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

Percentage Accurate: 74.9% → 90.8%

Time: 13.7s

Alternatives: 18

Speedup: 0.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 74.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 90.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{y \cdot z}{t}\\ t_2 := \frac{t\_1}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ t_3 := \mathsf{fma}\left(y, \frac{z}{\mathsf{fma}\left(t, \frac{\mathsf{fma}\left(a, t, y \cdot b\right)}{t}, t\right)}, \frac{x}{1 + \mathsf{fma}\left(y, \frac{b}{t}, a\right)}\right)\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+14}:\\ \;\;\;\;\frac{t\_1}{\mathsf{fma}\left(b, \frac{y}{t}, a + 1\right)}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ x (/ (* y z) t)))
        (t_2 (/ t_1 (+ (+ a 1.0) (/ (* y b) t))))
        (t_3
         (fma
          y
          (/ z (fma t (/ (fma a t (* y b)) t) t))
          (/ x (+ 1.0 (fma y (/ b t) a))))))
   (if (<= t_2 (- INFINITY))
     t_3
     (if (<= t_2 2e+14)
       (/ t_1 (fma b (/ y t) (+ a 1.0)))
       (if (<= t_2 INFINITY) t_3 (/ z b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x + ((y * z) / t);
	double t_2 = t_1 / ((a + 1.0) + ((y * b) / t));
	double t_3 = fma(y, (z / fma(t, (fma(a, t, (y * b)) / t), t)), (x / (1.0 + fma(y, (b / t), a))));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_3;
	} else if (t_2 <= 2e+14) {
		tmp = t_1 / fma(b, (y / t), (a + 1.0));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_3;
	} else {
		tmp = z / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x + Float64(Float64(y * z) / t))
	t_2 = Float64(t_1 / Float64(Float64(a + 1.0) + Float64(Float64(y * b) / t)))
	t_3 = fma(y, Float64(z / fma(t, Float64(fma(a, t, Float64(y * b)) / t), t)), Float64(x / Float64(1.0 + fma(y, Float64(b / t), a))))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_3;
	elseif (t_2 <= 2e+14)
		tmp = Float64(t_1 / fma(b, Float64(y / t), Float64(a + 1.0)));
	elseif (t_2 <= Inf)
		tmp = t_3;
	else
		tmp = Float64(z / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / N[(N[(a + 1.0), $MachinePrecision] + N[(N[(y * b), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(y * N[(z / N[(t * N[(N[(a * t + N[(y * b), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision] + N[(x / N[(1.0 + N[(y * N[(b / t), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$3, If[LessEqual[t$95$2, 2e+14], N[(t$95$1 / N[(b * N[(y / t), $MachinePrecision] + N[(a + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], t$95$3, 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 2e14 < (/.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 68.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-rgt-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-+.f64 N/A

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

   5. Applied rewrites 94.2%

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

   6. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 97.0

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

   8. Applied rewrites 97.0%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 2e14

   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

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 91.4

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

   4. Applied rewrites 91.4%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

Alternative 2: 90.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, \frac{b}{t}, a\right)\\ t_2 := x + \frac{y \cdot z}{t}\\ t_3 := \frac{t\_2}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ t_4 := \mathsf{fma}\left(y, \frac{z}{\mathsf{fma}\left(t, t\_1, t\right)}, \frac{x}{1 + t\_1}\right)\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{+14}:\\ \;\;\;\;\frac{t\_2}{\mathsf{fma}\left(b, \frac{y}{t}, a + 1\right)}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;t\_4\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma y (/ b t) a))
        (t_2 (+ x (/ (* y z) t)))
        (t_3 (/ t_2 (+ (+ a 1.0) (/ (* y b) t))))
        (t_4 (fma y (/ z (fma t t_1 t)) (/ x (+ 1.0 t_1)))))
   (if (<= t_3 (- INFINITY))
     t_4
     (if (<= t_3 2e+14)
       (/ t_2 (fma b (/ y t) (+ a 1.0)))
       (if (<= t_3 INFINITY) t_4 (/ z b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(y, (b / t), a);
	double t_2 = x + ((y * z) / t);
	double t_3 = t_2 / ((a + 1.0) + ((y * b) / t));
	double t_4 = fma(y, (z / fma(t, t_1, t)), (x / (1.0 + t_1)));
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = t_4;
	} else if (t_3 <= 2e+14) {
		tmp = t_2 / fma(b, (y / t), (a + 1.0));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = t_4;
	} else {
		tmp = z / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(y, Float64(b / t), a)
	t_2 = Float64(x + Float64(Float64(y * z) / t))
	t_3 = Float64(t_2 / Float64(Float64(a + 1.0) + Float64(Float64(y * b) / t)))
	t_4 = fma(y, Float64(z / fma(t, t_1, t)), Float64(x / Float64(1.0 + t_1)))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = t_4;
	elseif (t_3 <= 2e+14)
		tmp = Float64(t_2 / fma(b, Float64(y / t), Float64(a + 1.0)));
	elseif (t_3 <= Inf)
		tmp = t_4;
	else
		tmp = Float64(z / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(y * N[(b / t), $MachinePrecision] + a), $MachinePrecision]}, Block[{t$95$2 = N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 / N[(N[(a + 1.0), $MachinePrecision] + N[(N[(y * b), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(y * N[(z / N[(t * t$95$1 + t), $MachinePrecision]), $MachinePrecision] + N[(x / N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], t$95$4, If[LessEqual[t$95$3, 2e+14], N[(t$95$2 / N[(b * N[(y / t), $MachinePrecision] + N[(a + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], t$95$4, 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 2e14 < (/.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 68.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-rgt-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-+.f64 N/A

