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

Percentage Accurate: 74.8% → 88.8%

Time: 9.4s

Alternatives: 14

Speedup: 0.3×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 74.8% 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: 88.8% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t))))
        (t_2 (* z (/ y (fma t a (fma b y t))))))
   (if (<= t_1 (- INFINITY))
     t_2
     (if (<= t_1 1e+283)
       (/ (fma z (/ y t) x) (fma (/ y t) b (+ 1.0 a)))
       (if (<= t_1 INFINITY) t_2 (/ (fma t (/ x y) z) b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t));
	double t_2 = z * (y / fma(t, a, fma(b, y, t)));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_2;
	} else if (t_1 <= 1e+283) {
		tmp = fma(z, (y / t), x) / fma((y / t), b, (1.0 + a));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_2;
	} else {
		tmp = fma(t, (x / y), z) / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x + Float64(Float64(y * z) / t)) / Float64(Float64(a + 1.0) + Float64(Float64(y * b) / t)))
	t_2 = Float64(z * Float64(y / fma(t, a, fma(b, y, t))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = t_2;
	elseif (t_1 <= 1e+283)
		tmp = Float64(fma(z, Float64(y / t), x) / fma(Float64(y / t), b, Float64(1.0 + a)));
	elseif (t_1 <= Inf)
		tmp = t_2;
	else
		tmp = Float64(fma(t, Float64(x / y), z) / b);
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -inf.0 or 9.99999999999999955e282 < (/.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 31.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 55.3

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 55.3

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 55.3

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

   4. Applied rewrites 55.3%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. *-lft-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 40.8

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

   7. Applied rewrites 40.8%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 57.4%

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

   12. Applied rewrites 91.6%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 9.99999999999999955e282

   1. Initial program 90.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 88.1

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

   4. Applied rewrites 88.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lower-+.f64 92.9

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

   6. Applied rewrites 92.9%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-*l/ N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 0.6

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

   5. Applied rewrites 0.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 91.4%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 95.7%

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

4. Final simplification 93.0%

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

Alternative 2: 77.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{y \cdot z}{t}\\ t_2 := \frac{t\_1}{1 + a}\\ t_3 := \frac{t\_1}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ t_4 := z \cdot \frac{y}{\mathsf{fma}\left(t, a, \mathsf{fma}\left(b, y, t\right)\right)}\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_3 \leq -2 \cdot 10^{-306}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{b}, \frac{t}{y}, \frac{z}{b}\right)\\ \mathbf{elif}\;t\_3 \leq 4 \cdot 10^{+121}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 10^{+283}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{\mathsf{fma}\left(\frac{y}{t}, b, 1\right)}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;t\_4\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ x (/ (* y z) t)))
        (t_2 (/ t_1 (+ 1.0 a)))
        (t_3 (/ t_1 (+ (+ a 1.0) (/ (* y b) t))))
        (t_4 (* z (/ y (fma t a (fma b y t))))))
   (if (<= t_3 (- INFINITY))
     t_4
     (if (<= t_3 -2e-306)
       t_2
       (if (<= t_3 0.0)
         (fma (/ x b) (/ t y) (/ z b))
         (if (<= t_3 4e+121)
           t_2
           (if (<= t_3 1e+283)
             (/ (fma (/ y t) z x) (fma (/ y t) b 1.0))
             (if (<= t_3 INFINITY) t_4 (/ (fma t (/ x y) 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 / (1.0 + a);
	double t_3 = t_1 / ((a + 1.0) + ((y * b) / t));
	double t_4 = z * (y / fma(t, a, fma(b, y, t)));
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = t_4;
	} else if (t_3 <= -2e-306) {
		tmp = t_2;
	} else if (t_3 <= 0.0) {
		tmp = fma((x / b), (t / y), (z / b));
	} else if (t_3 <= 4e+121) {
		tmp = t_2;
	} else if (t_3 <= 1e+283) {
		tmp = fma((y / t), z, x) / fma((y / t), b, 1.0);
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = t_4;
	} else {
		tmp = fma(t, (x / y), 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(1.0 + a))
	t_3 = Float64(t_1 / Float64(Float64(a + 1.0) + Float64(Float64(y * b) / t)))
	t_4 = Float64(z * Float64(y / fma(t, a, fma(b, y, t))))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = t_4;
	elseif (t_3 <= -2e-306)
		tmp = t_2;
	elseif (t_3 <= 0.0)
		tmp = fma(Float64(x / b), Float64(t / y), Float64(z / b));
	elseif (t_3 <= 4e+121)
		tmp = t_2;
	elseif (t_3 <= 1e+283)
		tmp = Float64(fma(Float64(y / t), z, x) / fma(Float64(y / t), b, 1.0));
	elseif (t_3 <= Inf)
		tmp = t_4;
	else
		tmp = Float64(fma(t, Float64(x / y), z) / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / N[(1.0 + a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 / N[(N[(a + 1.0), $MachinePrecision] + N[(N[(y * b), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(z * N[(y / N[(t * a + N[(b * y + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], t$95$4, If[LessEqual[t$95$3, -2e-306], t$95$2, If[LessEqual[t$95$3, 0.0], N[(N[(x / b), $MachinePrecision] * N[(t / y), $MachinePrecision] + N[(z / b), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 4e+121], t$95$2, If[LessEqual[t$95$3, 1e+283], N[(N[(N[(y / t), $MachinePrecision] * z + x), $MachinePrecision] / N[(N[(y / t), $MachinePrecision] * b + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], t$95$4, N[(N[(t * N[(x / y), $MachinePrecision] + z), $MachinePrecision] / b), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -inf.0 or 9.99999999999999955e282 < (/.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 31.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 55.3

