64 72.6

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

   4. Applied rewrites 72.6%

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

   5. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 51.9

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

   7. Applied rewrites 51.9%

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

   8. Taylor expanded in a around 0

      \[\leadsto x + \color{blue}{-1 \cdot \left(a \cdot x\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 44.3%

       \[\leadsto x - \color{blue}{a \cdot x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 47.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.5 \cdot 10^{-17}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;y \leq 9.8 \cdot 10^{-83}:\\ \;\;\;\;\frac{x}{a}\\ \mathbf{elif}\;y \leq 9.4 \cdot 10^{+49}:\\ \;\;\;\;x - x \cdot a\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 56.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.7 \cdot 10^{+52}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;y \leq 10^{+50}:\\ \;\;\;\;\frac{x}{a + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -2.7e+52) (/ z b) (if (<= y 1e+50) (/ x (+ a 1.0)) (/ z b))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -2.7e+52) {
		tmp = z / b;
	} else if (y <= 1e+50) {
		tmp = x / (a + 1.0);
	} 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 <= (-2.7d+52)) then
        tmp = z / b
    else if (y <= 1d+50) then
        tmp = x / (a + 1.0d0)
    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 <= -2.7e+52) {
		tmp = z / b;
	} else if (y <= 1e+50) {
		tmp = x / (a + 1.0);
	} else {
		tmp = z / b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -2.7e+52:
		tmp = z / b
	elif y <= 1e+50:
		tmp = x / (a + 1.0)
	else:
		tmp = z / b
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -2.7e+52)
		tmp = Float64(z / b);
	elseif (y <= 1e+50)
		tmp = Float64(x / Float64(a + 1.0));
	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 <= -2.7e+52)
		tmp = z / b;
	elseif (y <= 1e+50)
		tmp = x / (a + 1.0);
	else
		tmp = z / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -2.7e+52], N[(z / b), $MachinePrecision], If[LessEqual[y, 1e+50], N[(x / N[(a + 1.0), $MachinePrecision]), $MachinePrecision], N[(z / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.7e52 or 1.0000000000000001e50 < y

   1. Initial program 50.2%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 61.9

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

   5. Applied rewrites 61.9%

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

   if -2.7e52 < y < 1.0000000000000001e50

   1. Initial program 91.5%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   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 58.9

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

   5. Applied rewrites 58.9%

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

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.7 \cdot 10^{+52}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;y \leq 10^{+50}:\\ \;\;\;\;\frac{x}{a + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 40.6% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - x \cdot a\\ \mathbf{if}\;t \leq -1.35 \cdot 10^{+35}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.95 \cdot 10^{+62}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (- x (* x a))))
   (if (<= t -1.35e+35) t_1 (if (<= t 1.95e+62) (/ z b) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x - (x * a);
	double tmp;
	if (t <= -1.35e+35) {
		tmp = t_1;
	} else if (t <= 1.95e+62) {
		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 = x - (x * a)
    if (t <= (-1.35d+35)) then
        tmp = t_1
    else if (t <= 1.95d+62) 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 = x - (x * a);
	double tmp;
	if (t <= -1.35e+35) {
		tmp = t_1;
	} else if (t <= 1.95e+62) {
		tmp = z / b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x - (x * a)
	tmp = 0
	if t <= -1.35e+35:
		tmp = t_1
	elif t <= 1.95e+62:
		tmp = z / b
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x - Float64(x * a))
	tmp = 0.0
	if (t <= -1.35e+35)
		tmp = t_1;
	elseif (t <= 1.95e+62)
		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 = x - (x * a);
	tmp = 0.0;
	if (t <= -1.35e+35)
		tmp = t_1;
	elseif (t <= 1.95e+62)
		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[(x - N[(x * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1.35e+35], t$95$1, If[LessEqual[t, 1.95e+62], N[(z / b), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.35000000000000001e35 or 1.95e62 < t

   1. Initial program 81.6%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   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. clear-num N/A

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

      5. inv-pow N/A

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

      6. div-inv N/A

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

      7. unpow-prod-down N/A

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

      8. inv-pow N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-/.f64 81.6

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

   4. Applied rewrites 81.6%

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

   5. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 72.5

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

   7. Applied rewrites 72.5%

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

   8. Taylor expanded in a around 0

      \[\leadsto x + \color{blue}{-1 \cdot \left(a \cdot x\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 37.4%

       \[\leadsto x - \color{blue}{a \cdot x} \]

   if -1.35000000000000001e35 < t < 1.95e62

   1. Initial program 69.9%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 48.6

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

   5. Applied rewrites 48.6%

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

4. Final simplification 44.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.35 \cdot 10^{+35}:\\ \;\;\;\;x - x \cdot a\\ \mathbf{elif}\;t \leq 1.95 \cdot 10^{+62}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;x - x \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 19.4% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ x - x \cdot a \end{array} \]

FPCore

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

C

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

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 - (x * a)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.2%

   \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
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. clear-num N/A

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

   5. inv-pow N/A

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

   6. div-inv N/A

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

   7. unpow-prod-down N/A

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

   8. inv-pow N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lower-pow.f64 N/A

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

   12. lower-/.f64 74.2

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

4. Applied rewrites 74.2%

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

5. Taylor expanded in t around inf

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

   1. lower-/.f64 N/A

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

   2. lower-+.f64 41.0

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

7. Applied rewrites 41.0%

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

8. Taylor expanded in a around 0

   \[\leadsto x + \color{blue}{-1 \cdot \left(a \cdot x\right)} \]
9. Step-by-step derivation

10. Applied rewrites 20.5%

    \[\leadsto x - \color{blue}{a \cdot x} \]

11. Final simplification 20.5%

    \[\leadsto x - x \cdot a \]
12. Add Preprocessing

Alternative 18: 4.1% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ a \cdot \left(-x\right) \end{array} \]

FPCore

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

C

double code(double x, double y, double z, double t, double a, double b) {
	return a * -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 = a * -x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.2%

   \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
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. clear-num N/A

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

   5. inv-pow N/A

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

   6. div-inv N/A

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

   7. unpow-prod-down N/A

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

   8. inv-pow N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lower-pow.f64 N/A

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

   12. lower-/.f64 74.2

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

4. Applied rewrites 74.2%

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

5. Taylor expanded in t around inf

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

   1. lower-/.f64 N/A

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

   2. lower-+.f64 41.0

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

7. Applied rewrites 41.0%

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

8. Taylor expanded in a around 0

   \[\leadsto x + \color{blue}{-1 \cdot \left(a \cdot x\right)} \]
9. Step-by-step derivation

10. Applied rewrites 20.5%

    \[\leadsto x - \color{blue}{a \cdot x} \]

11. Taylor expanded in a around inf

    \[\leadsto -1 \cdot \left(a \cdot \color{blue}{x}\right) \]
12. Step-by-step derivation

13. Applied rewrites 3.5%

    \[\leadsto x \cdot \left(-a\right) \]

14. Final simplification 3.5%

    \[\leadsto a \cdot \left(-x\right) \]
15. 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 2024238 
(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))))