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

   5. Applied rewrites 94.2%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 2e14

   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

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 91.4

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

   4. Applied rewrites 91.4%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

Alternative 3: 89.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{y \cdot z}{t}\\ t_2 := \frac{t\_1}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ t_3 := \mathsf{fma}\left(y, \frac{z}{\mathsf{fma}\left(a + 1, t, y \cdot b\right)}, \frac{x}{a}\right)\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+294}:\\ \;\;\;\;\frac{t\_1}{\mathsf{fma}\left(b, \frac{y}{t}, a + 1\right)}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ x (/ (* y z) t)))
        (t_2 (/ t_1 (+ (+ a 1.0) (/ (* y b) t))))
        (t_3 (fma y (/ z (fma (+ a 1.0) t (* y b))) (/ x a))))
   (if (<= t_2 (- INFINITY))
     t_3
     (if (<= t_2 5e+294)
       (/ t_1 (fma b (/ y t) (+ a 1.0)))
       (if (<= t_2 INFINITY) t_3 (/ z b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x + ((y * z) / t);
	double t_2 = t_1 / ((a + 1.0) + ((y * b) / t));
	double t_3 = fma(y, (z / fma((a + 1.0), t, (y * b))), (x / a));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_3;
	} else if (t_2 <= 5e+294) {
		tmp = t_1 / fma(b, (y / t), (a + 1.0));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_3;
	} else {
		tmp = z / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x + Float64(Float64(y * z) / t))
	t_2 = Float64(t_1 / Float64(Float64(a + 1.0) + Float64(Float64(y * b) / t)))
	t_3 = fma(y, Float64(z / fma(Float64(a + 1.0), t, Float64(y * b))), Float64(x / a))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_3;
	elseif (t_2 <= 5e+294)
		tmp = Float64(t_1 / fma(b, Float64(y / t), Float64(a + 1.0)));
	elseif (t_2 <= Inf)
		tmp = t_3;
	else
		tmp = Float64(z / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / N[(N[(a + 1.0), $MachinePrecision] + N[(N[(y * b), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(y * N[(z / N[(N[(a + 1.0), $MachinePrecision] * t + N[(y * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x / a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$3, If[LessEqual[t$95$2, 5e+294], N[(t$95$1 / N[(b * N[(y / t), $MachinePrecision] + N[(a + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], t$95$3, 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 4.9999999999999999e294 < (/.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 36.2%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-rgt-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-+.f64 N/A

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

   5. Applied rewrites 88.2%

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

   6. Taylor expanded in a around inf

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

      1. lower-/.f64 76.9

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

   8. Applied rewrites 76.9%

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

   9. Taylor expanded in t around 0

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

       1. *-commutative N/A

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

       2. distribute-rgt-in N/A

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

       3. *-lft-identity N/A

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

       4. lower-fma.f64 N/A

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

       5. +-commutative N/A

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

       6. lower-fma.f64 79.8

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

   11. Applied rewrites 79.8%

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

       1. lift-fma.f64 N/A

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

       2. +-commutative N/A

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

       3. lift-fma.f64 N/A

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

       4. distribute-lft1-in N/A

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

       5. lift-+.f64 N/A

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

       6. lower-fma.f64 N/A

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

       7. lower-*.f64 79.9

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

   13. Applied rewrites 79.9%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 4.9999999999999999e294

   1. Initial program 90.1%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 92.5

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

   4. Applied rewrites 92.5%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

Alternative 4: 86.9% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 42.4%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-rgt-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-+.f64 N/A

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

   5. Applied rewrites 88.7%

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

   6. Taylor expanded in a around inf

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

      1. lower-/.f64 72.7

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

   8. Applied rewrites 72.7%

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

   9. Taylor expanded in t around 0

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

       1. *-commutative N/A

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

       2. distribute-rgt-in N/A

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

       3. *-lft-identity N/A

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

       4. lower-fma.f64 N/A

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

       5. +-commutative N/A

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

       6. lower-fma.f64 72.7

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

   11. Applied rewrites 72.7%

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

       1. lift-fma.f64 N/A

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

       2. +-commutative N/A

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

       3. lift-fma.f64 N/A

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

       4. distribute-lft1-in N/A

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

       5. lift-+.f64 N/A

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

       6. lower-fma.f64 N/A

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

       7. lower-*.f64 72.8

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

   13. Applied rewrites 72.8%

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

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

   1. Initial program 85.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. div-inv N/A

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

      9. lower-*.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-/.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-/l* N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-/.f64 82.6

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lift-/.f64 N/A

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

      21. lift-*.f64 N/A

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

      22. associate-/l* N/A

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

   4. Applied rewrites 85.6%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. /-rgt-identity N/A

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

      7. un-div-inv N/A

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

      8. /-rgt-identity N/A

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

      9. lower-/.f64 85.7

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

      10. lift-fma.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. associate-*r/ N/A