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 55.3

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 55.3

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

   4. Applied rewrites 55.3%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. *-lft-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 40.8

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

   7. Applied rewrites 40.8%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 57.4%

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

   12. Applied rewrites 91.6%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -2.00000000000000006e-306 or 0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 4.00000000000000015e121

   1. Initial program 99.1%

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

   3. Taylor expanded in y around 0

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

      1. lower-+.f64 77.5

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

   5. Applied rewrites 77.5%

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

   if -2.00000000000000006e-306 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 0.0

   1. Initial program 50.9%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-*l/ N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 19.4

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

   5. Applied rewrites 19.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 73.4%

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

   if 4.00000000000000015e121 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 9.99999999999999955e282

   1. Initial program 99.6%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-*l/ N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 94.6

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

   5. Applied rewrites 94.6%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-*l/ N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 0.6

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

   5. Applied rewrites 0.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 91.4%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 95.7%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq -\infty:\\ \;\;\;\;z \cdot \frac{y}{\mathsf{fma}\left(t, a, \mathsf{fma}\left(b, y, t\right)\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq -2 \cdot 10^{-306}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{1 + a}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{b}, \frac{t}{y}, \frac{z}{b}\right)\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq 4 \cdot 10^{+121}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{1 + a}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq 10^{+283}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{\mathsf{fma}\left(\frac{y}{t}, b, 1\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq \infty:\\ \;\;\;\;z \cdot \frac{y}{\mathsf{fma}\left(t, a, \mathsf{fma}\left(b, y, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{y \cdot z}{t}\\ t_2 := \frac{t\_1}{1 + a}\\ t_3 := \frac{t\_1}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ t_4 := z \cdot \frac{y}{\mathsf{fma}\left(t, a, \mathsf{fma}\left(b, y, t\right)\right)}\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_3 \leq -2 \cdot 10^{-306}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{b}, \frac{t}{y}, \frac{z}{b}\right)\\ \mathbf{elif}\;t\_3 \leq 10^{+283}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;t\_4\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ x (/ (* y z) t)))
        (t_2 (/ t_1 (+ 1.0 a)))
        (t_3 (/ t_1 (+ (+ a 1.0) (/ (* y b) t))))
        (t_4 (* z (/ y (fma t a (fma b y t))))))
   (if (<= t_3 (- INFINITY))
     t_4
     (if (<= t_3 -2e-306)
       t_2
       (if (<= t_3 0.0)
         (fma (/ x b) (/ t y) (/ z b))
         (if (<= t_3 1e+283)
           t_2
           (if (<= t_3 INFINITY) t_4 (/ (fma t (/ x y) 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 / (1.0 + a);
	double t_3 = t_1 / ((a + 1.0) + ((y * b) / t));
	double t_4 = z * (y / fma(t, a, fma(b, y, t)));
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = t_4;
	} else if (t_3 <= -2e-306) {
		tmp = t_2;
	} else if (t_3 <= 0.0) {
		tmp = fma((x / b), (t / y), (z / b));
	} else if (t_3 <= 1e+283) {
		tmp = t_2;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = t_4;
	} else {
		tmp = fma(t, (x / y), 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(1.0 + a))
	t_3 = Float64(t_1 / Float64(Float64(a + 1.0) + Float64(Float64(y * b) / t)))
	t_4 = Float64(z * Float64(y / fma(t, a, fma(b, y, t))))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = t_4;
	elseif (t_3 <= -2e-306)
		tmp = t_2;
	elseif (t_3 <= 0.0)
		tmp = fma(Float64(x / b), Float64(t / y), Float64(z / b));
	elseif (t_3 <= 1e+283)
		tmp = t_2;
	elseif (t_3 <= Inf)
		tmp = t_4;
	else
		tmp = Float64(fma(t, Float64(x / y), z) / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / N[(1.0 + a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 / N[(N[(a + 1.0), $MachinePrecision] + N[(N[(y * b), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(z * N[(y / N[(t * a + N[(b * y + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], t$95$4, If[LessEqual[t$95$3, -2e-306], t$95$2, If[LessEqual[t$95$3, 0.0], N[(N[(x / b), $MachinePrecision] * N[(t / y), $MachinePrecision] + N[(z / b), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 1e+283], t$95$2, If[LessEqual[t$95$3, Infinity], t$95$4, N[(N[(t * N[(x / y), $MachinePrecision] + z), $MachinePrecision] / b), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -inf.0 or 9.99999999999999955e282 < (/.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 31.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 55.3

