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

Percentage Accurate: 74.3% → 89.0%

Time: 13.6s

Alternatives: 23

Speedup: 0.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 74.3% 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: 89.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y \cdot b}{t} + \left(a + 1\right)\\ t_2 := \mathsf{fma}\left(y, \frac{b}{t}, a\right)\\ t_3 := \frac{x + \frac{y \cdot z}{t}}{t\_1}\\ t_4 := t \cdot \left(\frac{x}{b} - \frac{\mathsf{fma}\left(z, a, z\right)}{b \cdot b}\right)\\ \mathbf{if}\;t\_3 \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{x + \frac{z}{\frac{t}{y}}}{\mathsf{fma}\left(b, \frac{y}{t}, a + 1\right)}\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;\frac{z}{b} + \frac{\mathsf{fma}\left(\left(-1 - a\right) \cdot t\_4, \frac{t}{y \cdot b}, t\_4\right)}{y}\\ \mathbf{elif}\;t\_3 \leq 10^{+263}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{t}, y \cdot z, x\right)}{t\_1}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;z \cdot \left(\frac{x}{\mathsf{fma}\left(z, t\_2, z\right)} + \frac{y}{\mathsf{fma}\left(t, t\_2, t\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (/ (* y b) t) (+ a 1.0)))
        (t_2 (fma y (/ b t) a))
        (t_3 (/ (+ x (/ (* y z) t)) t_1))
        (t_4 (* t (- (/ x b) (/ (fma z a z) (* b b))))))
   (if (<= t_3 -2e-310)
     (/ (+ x (/ z (/ t y))) (fma b (/ y t) (+ a 1.0)))
     (if (<= t_3 0.0)
       (+ (/ z b) (/ (fma (* (- -1.0 a) t_4) (/ t (* y b)) t_4) y))
       (if (<= t_3 1e+263)
         (/ (fma (/ 1.0 t) (* y z) x) t_1)
         (if (<= t_3 INFINITY)
           (* z (+ (/ x (fma z t_2 z)) (/ y (fma t t_2 t))))
           (/ z b)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((y * b) / t) + (a + 1.0);
	double t_2 = fma(y, (b / t), a);
	double t_3 = (x + ((y * z) / t)) / t_1;
	double t_4 = t * ((x / b) - (fma(z, a, z) / (b * b)));
	double tmp;
	if (t_3 <= -2e-310) {
		tmp = (x + (z / (t / y))) / fma(b, (y / t), (a + 1.0));
	} else if (t_3 <= 0.0) {
		tmp = (z / b) + (fma(((-1.0 - a) * t_4), (t / (y * b)), t_4) / y);
	} else if (t_3 <= 1e+263) {
		tmp = fma((1.0 / t), (y * z), x) / t_1;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = z * ((x / fma(z, t_2, z)) + (y / fma(t, t_2, t)));
	} else {
		tmp = z / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(y * b) / t) + Float64(a + 1.0))
	t_2 = fma(y, Float64(b / t), a)
	t_3 = Float64(Float64(x + Float64(Float64(y * z) / t)) / t_1)
	t_4 = Float64(t * Float64(Float64(x / b) - Float64(fma(z, a, z) / Float64(b * b))))
	tmp = 0.0
	if (t_3 <= -2e-310)
		tmp = Float64(Float64(x + Float64(z / Float64(t / y))) / fma(b, Float64(y / t), Float64(a + 1.0)));
	elseif (t_3 <= 0.0)
		tmp = Float64(Float64(z / b) + Float64(fma(Float64(Float64(-1.0 - a) * t_4), Float64(t / Float64(y * b)), t_4) / y));
	elseif (t_3 <= 1e+263)
		tmp = Float64(fma(Float64(1.0 / t), Float64(y * z), x) / t_1);
	elseif (t_3 <= Inf)
		tmp = Float64(z * Float64(Float64(x / fma(z, t_2, z)) + Float64(y / fma(t, t_2, t))));
	else
		tmp = Float64(z / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(y * b), $MachinePrecision] / t), $MachinePrecision] + N[(a + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(y * N[(b / t), $MachinePrecision] + a), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$4 = N[(t * N[(N[(x / b), $MachinePrecision] - N[(N[(z * a + z), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -2e-310], N[(N[(x + N[(z / N[(t / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * N[(y / t), $MachinePrecision] + N[(a + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.0], N[(N[(z / b), $MachinePrecision] + N[(N[(N[(N[(-1.0 - a), $MachinePrecision] * t$95$4), $MachinePrecision] * N[(t / N[(y * b), $MachinePrecision]), $MachinePrecision] + t$95$4), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 1e+263], N[(N[(N[(1.0 / t), $MachinePrecision] * N[(y * z), $MachinePrecision] + x), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(z * N[(N[(x / N[(z * t$95$2 + z), $MachinePrecision]), $MachinePrecision] + N[(y / N[(t * t$95$2 + t), $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))) < -1.999999999999994e-310

   1. Initial program 91.7%

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

      1. lift-+.f64 N/A

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

      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 89.4

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

   4. Applied rewrites 89.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 93.1

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

   6. Applied rewrites 93.1%

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

   if -1.999999999999994e-310 < (/.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 47.0%

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

   3. Taylor expanded in y around -inf

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

   5. Applied rewrites 82.0%

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

   if -0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 1.00000000000000002e263

   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. 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. associate-/r/ N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 98.6

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

   4. Applied rewrites 98.6%

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

   if 1.00000000000000002e263 < (/.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 50.4%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. *-rgt-identity N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