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

      13. associate-*l/ N/A

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

      14. lift-/.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-fma.f64 86.1

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

      17. lift-fma.f64 N/A

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

      18. lift-+.f64 N/A

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

      19. associate-+r+ N/A

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

      20. lift-fma.f64 N/A

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

      21. lower-+.f64 86.1

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

   6. Applied rewrites 88.2%

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

   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 y around inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 88.2%

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

Alternative 5: 83.2% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 82.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-/.f64 82.3

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/l* N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-/.f64 80.4

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lift-/.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. associate-/l* N/A

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

      21. lower-fma.f64 N/A

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

      22. lower-/.f64 83.1

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

   4. Applied rewrites 83.1%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

Alternative 6: 54.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{a + 1}\\ t_2 := \frac{\mathsf{fma}\left(x, t, y \cdot z\right)}{y \cdot b}\\ \mathbf{if}\;t \leq -3 \cdot 10^{+100}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq -1.8 \cdot 10^{-38}:\\ \;\;\;\;y \cdot \frac{z}{\mathsf{fma}\left(a, t, t\right)}\\ \mathbf{elif}\;t \leq -1.75 \cdot 10^{-180}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t \leq 2.4 \cdot 10^{-127}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;t \leq 5.6 \cdot 10^{-67}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ x (+ a 1.0))) (t_2 (/ (fma x t (* y z)) (* y b))))
   (if (<= t -3e+100)
     t_1
     (if (<= t -1.8e-38)
       (* y (/ z (fma a t t)))
       (if (<= t -1.75e-180)
         t_2
         (if (<= t 2.4e-127) (/ z b) (if (<= t 5.6e-67) t_2 t_1)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x / (a + 1.0);
	double t_2 = fma(x, t, (y * z)) / (y * b);
	double tmp;
	if (t <= -3e+100) {
		tmp = t_1;
	} else if (t <= -1.8e-38) {
		tmp = y * (z / fma(a, t, t));
	} else if (t <= -1.75e-180) {
		tmp = t_2;
	} else if (t <= 2.4e-127) {
		tmp = z / b;
	} else if (t <= 5.6e-67) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x / Float64(a + 1.0))
	t_2 = Float64(fma(x, t, Float64(y * z)) / Float64(y * b))
	tmp = 0.0
	if (t <= -3e+100)
		tmp = t_1;
	elseif (t <= -1.8e-38)
		tmp = Float64(y * Float64(z / fma(a, t, t)));
	elseif (t <= -1.75e-180)
		tmp = t_2;
	elseif (t <= 2.4e-127)
		tmp = Float64(z / b);
	elseif (t <= 5.6e-67)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x / N[(a + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * t + N[(y * z), $MachinePrecision]), $MachinePrecision] / N[(y * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -3e+100], t$95$1, If[LessEqual[t, -1.8e-38], N[(y * N[(z / N[(a * t + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -1.75e-180], t$95$2, If[LessEqual[t, 2.4e-127], N[(z / b), $MachinePrecision], If[LessEqual[t, 5.6e-67], t$95$2, t$95$1]]]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -2.99999999999999985e100 or 5.60000000000000021e-67 < t

   1. Initial program 85.9%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 60.9

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

   5. Applied rewrites 60.9%

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

   if -2.99999999999999985e100 < t < -1.8e-38

   1. Initial program 62.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-fma.f64 N/A

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

   5. Applied rewrites 51.7%

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

   6. Taylor expanded in z around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 46.0

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

   8. Applied rewrites 46.0%

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

   if -1.8e-38 < t < -1.75e-180 or 2.39999999999999982e-127 < t < 5.60000000000000021e-67

   1. Initial program 81.4%

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

   3. 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. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r/ N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 56.8

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

   5. Applied rewrites 56.8%

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

   6. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 70.3

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

   8. Applied rewrites 70.3%

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

   if -1.75e-180 < t < 2.39999999999999982e-127

   1. Initial program 53.5%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 68.6

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

   5. Applied rewrites 68.6%

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

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3 \cdot 10^{+100}:\\ \;\;\;\;\frac{x}{a + 1}\\ \mathbf{elif}\;t \leq -1.8 \cdot 10^{-38}:\\ \;\;\;\;y \cdot \frac{z}{\mathsf{fma}\left(a, t, t\right)}\\ \mathbf{elif}\;t \leq -1.75 \cdot 10^{-180}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, t, y \cdot z\right)}{y \cdot b}\\ \mathbf{elif}\;t \leq 2.4 \cdot 10^{-127}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;t \leq 5.6 \cdot 10^{-67}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, t, y \cdot z\right)}{y \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 73.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z, \frac{y}{\mathsf{fma}\left(y, b, \mathsf{fma}\left(t, a, t\right)\right)}, \frac{x}{a}\right)\\ \mathbf{if}\;a \leq -8.5 \cdot 10^{-19}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 5.2 \cdot 10^{+20}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}{\mathsf{fma}\left(y, \frac{b}{t}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma z (/ y (fma y b (fma t a t))) (/ x a))))
   (if (<= a -8.5e-19)
     t_1
     (if (<= a 5.2e+20) (/ (fma z (/ y t) x) (fma y (/ b t) 1.0)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(z, (y / fma(y, b, fma(t, a, t))), (x / a));
	double tmp;
	if (a <= -8.5e-19) {
		tmp = t_1;
	} else if (a <= 5.2e+20) {
		tmp = fma(z, (y / t), x) / fma(y, (b / t), 1.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(z, Float64(y / fma(y, b, fma(t, a, t))), Float64(x / a))
	tmp = 0.0
	if (a <= -8.5e-19)
		tmp = t_1;
	elseif (a <= 5.2e+20)
		tmp = Float64(fma(z, Float64(y / t), x) / fma(y, Float64(b / t), 1.0));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(z * N[(y / N[(y * b + N[(t * a + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x / a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -8.5e-19], t$95$1, If[LessEqual[a, 5.2e+20], N[(N[(z * N[(y / t), $MachinePrecision] + x), $MachinePrecision] / N[(y * N[(b / t), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -8.50000000000000003e-19 or 5.2e20 < a