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 55.3

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 55.3

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

   4. Applied rewrites 55.3%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. *-lft-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 40.8

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

   7. Applied rewrites 40.8%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 57.4%

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

   12. Applied rewrites 91.6%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -2.00000000000000006e-306 or 0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 9.99999999999999955e282

   1. Initial program 99.2%

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

   3. Taylor expanded in y around 0

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

      1. lower-+.f64 76.4

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

   5. Applied rewrites 76.4%

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

   if -2.00000000000000006e-306 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 0.0

   1. Initial program 50.9%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-*l/ N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 19.4

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

   5. Applied rewrites 19.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 73.4%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-*l/ N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 0.6

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

   5. Applied rewrites 0.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 91.4%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 95.7%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq -\infty:\\ \;\;\;\;z \cdot \frac{y}{\mathsf{fma}\left(t, a, \mathsf{fma}\left(b, y, t\right)\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq -2 \cdot 10^{-306}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{1 + a}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{b}, \frac{t}{y}, \frac{z}{b}\right)\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq 10^{+283}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{1 + a}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq \infty:\\ \;\;\;\;z \cdot \frac{y}{\mathsf{fma}\left(t, a, \mathsf{fma}\left(b, y, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{y \cdot z}{t}\\ t_2 := \frac{t\_1}{1 + a}\\ t_3 := \frac{t\_1}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ t_4 := z \cdot \frac{y}{\mathsf{fma}\left(t, a, \mathsf{fma}\left(b, y, t\right)\right)}\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_3 \leq -2 \cdot 10^{-306}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(\frac{y}{t}, b, 1 + a\right)}\\ \mathbf{elif}\;t\_3 \leq 10^{+283}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;t\_4\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ x (/ (* y z) t)))
        (t_2 (/ t_1 (+ 1.0 a)))
        (t_3 (/ t_1 (+ (+ a 1.0) (/ (* y b) t))))
        (t_4 (* z (/ y (fma t a (fma b y t))))))
   (if (<= t_3 (- INFINITY))
     t_4
     (if (<= t_3 -2e-306)
       t_2
       (if (<= t_3 0.0)
         (/ x (fma (/ y t) b (+ 1.0 a)))
         (if (<= t_3 1e+283)
           t_2
           (if (<= t_3 INFINITY) t_4 (/ (fma t (/ x y) 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 / (1.0 + a);
	double t_3 = t_1 / ((a + 1.0) + ((y * b) / t));
	double t_4 = z * (y / fma(t, a, fma(b, y, t)));
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = t_4;
	} else if (t_3 <= -2e-306) {
		tmp = t_2;
	} else if (t_3 <= 0.0) {
		tmp = x / fma((y / t), b, (1.0 + a));
	} else if (t_3 <= 1e+283) {
		tmp = t_2;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = t_4;
	} else {
		tmp = fma(t, (x / y), 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(1.0 + a))
	t_3 = Float64(t_1 / Float64(Float64(a + 1.0) + Float64(Float64(y * b) / t)))
	t_4 = Float64(z * Float64(y / fma(t, a, fma(b, y, t))))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = t_4;
	elseif (t_3 <= -2e-306)
		tmp = t_2;
	elseif (t_3 <= 0.0)
		tmp = Float64(x / fma(Float64(y / t), b, Float64(1.0 + a)));
	elseif (t_3 <= 1e+283)
		tmp = t_2;
	elseif (t_3 <= Inf)
		tmp = t_4;
	else
		tmp = Float64(fma(t, Float64(x / y), z) / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / N[(1.0 + a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 / N[(N[(a + 1.0), $MachinePrecision] + N[(N[(y * b), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(z * N[(y / N[(t * a + N[(b * y + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], t$95$4, If[LessEqual[t$95$3, -2e-306], t$95$2, If[LessEqual[t$95$3, 0.0], N[(x / N[(N[(y / t), $MachinePrecision] * b + N[(1.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 1e+283], t$95$2, If[LessEqual[t$95$3, Infinity], t$95$4, N[(N[(t * N[(x / y), $MachinePrecision] + z), $MachinePrecision] / b), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -inf.0 or 9.99999999999999955e282 < (/.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 31.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 55.3