   5. Applied rewrites 99.7%

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

   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 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{x + \frac{z}{\frac{t}{y}}}{\mathsf{fma}\left(b, \frac{y}{t}, a + 1\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq 0:\\ \;\;\;\;\frac{z}{b} + \frac{\mathsf{fma}\left(\left(-1 - a\right) \cdot \left(t \cdot \left(\frac{x}{b} - \frac{\mathsf{fma}\left(z, a, z\right)}{b \cdot b}\right)\right), \frac{t}{y \cdot b}, t \cdot \left(\frac{x}{b} - \frac{\mathsf{fma}\left(z, a, z\right)}{b \cdot b}\right)\right)}{y}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq 10^{+263}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{t}, y \cdot z, x\right)}{\frac{y \cdot b}{t} + \left(a + 1\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq \infty:\\ \;\;\;\;z \cdot \left(\frac{x}{\mathsf{fma}\left(z, \mathsf{fma}\left(y, \frac{b}{t}, a\right), z\right)} + \frac{y}{\mathsf{fma}\left(t, \mathsf{fma}\left(y, \frac{b}{t}, a\right), t\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 88.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y \cdot b}{t} + \left(a + 1\right)\\ t_2 := \mathsf{fma}\left(y, \frac{b}{t}, a\right)\\ t_3 := \frac{x + \frac{y \cdot z}{t}}{t\_1}\\ \mathbf{if}\;t\_3 \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{x + \frac{z}{\frac{t}{y}}}{\mathsf{fma}\left(b, \frac{y}{t}, a + 1\right)}\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;\frac{\frac{t \cdot \mathsf{fma}\left(y, \frac{z}{t}, x\right)}{y}}{b}\\ \mathbf{elif}\;t\_3 \leq 10^{+263}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{t}, y \cdot z, x\right)}{t\_1}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;z \cdot \left(\frac{x}{\mathsf{fma}\left(z, t\_2, z\right)} + \frac{y}{\mathsf{fma}\left(t, t\_2, t\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (/ (* y b) t) (+ a 1.0)))
        (t_2 (fma y (/ b t) a))
        (t_3 (/ (+ x (/ (* y z) t)) t_1)))
   (if (<= t_3 -2e-310)
     (/ (+ x (/ z (/ t y))) (fma b (/ y t) (+ a 1.0)))
     (if (<= t_3 0.0)
       (/ (/ (* t (fma y (/ z t) x)) y) b)
       (if (<= t_3 1e+263)
         (/ (fma (/ 1.0 t) (* y z) x) t_1)
         (if (<= t_3 INFINITY)
           (* z (+ (/ x (fma z t_2 z)) (/ y (fma t t_2 t))))
           (/ z b)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((y * b) / t) + (a + 1.0);
	double t_2 = fma(y, (b / t), a);
	double t_3 = (x + ((y * z) / t)) / t_1;
	double tmp;
	if (t_3 <= -2e-310) {
		tmp = (x + (z / (t / y))) / fma(b, (y / t), (a + 1.0));
	} else if (t_3 <= 0.0) {
		tmp = ((t * fma(y, (z / t), x)) / y) / b;
	} else if (t_3 <= 1e+263) {
		tmp = fma((1.0 / t), (y * z), x) / t_1;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = z * ((x / fma(z, t_2, z)) + (y / fma(t, t_2, t)));
	} else {
		tmp = z / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(y * b) / t) + Float64(a + 1.0))
	t_2 = fma(y, Float64(b / t), a)
	t_3 = Float64(Float64(x + Float64(Float64(y * z) / t)) / t_1)
	tmp = 0.0
	if (t_3 <= -2e-310)
		tmp = Float64(Float64(x + Float64(z / Float64(t / y))) / fma(b, Float64(y / t), Float64(a + 1.0)));
	elseif (t_3 <= 0.0)
		tmp = Float64(Float64(Float64(t * fma(y, Float64(z / t), x)) / y) / b);
	elseif (t_3 <= 1e+263)
		tmp = Float64(fma(Float64(1.0 / t), Float64(y * z), x) / t_1);
	elseif (t_3 <= Inf)
		tmp = Float64(z * Float64(Float64(x / fma(z, t_2, z)) + Float64(y / fma(t, t_2, t))));
	else
		tmp = Float64(z / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(y * b), $MachinePrecision] / t), $MachinePrecision] + N[(a + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(y * N[(b / t), $MachinePrecision] + a), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, If[LessEqual[t$95$3, -2e-310], N[(N[(x + N[(z / N[(t / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * N[(y / t), $MachinePrecision] + N[(a + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.0], N[(N[(N[(t * N[(y * N[(z / t), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] / b), $MachinePrecision], If[LessEqual[t$95$3, 1e+263], N[(N[(N[(1.0 / t), $MachinePrecision] * N[(y * z), $MachinePrecision] + x), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(z * N[(N[(x / N[(z * t$95$2 + z), $MachinePrecision]), $MachinePrecision] + N[(y / N[(t * t$95$2 + t), $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))) < -1.999999999999994e-310

   1. Initial program 91.7%

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

      1. lift-+.f64 N/A

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

      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 89.4

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

   4. Applied rewrites 89.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 93.1

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

   6. Applied rewrites 93.1%

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

   if -1.999999999999994e-310 < (/.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 47.0%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*r/ N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 27.3

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

   5. Applied rewrites 27.3%

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

   7. Applied rewrites 79.8%

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

   if -0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 1.00000000000000002e263

   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. 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. associate-/r/ N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 98.6

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

   4. Applied rewrites 98.6%

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

   if 1.00000000000000002e263 < (/.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 50.4%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. *-rgt-identity N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

   5. Applied rewrites 99.7%

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

   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 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{x + \frac{z}{\frac{t}{y}}}{\mathsf{fma}\left(b, \frac{y}{t}, a + 1\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq 0:\\ \;\;\;\;\frac{\frac{t \cdot \mathsf{fma}\left(y, \frac{z}{t}, x\right)}{y}}{b}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq 10^{+263}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{t}, y \cdot z, x\right)}{\frac{y \cdot b}{t} + \left(a + 1\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq \infty:\\ \;\;\;\;z \cdot \left(\frac{x}{\mathsf{fma}\left(z, \mathsf{fma}\left(y, \frac{b}{t}, a\right), z\right)} + \frac{y}{\mathsf{fma}\left(t, \mathsf{fma}\left(y, \frac{b}{t}, a\right), t\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 55.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)}\\ t_2 := \frac{x}{a + 1}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+177}:\\ \;\;\;\;\frac{y \cdot z}{\mathsf{fma}\left(t, a, t\right)}\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{-310}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{-302}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;t\_1 \leq 10^{+263}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

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 (/ x (+ a 1.0))))
   (if (<= t_1 -1e+177)
     (/ (* y z) (fma t a t))
     (if (<= t_1 -2e-310)
       t_2
       (if (<= t_1 4e-302) (/ z b) (if (<= t_1 1e+263) 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 = x / (a + 1.0);
	double tmp;
	if (t_1 <= -1e+177) {
		tmp = (y * z) / fma(t, a, t);
	} else if (t_1 <= -2e-310) {
		tmp = t_2;
	} else if (t_1 <= 4e-302) {
		tmp = z / b;
	} else if (t_1 <= 1e+263) {
		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(x / Float64(a + 1.0))
	tmp = 0.0
	if (t_1 <= -1e+177)
		tmp = Float64(Float64(y * z) / fma(t, a, t));
	elseif (t_1 <= -2e-310)
		tmp = t_2;
	elseif (t_1 <= 4e-302)
		tmp = Float64(z / b);
	elseif (t_1 <= 1e+263)
		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[(x / N[(a + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+177], N[(N[(y * z), $MachinePrecision] / N[(t * a + t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -2e-310], t$95$2, If[LessEqual[t$95$1, 4e-302], N[(z / b), $MachinePrecision], If[LessEqual[t$95$1, 1e+263], 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))) < -1e177

   1. Initial program 74.5%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 62.6

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

   5. Applied rewrites 62.6%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 51.1%

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

   if -1e177 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -1.999999999999994e-310 or 3.9999999999999999e-302 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 1.00000000000000002e263

   1. Initial program 98.8%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 57.4

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

   5. Applied rewrites 57.4%

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

   if -1.999999999999994e-310 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 3.9999999999999999e-302 or 1.00000000000000002e263 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t)))

   1. Initial program 38.4%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 72.4

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

   5. Applied rewrites 72.4%

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

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq -1 \cdot 10^{+177}:\\ \;\;\;\;\frac{y \cdot z}{\mathsf{fma}\left(t, a, t\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{x}{a + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq 4 \cdot 10^{-302}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq 10^{+263}:\\ \;\;\;\;\frac{x}{a + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 58.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)}\\ t_2 := \frac{x}{a + 1}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;y \cdot \frac{z}{\mathsf{fma}\left(t, a, t\right)}\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{-310}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{-302}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;t\_1 \leq 10^{+263}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

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 (/ x (+ a 1.0))))
   (if (<= t_1 (- INFINITY))
     (* y (/ z (fma t a t)))
     (if (<= t_1 -2e-310)
       t_2
       (if (<= t_1 4e-302) (/ z b) (if (<= t_1 1e+263) 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 = x / (a + 1.0);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = y * (z / fma(t, a, t));
	} else if (t_1 <= -2e-310) {
		tmp = t_2;
	} else if (t_1 <= 4e-302) {
		tmp = z / b;
	} else if (t_1 <= 1e+263) {
		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(x / Float64(a + 1.0))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(y * Float64(z / fma(t, a, t)));
	elseif (t_1 <= -2e-310)
		tmp = t_2;
	elseif (t_1 <= 4e-302)
		tmp = Float64(z / b);
	elseif (t_1 <= 1e+263)
		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[(x / N[(a + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(y * N[(z / N[(t * a + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -2e-310], t$95$2, If[LessEqual[t$95$1, 4e-302], N[(z / b), $MachinePrecision], If[LessEqual[t$95$1, 1e+263], 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