   1. Initial program 72.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-rgt-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-+.f64 N/A

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

   5. Applied rewrites 75.1%

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

   6. Taylor expanded in a around inf

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

      1. lower-/.f64 69.4

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

   8. Applied rewrites 69.4%

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

   9. Taylor expanded in t around 0

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

       1. *-commutative N/A

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

       2. distribute-rgt-in N/A

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

       3. *-lft-identity N/A

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

       4. lower-fma.f64 N/A

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

       5. +-commutative N/A

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

       6. lower-fma.f64 75.2

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

   11. Applied rewrites 75.2%

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

       1. lift-fma.f64 N/A

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

       2. lift-fma.f64 N/A

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

       3. lift-/.f64 N/A

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

       4. lift-/.f64 N/A

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

       5. lift-/.f64 N/A

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

       6. associate-*r/ N/A

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

       7. *-commutative N/A

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

       8. associate-/l* N/A

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

       9. lower-fma.f64 N/A

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

       10. lower-/.f64 77.4

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

       11. lift-fma.f64 N/A

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

       12. *-commutative N/A

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

       13. lower-fma.f64 77.4

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

   13. Applied rewrites 77.4%

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

   if -8.50000000000000003e-19 < a < 5.2e20

   1. Initial program 77.3%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 78.9

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

   5. Applied rewrites 78.9%

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

Alternative 8: 69.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.05 \cdot 10^{-7}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{a + 1}\\ \mathbf{elif}\;t \leq -2.25 \cdot 10^{-250}:\\ \;\;\;\;\frac{y \cdot z}{\mathsf{fma}\left(y, b, \mathsf{fma}\left(a, t, t\right)\right)}\\ \mathbf{elif}\;t \leq 3.05 \cdot 10^{-57}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}{a + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= t -2.05e-7)
   (/ (fma y (/ z t) x) (+ a 1.0))
   (if (<= t -2.25e-250)
     (/ (* y z) (fma y b (fma a t t)))
     (if (<= t 3.05e-57)
       (/ (fma t (/ x y) z) b)
       (/ (fma z (/ y t) x) (+ a 1.0))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= -2.05e-7) {
		tmp = fma(y, (z / t), x) / (a + 1.0);
	} else if (t <= -2.25e-250) {
		tmp = (y * z) / fma(y, b, fma(a, t, t));
	} else if (t <= 3.05e-57) {
		tmp = fma(t, (x / y), z) / b;
	} else {
		tmp = fma(z, (y / t), x) / (a + 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (t <= -2.05e-7)
		tmp = Float64(fma(y, Float64(z / t), x) / Float64(a + 1.0));
	elseif (t <= -2.25e-250)
		tmp = Float64(Float64(y * z) / fma(y, b, fma(a, t, t)));
	elseif (t <= 3.05e-57)
		tmp = Float64(fma(t, Float64(x / y), z) / b);
	else
		tmp = Float64(fma(z, Float64(y / t), x) / Float64(a + 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[t, -2.05e-7], N[(N[(y * N[(z / t), $MachinePrecision] + x), $MachinePrecision] / N[(a + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -2.25e-250], N[(N[(y * z), $MachinePrecision] / N[(y * b + N[(a * t + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3.05e-57], N[(N[(t * N[(x / y), $MachinePrecision] + z), $MachinePrecision] / b), $MachinePrecision], N[(N[(z * N[(y / t), $MachinePrecision] + x), $MachinePrecision] / N[(a + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if t < -2.05e-7

   1. Initial program 76.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. div-inv N/A

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

      9. lower-*.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-/.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-/l* N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-/.f64 82.3

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lift-/.f64 N/A

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

      21. lift-*.f64 N/A

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

      22. associate-/l* N/A

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

   4. Applied rewrites 93.8%

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

   5. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 74.2

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

   7. Applied rewrites 74.2%

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

   if -2.05e-7 < t < -2.24999999999999997e-250

   1. Initial program 73.3%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-rgt-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-+.f64 N/A

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

   5. Applied rewrites 78.9%

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

   6. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 80.7

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

   8. Applied rewrites 80.7%

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

   9. Taylor expanded in z around inf

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

       1. lower-/.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. associate-+r+ N/A

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

       4. *-lft-identity N/A

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

       5. distribute-rgt-in N/A

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

       6. +-commutative N/A

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

       7. *-commutative N/A

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

       8. distribute-rgt-in N/A

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

       9. *-lft-identity N/A

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

       10. lower-fma.f64 N/A

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

       11. +-commutative N/A

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

       12. lower-fma.f64 72.2

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

   11. Applied rewrites 72.2%

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

   if -2.24999999999999997e-250 < t < 3.0499999999999999e-57

   1. Initial program 58.5%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-rgt-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-+.f64 N/A

