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

Percentage Accurate: 76.0% → 91.3%

Time: 11.8s

Alternatives: 18

Speedup: 0.2×

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: 76.0% 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: 91.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{b \cdot y}{t} + \left(1 + a\right)\\ t_2 := \frac{\frac{z \cdot y}{t} + x}{t\_1}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{b}{t}, y, a\right), t, t\right)} \cdot z\\ \mathbf{elif}\;t\_2 \leq -4 \cdot 10^{-311}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{b} - \frac{z}{b \cdot b} \cdot \left(1 + a\right), t, \frac{z}{b} \cdot y\right)}{y}\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+288}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{t}, z \cdot y, x\right)}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (/ (* b y) t) (+ 1.0 a))) (t_2 (/ (+ (/ (* z y) t) x) t_1)))
   (if (<= t_2 (- INFINITY))
     (* (/ y (fma (fma (/ b t) y a) t t)) z)
     (if (<= t_2 -4e-311)
       t_2
       (if (<= t_2 0.0)
         (/ (fma (- (/ x b) (* (/ z (* b b)) (+ 1.0 a))) t (* (/ z b) y)) y)
         (if (<= t_2 2e+288)
           (/ (fma (/ 1.0 t) (* z y) x) t_1)
           (/ (fma t (/ x y) z) b)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((b * y) / t) + (1.0 + a);
	double t_2 = (((z * y) / t) + x) / t_1;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = (y / fma(fma((b / t), y, a), t, t)) * z;
	} else if (t_2 <= -4e-311) {
		tmp = t_2;
	} else if (t_2 <= 0.0) {
		tmp = fma(((x / b) - ((z / (b * b)) * (1.0 + a))), t, ((z / b) * y)) / y;
	} else if (t_2 <= 2e+288) {
		tmp = fma((1.0 / t), (z * y), x) / t_1;
	} else {
		tmp = fma(t, (x / y), z) / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(b * y) / t) + Float64(1.0 + a))
	t_2 = Float64(Float64(Float64(Float64(z * y) / t) + x) / t_1)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(Float64(y / fma(fma(Float64(b / t), y, a), t, t)) * z);
	elseif (t_2 <= -4e-311)
		tmp = t_2;
	elseif (t_2 <= 0.0)
		tmp = Float64(fma(Float64(Float64(x / b) - Float64(Float64(z / Float64(b * b)) * Float64(1.0 + a))), t, Float64(Float64(z / b) * y)) / y);
	elseif (t_2 <= 2e+288)
		tmp = Float64(fma(Float64(1.0 / t), Float64(z * y), x) / t_1);
	else
		tmp = Float64(fma(t, Float64(x / y), z) / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(b * y), $MachinePrecision] / t), $MachinePrecision] + N[(1.0 + a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(z * y), $MachinePrecision] / t), $MachinePrecision] + x), $MachinePrecision] / t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(N[(y / N[(N[(N[(b / t), $MachinePrecision] * y + a), $MachinePrecision] * t + t), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[t$95$2, -4e-311], t$95$2, If[LessEqual[t$95$2, 0.0], N[(N[(N[(N[(x / b), $MachinePrecision] - N[(N[(z / N[(b * b), $MachinePrecision]), $MachinePrecision] * N[(1.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t + N[(N[(z / b), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[t$95$2, 2e+288], N[(N[(N[(1.0 / t), $MachinePrecision] * N[(z * y), $MachinePrecision] + x), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(t * N[(x / y), $MachinePrecision] + z), $MachinePrecision] / b), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 27.4%

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

   3. Taylor expanded in z around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 93.7%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -3.99999999999979e-311

   1. Initial program 99.6%

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

   if -3.99999999999979e-311 < (/.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 39.0%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. distribute-lft-in N/A

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

      10. *-rgt-identity N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 63.5

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

   5. Applied rewrites 63.5%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 74.4%

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

   if 0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 2e288

   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. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. associate-/r/ N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 99.6

