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

Percentage Accurate: 75.3% → 90.6%

Time: 11.8s

Alternatives: 20

Speedup: 0.3×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 75.3% 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: 90.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)}\\ t_2 := \mathsf{fma}\left(\frac{y}{t}, b, 1 + a\right)\\ t_3 := \mathsf{fma}\left(\frac{b}{t}, y, a\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+277}:\\ \;\;\;\;\left(\frac{x}{\mathsf{fma}\left(z, t\_3, z\right)} + \frac{y}{\mathsf{fma}\left(t\_3, t, t\right)}\right) \cdot z\\ \mathbf{elif}\;t\_1 \leq -4 \cdot 10^{-321}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}{\frac{b}{\frac{t}{y}} + \left(1 + a\right)}\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{z}{b} + \frac{\left(\frac{x}{b} - \frac{\frac{\mathsf{fma}\left(z, a, z\right)}{b}}{b}\right) \cdot t}{y}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t\_2}, \frac{y}{t}, \frac{x}{t\_2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (+ (/ (* z y) t) x) (+ (/ (* b y) t) (+ 1.0 a))))
        (t_2 (fma (/ y t) b (+ 1.0 a)))
        (t_3 (fma (/ b t) y a)))
   (if (<= t_1 -5e+277)
     (* (+ (/ x (fma z t_3 z)) (/ y (fma t_3 t t))) z)
     (if (<= t_1 -4e-321)
       (/ (fma z (/ y t) x) (+ (/ b (/ t y)) (+ 1.0 a)))
       (if (<= t_1 0.0)
         (+ (/ z b) (/ (* (- (/ x b) (/ (/ (fma z a z) b) b)) t) y))
         (if (<= t_1 INFINITY)
           (fma (/ z t_2) (/ y t) (/ x t_2))
           (/ (fma t (/ x y) z) b)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (((z * y) / t) + x) / (((b * y) / t) + (1.0 + a));
	double t_2 = fma((y / t), b, (1.0 + a));
	double t_3 = fma((b / t), y, a);
	double tmp;
	if (t_1 <= -5e+277) {
		tmp = ((x / fma(z, t_3, z)) + (y / fma(t_3, t, t))) * z;
	} else if (t_1 <= -4e-321) {
		tmp = fma(z, (y / t), x) / ((b / (t / y)) + (1.0 + a));
	} else if (t_1 <= 0.0) {
		tmp = (z / b) + ((((x / b) - ((fma(z, a, z) / b) / b)) * t) / y);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = fma((z / t_2), (y / t), (x / t_2));
	} else {
		tmp = fma(t, (x / y), z) / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(Float64(z * y) / t) + x) / Float64(Float64(Float64(b * y) / t) + Float64(1.0 + a)))
	t_2 = fma(Float64(y / t), b, Float64(1.0 + a))
	t_3 = fma(Float64(b / t), y, a)
	tmp = 0.0
	if (t_1 <= -5e+277)
		tmp = Float64(Float64(Float64(x / fma(z, t_3, z)) + Float64(y / fma(t_3, t, t))) * z);
	elseif (t_1 <= -4e-321)
		tmp = Float64(fma(z, Float64(y / t), x) / Float64(Float64(b / Float64(t / y)) + Float64(1.0 + a)));
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(z / b) + Float64(Float64(Float64(Float64(x / b) - Float64(Float64(fma(z, a, z) / b) / b)) * t) / y));
	elseif (t_1 <= Inf)
		tmp = fma(Float64(z / t_2), Float64(y / t), Float64(x / t_2));
	else
		tmp = Float64(fma(t, Float64(x / y), z) / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(N[(z * y), $MachinePrecision] / t), $MachinePrecision] + x), $MachinePrecision] / N[(N[(N[(b * y), $MachinePrecision] / t), $MachinePrecision] + N[(1.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y / t), $MachinePrecision] * b + N[(1.0 + a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(b / t), $MachinePrecision] * y + a), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+277], N[(N[(N[(x / N[(z * t$95$3 + z), $MachinePrecision]), $MachinePrecision] + N[(y / N[(t$95$3 * t + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[t$95$1, -4e-321], N[(N[(z * N[(y / t), $MachinePrecision] + x), $MachinePrecision] / N[(N[(b / N[(t / y), $MachinePrecision]), $MachinePrecision] + N[(1.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(N[(z / b), $MachinePrecision] + N[(N[(N[(N[(x / b), $MachinePrecision] - N[(N[(N[(z * a + z), $MachinePrecision] / b), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(z / t$95$2), $MachinePrecision] * N[(y / t), $MachinePrecision] + N[(x / t$95$2), $MachinePrecision]), $MachinePrecision], N[(N[(t * N[(x / y), $MachinePrecision] + z), $MachinePrecision] / b), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -4.99999999999999982e277

   1. Initial program 27.4%

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

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 94.0%

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

   if -4.99999999999999982e277 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -4.00193e-321

   1. Initial program 98.3%

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

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 99.2

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

   4. Applied rewrites 99.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 99.3

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

   6. Applied rewrites 99.3%

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

   if -4.00193e-321 < (/.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 47.7%

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

   3. Taylor expanded in t around 0

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

      1. lower-+.f64 N/A

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

   5. Applied rewrites 71.5%

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

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

   1. Initial program 88.8%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 70.8

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

   5. Applied rewrites 70.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lift-/.f64 N/A

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

      7. lower-fma.f64 75.5

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

   7. Applied rewrites 75.5%

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

   8. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-rgt-in N/A

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

      7. *-lft-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. associate-/l* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

   10. Applied rewrites 93.3%

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

   12. Applied rewrites 96.0%

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

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

   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 b around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 5.1

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

   5. Applied rewrites 5.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lift-/.f64 N/A

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

      7. lower-fma.f64 15.9

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

   7. Applied rewrites 15.9%

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

   8. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-rgt-in N/A

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

      7. *-lft-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. associate-/l* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

   10. Applied rewrites 10.2%

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

   11. Taylor expanded in b around inf

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

   13. Applied rewrites 90.0%

       \[\leadsto \frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{\color{blue}{b}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq -5 \cdot 10^{+277}:\\ \;\;\;\;\left(\frac{x}{\mathsf{fma}\left(z, \mathsf{fma}\left(\frac{b}{t}, y, a\right), z\right)} + \frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{b}{t}, y, a\right), t, t\right)}\right) \cdot z\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq -4 \cdot 10^{-321}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}{\frac{b}{\frac{t}{y}} + \left(1 + a\right)}\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq 0:\\ \;\;\;\;\frac{z}{b} + \frac{\left(\frac{x}{b} - \frac{\frac{\mathsf{fma}\left(z, a, z\right)}{b}}{b}\right) \cdot t}{y}\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{\mathsf{fma}\left(\frac{y}{t}, b, 1 + a\right)}, \frac{y}{t}, \frac{x}{\mathsf{fma}\left(\frac{y}{t}, b, 1 + a\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 45.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)}\\ t_2 := \mathsf{fma}\left(-x, a, x\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{-197}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;t\_1 \leq 10^{+16}:\\ \;\;\;\;\frac{x}{a}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+288}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (+ (/ (* z y) t) x) (+ (/ (* b y) t) (+ 1.0 a))))
        (t_2 (fma (- x) a x)))
   (if (<= t_1 (- INFINITY))
     (/ z b)
     (if (<= t_1 -1e-197)
       t_2
       (if (<= t_1 0.0)
         (/ z b)
         (if (<= t_1 1e+16) (/ x a) (if (<= t_1 2e+288) t_2 (/ z b))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (((z * y) / t) + x) / (((b * y) / t) + (1.0 + a));
	double t_2 = fma(-x, a, x);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = z / b;
	} else if (t_1 <= -1e-197) {
		tmp = t_2;
	} else if (t_1 <= 0.0) {
		tmp = z / b;
	} else if (t_1 <= 1e+16) {
		tmp = x / a;
	} else if (t_1 <= 2e+288) {
		tmp = t_2;
	} else {
		tmp = z / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(Float64(z * y) / t) + x) / Float64(Float64(Float64(b * y) / t) + Float64(1.0 + a)))
	t_2 = fma(Float64(-x), a, x)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(z / b);
	elseif (t_1 <= -1e-197)
		tmp = t_2;
	elseif (t_1 <= 0.0)
		tmp = Float64(z / b);
	elseif (t_1 <= 1e+16)
		tmp = Float64(x / a);
	elseif (t_1 <= 2e+288)
		tmp = t_2;
	else
		tmp = Float64(z / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(N[(z * y), $MachinePrecision] / t), $MachinePrecision] + x), $MachinePrecision] / N[(N[(N[(b * y), $MachinePrecision] / t), $MachinePrecision] + N[(1.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[((-x) * a + x), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(z / b), $MachinePrecision], If[LessEqual[t$95$1, -1e-197], t$95$2, If[LessEqual[t$95$1, 0.0], N[(z / b), $MachinePrecision], If[LessEqual[t$95$1, 1e+16], N[(x / a), $MachinePrecision], If[LessEqual[t$95$1, 2e+288], t$95$2, N[(z / b), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -inf.0 or -9.9999999999999999e-198 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -0.0 or 2e288 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t)))

   1. Initial program 35.9%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 61.3

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

   5. Applied rewrites 61.3%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -9.9999999999999999e-198 or 1e16 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 2e288

   1. Initial program 98.7%

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

      1. lift-+.f64 N/A

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

      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. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 99.3