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

   5. Applied rewrites 50.8%

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

   6. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 74.6

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

   8. Applied rewrites 74.6%

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

   if 3.0499999999999999e-57 < t

   1. Initial program 86.2%

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

   3. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-+.f64 73.2

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

   5. Applied rewrites 73.2%

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

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.05 \cdot 10^{-7}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{a + 1}\\ \mathbf{elif}\;t \leq -2.25 \cdot 10^{-250}:\\ \;\;\;\;\frac{y \cdot z}{\mathsf{fma}\left(y, b, \mathsf{fma}\left(a, t, t\right)\right)}\\ \mathbf{elif}\;t \leq 3.05 \cdot 10^{-57}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}{a + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 69.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{a + 1}\\ \mathbf{if}\;t \leq -2.05 \cdot 10^{-7}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq -2.25 \cdot 10^{-250}:\\ \;\;\;\;\frac{y \cdot z}{\mathsf{fma}\left(y, b, \mathsf{fma}\left(a, t, t\right)\right)}\\ \mathbf{elif}\;t \leq 3.05 \cdot 10^{-57}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (fma y (/ z t) x) (+ a 1.0))))
   (if (<= t -2.05e-7)
     t_1
     (if (<= t -2.25e-250)
       (/ (* y z) (fma y b (fma a t t)))
       (if (<= t 3.05e-57) (/ (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 = fma(y, (z / t), x) / (a + 1.0);
	double tmp;
	if (t <= -2.05e-7) {
		tmp = t_1;
	} else if (t <= -2.25e-250) {
		tmp = (y * z) / fma(y, b, fma(a, t, t));
	} else if (t <= 3.05e-57) {
		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(fma(y, Float64(z / t), x) / Float64(a + 1.0))
	tmp = 0.0
	if (t <= -2.05e-7)
		tmp = t_1;
	elseif (t <= -2.25e-250)
		tmp = Float64(Float64(y * z) / fma(y, b, fma(a, t, t)));
	elseif (t <= 3.05e-57)
		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[(N[(y * N[(z / t), $MachinePrecision] + x), $MachinePrecision] / N[(a + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -2.05e-7], t$95$1, If[LessEqual[t, -2.25e-250], N[(N[(y * z), $MachinePrecision] / N[(y * b + N[(a * t + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3.05e-57], N[(N[(t * N[(x / y), $MachinePrecision] + z), $MachinePrecision] / b), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t < -2.05e-7 or 3.0499999999999999e-57 < t

   1. Initial program 82.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. div-inv N/A

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

      9. lower-*.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-/.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-/l* N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-/.f64 85.4

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lift-/.f64 N/A

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

      21. lift-*.f64 N/A

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

      22. associate-/l* N/A

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

   4. Applied rewrites 92.7%

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

   5. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 72.5

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

   7. Applied rewrites 72.5%

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

   if -2.05e-7 < t < -2.24999999999999997e-250

   1. Initial program 73.3%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-rgt-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-+.f64 N/A

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

   5. Applied rewrites 78.9%

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

   6. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 80.7

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

   8. Applied rewrites 80.7%

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

   9. Taylor expanded in z around inf

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

       1. lower-/.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. associate-+r+ N/A

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

       4. *-lft-identity N/A

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

       5. distribute-rgt-in N/A

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

       6. +-commutative N/A

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

       7. *-commutative N/A

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

       8. distribute-rgt-in N/A

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

       9. *-lft-identity N/A

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

       10. lower-fma.f64 N/A

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

       11. +-commutative N/A

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

       12. lower-fma.f64 72.2

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

   11. Applied rewrites 72.2%

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

   if -2.24999999999999997e-250 < t < 3.0499999999999999e-57

   1. Initial program 58.5%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-rgt-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-+.f64 N/A

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

   5. Applied rewrites 50.8%

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

   6. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 74.6

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

   8. Applied rewrites 74.6%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.05 \cdot 10^{-7}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{a + 1}\\ \mathbf{elif}\;t \leq -2.25 \cdot 10^{-250}:\\ \;\;\;\;\frac{y \cdot z}{\mathsf{fma}\left(y, b, \mathsf{fma}\left(a, t, t\right)\right)}\\ \mathbf{elif}\;t \leq 3.05 \cdot 10^{-57}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{a + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 62.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{1 + \mathsf{fma}\left(b, \frac{y}{t}, a\right)}\\ \mathbf{if}\;t \leq -2.2 \cdot 10^{+55}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1660000000000:\\ \;\;\;\;\frac{y \cdot z}{\mathsf{fma}\left(y, b, \mathsf{fma}\left(a, t, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ x (+ 1.0 (fma b (/ y t) a)))))
   (if (<= t -2.2e+55)
     t_1
     (if (<= t 1660000000000.0) (/ (* y z) (fma y b (fma a t t))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x / (1.0 + fma(b, (y / t), a));
	double tmp;
	if (t <= -2.2e+55) {
		tmp = t_1;
	} else if (t <= 1660000000000.0) {
		tmp = (y * z) / fma(y, b, fma(a, t, t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x / Float64(1.0 + fma(b, Float64(y / t), a)))
	tmp = 0.0
	if (t <= -2.2e+55)
		tmp = t_1;
	elseif (t <= 1660000000000.0)
		tmp = Float64(Float64(y * z) / fma(y, b, fma(a, t, t)));
	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 + N[(b * N[(y / t), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -2.2e+55], t$95$1, If[LessEqual[t, 1660000000000.0], N[(N[(y * z), $MachinePrecision] / N[(y * b + N[(a * t + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -2.2000000000000001e55 or 1.66e12 < t