   1. Initial program 49.4%

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

   3. Taylor expanded in y around 0

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

      1. lower-fma.f64 N/A

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

   5. Applied rewrites 62.3%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 62.3%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -1.999999999999994e-310 or 3.9999999999999999e-302 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 1.00000000000000002e263

   1. Initial program 98.8%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 55.5

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

   5. Applied rewrites 55.5%

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

   if -1.999999999999994e-310 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 3.9999999999999999e-302 or 1.00000000000000002e263 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t)))

   1. Initial program 38.4%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 72.4

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

   5. Applied rewrites 72.4%

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

4. Final simplification 61.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq -\infty:\\ \;\;\;\;y \cdot \frac{z}{\mathsf{fma}\left(t, a, t\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{x}{a + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq 4 \cdot 10^{-302}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq 10^{+263}:\\ \;\;\;\;\frac{x}{a + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 57.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)}\\ t_2 := \frac{x}{a + 1}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+201}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{t}, x\right)\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{-310}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{-302}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;t\_1 \leq 10^{+263}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

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 (/ x (+ a 1.0))))
   (if (<= t_1 -2e+201)
     (fma y (/ z t) x)
     (if (<= t_1 -2e-310)
       t_2
       (if (<= t_1 4e-302) (/ z b) (if (<= t_1 1e+263) 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 = x / (a + 1.0);
	double tmp;
	if (t_1 <= -2e+201) {
		tmp = fma(y, (z / t), x);
	} else if (t_1 <= -2e-310) {
		tmp = t_2;
	} else if (t_1 <= 4e-302) {
		tmp = z / b;
	} else if (t_1 <= 1e+263) {
		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(x / Float64(a + 1.0))
	tmp = 0.0
	if (t_1 <= -2e+201)
		tmp = fma(y, Float64(z / t), x);
	elseif (t_1 <= -2e-310)
		tmp = t_2;
	elseif (t_1 <= 4e-302)
		tmp = Float64(z / b);
	elseif (t_1 <= 1e+263)
		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[(x / N[(a + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+201], N[(y * N[(z / t), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, -2e-310], t$95$2, If[LessEqual[t$95$1, 4e-302], N[(z / b), $MachinePrecision], If[LessEqual[t$95$1, 1e+263], 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))) < -2.00000000000000008e201

   1. Initial program 73.5%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   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 61.5

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

   5. Applied rewrites 61.5%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 39.0%

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

   if -2.00000000000000008e201 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -1.999999999999994e-310 or 3.9999999999999999e-302 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 1.00000000000000002e263

   1. Initial program 98.8%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 57.0

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

   5. Applied rewrites 57.0%

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

   if -1.999999999999994e-310 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 3.9999999999999999e-302 or 1.00000000000000002e263 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t)))

   1. Initial program 38.4%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 72.4

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

   5. Applied rewrites 72.4%

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

4. Final simplification 60.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq -2 \cdot 10^{+201}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{t}, x\right)\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{x}{a + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq 4 \cdot 10^{-302}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq 10^{+263}:\\ \;\;\;\;\frac{x}{a + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y \cdot b}{t} + \left(a + 1\right)\\ t_2 := \frac{x + \frac{y \cdot z}{t}}{t\_1}\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{x + \frac{z}{\frac{t}{y}}}{\mathsf{fma}\left(b, \frac{y}{t}, a + 1\right)}\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;\frac{\frac{t \cdot \mathsf{fma}\left(y, \frac{z}{t}, x\right)}{y}}{b}\\ \mathbf{elif}\;t\_2 \leq 10^{+263}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{t}, y \cdot z, x\right)}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (/ (* y b) t) (+ a 1.0))) (t_2 (/ (+ x (/ (* y z) t)) t_1)))
   (if (<= t_2 -2e-310)
     (/ (+ x (/ z (/ t y))) (fma b (/ y t) (+ a 1.0)))
     (if (<= t_2 0.0)
       (/ (/ (* t (fma y (/ z t) x)) y) b)
       (if (<= t_2 1e+263) (/ (fma (/ 1.0 t) (* y z) x) t_1) (/ z b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((y * b) / t) + (a + 1.0);
	double t_2 = (x + ((y * z) / t)) / t_1;
	double tmp;
	if (t_2 <= -2e-310) {
		tmp = (x + (z / (t / y))) / fma(b, (y / t), (a + 1.0));
	} else if (t_2 <= 0.0) {
		tmp = ((t * fma(y, (z / t), x)) / y) / b;
	} else if (t_2 <= 1e+263) {
		tmp = fma((1.0 / t), (y * z), x) / t_1;
	} else {
		tmp = z / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(y * b) / t) + Float64(a + 1.0))
	t_2 = Float64(Float64(x + Float64(Float64(y * z) / t)) / t_1)
	tmp = 0.0
	if (t_2 <= -2e-310)
		tmp = Float64(Float64(x + Float64(z / Float64(t / y))) / fma(b, Float64(y / t), Float64(a + 1.0)));
	elseif (t_2 <= 0.0)
		tmp = Float64(Float64(Float64(t * fma(y, Float64(z / t), x)) / y) / b);
	elseif (t_2 <= 1e+263)
		tmp = Float64(fma(Float64(1.0 / t), Float64(y * z), x) / t_1);
	else
		tmp = Float64(z / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(y * b), $MachinePrecision] / t), $MachinePrecision] + N[(a + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, -2e-310], N[(N[(x + N[(z / N[(t / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * N[(y / t), $MachinePrecision] + N[(a + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.0], N[(N[(N[(t * N[(y * N[(z / t), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] / b), $MachinePrecision], If[LessEqual[t$95$2, 1e+263], N[(N[(N[(1.0 / t), $MachinePrecision] * N[(y * z), $MachinePrecision] + x), $MachinePrecision] / t$95$1), $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))) < -1.999999999999994e-310

   1. Initial program 91.7%

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

      1. lift-+.f64 N/A

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

      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 89.4

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

   4. Applied rewrites 89.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 93.1

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

   6. Applied rewrites 93.1%

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

   if -1.999999999999994e-310 < (/.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 47.0%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*r/ N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 27.3

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

   5. Applied rewrites 27.3%

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

   7. Applied rewrites 79.8%

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

   if -0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 1.00000000000000002e263

   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. 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. associate-/r/ N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 98.6

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

   4. Applied rewrites 98.6%

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

   if 1.00000000000000002e263 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t)))