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 59.1

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

   7. Applied rewrites 59.1%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 41.6%

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

   if -0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 1e16

   1. Initial program 96.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 49.6

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

   7. Applied rewrites 49.6%

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

   8. Taylor expanded in a around inf

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

   10. Applied rewrites 38.5%

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

4. Final simplification 49.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq -\infty:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq -1 \cdot 10^{-197}:\\ \;\;\;\;\mathsf{fma}\left(-x, a, x\right)\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq 0:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq 10^{+16}:\\ \;\;\;\;\frac{x}{a}\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq 2 \cdot 10^{+288}:\\ \;\;\;\;\mathsf{fma}\left(-x, a, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{b \cdot y}{t} + \left(1 + a\right)\\ t_2 := \frac{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}{t\_1}\\ t_3 := \frac{\frac{z \cdot y}{t} + x}{t\_1}\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{b}{t}, y, a\right), t, t\right)} \cdot z\\ \mathbf{elif}\;t\_3 \leq -4 \cdot 10^{-321}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{x}{b \cdot y}, \frac{z}{b}\right)\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (/ (* b y) t) (+ 1.0 a)))
        (t_2 (/ (fma z (/ y t) x) t_1))
        (t_3 (/ (+ (/ (* z y) t) x) t_1)))
   (if (<= t_3 (- INFINITY))
     (* (/ y (fma (fma (/ b t) y a) t t)) z)
     (if (<= t_3 -4e-321)
       t_2
       (if (<= t_3 0.0)
         (fma t (/ x (* b y)) (/ z b))
         (if (<= t_3 INFINITY) t_2 (/ (fma t (/ x y) z) b)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((b * y) / t) + (1.0 + a);
	double t_2 = fma(z, (y / t), x) / t_1;
	double t_3 = (((z * y) / t) + x) / t_1;
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = (y / fma(fma((b / t), y, a), t, t)) * z;
	} else if (t_3 <= -4e-321) {
		tmp = t_2;
	} else if (t_3 <= 0.0) {
		tmp = fma(t, (x / (b * y)), (z / b));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = t_2;
	} else {
		tmp = fma(t, (x / y), z) / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(b * y) / t) + Float64(1.0 + a))
	t_2 = Float64(fma(z, Float64(y / t), x) / t_1)
	t_3 = Float64(Float64(Float64(Float64(z * y) / t) + x) / t_1)
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(Float64(y / fma(fma(Float64(b / t), y, a), t, t)) * z);
	elseif (t_3 <= -4e-321)
		tmp = t_2;
	elseif (t_3 <= 0.0)
		tmp = fma(t, Float64(x / Float64(b * y)), Float64(z / b));
	elseif (t_3 <= Inf)
		tmp = t_2;
	else
		tmp = Float64(fma(t, Float64(x / y), z) / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(b * y), $MachinePrecision] / t), $MachinePrecision] + N[(1.0 + a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(z * N[(y / t), $MachinePrecision] + x), $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[(z * y), $MachinePrecision] / t), $MachinePrecision] + x), $MachinePrecision] / t$95$1), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], N[(N[(y / N[(N[(N[(b / t), $MachinePrecision] * y + a), $MachinePrecision] * t + t), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[t$95$3, -4e-321], t$95$2, If[LessEqual[t$95$3, 0.0], N[(t * N[(x / N[(b * y), $MachinePrecision]), $MachinePrecision] + N[(z / b), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], t$95$2, N[(N[(t * N[(x / y), $MachinePrecision] + z), $MachinePrecision] / b), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 17.8%

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

   3. Taylor expanded in z around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 86.9%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -4.00193e-321 or -0.0 < (/.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 92.6%

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

      1. lift-+.f64 N/A

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

      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. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 96.3

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

   4. Applied rewrites 96.3%

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

   if -4.00193e-321 < (/.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 47.7%

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

   3. Taylor expanded in b around 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-*l/ N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 28.4

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

   5. Applied rewrites 28.4%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 69.4%

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

   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 b around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 5.1

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

   5. Applied rewrites 5.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lift-/.f64 N/A

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

      7. lower-fma.f64 15.9

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

   7. Applied rewrites 15.9%

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

   8. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-rgt-in N/A

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

      7. *-lft-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. associate-/l* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

   10. Applied rewrites 10.2%

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

   11. Taylor expanded in b around inf

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

   13. Applied rewrites 90.0%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq -\infty:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{b}{t}, y, a\right), t, t\right)} \cdot z\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq -4 \cdot 10^{-321}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}{\frac{b \cdot y}{t} + \left(1 + a\right)}\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq 0:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{x}{b \cdot y}, \frac{z}{b}\right)\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq \infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}{\frac{b \cdot y}{t} + \left(1 + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{b}{t}, y, a\right), t, t\right)} \cdot z\\ \mathbf{elif}\;t\_1 \leq -4 \cdot 10^{-321}:\\ \;\;\;\;\frac{\frac{z}{\frac{t}{y}} + x}{1 + a}\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{x}{b \cdot y}, \frac{z}{b}\right)\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{1 + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (+ (/ (* z y) t) x) (+ (/ (* b y) t) (+ 1.0 a)))))
   (if (<= t_1 (- INFINITY))
     (* (/ y (fma (fma (/ b t) y a) t t)) z)
     (if (<= t_1 -4e-321)
       (/ (+ (/ z (/ t y)) x) (+ 1.0 a))
       (if (<= t_1 0.0)
         (fma t (/ x (* b y)) (/ z b))
         (if (<= t_1 INFINITY)
           (/ (fma (/ y t) z x) (+ 1.0 a))
           (/ (fma t (/ x y) z) b)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (((z * y) / t) + x) / (((b * y) / t) + (1.0 + a));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (y / fma(fma((b / t), y, a), t, t)) * z;
	} else if (t_1 <= -4e-321) {
		tmp = ((z / (t / y)) + x) / (1.0 + a);
	} else if (t_1 <= 0.0) {
		tmp = fma(t, (x / (b * y)), (z / b));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = fma((y / t), z, x) / (1.0 + a);
	} else {
		tmp = fma(t, (x / y), z) / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(Float64(z * y) / t) + x) / Float64(Float64(Float64(b * y) / t) + Float64(1.0 + a)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(y / fma(fma(Float64(b / t), y, a), t, t)) * z);
	elseif (t_1 <= -4e-321)
		tmp = Float64(Float64(Float64(z / Float64(t / y)) + x) / Float64(1.0 + a));
	elseif (t_1 <= 0.0)
		tmp = fma(t, Float64(x / Float64(b * y)), Float64(z / b));
	elseif (t_1 <= Inf)
		tmp = Float64(fma(Float64(y / t), z, x) / Float64(1.0 + a));
	else
		tmp = Float64(fma(t, Float64(x / y), z) / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(N[(z * y), $MachinePrecision] / t), $MachinePrecision] + x), $MachinePrecision] / N[(N[(N[(b * y), $MachinePrecision] / t), $MachinePrecision] + N[(1.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(y / N[(N[(N[(b / t), $MachinePrecision] * y + a), $MachinePrecision] * t + t), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[t$95$1, -4e-321], N[(N[(N[(z / N[(t / y), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] / N[(1.0 + a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(t * N[(x / N[(b * y), $MachinePrecision]), $MachinePrecision] + N[(z / b), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(N[(y / t), $MachinePrecision] * z + x), $MachinePrecision] / N[(1.0 + a), $MachinePrecision]), $MachinePrecision], N[(N[(t * N[(x / y), $MachinePrecision] + z), $MachinePrecision] / b), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 17.8%

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

   3. Taylor expanded in z around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 86.9%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -4.00193e-321

   1. Initial program 98.4%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 77.7

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

   5. Applied rewrites 77.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 78.7

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

   7. Applied rewrites 78.7%

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

   if -4.00193e-321 < (/.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 47.7%

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

   3. Taylor expanded in b around 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-*l/ N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 28.4

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

   5. Applied rewrites 28.4%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 69.4%

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

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

   1. Initial program 88.8%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 70.8

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

   5. Applied rewrites 70.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lift-/.f64 N/A

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

      7. lower-fma.f64 75.5

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

   7. Applied rewrites 75.5%

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

   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 b around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 5.1

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

   5. Applied rewrites 5.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lift-/.f64 N/A

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

      7. lower-fma.f64 15.9

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

   7. Applied rewrites 15.9%

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

   8. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-rgt-in N/A

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

      7. *-lft-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. associate-/l* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