   1. Initial program 80.8%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-rgt-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-+.f64 N/A

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

   5. Applied rewrites 88.2%

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

   6. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 73.7

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

   8. Applied rewrites 73.7%

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

   if -2.2000000000000001e55 < t < 1.66e12

   1. Initial program 70.4%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-rgt-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-+.f64 N/A

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

   5. Applied rewrites 68.3%

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

   6. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 71.0

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

   8. Applied rewrites 71.0%

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

   9. Taylor expanded in z around inf

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

       1. lower-/.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. associate-+r+ N/A

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

       4. *-lft-identity N/A

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

       5. distribute-rgt-in N/A

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

       6. +-commutative N/A

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

       7. *-commutative N/A

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

       8. distribute-rgt-in N/A

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

       9. *-lft-identity N/A

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

       10. lower-fma.f64 N/A

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

       11. +-commutative N/A

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

       12. lower-fma.f64 63.2

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

   11. Applied rewrites 63.2%

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

Alternative 11: 58.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{a + 1}\\ \mathbf{if}\;t \leq -8.5 \cdot 10^{+100}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 2.25 \cdot 10^{+17}:\\ \;\;\;\;\frac{y \cdot z}{\mathsf{fma}\left(y, b, \mathsf{fma}\left(a, t, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ x (+ a 1.0))))
   (if (<= t -8.5e+100)
     t_1
     (if (<= t 2.25e+17) (/ (* y z) (fma y b (fma a t t))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x / (a + 1.0);
	double tmp;
	if (t <= -8.5e+100) {
		tmp = t_1;
	} else if (t <= 2.25e+17) {
		tmp = (y * z) / fma(y, b, fma(a, t, t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x / Float64(a + 1.0))
	tmp = 0.0
	if (t <= -8.5e+100)
		tmp = t_1;
	elseif (t <= 2.25e+17)
		tmp = Float64(Float64(y * z) / fma(y, b, fma(a, t, t)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x / N[(a + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -8.5e+100], t$95$1, If[LessEqual[t, 2.25e+17], N[(N[(y * z), $MachinePrecision] / N[(y * b + N[(a * t + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -8.50000000000000043e100 or 2.25e17 < t

   1. Initial program 84.9%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 68.5

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

   5. Applied rewrites 68.5%

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

   if -8.50000000000000043e100 < t < 2.25e17

   1. Initial program 68.8%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-rgt-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-+.f64 N/A

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

   5. Applied rewrites 69.2%

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

   6. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 71.7

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

   8. Applied rewrites 71.7%

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

   9. Taylor expanded in z around inf

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

       1. lower-/.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. associate-+r+ N/A

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

       4. *-lft-identity N/A

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

       5. distribute-rgt-in N/A

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

       6. +-commutative N/A

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

       7. *-commutative N/A

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

       8. distribute-rgt-in N/A

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

       9. *-lft-identity N/A

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

       10. lower-fma.f64 N/A

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

       11. +-commutative N/A

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

       12. lower-fma.f64 60.0

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

   11. Applied rewrites 60.0%

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

4. Final simplification 63.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -8.5 \cdot 10^{+100}:\\ \;\;\;\;\frac{x}{a + 1}\\ \mathbf{elif}\;t \leq 2.25 \cdot 10^{+17}:\\ \;\;\;\;\frac{y \cdot z}{\mathsf{fma}\left(y, b, \mathsf{fma}\left(a, t, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 54.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{a + 1}\\ \mathbf{if}\;t \leq -3 \cdot 10^{+100}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq -1.3 \cdot 10^{-95}:\\ \;\;\;\;y \cdot \frac{z}{\mathsf{fma}\left(a, t, t\right)}\\ \mathbf{elif}\;t \leq 3.2 \cdot 10^{-56}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ x (+ a 1.0))))
   (if (<= t -3e+100)
     t_1
     (if (<= t -1.3e-95)
       (* y (/ z (fma a t t)))
       (if (<= t 3.2e-56) (/ z b) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x / (a + 1.0);
	double tmp;
	if (t <= -3e+100) {
		tmp = t_1;
	} else if (t <= -1.3e-95) {
		tmp = y * (z / fma(a, t, t));
	} else if (t <= 3.2e-56) {
		tmp = z / b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x / Float64(a + 1.0))
	tmp = 0.0
	if (t <= -3e+100)
		tmp = t_1;
	elseif (t <= -1.3e-95)
		tmp = Float64(y * Float64(z / fma(a, t, t)));
	elseif (t <= 3.2e-56)
		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 / N[(a + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -3e+100], t$95$1, If[LessEqual[t, -1.3e-95], N[(y * N[(z / N[(a * t + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3.2e-56], N[(z / b), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t < -2.99999999999999985e100 or 3.19999999999999986e-56 < t