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 55.3

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 55.3

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

   4. Applied rewrites 55.3%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. *-lft-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 40.8

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

   7. Applied rewrites 40.8%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 57.4%

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

   12. Applied rewrites 91.6%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -2.00000000000000006e-306 or 0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 9.99999999999999955e282

   1. Initial program 99.2%

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

   3. Taylor expanded in y around 0

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

      1. lower-+.f64 76.4

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

   5. Applied rewrites 76.4%

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

   if -2.00000000000000006e-306 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 0.0

   1. Initial program 50.9%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-+.f64 70.5

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

   5. Applied rewrites 70.5%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-*l/ N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 0.6

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

   5. Applied rewrites 0.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 91.4%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 95.7%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq -\infty:\\ \;\;\;\;z \cdot \frac{y}{\mathsf{fma}\left(t, a, \mathsf{fma}\left(b, y, t\right)\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq -2 \cdot 10^{-306}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{1 + a}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq 0:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(\frac{y}{t}, b, 1 + a\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq 10^{+283}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{1 + a}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq \infty:\\ \;\;\;\;z \cdot \frac{y}{\mathsf{fma}\left(t, a, \mathsf{fma}\left(b, y, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{1 + a}\\ t_2 := \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ t_3 := z \cdot \frac{y}{\mathsf{fma}\left(t, a, \mathsf{fma}\left(b, y, t\right)\right)}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq -2 \cdot 10^{-306}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(\frac{y}{t}, b, 1 + a\right)}\\ \mathbf{elif}\;t\_2 \leq 10^{+283}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (fma (/ y t) z x) (+ 1.0 a)))
        (t_2 (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t))))
        (t_3 (* z (/ y (fma t a (fma b y t))))))
   (if (<= t_2 (- INFINITY))
     t_3
     (if (<= t_2 -2e-306)
       t_1
       (if (<= t_2 0.0)
         (/ x (fma (/ y t) b (+ 1.0 a)))
         (if (<= t_2 1e+283)
           t_1
           (if (<= t_2 INFINITY) t_3 (/ (fma t (/ x y) z) b))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma((y / t), z, x) / (1.0 + a);
	double t_2 = (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t));
	double t_3 = z * (y / fma(t, a, fma(b, y, t)));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_3;
	} else if (t_2 <= -2e-306) {
		tmp = t_1;
	} else if (t_2 <= 0.0) {
		tmp = x / fma((y / t), b, (1.0 + a));
	} else if (t_2 <= 1e+283) {
		tmp = t_1;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_3;
	} else {
		tmp = fma(t, (x / y), z) / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(fma(Float64(y / t), z, x) / Float64(1.0 + a))
	t_2 = Float64(Float64(x + Float64(Float64(y * z) / t)) / Float64(Float64(a + 1.0) + Float64(Float64(y * b) / t)))
	t_3 = Float64(z * Float64(y / fma(t, a, fma(b, y, t))))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_3;
	elseif (t_2 <= -2e-306)
		tmp = t_1;
	elseif (t_2 <= 0.0)
		tmp = Float64(x / fma(Float64(y / t), b, Float64(1.0 + a)));
	elseif (t_2 <= 1e+283)
		tmp = t_1;
	elseif (t_2 <= Inf)
		tmp = t_3;
	else
		tmp = Float64(fma(t, Float64(x / y), z) / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(y / t), $MachinePrecision] * z + x), $MachinePrecision] / N[(1.0 + a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = 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]}, Block[{t$95$3 = N[(z * N[(y / N[(t * a + N[(b * y + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$3, If[LessEqual[t$95$2, -2e-306], t$95$1, If[LessEqual[t$95$2, 0.0], N[(x / N[(N[(y / t), $MachinePrecision] * b + N[(1.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e+283], t$95$1, If[LessEqual[t$95$2, Infinity], t$95$3, N[(N[(t * N[(x / y), $MachinePrecision] + z), $MachinePrecision] / b), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -inf.0 or 9.99999999999999955e282 < (/.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 31.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 55.3