   1. Initial program 23.3%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 80.8

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

   5. Applied rewrites 80.8%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{x + \frac{z}{\frac{t}{y}}}{\mathsf{fma}\left(b, \frac{y}{t}, a + 1\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq 0:\\ \;\;\;\;\frac{\frac{t \cdot \mathsf{fma}\left(y, \frac{z}{t}, x\right)}{y}}{b}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq 10^{+263}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{t}, y \cdot z, x\right)}{\frac{y \cdot b}{t} + \left(a + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 86.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y \cdot b}{t} + \left(a + 1\right)\\ t_2 := \frac{x + \frac{y \cdot z}{t}}{t\_1}\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{-310}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;\frac{\frac{t \cdot \mathsf{fma}\left(y, \frac{z}{t}, x\right)}{y}}{b}\\ \mathbf{elif}\;t\_2 \leq 10^{+263}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{t}, y \cdot z, x\right)}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (/ (* y b) t) (+ a 1.0))) (t_2 (/ (+ x (/ (* y z) t)) t_1)))
   (if (<= t_2 -2e-310)
     t_2
     (if (<= t_2 0.0)
       (/ (/ (* t (fma y (/ z t) x)) y) b)
       (if (<= t_2 1e+263) (/ (fma (/ 1.0 t) (* y z) x) t_1) (/ z b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((y * b) / t) + (a + 1.0);
	double t_2 = (x + ((y * z) / t)) / t_1;
	double tmp;
	if (t_2 <= -2e-310) {
		tmp = t_2;
	} else if (t_2 <= 0.0) {
		tmp = ((t * fma(y, (z / t), x)) / y) / b;
	} else if (t_2 <= 1e+263) {
		tmp = fma((1.0 / t), (y * z), x) / t_1;
	} else {
		tmp = z / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(y * b) / t) + Float64(a + 1.0))
	t_2 = Float64(Float64(x + Float64(Float64(y * z) / t)) / t_1)
	tmp = 0.0
	if (t_2 <= -2e-310)
		tmp = t_2;
	elseif (t_2 <= 0.0)
		tmp = Float64(Float64(Float64(t * fma(y, Float64(z / t), x)) / y) / b);
	elseif (t_2 <= 1e+263)
		tmp = Float64(fma(Float64(1.0 / t), Float64(y * z), x) / t_1);
	else
		tmp = Float64(z / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(y * b), $MachinePrecision] / t), $MachinePrecision] + N[(a + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, -2e-310], t$95$2, If[LessEqual[t$95$2, 0.0], N[(N[(N[(t * N[(y * N[(z / t), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] / b), $MachinePrecision], If[LessEqual[t$95$2, 1e+263], N[(N[(N[(1.0 / t), $MachinePrecision] * N[(y * z), $MachinePrecision] + x), $MachinePrecision] / t$95$1), $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))) < -1.999999999999994e-310

   1. Initial program 91.7%

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

   if -1.999999999999994e-310 < (/.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 47.0%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*r/ N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 27.3

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

   5. Applied rewrites 27.3%

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

   7. Applied rewrites 79.8%

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

   if -0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 1.00000000000000002e263

   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. 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. associate-/r/ N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 98.6

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

   4. Applied rewrites 98.6%

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

   if 1.00000000000000002e263 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t)))

   1. Initial program 23.3%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 80.8

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

   5. Applied rewrites 80.8%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq 0:\\ \;\;\;\;\frac{\frac{t \cdot \mathsf{fma}\left(y, \frac{z}{t}, x\right)}{y}}{b}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq 10^{+263}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{t}, y \cdot z, x\right)}{\frac{y \cdot b}{t} + \left(a + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 86.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-310}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{\frac{t \cdot \mathsf{fma}\left(y, \frac{z}{t}, x\right)}{y}}{b}\\ \mathbf{elif}\;t\_1 \leq 10^{+263}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (* y z) t)) (+ (/ (* y b) t) (+ a 1.0)))))
   (if (<= t_1 -2e-310)
     t_1
     (if (<= t_1 0.0)
       (/ (/ (* t (fma y (/ z t) x)) y) b)
       (if (<= t_1 1e+263) t_1 (/ 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 tmp;
	if (t_1 <= -2e-310) {
		tmp = t_1;
	} else if (t_1 <= 0.0) {
		tmp = ((t * fma(y, (z / t), x)) / y) / b;
	} else if (t_1 <= 1e+263) {
		tmp = t_1;
	} 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)))
	tmp = 0.0
	if (t_1 <= -2e-310)
		tmp = t_1;
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(Float64(t * fma(y, Float64(z / t), x)) / y) / b);
	elseif (t_1 <= 1e+263)
		tmp = t_1;
	else
		tmp = Float64(z / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(y * b), $MachinePrecision] / t), $MachinePrecision] + N[(a + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-310], t$95$1, If[LessEqual[t$95$1, 0.0], N[(N[(N[(t * N[(y * N[(z / t), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] / b), $MachinePrecision], If[LessEqual[t$95$1, 1e+263], t$95$1, N[(z / b), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -1.999999999999994e-310 or -0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 1.00000000000000002e263

   1. Initial program 95.1%

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

   if -1.999999999999994e-310 < (/.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 47.0%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*r/ N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 27.3

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

   5. Applied rewrites 27.3%

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

   7. Applied rewrites 79.8%

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

   if 1.00000000000000002e263 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t)))

   1. Initial program 23.3%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 80.8

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

   5. Applied rewrites 80.8%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq 0:\\ \;\;\;\;\frac{\frac{t \cdot \mathsf{fma}\left(y, \frac{z}{t}, x\right)}{y}}{b}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq 10^{+263}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 71.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{x + \frac{z}{\frac{t}{y}}}{a + 1}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-319}:\\ \;\;\;\;\frac{\frac{t \cdot \mathsf{fma}\left(y, \frac{z}{t}, x\right)}{y}}{b}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+251}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{t}, y \cdot z, x\right)}{a + 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{x}{y \cdot b}, \frac{z}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (* y z) t)) (+ (/ (* y b) t) (+ a 1.0)))))
   (if (<= t_1 -2e-310)
     (/ (+ x (/ z (/ t y))) (+ a 1.0))
     (if (<= t_1 5e-319)
       (/ (/ (* t (fma y (/ z t) x)) y) b)
       (if (<= t_1 2e+251)
         (/ (fma (/ 1.0 t) (* y z) x) (+ a 1.0))
         (fma t (/ x (* y b)) (/ 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 tmp;
	if (t_1 <= -2e-310) {
		tmp = (x + (z / (t / y))) / (a + 1.0);
	} else if (t_1 <= 5e-319) {
		tmp = ((t * fma(y, (z / t), x)) / y) / b;
	} else if (t_1 <= 2e+251) {
		tmp = fma((1.0 / t), (y * z), x) / (a + 1.0);
	} else {
		tmp = fma(t, (x / (y * b)), (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)))
	tmp = 0.0
	if (t_1 <= -2e-310)
		tmp = Float64(Float64(x + Float64(z / Float64(t / y))) / Float64(a + 1.0));
	elseif (t_1 <= 5e-319)
		tmp = Float64(Float64(Float64(t * fma(y, Float64(z / t), x)) / y) / b);
	elseif (t_1 <= 2e+251)
		tmp = Float64(fma(Float64(1.0 / t), Float64(y * z), x) / Float64(a + 1.0));
	else
		tmp = fma(t, Float64(x / Float64(y * b)), 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]}, If[LessEqual[t$95$1, -2e-310], N[(N[(x + N[(z / N[(t / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e-319], N[(N[(N[(t * N[(y * N[(z / t), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] / b), $MachinePrecision], If[LessEqual[t$95$1, 2e+251], N[(N[(N[(1.0 / t), $MachinePrecision] * N[(y * z), $MachinePrecision] + x), $MachinePrecision] / N[(a + 1.0), $MachinePrecision]), $MachinePrecision], N[(t * N[(x / N[(y * b), $MachinePrecision]), $MachinePrecision] + N[(z / b), $MachinePrecision]), $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))) < -1.999999999999994e-310

   1. Initial program 91.7%

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

      1. lift-+.f64 N/A

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

      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 89.4

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

   4. Applied rewrites 89.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 93.1

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

   6. Applied rewrites 93.1%

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

   7. Taylor expanded in b around 0

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

      1. lower-+.f64 78.6

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

   9. Applied rewrites 78.6%

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

   if -1.999999999999994e-310 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 4.9999937e-319

   1. Initial program 48.1%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*r/ N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 28.9

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

   5. Applied rewrites 28.9%

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

   7. Applied rewrites 80.3%

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

   if 4.9999937e-319 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 2.0000000000000001e251

   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. 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 96.2