   10. Applied rewrites 10.2%

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

   11. Taylor expanded in b around inf

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

   13. Applied rewrites 90.0%

       \[\leadsto \frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{\color{blue}{b}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq -\infty:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{b}{t}, y, a\right), t, t\right)} \cdot z\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq -4 \cdot 10^{-321}:\\ \;\;\;\;\frac{\frac{z}{\frac{t}{y}} + x}{1 + a}\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq 0:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{x}{b \cdot y}, \frac{z}{b}\right)\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq \infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{1 + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{1 + a}\\ t_2 := \frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{b}{t}, y, a\right), t, t\right)} \cdot z\\ \mathbf{elif}\;t\_2 \leq -4 \cdot 10^{-321}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{x}{b \cdot y}, \frac{z}{b}\right)\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (fma (/ y t) z x) (+ 1.0 a)))
        (t_2 (/ (+ (/ (* z y) t) x) (+ (/ (* b y) t) (+ 1.0 a)))))
   (if (<= t_2 (- INFINITY))
     (* (/ y (fma (fma (/ b t) y a) t t)) z)
     (if (<= t_2 -4e-321)
       t_1
       (if (<= t_2 0.0)
         (fma t (/ x (* b y)) (/ z b))
         (if (<= t_2 INFINITY) t_1 (/ (fma t (/ x y) z) b)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma((y / t), z, x) / (1.0 + a);
	double t_2 = (((z * y) / t) + x) / (((b * y) / t) + (1.0 + a));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = (y / fma(fma((b / t), y, a), t, t)) * z;
	} else if (t_2 <= -4e-321) {
		tmp = t_1;
	} else if (t_2 <= 0.0) {
		tmp = fma(t, (x / (b * y)), (z / b));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = fma(t, (x / y), z) / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(fma(Float64(y / t), z, x) / Float64(1.0 + a))
	t_2 = Float64(Float64(Float64(Float64(z * y) / t) + x) / Float64(Float64(Float64(b * y) / t) + Float64(1.0 + a)))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(Float64(y / fma(fma(Float64(b / t), y, a), t, t)) * z);
	elseif (t_2 <= -4e-321)
		tmp = t_1;
	elseif (t_2 <= 0.0)
		tmp = fma(t, Float64(x / Float64(b * y)), Float64(z / b));
	elseif (t_2 <= Inf)
		tmp = t_1;
	else
		tmp = Float64(fma(t, Float64(x / y), z) / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(y / t), $MachinePrecision] * z + x), $MachinePrecision] / N[(1.0 + a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(z * y), $MachinePrecision] / t), $MachinePrecision] + x), $MachinePrecision] / N[(N[(N[(b * y), $MachinePrecision] / t), $MachinePrecision] + N[(1.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(N[(y / N[(N[(N[(b / t), $MachinePrecision] * y + a), $MachinePrecision] * t + t), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[t$95$2, -4e-321], t$95$1, If[LessEqual[t$95$2, 0.0], N[(t * N[(x / N[(b * y), $MachinePrecision]), $MachinePrecision] + N[(z / b), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], t$95$1, N[(N[(t * N[(x / y), $MachinePrecision] + z), $MachinePrecision] / b), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 17.8%

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

   3. Taylor expanded in z around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 86.9%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -4.00193e-321 or -0.0 < (/.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 92.6%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 73.6

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

   5. Applied rewrites 73.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lift-/.f64 N/A

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

      7. lower-fma.f64 76.7

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

   7. Applied rewrites 76.7%

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

   if -4.00193e-321 < (/.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 47.7%

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

   3. Taylor expanded in b around 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-*l/ N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 28.4

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

   5. Applied rewrites 28.4%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 69.4%

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

   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 b around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 5.1

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

   5. Applied rewrites 5.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lift-/.f64 N/A

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

      7. lower-fma.f64 15.9

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

   7. Applied rewrites 15.9%

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

   8. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-rgt-in N/A

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

      7. *-lft-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. associate-/l* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

   10. Applied rewrites 10.2%

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

   11. Taylor expanded in b around inf

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

   13. Applied rewrites 90.0%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq -\infty:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{b}{t}, y, a\right), t, t\right)} \cdot z\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq -4 \cdot 10^{-321}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{1 + a}\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq 0:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{x}{b \cdot y}, \frac{z}{b}\right)\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq \infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{1 + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 90.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{b}{t}, y, a\right)\\ t_2 := \frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)}\\ t_3 := \left(\frac{x}{\mathsf{fma}\left(z, t\_1, z\right)} + \frac{y}{\mathsf{fma}\left(t\_1, t, t\right)}\right) \cdot z\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+277}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+288}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}{\frac{b}{\frac{t}{y}} + \left(1 + a\right)}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma (/ b t) y a))
        (t_2 (/ (+ (/ (* z y) t) x) (+ (/ (* b y) t) (+ 1.0 a))))
        (t_3 (* (+ (/ x (fma z t_1 z)) (/ y (fma t_1 t t))) z)))
   (if (<= t_2 -5e+277)
     t_3
     (if (<= t_2 2e+288)
       (/ (fma z (/ y t) x) (+ (/ b (/ t y)) (+ 1.0 a)))
       (if (<= t_2 INFINITY) t_3 (/ (fma t (/ x y) z) b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma((b / t), y, a);
	double t_2 = (((z * y) / t) + x) / (((b * y) / t) + (1.0 + a));
	double t_3 = ((x / fma(z, t_1, z)) + (y / fma(t_1, t, t))) * z;
	double tmp;
	if (t_2 <= -5e+277) {
		tmp = t_3;
	} else if (t_2 <= 2e+288) {
		tmp = fma(z, (y / t), x) / ((b / (t / y)) + (1.0 + a));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_3;
	} else {
		tmp = fma(t, (x / y), z) / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(Float64(b / t), y, a)
	t_2 = Float64(Float64(Float64(Float64(z * y) / t) + x) / Float64(Float64(Float64(b * y) / t) + Float64(1.0 + a)))
	t_3 = Float64(Float64(Float64(x / fma(z, t_1, z)) + Float64(y / fma(t_1, t, t))) * z)
	tmp = 0.0
	if (t_2 <= -5e+277)
		tmp = t_3;
	elseif (t_2 <= 2e+288)
		tmp = Float64(fma(z, Float64(y / t), x) / Float64(Float64(b / Float64(t / y)) + Float64(1.0 + a)));
	elseif (t_2 <= Inf)
		tmp = t_3;
	else
		tmp = Float64(fma(t, Float64(x / y), z) / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(b / t), $MachinePrecision] * y + a), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(z * y), $MachinePrecision] / t), $MachinePrecision] + x), $MachinePrecision] / N[(N[(N[(b * y), $MachinePrecision] / t), $MachinePrecision] + N[(1.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(x / N[(z * t$95$1 + z), $MachinePrecision]), $MachinePrecision] + N[(y / N[(t$95$1 * t + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+277], t$95$3, If[LessEqual[t$95$2, 2e+288], N[(N[(z * N[(y / t), $MachinePrecision] + x), $MachinePrecision] / N[(N[(b / N[(t / y), $MachinePrecision]), $MachinePrecision] + N[(1.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], t$95$3, N[(N[(t * N[(x / y), $MachinePrecision] + z), $MachinePrecision] / b), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -4.99999999999999982e277 or 2e288 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < +inf.0

   1. Initial program 37.3%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 94.1%

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

   if -4.99999999999999982e277 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 2e288

   1. Initial program 88.0%

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

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 87.1

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

   4. Applied rewrites 87.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 90.1

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

   6. Applied rewrites 90.1%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 5.1

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

   5. Applied rewrites 5.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lift-/.f64 N/A

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

      7. lower-fma.f64 15.9

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

   7. Applied rewrites 15.9%

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

   8. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-rgt-in N/A

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

      7. *-lft-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. associate-/l* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

   10. Applied rewrites 10.2%

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

   11. Taylor expanded in b around inf

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

   13. Applied rewrites 90.0%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq -5 \cdot 10^{+277}:\\ \;\;\;\;\left(\frac{x}{\mathsf{fma}\left(z, \mathsf{fma}\left(\frac{b}{t}, y, a\right), z\right)} + \frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{b}{t}, y, a\right), t, t\right)}\right) \cdot z\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq 2 \cdot 10^{+288}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}{\frac{b}{\frac{t}{y}} + \left(1 + a\right)}\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq \infty:\\ \;\;\;\;\left(\frac{x}{\mathsf{fma}\left(z, \mathsf{fma}\left(\frac{b}{t}, y, a\right), z\right)} + \frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{b}{t}, y, a\right), t, t\right)}\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 88.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)}\\ t_2 := \mathsf{fma}\left(b, \frac{y}{t}, a\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{b}{t}, y, a\right), t, t\right)} \cdot z\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+161}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}{\frac{b}{\frac{t}{y}} + \left(1 + a\right)}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{\mathsf{fma}\left(t\_2, t, t\right)}, \frac{x}{t\_2 + 1}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (+ (/ (* z y) t) x) (+ (/ (* b y) t) (+ 1.0 a))))
        (t_2 (fma b (/ y t) a)))
   (if (<= t_1 (- INFINITY))
     (* (/ y (fma (fma (/ b t) y a) t t)) z)
     (if (<= t_1 2e+161)
       (/ (fma z (/ y t) x) (+ (/ b (/ t y)) (+ 1.0 a)))
       (if (<= t_1 INFINITY)
         (fma y (/ z (fma t_2 t t)) (/ x (+ t_2 1.0)))
         (/ (fma t (/ x y) z) b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (((z * y) / t) + x) / (((b * y) / t) + (1.0 + a));
	double t_2 = fma(b, (y / t), a);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (y / fma(fma((b / t), y, a), t, t)) * z;
	} else if (t_1 <= 2e+161) {
		tmp = fma(z, (y / t), x) / ((b / (t / y)) + (1.0 + a));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = fma(y, (z / fma(t_2, t, t)), (x / (t_2 + 1.0)));
	} else {
		tmp = fma(t, (x / y), z) / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(Float64(z * y) / t) + x) / Float64(Float64(Float64(b * y) / t) + Float64(1.0 + a)))
	t_2 = fma(b, Float64(y / t), a)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(y / fma(fma(Float64(b / t), y, a), t, t)) * z);
	elseif (t_1 <= 2e+161)
		tmp = Float64(fma(z, Float64(y / t), x) / Float64(Float64(b / Float64(t / y)) + Float64(1.0 + a)));
	elseif (t_1 <= Inf)
		tmp = fma(y, Float64(z / fma(t_2, t, t)), Float64(x / Float64(t_2 + 1.0)));
	else
		tmp = Float64(fma(t, Float64(x / y), z) / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(N[(z * y), $MachinePrecision] / t), $MachinePrecision] + x), $MachinePrecision] / N[(N[(N[(b * y), $MachinePrecision] / t), $MachinePrecision] + N[(1.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(b * N[(y / t), $MachinePrecision] + a), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(y / N[(N[(N[(b / t), $MachinePrecision] * y + a), $MachinePrecision] * t + t), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[t$95$1, 2e+161], N[(N[(z * N[(y / t), $MachinePrecision] + x), $MachinePrecision] / N[(N[(b / N[(t / y), $MachinePrecision]), $MachinePrecision] + N[(1.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(y * N[(z / N[(t$95$2 * t + t), $MachinePrecision]), $MachinePrecision] + N[(x / N[(t$95$2 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t * N[(x / y), $MachinePrecision] + z), $MachinePrecision] / b), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 17.8%