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 55.3

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 55.3

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

   4. Applied rewrites 55.3%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. *-lft-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 40.8

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

   7. Applied rewrites 40.8%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 57.4%

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

   12. Applied rewrites 91.6%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -2.00000000000000006e-306 or 0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 9.99999999999999955e282

   1. Initial program 99.2%

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

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 74.9

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

   5. Applied rewrites 74.9%

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

   if -2.00000000000000006e-306 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 0.0

   1. Initial program 50.9%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-+.f64 70.5

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

   5. Applied rewrites 70.5%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-*l/ N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 0.6

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

   5. Applied rewrites 0.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 91.4%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 95.7%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq -\infty:\\ \;\;\;\;z \cdot \frac{y}{\mathsf{fma}\left(t, a, \mathsf{fma}\left(b, y, t\right)\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq -2 \cdot 10^{-306}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{1 + a}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq 0:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(\frac{y}{t}, b, 1 + a\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq 10^{+283}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{1 + a}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq \infty:\\ \;\;\;\;z \cdot \frac{y}{\mathsf{fma}\left(t, a, \mathsf{fma}\left(b, y, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 70.0% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t))))
        (t_2 (* z (/ y (fma t a (fma b y t))))))
   (if (<= t_1 (- INFINITY))
     t_2
     (if (<= t_1 1e+283)
       (/ x (fma (/ y t) b (+ 1.0 a)))
       (if (<= t_1 INFINITY) t_2 (/ (fma t (/ x y) z) b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t));
	double t_2 = z * (y / fma(t, a, fma(b, y, t)));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_2;
	} else if (t_1 <= 1e+283) {
		tmp = x / fma((y / t), b, (1.0 + a));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_2;
	} else {
		tmp = fma(t, (x / y), z) / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x + Float64(Float64(y * z) / t)) / Float64(Float64(a + 1.0) + Float64(Float64(y * b) / t)))
	t_2 = Float64(z * Float64(y / fma(t, a, fma(b, y, t))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = t_2;
	elseif (t_1 <= 1e+283)
		tmp = Float64(x / fma(Float64(y / t), b, Float64(1.0 + a)));
	elseif (t_1 <= Inf)
		tmp = t_2;
	else
		tmp = Float64(fma(t, Float64(x / y), z) / b);
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -inf.0 or 9.99999999999999955e282 < (/.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 31.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 55.3

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 55.3

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 55.3

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

   4. Applied rewrites 55.3%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. *-lft-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 40.8

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

   7. Applied rewrites 40.8%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 57.4%

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

   12. Applied rewrites 91.6%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 9.99999999999999955e282

   1. Initial program 90.8%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-+.f64 67.6

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

   5. Applied rewrites 67.6%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-*l/ N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 0.6

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

   5. Applied rewrites 0.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 91.4%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 95.7%

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

4. Final simplification 73.5%

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

Alternative 7: 55.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{if}\;t\_1 \leq -\infty \lor \neg \left(t\_1 \leq 10^{+283}\right):\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t)))))
   (if (or (<= t_1 (- INFINITY)) (not (<= t_1 1e+283)))
     (/ z b)
     (/ x (+ 1.0 a)))))