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

   4. Applied rewrites 96.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 95.2

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

   6. Applied rewrites 95.2%

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

   7. Taylor expanded in b around 0

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

      1. lower-+.f64 80.2

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

   9. Applied rewrites 80.2%

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

       1. lift-+.f64 N/A

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

       2. +-commutative N/A

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

       3. lift-/.f64 N/A

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

       4. clear-num N/A

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

       5. lift-/.f64 N/A

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

       6. associate-/r* N/A

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

       7. lift-*.f64 N/A

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

       8. associate-/r/ N/A

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

       9. lower-fma.f64 N/A

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

       10. lower-/.f64 81.8

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

       11. lift-*.f64 N/A

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

       12. *-commutative N/A

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

       13. lower-*.f64 81.8

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

   11. Applied rewrites 81.8%

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

   if 2.0000000000000001e251 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t)))

   1. Initial program 25.2%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*r/ N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 26.7

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

   5. Applied rewrites 26.7%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 80.9%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{x + \frac{z}{\frac{t}{y}}}{a + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq 5 \cdot 10^{-319}:\\ \;\;\;\;\frac{\frac{t \cdot \mathsf{fma}\left(y, \frac{z}{t}, x\right)}{y}}{b}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq 2 \cdot 10^{+251}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{t}, y \cdot z, x\right)}{a + 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{x}{y \cdot b}, \frac{z}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 70.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)}\\ t_2 := \mathsf{fma}\left(t, \frac{x}{y \cdot b}, \frac{z}{b}\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{x + \frac{z}{\frac{t}{y}}}{a + 1}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-319}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+251}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{t}, y \cdot z, x\right)}{a + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

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 t (/ x (* y b)) (/ z b))))
   (if (<= t_1 -2e-310)
     (/ (+ x (/ z (/ t y))) (+ a 1.0))
     (if (<= t_1 5e-319)
       t_2
       (if (<= t_1 2e+251) (/ (fma (/ 1.0 t) (* y z) x) (+ a 1.0)) t_2)))))

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(t, (x / (y * b)), (z / b));
	double tmp;
	if (t_1 <= -2e-310) {
		tmp = (x + (z / (t / y))) / (a + 1.0);
	} else if (t_1 <= 5e-319) {
		tmp = t_2;
	} else if (t_1 <= 2e+251) {
		tmp = fma((1.0 / t), (y * z), x) / (a + 1.0);
	} else {
		tmp = t_2;
	}
	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(t, Float64(x / Float64(y * b)), Float64(z / b))
	tmp = 0.0
	if (t_1 <= -2e-310)
		tmp = Float64(Float64(x + Float64(z / Float64(t / y))) / Float64(a + 1.0));
	elseif (t_1 <= 5e-319)
		tmp = t_2;
	elseif (t_1 <= 2e+251)
		tmp = Float64(fma(Float64(1.0 / t), Float64(y * z), x) / Float64(a + 1.0));
	else
		tmp = t_2;
	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[(t * N[(x / N[(y * b), $MachinePrecision]), $MachinePrecision] + N[(z / b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-310], N[(N[(x + N[(z / N[(t / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e-319], t$95$2, If[LessEqual[t$95$1, 2e+251], N[(N[(N[(1.0 / t), $MachinePrecision] * N[(y * z), $MachinePrecision] + x), $MachinePrecision] / N[(a + 1.0), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

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))) < -1.999999999999994e-310

   1. Initial program 91.7%

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

      1. lift-+.f64 N/A

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

      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 89.4

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

   4. Applied rewrites 89.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 93.1

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

   6. Applied rewrites 93.1%

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

   7. Taylor expanded in b around 0

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

      1. lower-+.f64 78.6

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

   9. Applied rewrites 78.6%

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

   if -1.999999999999994e-310 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 4.9999937e-319 or 2.0000000000000001e251 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t)))

   1. Initial program 37.0%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*r/ N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 27.8

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

   5. Applied rewrites 27.8%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 77.4%

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

   if 4.9999937e-319 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 2.0000000000000001e251

   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. 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 96.2

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

   4. Applied rewrites 96.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 95.2

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

   6. Applied rewrites 95.2%

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

   7. Taylor expanded in b around 0

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

      1. lower-+.f64 80.2

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

   9. Applied rewrites 80.2%

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

       1. lift-+.f64 N/A

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

       2. +-commutative N/A

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

       3. lift-/.f64 N/A

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

       4. clear-num N/A

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

       5. lift-/.f64 N/A

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

       6. associate-/r* N/A

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

       7. lift-*.f64 N/A

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

       8. associate-/r/ N/A

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

       9. lower-fma.f64 N/A

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

       10. lower-/.f64 81.8

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

       11. lift-*.f64 N/A

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

       12. *-commutative N/A

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

       13. lower-*.f64 81.8

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

   11. Applied rewrites 81.8%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{x + \frac{z}{\frac{t}{y}}}{a + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq 5 \cdot 10^{-319}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{x}{y \cdot b}, \frac{z}{b}\right)\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq 2 \cdot 10^{+251}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{t}, y \cdot z, x\right)}{a + 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{x}{y \cdot b}, \frac{z}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 70.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)}\\ t_2 := \mathsf{fma}\left(t, \frac{x}{y \cdot b}, \frac{z}{b}\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}{a + 1}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-319}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+251}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{t}, y \cdot z, x\right)}{a + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

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 t (/ x (* y b)) (/ z b))))
   (if (<= t_1 -2e-310)
     (/ (fma z (/ y t) x) (+ a 1.0))
     (if (<= t_1 5e-319)
       t_2
       (if (<= t_1 2e+251) (/ (fma (/ 1.0 t) (* y z) x) (+ a 1.0)) t_2)))))

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(t, (x / (y * b)), (z / b));
	double tmp;
	if (t_1 <= -2e-310) {
		tmp = fma(z, (y / t), x) / (a + 1.0);
	} else if (t_1 <= 5e-319) {
		tmp = t_2;
	} else if (t_1 <= 2e+251) {
		tmp = fma((1.0 / t), (y * z), x) / (a + 1.0);
	} else {
		tmp = t_2;
	}
	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(t, Float64(x / Float64(y * b)), Float64(z / b))
	tmp = 0.0
	if (t_1 <= -2e-310)
		tmp = Float64(fma(z, Float64(y / t), x) / Float64(a + 1.0));
	elseif (t_1 <= 5e-319)
		tmp = t_2;
	elseif (t_1 <= 2e+251)
		tmp = Float64(fma(Float64(1.0 / t), Float64(y * z), x) / Float64(a + 1.0));
	else
		tmp = t_2;
	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[(t * N[(x / N[(y * b), $MachinePrecision]), $MachinePrecision] + N[(z / b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-310], N[(N[(z * N[(y / t), $MachinePrecision] + x), $MachinePrecision] / N[(a + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e-319], t$95$2, If[LessEqual[t$95$1, 2e+251], N[(N[(N[(1.0 / t), $MachinePrecision] * N[(y * z), $MachinePrecision] + x), $MachinePrecision] / N[(a + 1.0), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

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))) < -1.999999999999994e-310

   1. Initial program 91.7%

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

   3. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-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 78.5

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

   5. Applied rewrites 78.5%

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

   if -1.999999999999994e-310 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 4.9999937e-319 or 2.0000000000000001e251 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t)))

   1. Initial program 37.0%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*r/ N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 27.8

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

   5. Applied rewrites 27.8%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 77.4%

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

   if 4.9999937e-319 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 2.0000000000000001e251

   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. 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 96.2