   1. Initial program 86.4%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 61.5

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

   5. Applied rewrites 61.5%

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

   if -2.99999999999999985e100 < t < -1.3e-95

   1. Initial program 67.3%

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

   3. Taylor expanded in y around 0

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

      1. lower-fma.f64 N/A

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

   5. Applied rewrites 50.1%

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

   6. Taylor expanded in z around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 45.5

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

   8. Applied rewrites 45.5%

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

   if -1.3e-95 < t < 3.19999999999999986e-56

   1. Initial program 63.6%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 58.6

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

   5. Applied rewrites 58.6%

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

4. Final simplification 58.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3 \cdot 10^{+100}:\\ \;\;\;\;\frac{x}{a + 1}\\ \mathbf{elif}\;t \leq -1.3 \cdot 10^{-95}:\\ \;\;\;\;y \cdot \frac{z}{\mathsf{fma}\left(a, t, t\right)}\\ \mathbf{elif}\;t \leq 3.2 \cdot 10^{-56}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 39.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3.5 \cdot 10^{+236}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq -6.5 \cdot 10^{+77}:\\ \;\;\;\;\frac{x}{a}\\ \mathbf{elif}\;t \leq 3.75 \cdot 10^{-56}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= t -3.5e+236)
   x
   (if (<= t -6.5e+77) (/ x a) (if (<= t 3.75e-56) (/ z b) x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= -3.5e+236) {
		tmp = x;
	} else if (t <= -6.5e+77) {
		tmp = x / a;
	} else if (t <= 3.75e-56) {
		tmp = z / b;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (t <= (-3.5d+236)) then
        tmp = x
    else if (t <= (-6.5d+77)) then
        tmp = x / a
    else if (t <= 3.75d-56) then
        tmp = z / b
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= -3.5e+236) {
		tmp = x;
	} else if (t <= -6.5e+77) {
		tmp = x / a;
	} else if (t <= 3.75e-56) {
		tmp = z / b;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if t <= -3.5e+236:
		tmp = x
	elif t <= -6.5e+77:
		tmp = x / a
	elif t <= 3.75e-56:
		tmp = z / b
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (t <= -3.5e+236)
		tmp = x;
	elseif (t <= -6.5e+77)
		tmp = Float64(x / a);
	elseif (t <= 3.75e-56)
		tmp = Float64(z / b);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (t <= -3.5e+236)
		tmp = x;
	elseif (t <= -6.5e+77)
		tmp = x / a;
	elseif (t <= 3.75e-56)
		tmp = z / b;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[t, -3.5e+236], x, If[LessEqual[t, -6.5e+77], N[(x / a), $MachinePrecision], If[LessEqual[t, 3.75e-56], N[(z / b), $MachinePrecision], x]]]

Derivation

1. Split input into 3 regimes

2. if t < -3.49999999999999979e236 or 3.75000000000000021e-56 < t

   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 y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 57.9

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

   5. Applied rewrites 57.9%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 35.3%

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

   if -3.49999999999999979e236 < t < -6.5e77

   1. Initial program 80.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 62.5

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

   5. Applied rewrites 62.5%

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

   6. Taylor expanded in a around inf

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

      1. lower-/.f64 43.7

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

   8. Applied rewrites 43.7%

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

   if -6.5e77 < t < 3.75000000000000021e-56

   1. Initial program 64.8%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 54.0

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

   5. Applied rewrites 54.0%

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

4. Final simplification 45.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.5 \cdot 10^{+236}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq -6.5 \cdot 10^{+77}:\\ \;\;\;\;\frac{x}{a}\\ \mathbf{elif}\;t \leq 3.75 \cdot 10^{-56}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 41.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a + 1 \leq -5000000:\\ \;\;\;\;\frac{x}{a}\\ \mathbf{elif}\;a + 1 \leq 1.00005:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (+ a 1.0) -5000000.0) (/ x a) (if (<= (+ a 1.0) 1.00005) x (/ x a))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((a + 1.0) <= -5000000.0) {
		tmp = x / a;
	} else if ((a + 1.0) <= 1.00005) {
		tmp = x;
	} else {
		tmp = x / a;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((a + 1.0d0) <= (-5000000.0d0)) then
        tmp = x / a
    else if ((a + 1.0d0) <= 1.00005d0) then
        tmp = x
    else
        tmp = x / a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((a + 1.0) <= -5000000.0) {
		tmp = x / a;
	} else if ((a + 1.0) <= 1.00005) {
		tmp = x;
	} else {
		tmp = x / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (a + 1.0) <= -5000000.0:
		tmp = x / a
	elif (a + 1.0) <= 1.00005:
		tmp = x
	else:
		tmp = x / a
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(a + 1.0) <= -5000000.0)
		tmp = Float64(x / a);
	elseif (Float64(a + 1.0) <= 1.00005)
		tmp = x;
	else
		tmp = Float64(x / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((a + 1.0) <= -5000000.0)
		tmp = x / a;
	elseif ((a + 1.0) <= 1.00005)
		tmp = x;
	else
		tmp = x / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(a + 1.0), $MachinePrecision], -5000000.0], N[(x / a), $MachinePrecision], If[LessEqual[N[(a + 1.0), $MachinePrecision], 1.00005], x, N[(x / a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 a #s(literal 1 binary64)) < -5e6 or 1.00005000000000011 < (+.f64 a #s(literal 1 binary64))