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)) || !(t_1 <= 1e+283)) {
		tmp = z / b;
	} else {
		tmp = x / (1.0 + a);
	}
	return tmp;
}

Java

public static 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.POSITIVE_INFINITY) || !(t_1 <= 1e+283)) {
		tmp = z / b;
	} else {
		tmp = x / (1.0 + a);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t))
	tmp = 0
	if (t_1 <= -math.inf) or not (t_1 <= 1e+283):
		tmp = z / b
	else:
		tmp = x / (1.0 + a)
	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)) || !(t_1 <= 1e+283))
		tmp = Float64(z / b);
	else
		tmp = Float64(x / Float64(1.0 + a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t));
	tmp = 0.0;
	if ((t_1 <= -Inf) || ~((t_1 <= 1e+283)))
		tmp = z / b;
	else
		tmp = x / (1.0 + a);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[(N[(a + 1.0), $MachinePrecision] + N[(N[(y * b), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, (-Infinity)], N[Not[LessEqual[t$95$1, 1e+283]], $MachinePrecision]], N[(z / b), $MachinePrecision], N[(x / N[(1.0 + a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -inf.0 or 9.99999999999999955e282 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t)))

   1. Initial program 19.3%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 75.3

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

   5. Applied rewrites 75.3%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 9.99999999999999955e282

   1. Initial program 90.8%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 54.5

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

   5. Applied rewrites 54.5%

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

4. Final simplification 59.3%

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

Alternative 8: 56.1% accurate, 0.4× speedup

Math

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

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))
     (* z (/ y (fma b y t)))
     (if (<= t_1 1e+283) (/ x (+ 1.0 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 = z * (y / fma(b, y, t));
	} else if (t_1 <= 1e+283) {
		tmp = x / (1.0 + 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 = Float64(z * Float64(y / fma(b, y, t)));
	elseif (t_1 <= 1e+283)
		tmp = Float64(x / Float64(1.0 + a));
	else
		tmp = Float64(z / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(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[(z * N[(y / N[(b * y + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+283], N[(x / N[(1.0 + a), $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 41.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 56.1

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 56.1

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 56.1

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

   4. Applied rewrites 56.1%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. *-lft-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 47.2

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

   7. Applied rewrites 47.2%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 84.9%

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

   12. Applied rewrites 85.0%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 9.99999999999999955e282

   1. Initial program 90.8%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 54.5

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

   5. Applied rewrites 54.5%

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

   if 9.99999999999999955e282 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t)))

   1. Initial program 13.1%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 72.6

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

   5. Applied rewrites 72.6%

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

4. Final simplification 59.3%

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

Alternative 9: 56.0% accurate, 0.4× speedup

Math

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

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))
     (* y (/ z (fma b y t)))
     (if (<= t_1 1e+283) (/ x (+ 1.0 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 = y * (z / fma(b, y, t));
	} else if (t_1 <= 1e+283) {
		tmp = x / (1.0 + 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 = Float64(y * Float64(z / fma(b, y, t)));
	elseif (t_1 <= 1e+283)
		tmp = Float64(x / Float64(1.0 + a));
	else
		tmp = Float64(z / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(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[(b * y + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+283], N[(x / N[(1.0 + a), $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 41.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 56.1

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 56.1

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 56.1

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

   4. Applied rewrites 56.1%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. *-lft-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 47.2

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

   7. Applied rewrites 47.2%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 84.9%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 9.99999999999999955e282

   1. Initial program 90.8%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 54.5

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

   5. Applied rewrites 54.5%

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

   if 9.99999999999999955e282 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t)))

   1. Initial program 13.1%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 72.6

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

   5. Applied rewrites 72.6%

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

4. Final simplification 59.3%

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

Alternative 10: 59.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.15 \cdot 10^{-7} \lor \neg \left(y \leq 1.72 \cdot 10^{+143}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -1.15e-7) (not (<= y 1.72e+143)))
   (/ (fma t (/ x y) z) b)
   (/ x (+ 1.0 a))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -1.15e-7) || !(y <= 1.72e+143)) {
		tmp = fma(t, (x / y), z) / b;
	} else {
		tmp = x / (1.0 + a);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -1.15e-7) || !(y <= 1.72e+143))
		tmp = Float64(fma(t, Float64(x / y), z) / b);
	else
		tmp = Float64(x / Float64(1.0 + a));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -1.15e-7], N[Not[LessEqual[y, 1.72e+143]], $MachinePrecision]], N[(N[(t * N[(x / y), $MachinePrecision] + z), $MachinePrecision] / b), $MachinePrecision], N[(x / N[(1.0 + a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.14999999999999997e-7 or 1.72000000000000005e143 < y