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

   4. Applied rewrites 96.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 95.2

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

   6. Applied rewrites 95.2%

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

   7. Taylor expanded in b around 0

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

      1. lower-+.f64 80.2

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

   9. Applied rewrites 80.2%

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

       1. lift-+.f64 N/A

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

       2. +-commutative N/A

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

       3. lift-/.f64 N/A

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

       4. clear-num N/A

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

       5. lift-/.f64 N/A

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

       6. associate-/r* N/A

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

       7. lift-*.f64 N/A

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

       8. associate-/r/ N/A

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

       9. lower-fma.f64 N/A

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

       10. lower-/.f64 81.8

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

       11. lift-*.f64 N/A

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

       12. *-commutative N/A

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

       13. lower-*.f64 81.8

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

   11. Applied rewrites 81.8%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}{a + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq 5 \cdot 10^{-319}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{x}{y \cdot b}, \frac{z}{b}\right)\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq 2 \cdot 10^{+251}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{t}, y \cdot z, x\right)}{a + 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{x}{y \cdot b}, \frac{z}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 70.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{y \cdot z}{t}\\ t_2 := \frac{t\_1}{\frac{y \cdot b}{t} + \left(a + 1\right)}\\ t_3 := \mathsf{fma}\left(t, \frac{x}{y \cdot b}, \frac{z}{b}\right)\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}{a + 1}\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{-319}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+251}:\\ \;\;\;\;\frac{t\_1}{a + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

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 t (/ x (* y b)) (/ z b))))
   (if (<= t_2 -2e-310)
     (/ (fma z (/ y t) x) (+ a 1.0))
     (if (<= t_2 5e-319) t_3 (if (<= t_2 2e+251) (/ t_1 (+ a 1.0)) t_3)))))

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(t, (x / (y * b)), (z / b));
	double tmp;
	if (t_2 <= -2e-310) {
		tmp = fma(z, (y / t), x) / (a + 1.0);
	} else if (t_2 <= 5e-319) {
		tmp = t_3;
	} else if (t_2 <= 2e+251) {
		tmp = t_1 / (a + 1.0);
	} else {
		tmp = t_3;
	}
	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(t, Float64(x / Float64(y * b)), Float64(z / b))
	tmp = 0.0
	if (t_2 <= -2e-310)
		tmp = Float64(fma(z, Float64(y / t), x) / Float64(a + 1.0));
	elseif (t_2 <= 5e-319)
		tmp = t_3;
	elseif (t_2 <= 2e+251)
		tmp = Float64(t_1 / Float64(a + 1.0));
	else
		tmp = t_3;
	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[(t * N[(x / N[(y * b), $MachinePrecision]), $MachinePrecision] + N[(z / b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e-310], N[(N[(z * N[(y / t), $MachinePrecision] + x), $MachinePrecision] / N[(a + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e-319], t$95$3, If[LessEqual[t$95$2, 2e+251], N[(t$95$1 / N[(a + 1.0), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]

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))) < -1.999999999999994e-310

   1. Initial program 91.7%

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

   3. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-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 78.5

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

   5. Applied rewrites 78.5%

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

   if -1.999999999999994e-310 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 4.9999937e-319 or 2.0000000000000001e251 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t)))

   1. Initial program 37.0%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*r/ N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 27.8

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

   5. Applied rewrites 27.8%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 77.4%

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

   if 4.9999937e-319 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 2.0000000000000001e251

   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 81.8

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

   5. Applied rewrites 81.8%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}{a + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq 5 \cdot 10^{-319}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{x}{y \cdot b}, \frac{z}{b}\right)\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq 2 \cdot 10^{+251}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{a + 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{x}{y \cdot b}, \frac{z}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 70.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(t, \frac{x}{y \cdot b}, \frac{z}{b}\right)\\ t_2 := \frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)}\\ t_3 := \frac{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}{a + 1}\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{-310}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{-319}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+251}:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma t (/ x (* y b)) (/ z b)))
        (t_2 (/ (+ x (/ (* y z) t)) (+ (/ (* y b) t) (+ a 1.0))))
        (t_3 (/ (fma z (/ y t) x) (+ a 1.0))))
   (if (<= t_2 -2e-310)
     t_3
     (if (<= t_2 5e-319) t_1 (if (<= t_2 2e+251) t_3 t_1)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(t, (x / (y * b)), (z / b));
	double t_2 = (x + ((y * z) / t)) / (((y * b) / t) + (a + 1.0));
	double t_3 = fma(z, (y / t), x) / (a + 1.0);
	double tmp;
	if (t_2 <= -2e-310) {
		tmp = t_3;
	} else if (t_2 <= 5e-319) {
		tmp = t_1;
	} else if (t_2 <= 2e+251) {
		tmp = t_3;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(t, Float64(x / Float64(y * b)), Float64(z / b))
	t_2 = Float64(Float64(x + Float64(Float64(y * z) / t)) / Float64(Float64(Float64(y * b) / t) + Float64(a + 1.0)))
	t_3 = Float64(fma(z, Float64(y / t), x) / Float64(a + 1.0))
	tmp = 0.0
	if (t_2 <= -2e-310)
		tmp = t_3;
	elseif (t_2 <= 5e-319)
		tmp = t_1;
	elseif (t_2 <= 2e+251)
		tmp = t_3;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(t * N[(x / N[(y * b), $MachinePrecision]), $MachinePrecision] + N[(z / b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = 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$3 = N[(N[(z * N[(y / t), $MachinePrecision] + x), $MachinePrecision] / N[(a + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e-310], t$95$3, If[LessEqual[t$95$2, 5e-319], t$95$1, If[LessEqual[t$95$2, 2e+251], t$95$3, t$95$1]]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -1.999999999999994e-310 or 4.9999937e-319 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 2.0000000000000001e251

   1. Initial program 95.0%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   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 80.1

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

   5. Applied rewrites 80.1%

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

   if -1.999999999999994e-310 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 4.9999937e-319 or 2.0000000000000001e251 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t)))

   1. Initial program 37.0%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*r/ N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 27.8

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

   5. Applied rewrites 27.8%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 77.4%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}{a + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq 5 \cdot 10^{-319}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{x}{y \cdot b}, \frac{z}{b}\right)\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq 2 \cdot 10^{+251}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}{a + 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{x}{y \cdot b}, \frac{z}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 70.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)}\\ t_2 := \frac{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}{a + 1}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-310}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-319}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;t\_1 \leq 10^{+263}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

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 z (/ y t) x) (+ a 1.0))))
   (if (<= t_1 -2e-310)
     t_2
     (if (<= t_1 5e-319) (/ z b) (if (<= t_1 1e+263) 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(z, (y / t), x) / (a + 1.0);
	double tmp;
	if (t_1 <= -2e-310) {
		tmp = t_2;
	} else if (t_1 <= 5e-319) {
		tmp = z / b;
	} else if (t_1 <= 1e+263) {
		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(fma(z, Float64(y / t), x) / Float64(a + 1.0))
	tmp = 0.0
	if (t_1 <= -2e-310)
		tmp = t_2;
	elseif (t_1 <= 5e-319)
		tmp = Float64(z / b);
	elseif (t_1 <= 1e+263)
		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[(N[(z * N[(y / t), $MachinePrecision] + x), $MachinePrecision] / N[(a + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-310], t$95$2, If[LessEqual[t$95$1, 5e-319], N[(z / b), $MachinePrecision], If[LessEqual[t$95$1, 1e+263], t$95$2, N[(z / b), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -1.999999999999994e-310 or 4.9999937e-319 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 1.00000000000000002e263

   1. Initial program 95.1%

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

   3. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-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 79.7

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

   5. Applied rewrites 79.7%

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

   if -1.999999999999994e-310 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 4.9999937e-319 or 1.00000000000000002e263 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t)))