   1. Initial program 73.2%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 41.2

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

   5. Applied rewrites 41.2%

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

   6. Taylor expanded in a around inf

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

      1. lower-/.f64 40.0

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

   8. Applied rewrites 40.0%

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

   if -5e6 < (+.f64 a #s(literal 1 binary64)) < 1.00005000000000011

   1. Initial program 76.4%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 30.6

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

   5. Applied rewrites 30.6%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 30.6%

      \[\leadsto \frac{x}{\color{blue}{1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 35.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a + 1 \leq -5000000:\\ \;\;\;\;\frac{x}{a}\\ \mathbf{elif}\;a + 1 \leq 1.00005:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 54.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{a + 1}\\ \mathbf{if}\;t \leq -2.85 \cdot 10^{+76}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 3.2 \cdot 10^{-56}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ x (+ a 1.0))))
   (if (<= t -2.85e+76) t_1 (if (<= t 3.2e-56) (/ z b) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x / (a + 1.0);
	double tmp;
	if (t <= -2.85e+76) {
		tmp = t_1;
	} else if (t <= 3.2e-56) {
		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 / (a + 1.0d0)
    if (t <= (-2.85d+76)) then
        tmp = t_1
    else if (t <= 3.2d-56) 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 / (a + 1.0);
	double tmp;
	if (t <= -2.85e+76) {
		tmp = t_1;
	} else if (t <= 3.2e-56) {
		tmp = z / b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x / (a + 1.0)
	tmp = 0
	if t <= -2.85e+76:
		tmp = t_1
	elif t <= 3.2e-56:
		tmp = z / b
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x / Float64(a + 1.0))
	tmp = 0.0
	if (t <= -2.85e+76)
		tmp = t_1;
	elseif (t <= 3.2e-56)
		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 / (a + 1.0);
	tmp = 0.0;
	if (t <= -2.85e+76)
		tmp = t_1;
	elseif (t <= 3.2e-56)
		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[(a + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -2.85e+76], t$95$1, If[LessEqual[t, 3.2e-56], N[(z / b), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -2.85000000000000002e76 or 3.19999999999999986e-56 < t

   1. Initial program 85.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 59.1

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

   5. Applied rewrites 59.1%

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

   if -2.85000000000000002e76 < t < 3.19999999999999986e-56

   1. Initial program 64.8%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 54.0

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

   5. Applied rewrites 54.0%

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

4. Final simplification 56.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.85 \cdot 10^{+76}:\\ \;\;\;\;\frac{x}{a + 1}\\ \mathbf{elif}\;t \leq 3.2 \cdot 10^{-56}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 18.7% accurate, 5.9× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = x - (x * a)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.9%

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

3. Taylor expanded in y around 0

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

   1. lower-/.f64 N/A

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

   2. lower-+.f64 35.7

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

5. Applied rewrites 35.7%

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

6. Taylor expanded in a around 0

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

   1. mul-1-neg N/A

      \[\leadsto x + \color{blue}{\left(\mathsf{neg}\left(a \cdot x\right)\right)} \]

   2. unsub-neg N/A

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

   3. lower--.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 17.5

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

8. Applied rewrites 17.5%

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

Alternative 17: 3.9% accurate, 6.6× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = -(x * a)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.9%

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

3. Taylor expanded in y around 0

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

   1. lower-/.f64 N/A

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

   2. lower-+.f64 35.7

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

5. Applied rewrites 35.7%

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

6. Taylor expanded in a around 0

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

   1. mul-1-neg N/A

      \[\leadsto x + \color{blue}{\left(\mathsf{neg}\left(a \cdot x\right)\right)} \]

   2. unsub-neg N/A

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

   3. lower--.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 17.5

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

8. Applied rewrites 17.5%

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

9. Taylor expanded in a around inf

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

    1. mul-1-neg N/A

       \[\leadsto \color{blue}{\mathsf{neg}\left(a \cdot x\right)} \]

    2. *-commutative N/A

       \[\leadsto \mathsf{neg}\left(\color{blue}{x \cdot a}\right) \]

    3. distribute-rgt-neg-in N/A

       \[\leadsto \color{blue}{x \cdot \left(\mathsf{neg}\left(a\right)\right)} \]

    4. mul-1-neg N/A

       \[\leadsto x \cdot \color{blue}{\left(-1 \cdot a\right)} \]

    5. lower-*.f64 N/A

       \[\leadsto \color{blue}{x \cdot \left(-1 \cdot a\right)} \]

    6. mul-1-neg N/A

       \[\leadsto x \cdot \color{blue}{\left(\mathsf{neg}\left(a\right)\right)} \]

    7. lower-neg.f64 5.1

       \[\leadsto x \cdot \color{blue}{\left(-a\right)} \]

11. Applied rewrites 5.1%

    \[\leadsto \color{blue}{x \cdot \left(-a\right)} \]

12. Final simplification 5.1%

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

Alternative 18: 19.2% accurate, 53.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_] := x

Derivation

1. Initial program 74.9%

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

3. Taylor expanded in y around 0

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

   1. lower-/.f64 N/A

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

   2. lower-+.f64 35.7

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

5. Applied rewrites 35.7%

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

6. Taylor expanded in a around 0

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

8. Applied rewrites 17.9%

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

9. Final simplification 17.9%

   \[\leadsto x \]
10. Add Preprocessing

Developer Target 1: 79.5% 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 2024219 
(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))))