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

   3. Taylor expanded in z around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 86.9%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 2.0000000000000001e161

   1. Initial program 87.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 86.3

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

   4. Applied rewrites 86.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 89.4

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

   6. Applied rewrites 89.4%

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

   if 2.0000000000000001e161 < (/.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 69.9%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 59.9

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

   5. Applied rewrites 59.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lift-/.f64 N/A

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

      7. lower-fma.f64 72.1

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

   7. Applied rewrites 72.1%

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

   8. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-rgt-in N/A

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

      7. *-lft-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. associate-/l* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

   10. Applied rewrites 93.7%

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

   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 b around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 5.1

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

   5. Applied rewrites 5.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lift-/.f64 N/A

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

      7. lower-fma.f64 15.9

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

   7. Applied rewrites 15.9%

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

   8. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-rgt-in N/A

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

      7. *-lft-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. associate-/l* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

   10. Applied rewrites 10.2%

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

   11. Taylor expanded in b around inf

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

   13. Applied rewrites 90.0%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq -\infty:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{b}{t}, y, a\right), t, t\right)} \cdot z\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq 2 \cdot 10^{+161}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}{\frac{b}{\frac{t}{y}} + \left(1 + a\right)}\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{\mathsf{fma}\left(\mathsf{fma}\left(b, \frac{y}{t}, a\right), t, t\right)}, \frac{x}{\mathsf{fma}\left(b, \frac{y}{t}, a\right) + 1}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 88.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{b}{t}, y, a\right), t, t\right)} \cdot z\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+161}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}{\frac{b}{\frac{t}{y}} + \left(1 + a\right)}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{\mathsf{fma}\left(\mathsf{fma}\left(b, \frac{y}{t}, a\right), t, t\right)}, \frac{x}{\mathsf{fma}\left(b, \frac{y}{t}, 1 + a\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (+ (/ (* z y) t) x) (+ (/ (* b y) t) (+ 1.0 a)))))
   (if (<= t_1 (- INFINITY))
     (* (/ y (fma (fma (/ b t) y a) t t)) z)
     (if (<= t_1 2e+161)
       (/ (fma z (/ y t) x) (+ (/ b (/ t y)) (+ 1.0 a)))
       (if (<= t_1 INFINITY)
         (fma
          y
          (/ z (fma (fma b (/ y t) a) t t))
          (/ x (fma b (/ y t) (+ 1.0 a))))
         (/ (fma t (/ x y) z) b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (((z * y) / t) + x) / (((b * y) / t) + (1.0 + a));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (y / fma(fma((b / t), y, a), t, t)) * z;
	} else if (t_1 <= 2e+161) {
		tmp = fma(z, (y / t), x) / ((b / (t / y)) + (1.0 + a));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = fma(y, (z / fma(fma(b, (y / t), a), t, t)), (x / fma(b, (y / t), (1.0 + a))));
	} else {
		tmp = fma(t, (x / y), z) / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(Float64(z * y) / t) + x) / Float64(Float64(Float64(b * y) / t) + Float64(1.0 + a)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(y / fma(fma(Float64(b / t), y, a), t, t)) * z);
	elseif (t_1 <= 2e+161)
		tmp = Float64(fma(z, Float64(y / t), x) / Float64(Float64(b / Float64(t / y)) + Float64(1.0 + a)));
	elseif (t_1 <= Inf)
		tmp = fma(y, Float64(z / fma(fma(b, Float64(y / t), a), t, t)), Float64(x / fma(b, Float64(y / t), Float64(1.0 + a))));
	else
		tmp = Float64(fma(t, Float64(x / y), z) / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(N[(z * y), $MachinePrecision] / t), $MachinePrecision] + x), $MachinePrecision] / N[(N[(N[(b * y), $MachinePrecision] / t), $MachinePrecision] + N[(1.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(y / N[(N[(N[(b / t), $MachinePrecision] * y + a), $MachinePrecision] * t + t), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[t$95$1, 2e+161], N[(N[(z * N[(y / t), $MachinePrecision] + x), $MachinePrecision] / N[(N[(b / N[(t / y), $MachinePrecision]), $MachinePrecision] + N[(1.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(y * N[(z / N[(N[(b * N[(y / t), $MachinePrecision] + a), $MachinePrecision] * t + t), $MachinePrecision]), $MachinePrecision] + N[(x / N[(b * N[(y / t), $MachinePrecision] + N[(1.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t * N[(x / y), $MachinePrecision] + z), $MachinePrecision] / b), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 17.8%

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

   3. Taylor expanded in z around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 86.9%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 2.0000000000000001e161

   1. Initial program 87.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 86.3

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

   4. Applied rewrites 86.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 89.4

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

   6. Applied rewrites 89.4%

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

   if 2.0000000000000001e161 < (/.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 69.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 82.1

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

   4. Applied rewrites 82.1%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-rgt-in N/A

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

      7. *-lft-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. associate-/l* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

   7. Applied rewrites 93.7%

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

   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 b around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 5.1

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

   5. Applied rewrites 5.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lift-/.f64 N/A

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

      7. lower-fma.f64 15.9

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

   7. Applied rewrites 15.9%

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

   8. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-rgt-in N/A

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

      7. *-lft-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. associate-/l* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

   10. Applied rewrites 10.2%

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

   11. Taylor expanded in b around inf

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

   13. Applied rewrites 90.0%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq -\infty:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{b}{t}, y, a\right), t, t\right)} \cdot z\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq 2 \cdot 10^{+161}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}{\frac{b}{\frac{t}{y}} + \left(1 + a\right)}\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{\mathsf{fma}\left(\mathsf{fma}\left(b, \frac{y}{t}, a\right), t, t\right)}, \frac{x}{\mathsf{fma}\left(b, \frac{y}{t}, 1 + a\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 72.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)}\\ t_2 := \frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{1 + a}\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{-321}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{x}{b \cdot y}, \frac{z}{b}\right)\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (+ (/ (* z y) t) x) (+ (/ (* b y) t) (+ 1.0 a))))
        (t_2 (/ (fma (/ y t) z x) (+ 1.0 a))))
   (if (<= t_1 -4e-321)
     t_2
     (if (<= t_1 0.0)
       (fma t (/ x (* b y)) (/ z b))
       (if (<= t_1 INFINITY) t_2 (/ (fma t (/ x y) z) b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (((z * y) / t) + x) / (((b * y) / t) + (1.0 + a));
	double t_2 = fma((y / t), z, x) / (1.0 + a);
	double tmp;
	if (t_1 <= -4e-321) {
		tmp = t_2;
	} else if (t_1 <= 0.0) {
		tmp = fma(t, (x / (b * y)), (z / b));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_2;
	} else {
		tmp = fma(t, (x / y), z) / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(Float64(z * y) / t) + x) / Float64(Float64(Float64(b * y) / t) + Float64(1.0 + a)))
	t_2 = Float64(fma(Float64(y / t), z, x) / Float64(1.0 + a))
	tmp = 0.0
	if (t_1 <= -4e-321)
		tmp = t_2;
	elseif (t_1 <= 0.0)
		tmp = fma(t, Float64(x / Float64(b * y)), Float64(z / b));
	elseif (t_1 <= Inf)
		tmp = t_2;
	else
		tmp = Float64(fma(t, Float64(x / y), z) / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(N[(z * y), $MachinePrecision] / t), $MachinePrecision] + x), $MachinePrecision] / N[(N[(N[(b * y), $MachinePrecision] / t), $MachinePrecision] + N[(1.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(y / t), $MachinePrecision] * z + x), $MachinePrecision] / N[(1.0 + a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -4e-321], t$95$2, If[LessEqual[t$95$1, 0.0], N[(t * N[(x / N[(b * y), $MachinePrecision]), $MachinePrecision] + N[(z / b), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], t$95$2, N[(N[(t * N[(x / y), $MachinePrecision] + z), $MachinePrecision] / b), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -4.00193e-321 or -0.0 < (/.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 86.7%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 69.0

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

   5. Applied rewrites 69.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lift-/.f64 N/A

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

      7. lower-fma.f64 73.3

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

   7. Applied rewrites 73.3%

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

   if -4.00193e-321 < (/.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 47.7%

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

   3. Taylor expanded in b around 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-*l/ N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 28.4

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

   5. Applied rewrites 28.4%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 69.4%

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

   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 b around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 5.1