   1. Initial program 49.4%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-*l/ N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 26.7

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

   5. Applied rewrites 26.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 64.9%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 68.0%

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

   if -1.14999999999999997e-7 < y < 1.72000000000000005e143

   1. Initial program 90.9%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   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 59.2

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

   5. Applied rewrites 59.2%

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

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.15 \cdot 10^{-7} \lor \neg \left(y \leq 1.72 \cdot 10^{+143}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + a}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 42.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5.9 \cdot 10^{+81}:\\ \;\;\;\;\frac{x}{a}\\ \mathbf{elif}\;a \leq -2.2 \cdot 10^{-220}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;a \leq 0.75:\\ \;\;\;\;\mathsf{fma}\left(-a, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= a -5.9e+81)
   (/ x a)
   (if (<= a -2.2e-220) (/ z b) (if (<= a 0.75) (fma (- a) x x) (/ x a)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= -5.9e+81) {
		tmp = x / a;
	} else if (a <= -2.2e-220) {
		tmp = z / b;
	} else if (a <= 0.75) {
		tmp = fma(-a, x, x);
	} else {
		tmp = x / a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (a <= -5.9e+81)
		tmp = Float64(x / a);
	elseif (a <= -2.2e-220)
		tmp = Float64(z / b);
	elseif (a <= 0.75)
		tmp = fma(Float64(-a), x, x);
	else
		tmp = Float64(x / a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[a, -5.9e+81], N[(x / a), $MachinePrecision], If[LessEqual[a, -2.2e-220], N[(z / b), $MachinePrecision], If[LessEqual[a, 0.75], N[((-a) * x + x), $MachinePrecision], N[(x / a), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < -5.9000000000000004e81 or 0.75 < a

   1. Initial program 75.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 75.5

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 78.4

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 78.4

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

   4. Applied rewrites 78.4%

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

   5. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 51.5

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

   7. Applied rewrites 51.5%

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

   8. Taylor expanded in a around inf

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

   10. Applied rewrites 51.2%

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

   if -5.9000000000000004e81 < a < -2.19999999999999987e-220

   1. Initial program 61.5%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 52.1

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

   5. Applied rewrites 52.1%

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

   if -2.19999999999999987e-220 < a < 0.75

   1. Initial program 79.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 77.0

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 80.2

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 80.2

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

   4. Applied rewrites 80.2%

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

   5. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 44.2

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

   7. Applied rewrites 44.2%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 43.4%

       \[\leadsto \mathsf{fma}\left(-a, \color{blue}{x}, x\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 48.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5.9 \cdot 10^{+81}:\\ \;\;\;\;\frac{x}{a}\\ \mathbf{elif}\;a \leq -2.2 \cdot 10^{-220}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;a \leq 0.75:\\ \;\;\;\;\mathsf{fma}\left(-a, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 39.3% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -4 \cdot 10^{+176} \lor \neg \left(t \leq 53\right):\\ \;\;\;\;\mathsf{fma}\left(-a, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= t -4e+176) (not (<= t 53.0))) (fma (- a) x x) (/ z b)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((t <= -4e+176) || !(t <= 53.0)) {
		tmp = fma(-a, x, x);
	} else {
		tmp = z / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((t <= -4e+176) || !(t <= 53.0))
		tmp = fma(Float64(-a), x, x);
	else
		tmp = Float64(z / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[t, -4e+176], N[Not[LessEqual[t, 53.0]], $MachinePrecision]], N[((-a) * x + x), $MachinePrecision], N[(z / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -4e176 or 53 < t

   1. Initial program 77.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 84.0

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 94.9

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 94.9

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

   4. Applied rewrites 94.9%

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

   5. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 67.2

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

   7. Applied rewrites 67.2%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 35.9%

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

   if -4e176 < t < 53

   1. Initial program 72.6%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 45.4

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

   5. Applied rewrites 45.4%

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

4. Final simplification 42.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4 \cdot 10^{+176} \lor \neg \left(t \leq 53\right):\\ \;\;\;\;\mathsf{fma}\left(-a, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 19.4% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-a, x, x\right) \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 74.3%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-/l* N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-/.f64 73.4