   1. Initial program 36.3%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 73.8

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

   5. Applied rewrites 73.8%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}{a + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq 5 \cdot 10^{-319}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq 10^{+263}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}{a + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 56.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)}\\ t_2 := \frac{x}{a + 1}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-310}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{-302}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;t\_1 \leq 10^{+263}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

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 (/ x (+ a 1.0))))
   (if (<= t_1 -2e-310)
     t_2
     (if (<= t_1 4e-302) (/ z b) (if (<= t_1 1e+263) 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 = x / (a + 1.0);
	double tmp;
	if (t_1 <= -2e-310) {
		tmp = t_2;
	} else if (t_1 <= 4e-302) {
		tmp = z / b;
	} else if (t_1 <= 1e+263) {
		tmp = t_2;
	} 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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (x + ((y * z) / t)) / (((y * b) / t) + (a + 1.0d0))
    t_2 = x / (a + 1.0d0)
    if (t_1 <= (-2d-310)) then
        tmp = t_2
    else if (t_1 <= 4d-302) then
        tmp = z / b
    else if (t_1 <= 1d+263) then
        tmp = t_2
    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 t_1 = (x + ((y * z) / t)) / (((y * b) / t) + (a + 1.0));
	double t_2 = x / (a + 1.0);
	double tmp;
	if (t_1 <= -2e-310) {
		tmp = t_2;
	} else if (t_1 <= 4e-302) {
		tmp = z / b;
	} else if (t_1 <= 1e+263) {
		tmp = t_2;
	} else {
		tmp = z / b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (x + ((y * z) / t)) / (((y * b) / t) + (a + 1.0))
	t_2 = x / (a + 1.0)
	tmp = 0
	if t_1 <= -2e-310:
		tmp = t_2
	elif t_1 <= 4e-302:
		tmp = z / b
	elif t_1 <= 1e+263:
		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(x / Float64(a + 1.0))
	tmp = 0.0
	if (t_1 <= -2e-310)
		tmp = t_2;
	elseif (t_1 <= 4e-302)
		tmp = Float64(z / b);
	elseif (t_1 <= 1e+263)
		tmp = t_2;
	else
		tmp = Float64(z / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (x + ((y * z) / t)) / (((y * b) / t) + (a + 1.0));
	t_2 = x / (a + 1.0);
	tmp = 0.0;
	if (t_1 <= -2e-310)
		tmp = t_2;
	elseif (t_1 <= 4e-302)
		tmp = z / b;
	elseif (t_1 <= 1e+263)
		tmp = t_2;
	else
		tmp = z / b;
	end
	tmp_2 = 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[(x / N[(a + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-310], t$95$2, If[LessEqual[t$95$1, 4e-302], N[(z / b), $MachinePrecision], If[LessEqual[t$95$1, 1e+263], t$95$2, N[(z / b), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -1.999999999999994e-310 or 3.9999999999999999e-302 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 1.00000000000000002e263

   1. Initial program 95.0%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   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 51.4

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

   5. Applied rewrites 51.4%

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

   if -1.999999999999994e-310 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 3.9999999999999999e-302 or 1.00000000000000002e263 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t)))

   1. Initial program 38.4%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 72.4

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

   5. Applied rewrites 72.4%

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

4. Final simplification 58.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{x}{a + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq 4 \cdot 10^{-302}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq 10^{+263}:\\ \;\;\;\;\frac{x}{a + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 39.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-272}:\\ \;\;\;\;\frac{x}{a}\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{-302}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;t\_1 \leq 10^{+263}:\\ \;\;\;\;\frac{x}{1}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (* y z) t)) (+ (/ (* y b) t) (+ a 1.0)))))
   (if (<= t_1 -2e-272)
     (/ x a)
     (if (<= t_1 4e-302) (/ z b) (if (<= t_1 1e+263) (/ x 1.0) (/ 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 tmp;
	if (t_1 <= -2e-272) {
		tmp = x / a;
	} else if (t_1 <= 4e-302) {
		tmp = z / b;
	} else if (t_1 <= 1e+263) {
		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) :: t_1
    real(8) :: tmp
    t_1 = (x + ((y * z) / t)) / (((y * b) / t) + (a + 1.0d0))
    if (t_1 <= (-2d-272)) then
        tmp = x / a
    else if (t_1 <= 4d-302) then
        tmp = z / b
    else if (t_1 <= 1d+263) 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 t_1 = (x + ((y * z) / t)) / (((y * b) / t) + (a + 1.0));
	double tmp;
	if (t_1 <= -2e-272) {
		tmp = x / a;
	} else if (t_1 <= 4e-302) {
		tmp = z / b;
	} else if (t_1 <= 1e+263) {
		tmp = x / 1.0;
	} else {
		tmp = z / b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (x + ((y * z) / t)) / (((y * b) / t) + (a + 1.0))
	tmp = 0
	if t_1 <= -2e-272:
		tmp = x / a
	elif t_1 <= 4e-302:
		tmp = z / b
	elif t_1 <= 1e+263:
		tmp = x / 1.0
	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)))
	tmp = 0.0
	if (t_1 <= -2e-272)
		tmp = Float64(x / a);
	elseif (t_1 <= 4e-302)
		tmp = Float64(z / b);
	elseif (t_1 <= 1e+263)
		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)
	t_1 = (x + ((y * z) / t)) / (((y * b) / t) + (a + 1.0));
	tmp = 0.0;
	if (t_1 <= -2e-272)
		tmp = x / a;
	elseif (t_1 <= 4e-302)
		tmp = z / b;
	elseif (t_1 <= 1e+263)
		tmp = x / 1.0;
	else
		tmp = z / b;
	end
	tmp_2 = 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]}, If[LessEqual[t$95$1, -2e-272], N[(x / a), $MachinePrecision], If[LessEqual[t$95$1, 4e-302], N[(z / b), $MachinePrecision], If[LessEqual[t$95$1, 1e+263], N[(x / 1.0), $MachinePrecision], 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))) < -1.99999999999999986e-272

   1. Initial program 91.6%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 13.1

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

   5. Applied rewrites 13.1%

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

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)}} \]
   7. 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. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 47.0

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

   8. Applied rewrites 47.0%

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

   9. Taylor expanded in a around inf

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

   11. Applied rewrites 26.9%

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

   if -1.99999999999999986e-272 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 3.9999999999999999e-302 or 1.00000000000000002e263 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t)))

   1. Initial program 39.1%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 71.7

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

   5. Applied rewrites 71.7%

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

   if 3.9999999999999999e-302 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 1.00000000000000002e263

   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

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 62.3

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

   5. Applied rewrites 62.3%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 35.5%

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

4. Final simplification 45.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq -2 \cdot 10^{-272}:\\ \;\;\;\;\frac{x}{a}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq 4 \cdot 10^{-302}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq 10^{+263}:\\ \;\;\;\;\frac{x}{1}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 66.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+243}:\\ \;\;\;\;\frac{y \cdot z}{\mathsf{fma}\left(y, b, \mathsf{fma}\left(t, a, t\right)\right)}\\ \mathbf{elif}\;t\_1 \leq 10^{+263}:\\ \;\;\;\;\frac{x}{1 + \mathsf{fma}\left(b, \frac{y}{t}, a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (* y z) t)) (+ (/ (* y b) t) (+ a 1.0)))))
   (if (<= t_1 -2e+243)
     (/ (* y z) (fma y b (fma t a t)))
     (if (<= t_1 1e+263) (/ x (+ 1.0 (fma b (/ y 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 tmp;
	if (t_1 <= -2e+243) {
		tmp = (y * z) / fma(y, b, fma(t, a, t));
	} else if (t_1 <= 1e+263) {
		tmp = x / (1.0 + fma(b, (y / 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)))
	tmp = 0.0
	if (t_1 <= -2e+243)
		tmp = Float64(Float64(y * z) / fma(y, b, fma(t, a, t)));
	elseif (t_1 <= 1e+263)
		tmp = Float64(x / Float64(1.0 + fma(b, Float64(y / 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]}, If[LessEqual[t$95$1, -2e+243], N[(N[(y * z), $MachinePrecision] / N[(y * b + N[(t * a + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+263], N[(x / N[(1.0 + N[(b * N[(y / t), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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))) < -2.0000000000000001e243