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

   5. Applied rewrites 5.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lift-/.f64 N/A

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

      7. lower-fma.f64 15.9

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

   7. Applied rewrites 15.9%

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

   8. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-rgt-in N/A

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

      7. *-lft-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. associate-/l* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

   10. Applied rewrites 10.2%

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

   11. Taylor expanded in b around inf

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

   13. Applied rewrites 90.0%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq -4 \cdot 10^{-321}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{1 + a}\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq 0:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{x}{b \cdot y}, \frac{z}{b}\right)\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq \infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{1 + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 74.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ t_2 := \frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)}\\ t_3 := \frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{1 + a}\\ \mathbf{if}\;t\_2 \leq -4 \cdot 10^{-321}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (fma t (/ x y) z) b))
        (t_2 (/ (+ (/ (* z y) t) x) (+ (/ (* b y) t) (+ 1.0 a))))
        (t_3 (/ (fma (/ y t) z x) (+ 1.0 a))))
   (if (<= t_2 -4e-321)
     t_3
     (if (<= t_2 0.0) t_1 (if (<= t_2 INFINITY) 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), z) / b;
	double t_2 = (((z * y) / t) + x) / (((b * y) / t) + (1.0 + a));
	double t_3 = fma((y / t), z, x) / (1.0 + a);
	double tmp;
	if (t_2 <= -4e-321) {
		tmp = t_3;
	} else if (t_2 <= 0.0) {
		tmp = t_1;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_3;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(fma(t, Float64(x / y), z) / b)
	t_2 = Float64(Float64(Float64(Float64(z * y) / t) + x) / Float64(Float64(Float64(b * y) / t) + Float64(1.0 + a)))
	t_3 = Float64(fma(Float64(y / t), z, x) / Float64(1.0 + a))
	tmp = 0.0
	if (t_2 <= -4e-321)
		tmp = t_3;
	elseif (t_2 <= 0.0)
		tmp = t_1;
	elseif (t_2 <= Inf)
		tmp = t_3;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t * N[(x / y), $MachinePrecision] + z), $MachinePrecision] / b), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(z * y), $MachinePrecision] / t), $MachinePrecision] + x), $MachinePrecision] / N[(N[(N[(b * y), $MachinePrecision] / t), $MachinePrecision] + N[(1.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(y / t), $MachinePrecision] * z + x), $MachinePrecision] / N[(1.0 + a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -4e-321], t$95$3, If[LessEqual[t$95$2, 0.0], t$95$1, If[LessEqual[t$95$2, Infinity], 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))) < -4.00193e-321 or -0.0 < (/.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 86.7%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 69.0

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

   5. Applied rewrites 69.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lift-/.f64 N/A

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

      7. lower-fma.f64 73.3

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

   7. Applied rewrites 73.3%

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

   if -4.00193e-321 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -0.0 or +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 27.5%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 21.7

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

   5. Applied rewrites 21.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lift-/.f64 N/A

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

      7. lower-fma.f64 24.8

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

   7. Applied rewrites 24.8%

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

   8. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-rgt-in N/A

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

      7. *-lft-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. associate-/l* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

   10. Applied rewrites 39.1%

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

   11. Taylor expanded in b around inf

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

   13. Applied rewrites 77.8%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq -4 \cdot 10^{-321}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{1 + a}\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq \infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{1 + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 71.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ t_2 := \frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)}\\ t_3 := \frac{\mathsf{fma}\left(\frac{z}{t}, y, x\right)}{1 + a}\\ \mathbf{if}\;t\_2 \leq -4 \cdot 10^{-321}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (fma t (/ x y) z) b))
        (t_2 (/ (+ (/ (* z y) t) x) (+ (/ (* b y) t) (+ 1.0 a))))
        (t_3 (/ (fma (/ z t) y x) (+ 1.0 a))))
   (if (<= t_2 -4e-321)
     t_3
     (if (<= t_2 0.0) t_1 (if (<= t_2 INFINITY) 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), z) / b;
	double t_2 = (((z * y) / t) + x) / (((b * y) / t) + (1.0 + a));
	double t_3 = fma((z / t), y, x) / (1.0 + a);
	double tmp;
	if (t_2 <= -4e-321) {
		tmp = t_3;
	} else if (t_2 <= 0.0) {
		tmp = t_1;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_3;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(fma(t, Float64(x / y), z) / b)
	t_2 = Float64(Float64(Float64(Float64(z * y) / t) + x) / Float64(Float64(Float64(b * y) / t) + Float64(1.0 + a)))
	t_3 = Float64(fma(Float64(z / t), y, x) / Float64(1.0 + a))
	tmp = 0.0
	if (t_2 <= -4e-321)
		tmp = t_3;
	elseif (t_2 <= 0.0)
		tmp = t_1;
	elseif (t_2 <= Inf)
		tmp = t_3;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t * N[(x / y), $MachinePrecision] + z), $MachinePrecision] / b), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(z * y), $MachinePrecision] / t), $MachinePrecision] + x), $MachinePrecision] / N[(N[(N[(b * y), $MachinePrecision] / t), $MachinePrecision] + N[(1.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(z / t), $MachinePrecision] * y + x), $MachinePrecision] / N[(1.0 + a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -4e-321], t$95$3, If[LessEqual[t$95$2, 0.0], t$95$1, If[LessEqual[t$95$2, Infinity], 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))) < -4.00193e-321 or -0.0 < (/.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 86.7%

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

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 68.9

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

   5. Applied rewrites 68.9%

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

   if -4.00193e-321 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -0.0 or +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 27.5%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 21.7

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

   5. Applied rewrites 21.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lift-/.f64 N/A

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

      7. lower-fma.f64 24.8

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

   7. Applied rewrites 24.8%

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

   8. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-rgt-in N/A

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

      7. *-lft-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. associate-/l* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

   10. Applied rewrites 39.1%

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

   11. Taylor expanded in b around inf

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

   13. Applied rewrites 77.8%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq -4 \cdot 10^{-321}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{z}{t}, y, x\right)}{1 + a}\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq \infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{z}{t}, y, x\right)}{1 + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 88.2% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(N[(z * y), $MachinePrecision] / t), $MachinePrecision] + x), $MachinePrecision] / N[(N[(N[(b * y), $MachinePrecision] / t), $MachinePrecision] + N[(1.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(y / N[(N[(N[(b / t), $MachinePrecision] * y + a), $MachinePrecision] * t + t), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(z * N[(y / t), $MachinePrecision] + x), $MachinePrecision] / N[(N[(b / N[(t / y), $MachinePrecision]), $MachinePrecision] + N[(1.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t * N[(x / y), $MachinePrecision] + z), $MachinePrecision] / 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 17.8%

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

   3. Taylor expanded in z around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 86.9%

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

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

   1. Initial program 84.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 85.6

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

   4. Applied rewrites 85.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 87.9

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

   6. Applied rewrites 87.9%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 5.1

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

   5. Applied rewrites 5.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lift-/.f64 N/A

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

      7. lower-fma.f64 15.9

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

   7. Applied rewrites 15.9%

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

   8. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-rgt-in N/A

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

      7. *-lft-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. associate-/l* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

   10. Applied rewrites 10.2%

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

   11. Taylor expanded in b around inf

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

   13. Applied rewrites 90.0%

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

4. Final simplification 88.0%

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

Alternative 13: 85.6% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(N[(z * y), $MachinePrecision] / t), $MachinePrecision] + x), $MachinePrecision] / N[(N[(N[(b * y), $MachinePrecision] / t), $MachinePrecision] + N[(1.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(y / N[(N[(N[(b / t), $MachinePrecision] * y + a), $MachinePrecision] * t + t), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(N[(z / t), $MachinePrecision] * y + x), $MachinePrecision] / N[(N[(b / t), $MachinePrecision] * y + N[(1.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t * N[(x / y), $MachinePrecision] + z), $MachinePrecision] / 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 17.8%

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

   3. Taylor expanded in z around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 86.9%

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

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

   1. Initial program 84.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 82.3

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 83.2

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 83.2

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

   4. Applied rewrites 83.2%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 5.1

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

   5. Applied rewrites 5.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lift-/.f64 N/A

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

      7. lower-fma.f64 15.9

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

   7. Applied rewrites 15.9%

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

   8. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-rgt-in N/A

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

      7. *-lft-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. associate-/l* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(y, \frac{z}{\mathsf{fma}\left(\mathsf{fma}\left(b, \frac{y}{t}, a\right), t, t\right)}, \frac{x}{\color{blue}{\left(a + \frac{b \cdot y}{t}\right) + 1}}\right) \]

      15. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, \frac{z}{\mathsf{fma}\left(\mathsf{fma}\left(b, \frac{y}{t}, a\right), t, t\right)}, \frac{x}{\color{blue}{\left(a + \frac{b \cdot y}{t}\right) + 1}}\right) \]

   10. Applied rewrites 10.2%

       \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{z}{\mathsf{fma}\left(\mathsf{fma}\left(b, \frac{y}{t}, a\right), t, t\right)}, \frac{x}{\mathsf{fma}\left(b, \frac{y}{t}, a\right) + 1}\right)} \]

   11. Taylor expanded in b around inf

       \[\leadsto \frac{z + \frac{t \cdot x}{y}}{\color{blue}{b}} \]
   12. Step-by-step derivation