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lift-/.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. associate-/l* N/A

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

   14. *-commutative N/A

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

   15. lower-fma.f64 N/A

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

   16. lower-/.f64 76.6

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

   17. lift-+.f64 N/A

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

   18. +-commutative N/A

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

   19. lower-+.f64 76.6

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

4. Applied rewrites 76.6%

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

5. Taylor expanded in y around 0

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

   1. lower-/.f64 N/A

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

   2. lower-+.f64 43.3

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

7. Applied rewrites 43.3%

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

8. Taylor expanded in a around 0

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

10. Applied rewrites 19.6%

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

Alternative 14: 4.2% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \left(-a\right) \cdot x \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = -a * x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.3%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-/l* N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-/.f64 73.4

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lift-/.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. associate-/l* N/A

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

   14. *-commutative N/A

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

   15. lower-fma.f64 N/A

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

   16. lower-/.f64 76.6

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

   17. lift-+.f64 N/A

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

   18. +-commutative N/A

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

   19. lower-+.f64 76.6

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

4. Applied rewrites 76.6%

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

5. Taylor expanded in y around 0

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

   1. lower-/.f64 N/A

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

   2. lower-+.f64 43.3

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

7. Applied rewrites 43.3%

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

8. Taylor expanded in a around 0

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

10. Applied rewrites 19.6%

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

11. Taylor expanded in a around inf

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

13. Applied rewrites 3.0%

    \[\leadsto \left(-a\right) \cdot x \]
14. Add Preprocessing

Developer Target 1: 79.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \mathbf{if}\;t < -1.3659085366310088 \cdot 10^{-271}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t < 3.036967103737246 \cdot 10^{-130}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1
         (* 1.0 (* (+ x (* (/ y t) z)) (/ 1.0 (+ (+ a 1.0) (* (/ y t) b)))))))
   (if (< t -1.3659085366310088e-271)
     t_1
     (if (< t 3.036967103737246e-130) (/ z b) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = 1.0 * ((x + ((y / t) * z)) * (1.0 / ((a + 1.0) + ((y / t) * b))));
	double tmp;
	if (t < -1.3659085366310088e-271) {
		tmp = t_1;
	} else if (t < 3.036967103737246e-130) {
		tmp = z / b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 1.0d0 * ((x + ((y / t) * z)) * (1.0d0 / ((a + 1.0d0) + ((y / t) * b))))
    if (t < (-1.3659085366310088d-271)) then
        tmp = t_1
    else if (t < 3.036967103737246d-130) then
        tmp = z / b
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = 1.0 * ((x + ((y / t) * z)) * (1.0 / ((a + 1.0) + ((y / t) * b))));
	double tmp;
	if (t < -1.3659085366310088e-271) {
		tmp = t_1;
	} else if (t < 3.036967103737246e-130) {
		tmp = z / b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = 1.0 * ((x + ((y / t) * z)) * (1.0 / ((a + 1.0) + ((y / t) * b))))
	tmp = 0
	if t < -1.3659085366310088e-271:
		tmp = t_1
	elif t < 3.036967103737246e-130:
		tmp = z / b
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(1.0 * Float64(Float64(x + Float64(Float64(y / t) * z)) * Float64(1.0 / Float64(Float64(a + 1.0) + Float64(Float64(y / t) * b)))))
	tmp = 0.0
	if (t < -1.3659085366310088e-271)
		tmp = t_1;
	elseif (t < 3.036967103737246e-130)
		tmp = Float64(z / b);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = 1.0 * ((x + ((y / t) * z)) * (1.0 / ((a + 1.0) + ((y / t) * b))));
	tmp = 0.0;
	if (t < -1.3659085366310088e-271)
		tmp = t_1;
	elseif (t < 3.036967103737246e-130)
		tmp = z / b;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(1.0 * N[(N[(x + N[(N[(y / t), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(N[(a + 1.0), $MachinePrecision] + N[(N[(y / t), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Less[t, -1.3659085366310088e-271], t$95$1, If[Less[t, 3.036967103737246e-130], N[(z / b), $MachinePrecision], t$95$1]]]

Reproduce

herbie shell --seed 2024318 
(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))))