   1. Initial program 67.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 66.3

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

   5. Applied rewrites 66.3%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 66.3%

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

   if -2.0000000000000001e243 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 1.00000000000000002e263

   1. Initial program 87.1%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 24.3

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

   5. Applied rewrites 24.3%

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

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)}} \]
   7. 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. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 62.7

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

   8. Applied rewrites 62.7%

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

   if 1.00000000000000002e263 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t)))

   1. Initial program 23.3%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 80.8

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

   5. Applied rewrites 80.8%

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

4. Final simplification 65.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq -2 \cdot 10^{+243}:\\ \;\;\;\;\frac{y \cdot z}{\mathsf{fma}\left(y, b, \mathsf{fma}\left(t, a, t\right)\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq 10^{+263}:\\ \;\;\;\;\frac{x}{1 + \mathsf{fma}\left(b, \frac{y}{t}, a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 85.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{y \cdot z}{t}\\ \mathbf{if}\;\frac{t\_1}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq 10^{+263}:\\ \;\;\;\;\frac{t\_1}{\mathsf{fma}\left(b, \frac{y}{t}, a + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ x (/ (* y z) t))))
   (if (<= (/ t_1 (+ (/ (* y b) t) (+ a 1.0))) 1e+263)
     (/ t_1 (fma b (/ y t) (+ a 1.0)))
     (/ 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 tmp;
	if ((t_1 / (((y * b) / t) + (a + 1.0))) <= 1e+263) {
		tmp = t_1 / fma(b, (y / t), (a + 1.0));
	} else {
		tmp = z / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x + Float64(Float64(y * z) / t))
	tmp = 0.0
	if (Float64(t_1 / Float64(Float64(Float64(y * b) / t) + Float64(a + 1.0))) <= 1e+263)
		tmp = Float64(t_1 / fma(b, Float64(y / t), Float64(a + 1.0)));
	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]}, If[LessEqual[N[(t$95$1 / N[(N[(N[(y * b), $MachinePrecision] / t), $MachinePrecision] + N[(a + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+263], N[(t$95$1 / N[(b * N[(y / t), $MachinePrecision] + N[(a + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 1.00000000000000002e263

   1. Initial program 85.2%

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

      1. lift-+.f64 N/A

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

      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 86.9

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

   4. Applied rewrites 86.9%

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

   if 1.00000000000000002e263 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t)))

   1. Initial program 23.3%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 80.8

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

   5. Applied rewrites 80.8%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq 10^{+263}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\mathsf{fma}\left(b, \frac{y}{t}, a + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 83.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 82.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. 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 80.5

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

      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 81.3

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

   4. Applied rewrites 81.3%

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

   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 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 82.9%

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

Alternative 20: 58.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{a + 1}\\ \mathbf{if}\;t \leq -1.24 \cdot 10^{-20}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 820:\\ \;\;\;\;\frac{y \cdot z}{\mathsf{fma}\left(y, b, \mathsf{fma}\left(t, a, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ x (+ a 1.0))))
   (if (<= t -1.24e-20)
     t_1
     (if (<= t 820.0) (/ (* y z) (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 <= -1.24e-20) {
		tmp = t_1;
	} else if (t <= 820.0) {
		tmp = (y * z) / 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 <= -1.24e-20)
		tmp = t_1;
	elseif (t <= 820.0)
		tmp = Float64(Float64(y * z) / 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, -1.24e-20], t$95$1, If[LessEqual[t, 820.0], N[(N[(y * z), $MachinePrecision] / N[(y * b + N[(t * a + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.2399999999999999e-20 or 820 < t

   1. Initial program 84.5%

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

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

   if -1.2399999999999999e-20 < t < 820

   1. Initial program 66.8%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 47.9

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

   5. Applied rewrites 47.9%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 63.5%

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

4. Final simplification 61.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.24 \cdot 10^{-20}:\\ \;\;\;\;\frac{x}{a + 1}\\ \mathbf{elif}\;t \leq 820:\\ \;\;\;\;\frac{y \cdot z}{\mathsf{fma}\left(y, b, \mathsf{fma}\left(t, a, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 40.4% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.00055:\\ \;\;\;\;\frac{x}{a}\\ \mathbf{elif}\;a \leq 4.7 \cdot 10^{-20}:\\ \;\;\;\;x - x \cdot a\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= a -0.00055) (/ x a) (if (<= a 4.7e-20) (- x (* x a)) (/ x a))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= -0.00055) {
		tmp = x / a;
	} else if (a <= 4.7e-20) {
		tmp = x - (x * a);
	} else {
		tmp = x / a;
	}
	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 (a <= (-0.00055d0)) then
        tmp = x / a
    else if (a <= 4.7d-20) then
        tmp = x - (x * a)
    else
        tmp = x / a
    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 (a <= -0.00055) {
		tmp = x / a;
	} else if (a <= 4.7e-20) {
		tmp = x - (x * a);
	} else {
		tmp = x / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if a <= -0.00055:
		tmp = x / a
	elif a <= 4.7e-20:
		tmp = x - (x * a)
	else:
		tmp = x / a
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (a <= -0.00055)
		tmp = Float64(x / a);
	elseif (a <= 4.7e-20)
		tmp = Float64(x - Float64(x * a));
	else
		tmp = Float64(x / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (a <= -0.00055)
		tmp = x / a;
	elseif (a <= 4.7e-20)
		tmp = x - (x * a);
	else
		tmp = x / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[a, -0.00055], N[(x / a), $MachinePrecision], If[LessEqual[a, 4.7e-20], N[(x - N[(x * a), $MachinePrecision]), $MachinePrecision], N[(x / a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -5.50000000000000033e-4 or 4.70000000000000015e-20 < a

   1. Initial program 70.9%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 29.6

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

   5. Applied rewrites 29.6%

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

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)}} \]
   7. 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. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 49.2

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

   8. Applied rewrites 49.2%

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

   9. Taylor expanded in a around inf

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

   11. Applied rewrites 39.1%

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

   if -5.50000000000000033e-4 < a < 4.70000000000000015e-20

   1. Initial program 79.9%

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

   3. Taylor expanded in y around 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 35.3

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

   5. Applied rewrites 35.3%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 35.3%

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

Alternative 22: 19.0% 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 75.3%

   \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
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 37.6

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

5. Applied rewrites 37.6%

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

6. Taylor expanded in a around 0

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

8. Applied rewrites 19.6%

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

Alternative 23: 4.2% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ -x \cdot a \end{array} \]

FPCore

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

C

double code(double x, double y, double z, double t, double a, double b) {
	return -(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 * a)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.3%

   \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
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 37.6

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

5. Applied rewrites 37.6%

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

6. Taylor expanded in a around 0

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

8. Applied rewrites 19.6%

   \[\leadsto x - \color{blue}{x \cdot a} \]

9. Taylor expanded in a around inf

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

11. Applied rewrites 6.8%

    \[\leadsto x \cdot \left(-a\right) \]

12. Final simplification 6.8%

    \[\leadsto -x \cdot a \]
13. Add Preprocessing

Developer Target 1: 79.0% 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 2024232 
(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))))