   13. Applied rewrites 90.0%

       \[\leadsto \frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{\color{blue}{b}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq -\infty:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{b}{t}, y, a\right), t, t\right)} \cdot z\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq \infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{z}{t}, y, x\right)}{\mathsf{fma}\left(\frac{b}{t}, y, 1 + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 66.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{\mathsf{fma}\left(\frac{b}{t}, y, 1 + a\right)}\\ \mathbf{if}\;t \leq -6 \cdot 10^{-78}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 6.6 \cdot 10^{-42}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ x (fma (/ b t) y (+ 1.0 a)))))
   (if (<= t -6e-78) t_1 (if (<= t 6.6e-42) (/ (fma t (/ x y) z) b) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x / fma((b / t), y, (1.0 + a));
	double tmp;
	if (t <= -6e-78) {
		tmp = t_1;
	} else if (t <= 6.6e-42) {
		tmp = fma(t, (x / y), z) / b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x / fma(Float64(b / t), y, Float64(1.0 + a)))
	tmp = 0.0
	if (t <= -6e-78)
		tmp = t_1;
	elseif (t <= 6.6e-42)
		tmp = Float64(fma(t, Float64(x / y), z) / b);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x / N[(N[(b / t), $MachinePrecision] * y + N[(1.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -6e-78], t$95$1, If[LessEqual[t, 6.6e-42], N[(N[(t * N[(x / y), $MachinePrecision] + z), $MachinePrecision] / b), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -5.99999999999999975e-78 or 6.6000000000000005e-42 < t

   1. Initial program 81.8%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)}} \]

      2. associate-+r+ N/A

         \[\leadsto \frac{x}{\color{blue}{\left(1 + a\right) + \frac{b \cdot y}{t}}} \]

      3. +-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{\frac{b \cdot y}{t} + \left(1 + a\right)}} \]

      4. remove-double-neg N/A

         \[\leadsto \frac{x}{\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{b \cdot y}{t}\right)\right)\right)\right)} + \left(1 + a\right)} \]

      5. associate-/l* N/A

         \[\leadsto \frac{x}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{b \cdot \frac{y}{t}}\right)\right)\right)\right) + \left(1 + a\right)} \]

      6. distribute-rgt-neg-out N/A

         \[\leadsto \frac{x}{\left(\mathsf{neg}\left(\color{blue}{b \cdot \left(\mathsf{neg}\left(\frac{y}{t}\right)\right)}\right)\right) + \left(1 + a\right)} \]

      7. mul-1-neg N/A

         \[\leadsto \frac{x}{\left(\mathsf{neg}\left(b \cdot \color{blue}{\left(-1 \cdot \frac{y}{t}\right)}\right)\right) + \left(1 + a\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{x}{\left(\mathsf{neg}\left(b \cdot \color{blue}{\left(\frac{y}{t} \cdot -1\right)}\right)\right) + \left(1 + a\right)} \]

      9. associate-*r* N/A

         \[\leadsto \frac{x}{\left(\mathsf{neg}\left(\color{blue}{\left(b \cdot \frac{y}{t}\right) \cdot -1}\right)\right) + \left(1 + a\right)} \]

      10. associate-/l* N/A

          \[\leadsto \frac{x}{\left(\mathsf{neg}\left(\color{blue}{\frac{b \cdot y}{t}} \cdot -1\right)\right) + \left(1 + a\right)} \]

      11. *-commutative N/A

          \[\leadsto \frac{x}{\left(\mathsf{neg}\left(\frac{\color{blue}{y \cdot b}}{t} \cdot -1\right)\right) + \left(1 + a\right)} \]

      12. associate-/l* N/A

          \[\leadsto \frac{x}{\left(\mathsf{neg}\left(\color{blue}{\left(y \cdot \frac{b}{t}\right)} \cdot -1\right)\right) + \left(1 + a\right)} \]

      13. associate-*r* N/A

          \[\leadsto \frac{x}{\left(\mathsf{neg}\left(\color{blue}{y \cdot \left(\frac{b}{t} \cdot -1\right)}\right)\right) + \left(1 + a\right)} \]

      14. *-commutative N/A

          \[\leadsto \frac{x}{\left(\mathsf{neg}\left(y \cdot \color{blue}{\left(-1 \cdot \frac{b}{t}\right)}\right)\right) + \left(1 + a\right)} \]

      15. *-commutative N/A

          \[\leadsto \frac{x}{\left(\mathsf{neg}\left(\color{blue}{\left(-1 \cdot \frac{b}{t}\right) \cdot y}\right)\right) + \left(1 + a\right)} \]

      16. mul-1-neg N/A

          \[\leadsto \frac{x}{\left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\frac{b}{t}\right)\right)} \cdot y\right)\right) + \left(1 + a\right)} \]

      17. distribute-lft-neg-out N/A

          \[\leadsto \frac{x}{\left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\frac{b}{t} \cdot y\right)\right)}\right)\right) + \left(1 + a\right)} \]

      18. remove-double-neg N/A

          \[\leadsto \frac{x}{\color{blue}{\frac{b}{t} \cdot y} + \left(1 + a\right)} \]

      19. lower-fma.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(\frac{b}{t}, y, 1 + a\right)}} \]

      20. lower-/.f64 N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(\color{blue}{\frac{b}{t}}, y, 1 + a\right)} \]

      21. +-commutative N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(\frac{b}{t}, y, \color{blue}{a + 1}\right)} \]

      22. lower-+.f64 68.8

          \[\leadsto \frac{x}{\mathsf{fma}\left(\frac{b}{t}, y, \color{blue}{a + 1}\right)} \]

   5. Applied rewrites 68.8%

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(\frac{b}{t}, y, a + 1\right)}} \]

   if -5.99999999999999975e-78 < t < 6.6000000000000005e-42

   1. Initial program 56.5%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0

      \[\leadsto \frac{x + \frac{y \cdot z}{t}}{\color{blue}{1 + a}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{x + \frac{y \cdot z}{t}}{\color{blue}{a + 1}} \]

      2. lower-+.f64 37.9

         \[\leadsto \frac{x + \frac{y \cdot z}{t}}{\color{blue}{a + 1}} \]

   5. Applied rewrites 37.9%

      \[\leadsto \frac{x + \frac{y \cdot z}{t}}{\color{blue}{a + 1}} \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{x + \frac{y \cdot z}{t}}}{a + 1} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z}{t} + x}}{a + 1} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z}{t}} + x}{a + 1} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{y \cdot z}}{t} + x}{a + 1} \]

      5. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{t} \cdot z} + x}{a + 1} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{t}} \cdot z + x}{a + 1} \]

      7. lower-fma.f64 38.9

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}}{a + 1} \]

   7. Applied rewrites 38.9%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}}{a + 1} \]

   8. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)} + \frac{y \cdot z}{t \cdot \left(1 + \left(a + \frac{b \cdot y}{t}\right)\right)}} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{y \cdot z}{t \cdot \left(1 + \left(a + \frac{b \cdot y}{t}\right)\right)} + \frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)}} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{y \cdot \frac{z}{t \cdot \left(1 + \left(a + \frac{b \cdot y}{t}\right)\right)}} + \frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{z}{t \cdot \left(1 + \left(a + \frac{b \cdot y}{t}\right)\right)}, \frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)}\right)} \]

      4. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{z}{t \cdot \left(1 + \left(a + \frac{b \cdot y}{t}\right)\right)}}, \frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)}\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{z}{t \cdot \color{blue}{\left(\left(a + \frac{b \cdot y}{t}\right) + 1\right)}}, \frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)}\right) \]

      6. distribute-rgt-in N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{z}{\color{blue}{\left(a + \frac{b \cdot y}{t}\right) \cdot t + 1 \cdot t}}, \frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)}\right) \]

      7. *-lft-identity N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{z}{\left(a + \frac{b \cdot y}{t}\right) \cdot t + \color{blue}{t}}, \frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)}\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{z}{\color{blue}{\mathsf{fma}\left(a + \frac{b \cdot y}{t}, t, t\right)}}, \frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)}\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{z}{\mathsf{fma}\left(\color{blue}{\frac{b \cdot y}{t} + a}, t, t\right)}, \frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)}\right) \]

      10. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(y, \frac{z}{\mathsf{fma}\left(\color{blue}{b \cdot \frac{y}{t}} + a, t, t\right)}, \frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)}\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, \frac{z}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(b, \frac{y}{t}, a\right)}, t, t\right)}, \frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)}\right) \]

      12. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, \frac{z}{\mathsf{fma}\left(\mathsf{fma}\left(b, \color{blue}{\frac{y}{t}}, a\right), t, t\right)}, \frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)}\right) \]

      13. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, \frac{z}{\mathsf{fma}\left(\mathsf{fma}\left(b, \frac{y}{t}, a\right), t, t\right)}, \color{blue}{\frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)}}\right) \]

      14. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(y, \frac{z}{\mathsf{fma}\left(\mathsf{fma}\left(b, \frac{y}{t}, a\right), t, t\right)}, \frac{x}{\color{blue}{\left(a + \frac{b \cdot y}{t}\right) + 1}}\right) \]

      15. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, \frac{z}{\mathsf{fma}\left(\mathsf{fma}\left(b, \frac{y}{t}, a\right), t, t\right)}, \frac{x}{\color{blue}{\left(a + \frac{b \cdot y}{t}\right) + 1}}\right) \]

   10. Applied rewrites 60.8%

       \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{z}{\mathsf{fma}\left(\mathsf{fma}\left(b, \frac{y}{t}, a\right), t, t\right)}, \frac{x}{\mathsf{fma}\left(b, \frac{y}{t}, a\right) + 1}\right)} \]

   11. Taylor expanded in b around inf

       \[\leadsto \frac{z + \frac{t \cdot x}{y}}{\color{blue}{b}} \]
   12. Step-by-step derivation

   13. Applied rewrites 60.4%

       \[\leadsto \frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{\color{blue}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -6 \cdot 10^{-78}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(\frac{b}{t}, y, 1 + a\right)}\\ \mathbf{elif}\;t \leq 6.6 \cdot 10^{-42}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(\frac{b}{t}, y, 1 + a\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 57.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{y}{t}, z, x\right)\\ t_2 := \frac{t\_1}{a}\\ \mathbf{if}\;a \leq -1:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;a \leq 1:\\ \;\;\;\;\frac{t\_1}{1}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma (/ y t) z x)) (t_2 (/ t_1 a)))
   (if (<= a -1.0) t_2 (if (<= a 1.0) (/ t_1 1.0) t_2))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma((y / t), z, x);
	double t_2 = t_1 / a;
	double tmp;
	if (a <= -1.0) {
		tmp = t_2;
	} else if (a <= 1.0) {
		tmp = t_1 / 1.0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(Float64(y / t), z, x)
	t_2 = Float64(t_1 / a)
	tmp = 0.0
	if (a <= -1.0)
		tmp = t_2;
	elseif (a <= 1.0)
		tmp = Float64(t_1 / 1.0);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(y / t), $MachinePrecision] * z + x), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / a), $MachinePrecision]}, If[LessEqual[a, -1.0], t$95$2, If[LessEqual[a, 1.0], N[(t$95$1 / 1.0), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if a < -1 or 1 < a

   1. Initial program 70.2%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0

      \[\leadsto \frac{x + \frac{y \cdot z}{t}}{\color{blue}{1 + a}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{x + \frac{y \cdot z}{t}}{\color{blue}{a + 1}} \]

      2. lower-+.f64 59.9

         \[\leadsto \frac{x + \frac{y \cdot z}{t}}{\color{blue}{a + 1}} \]

   5. Applied rewrites 59.9%

      \[\leadsto \frac{x + \frac{y \cdot z}{t}}{\color{blue}{a + 1}} \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{x + \frac{y \cdot z}{t}}}{a + 1} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z}{t} + x}}{a + 1} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z}{t}} + x}{a + 1} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{y \cdot z}}{t} + x}{a + 1} \]

      5. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{t} \cdot z} + x}{a + 1} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{t}} \cdot z + x}{a + 1} \]

      7. lower-fma.f64 64.0

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}}{a + 1} \]

   7. Applied rewrites 64.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}}{a + 1} \]

   8. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\frac{x + \frac{y \cdot z}{t}}{a}} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x + \frac{y \cdot z}{t}}{a}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z}{t} + x}}{a} \]

      3. associate-/l* N/A

         \[\leadsto \frac{\color{blue}{y \cdot \frac{z}{t}} + x}{a} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}}{a} \]

      5. lower-/.f64 58.1

         \[\leadsto \frac{\mathsf{fma}\left(y, \color{blue}{\frac{z}{t}}, x\right)}{a} \]

   10. Applied rewrites 58.1%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{a}} \]
   11. Step-by-step derivation

   12. Applied rewrites 63.8%

       \[\leadsto \frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{a} \]

   if -1 < a < 1

   1. Initial program 72.8%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0

      \[\leadsto \frac{x + \frac{y \cdot z}{t}}{\color{blue}{1 + a}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{x + \frac{y \cdot z}{t}}{\color{blue}{a + 1}} \]

      2. lower-+.f64 53.4

         \[\leadsto \frac{x + \frac{y \cdot z}{t}}{\color{blue}{a + 1}} \]

   5. Applied rewrites 53.4%

      \[\leadsto \frac{x + \frac{y \cdot z}{t}}{\color{blue}{a + 1}} \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{x + \frac{y \cdot z}{t}}}{a + 1} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z}{t} + x}}{a + 1} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z}{t}} + x}{a + 1} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{y \cdot z}}{t} + x}{a + 1} \]

      5. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{t} \cdot z} + x}{a + 1} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{t}} \cdot z + x}{a + 1} \]

      7. lower-fma.f64 57.2

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}}{a + 1} \]

   7. Applied rewrites 57.2%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}}{a + 1} \]

   8. Taylor expanded in a around 0

      \[\leadsto \frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{1} \]
   9. Step-by-step derivation

   10. Applied rewrites 55.9%

       \[\leadsto \frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 60.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{1 + a}\\ \mathbf{if}\;t \leq -9.2 \cdot 10^{-6}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.9 \cdot 10^{-18}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ x (+ 1.0 a))))
   (if (<= t -9.2e-6) t_1 (if (<= t 1.9e-18) (/ (fma t (/ x y) z) b) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x / (1.0 + a);
	double tmp;
	if (t <= -9.2e-6) {
		tmp = t_1;
	} else if (t <= 1.9e-18) {
		tmp = fma(t, (x / y), z) / b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x / Float64(1.0 + a))
	tmp = 0.0
	if (t <= -9.2e-6)
		tmp = t_1;
	elseif (t <= 1.9e-18)
		tmp = Float64(fma(t, Float64(x / y), z) / b);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x / N[(1.0 + a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -9.2e-6], t$95$1, If[LessEqual[t, 1.9e-18], N[(N[(t * N[(x / y), $MachinePrecision] + z), $MachinePrecision] / b), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -9.2e-6 or 1.8999999999999999e-18 < t

   1. Initial program 81.7%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\frac{x}{1 + a}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{1 + a}} \]

      2. +-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{a + 1}} \]

      3. lower-+.f64 59.9

         \[\leadsto \frac{x}{\color{blue}{a + 1}} \]

   5. Applied rewrites 59.9%

      \[\leadsto \color{blue}{\frac{x}{a + 1}} \]

   if -9.2e-6 < t < 1.8999999999999999e-18

   1. Initial program 60.1%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0

      \[\leadsto \frac{x + \frac{y \cdot z}{t}}{\color{blue}{1 + a}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{x + \frac{y \cdot z}{t}}{\color{blue}{a + 1}} \]

      2. lower-+.f64 40.2

         \[\leadsto \frac{x + \frac{y \cdot z}{t}}{\color{blue}{a + 1}} \]

   5. Applied rewrites 40.2%

      \[\leadsto \frac{x + \frac{y \cdot z}{t}}{\color{blue}{a + 1}} \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{x + \frac{y \cdot z}{t}}}{a + 1} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z}{t} + x}}{a + 1} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z}{t}} + x}{a + 1} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{y \cdot z}}{t} + x}{a + 1} \]

      5. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{t} \cdot z} + x}{a + 1} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{t}} \cdot z + x}{a + 1} \]

      7. lower-fma.f64 41.0

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}}{a + 1} \]

   7. Applied rewrites 41.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}}{a + 1} \]

   8. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)} + \frac{y \cdot z}{t \cdot \left(1 + \left(a + \frac{b \cdot y}{t}\right)\right)}} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{y \cdot z}{t \cdot \left(1 + \left(a + \frac{b \cdot y}{t}\right)\right)} + \frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)}} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{y \cdot \frac{z}{t \cdot \left(1 + \left(a + \frac{b \cdot y}{t}\right)\right)}} + \frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{z}{t \cdot \left(1 + \left(a + \frac{b \cdot y}{t}\right)\right)}, \frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)}\right)} \]

      4. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{z}{t \cdot \left(1 + \left(a + \frac{b \cdot y}{t}\right)\right)}}, \frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)}\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{z}{t \cdot \color{blue}{\left(\left(a + \frac{b \cdot y}{t}\right) + 1\right)}}, \frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)}\right) \]

      6. distribute-rgt-in N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{z}{\color{blue}{\left(a + \frac{b \cdot y}{t}\right) \cdot t + 1 \cdot t}}, \frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)}\right) \]

      7. *-lft-identity N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{z}{\left(a + \frac{b \cdot y}{t}\right) \cdot t + \color{blue}{t}}, \frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)}\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{z}{\color{blue}{\mathsf{fma}\left(a + \frac{b \cdot y}{t}, t, t\right)}}, \frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)}\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{z}{\mathsf{fma}\left(\color{blue}{\frac{b \cdot y}{t} + a}, t, t\right)}, \frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)}\right) \]

      10. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(y, \frac{z}{\mathsf{fma}\left(\color{blue}{b \cdot \frac{y}{t}} + a, t, t\right)}, \frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)}\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, \frac{z}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(b, \frac{y}{t}, a\right)}, t, t\right)}, \frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)}\right) \]

      12. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, \frac{z}{\mathsf{fma}\left(\mathsf{fma}\left(b, \color{blue}{\frac{y}{t}}, a\right), t, t\right)}, \frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)}\right) \]

      13. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, \frac{z}{\mathsf{fma}\left(\mathsf{fma}\left(b, \frac{y}{t}, a\right), t, t\right)}, \color{blue}{\frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)}}\right) \]

      14. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(y, \frac{z}{\mathsf{fma}\left(\mathsf{fma}\left(b, \frac{y}{t}, a\right), t, t\right)}, \frac{x}{\color{blue}{\left(a + \frac{b \cdot y}{t}\right) + 1}}\right) \]

      15. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, \frac{z}{\mathsf{fma}\left(\mathsf{fma}\left(b, \frac{y}{t}, a\right), t, t\right)}, \frac{x}{\color{blue}{\left(a + \frac{b \cdot y}{t}\right) + 1}}\right) \]

   10. Applied rewrites 64.6%

       \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{z}{\mathsf{fma}\left(\mathsf{fma}\left(b, \frac{y}{t}, a\right), t, t\right)}, \frac{x}{\mathsf{fma}\left(b, \frac{y}{t}, a\right) + 1}\right)} \]

   11. Taylor expanded in b around inf

       \[\leadsto \frac{z + \frac{t \cdot x}{y}}{\color{blue}{b}} \]
   12. Step-by-step derivation

   13. Applied rewrites 57.2%

       \[\leadsto \frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{\color{blue}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -9.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{x}{1 + a}\\ \mathbf{elif}\;t \leq 1.9 \cdot 10^{-18}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + a}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 57.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{1 + a}\\ \mathbf{if}\;t \leq -6 \cdot 10^{-78}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.85 \cdot 10^{-75}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ x (+ 1.0 a))))
   (if (<= t -6e-78) t_1 (if (<= t 1.85e-75) (/ z b) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x / (1.0 + a);
	double tmp;
	if (t <= -6e-78) {
		tmp = t_1;
	} else if (t <= 1.85e-75) {
		tmp = z / b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x / (1.0d0 + a)
    if (t <= (-6d-78)) then
        tmp = t_1
    else if (t <= 1.85d-75) then
        tmp = z / b
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x / (1.0 + a);
	double tmp;
	if (t <= -6e-78) {
		tmp = t_1;
	} else if (t <= 1.85e-75) {
		tmp = z / b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x / (1.0 + a)
	tmp = 0
	if t <= -6e-78:
		tmp = t_1
	elif t <= 1.85e-75:
		tmp = z / b
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x / Float64(1.0 + a))
	tmp = 0.0
	if (t <= -6e-78)
		tmp = t_1;
	elseif (t <= 1.85e-75)
		tmp = Float64(z / b);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x / (1.0 + a);
	tmp = 0.0;
	if (t <= -6e-78)
		tmp = t_1;
	elseif (t <= 1.85e-75)
		tmp = z / b;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x / N[(1.0 + a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -6e-78], t$95$1, If[LessEqual[t, 1.85e-75], N[(z / b), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -5.99999999999999975e-78 or 1.85000000000000012e-75 < t

   1. Initial program 80.6%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\frac{x}{1 + a}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{1 + a}} \]

      2. +-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{a + 1}} \]

      3. lower-+.f64 56.5

         \[\leadsto \frac{x}{\color{blue}{a + 1}} \]

   5. Applied rewrites 56.5%

      \[\leadsto \color{blue}{\frac{x}{a + 1}} \]

   if -5.99999999999999975e-78 < t < 1.85000000000000012e-75

   1. Initial program 57.2%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{z}{b}} \]
   4. Step-by-step derivation

      1. lower-/.f64 56.8

         \[\leadsto \color{blue}{\frac{z}{b}} \]

   5. Applied rewrites 56.8%

      \[\leadsto \color{blue}{\frac{z}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -6 \cdot 10^{-78}:\\ \;\;\;\;\frac{x}{1 + a}\\ \mathbf{elif}\;t \leq 1.85 \cdot 10^{-75}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + a}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 40.3% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-x, a, x\right)\\ \mathbf{if}\;t \leq -5.2 \cdot 10^{+182}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.9 \cdot 10^{+78}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma (- x) a x)))
   (if (<= t -5.2e+182) t_1 (if (<= t 1.9e+78) (/ z b) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(-x, a, x);
	double tmp;
	if (t <= -5.2e+182) {
		tmp = t_1;
	} else if (t <= 1.9e+78) {
		tmp = z / b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(Float64(-x), a, x)
	tmp = 0.0
	if (t <= -5.2e+182)
		tmp = t_1;
	elseif (t <= 1.9e+78)
		tmp = Float64(z / b);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[((-x) * a + x), $MachinePrecision]}, If[LessEqual[t, -5.2e+182], t$95$1, If[LessEqual[t, 1.9e+78], N[(z / b), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -5.2e182 or 1.9e78 < t

   1. Initial program 79.7%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{x + \frac{y \cdot z}{t}}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z}{t} + x}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z}{t}} + x}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{y \cdot z}}{t} + x}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

      5. *-commutative N/A

         \[\leadsto \frac{\frac{\color{blue}{z \cdot y}}{t} + x}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

      6. associate-/l* N/A

         \[\leadsto \frac{\color{blue}{z \cdot \frac{y}{t}} + x}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

      8. lower-/.f64 87.0

         \[\leadsto \frac{\mathsf{fma}\left(z, \color{blue}{\frac{y}{t}}, x\right)}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

   4. Applied rewrites 87.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

   5. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\frac{x}{1 + a}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{1 + a}} \]

      2. +-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{a + 1}} \]

      3. lower-+.f64 72.1

         \[\leadsto \frac{x}{\color{blue}{a + 1}} \]

   7. Applied rewrites 72.1%

      \[\leadsto \color{blue}{\frac{x}{a + 1}} \]

   8. Taylor expanded in a around 0

      \[\leadsto x + \color{blue}{-1 \cdot \left(a \cdot x\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 42.2%

       \[\leadsto \mathsf{fma}\left(-x, \color{blue}{a}, x\right) \]

   if -5.2e182 < t < 1.9e78

   1. Initial program 67.9%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{z}{b}} \]
   4. Step-by-step derivation

      1. lower-/.f64 43.0

         \[\leadsto \color{blue}{\frac{z}{b}} \]

   5. Applied rewrites 43.0%

      \[\leadsto \color{blue}{\frac{z}{b}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 19.4% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-x, a, x\right) \end{array} \]

FPCore

(FPCore (x y z t a b) :precision binary64 (fma (- x) a x))

C

double code(double x, double y, double z, double t, double a, double b) {
	return fma(-x, a, x);
}

Julia

function code(x, y, z, t, a, b)
	return fma(Float64(-x), a, x)
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[((-x) * a + x), $MachinePrecision]

Derivation

1. Initial program 71.4%

   \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
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. *-commutative N/A

      \[\leadsto \frac{\frac{\color{blue}{z \cdot y}}{t} + x}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

   6. associate-/l* N/A

      \[\leadsto \frac{\color{blue}{z \cdot \frac{y}{t}} + x}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

   7. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

   8. lower-/.f64 73.9

      \[\leadsto \frac{\mathsf{fma}\left(z, \color{blue}{\frac{y}{t}}, x\right)}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

4. Applied rewrites 73.9%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

5. Taylor expanded in t around inf

   \[\leadsto \color{blue}{\frac{x}{1 + a}} \]
6. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{x}{1 + a}} \]

   2. +-commutative N/A

      \[\leadsto \frac{x}{\color{blue}{a + 1}} \]

   3. lower-+.f64 41.1

      \[\leadsto \frac{x}{\color{blue}{a + 1}} \]

7. Applied rewrites 41.1%

   \[\leadsto \color{blue}{\frac{x}{a + 1}} \]

8. Taylor expanded in a around 0

   \[\leadsto x + \color{blue}{-1 \cdot \left(a \cdot x\right)} \]
9. Step-by-step derivation

10. Applied rewrites 19.9%

    \[\leadsto \mathsf{fma}\left(-x, \color{blue}{a}, x\right) \]
11. Add Preprocessing

Alternative 20: 4.4% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \left(-a\right) \cdot x \end{array} \]

FPCore

(FPCore (x y z t a b) :precision binary64 (* (- a) x))

C

double code(double x, double y, double z, double t, double a, double b) {
	return -a * x;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = -a * x
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return -a * x;
}

Python

def code(x, y, z, t, a, b):
	return -a * x

Julia

function code(x, y, z, t, a, b)
	return Float64(Float64(-a) * x)
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = -a * x;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[((-a) * x), $MachinePrecision]

Derivation

1. Initial program 71.4%

   \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
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. *-commutative N/A

      \[\leadsto \frac{\frac{\color{blue}{z \cdot y}}{t} + x}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

   6. associate-/l* N/A

      \[\leadsto \frac{\color{blue}{z \cdot \frac{y}{t}} + x}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

   7. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

   8. lower-/.f64 73.9

      \[\leadsto \frac{\mathsf{fma}\left(z, \color{blue}{\frac{y}{t}}, x\right)}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

4. Applied rewrites 73.9%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

5. Taylor expanded in t around inf

   \[\leadsto \color{blue}{\frac{x}{1 + a}} \]
6. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{x}{1 + a}} \]

   2. +-commutative N/A

      \[\leadsto \frac{x}{\color{blue}{a + 1}} \]

   3. lower-+.f64 41.1

      \[\leadsto \frac{x}{\color{blue}{a + 1}} \]

7. Applied rewrites 41.1%

   \[\leadsto \color{blue}{\frac{x}{a + 1}} \]

8. Taylor expanded in a around 0

   \[\leadsto x + \color{blue}{-1 \cdot \left(a \cdot x\right)} \]
9. Step-by-step derivation

10. Applied rewrites 19.9%

    \[\leadsto \mathsf{fma}\left(-x, \color{blue}{a}, x\right) \]

11. Taylor expanded in a around inf

    \[\leadsto -1 \cdot \left(a \cdot \color{blue}{x}\right) \]
12. Step-by-step derivation

13. Applied rewrites 3.3%

    \[\leadsto \left(-a\right) \cdot x \]
14. Add Preprocessing

Developer Target 1: 79.3% 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 2024268 
(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))))