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 99.6

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

   4. Applied rewrites 99.6%

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

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

   1. Initial program 5.7%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. distribute-lft-in N/A

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

      10. *-rgt-identity N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 75.0

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

   5. Applied rewrites 75.0%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 87.1%

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

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq -\infty:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{b}{t}, y, a\right), t, t\right)} \cdot z\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq -4 \cdot 10^{-311}:\\ \;\;\;\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)}\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{b} - \frac{z}{b \cdot b} \cdot \left(1 + a\right), t, \frac{z}{b} \cdot y\right)}{y}\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq 2 \cdot 10^{+288}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{t}, z \cdot y, x\right)}{\frac{b \cdot y}{t} + \left(1 + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 75.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot y}{t} + x\\ t_2 := \frac{t\_1}{\frac{b \cdot y}{t} + \left(1 + a\right)}\\ t_3 := \frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{b}{t}, y, a\right), t, t\right)} \cdot z\\ \mathbf{elif}\;t\_2 \leq -2 \cdot 10^{+62}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(b, \frac{y}{t}, 1 + a\right)}\\ \mathbf{elif}\;t\_2 \leq -4 \cdot 10^{-311}:\\ \;\;\;\;\frac{t\_1}{1 + a}\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+288}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{t}, z \cdot y, x\right)}{1 + a}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (/ (* z y) t) x))
        (t_2 (/ t_1 (+ (/ (* b y) t) (+ 1.0 a))))
        (t_3 (/ (fma t (/ x y) z) b)))
   (if (<= t_2 (- INFINITY))
     (* (/ y (fma (fma (/ b t) y a) t t)) z)
     (if (<= t_2 -2e+62)
       (/ x (fma b (/ y t) (+ 1.0 a)))
       (if (<= t_2 -4e-311)
         (/ t_1 (+ 1.0 a))
         (if (<= t_2 0.0)
           t_3
           (if (<= t_2 2e+288)
             (/ (fma (/ 1.0 t) (* z y) x) (+ 1.0 a))
             t_3)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((z * y) / t) + x;
	double t_2 = t_1 / (((b * y) / t) + (1.0 + a));
	double t_3 = fma(t, (x / y), z) / b;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = (y / fma(fma((b / t), y, a), t, t)) * z;
	} else if (t_2 <= -2e+62) {
		tmp = x / fma(b, (y / t), (1.0 + a));
	} else if (t_2 <= -4e-311) {
		tmp = t_1 / (1.0 + a);
	} else if (t_2 <= 0.0) {
		tmp = t_3;
	} else if (t_2 <= 2e+288) {
		tmp = fma((1.0 / t), (z * y), x) / (1.0 + a);
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(z * y) / t) + x)
	t_2 = Float64(t_1 / Float64(Float64(Float64(b * y) / t) + Float64(1.0 + a)))
	t_3 = Float64(fma(t, Float64(x / y), z) / b)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(Float64(y / fma(fma(Float64(b / t), y, a), t, t)) * z);
	elseif (t_2 <= -2e+62)
		tmp = Float64(x / fma(b, Float64(y / t), Float64(1.0 + a)));
	elseif (t_2 <= -4e-311)
		tmp = Float64(t_1 / Float64(1.0 + a));
	elseif (t_2 <= 0.0)
		tmp = t_3;
	elseif (t_2 <= 2e+288)
		tmp = Float64(fma(Float64(1.0 / t), Float64(z * y), x) / Float64(1.0 + a));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(z * y), $MachinePrecision] / t), $MachinePrecision] + x), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / N[(N[(N[(b * y), $MachinePrecision] / t), $MachinePrecision] + N[(1.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(t * N[(x / y), $MachinePrecision] + z), $MachinePrecision] / b), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(N[(y / N[(N[(N[(b / t), $MachinePrecision] * y + a), $MachinePrecision] * t + t), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[t$95$2, -2e+62], N[(x / N[(b * N[(y / t), $MachinePrecision] + N[(1.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -4e-311], N[(t$95$1 / N[(1.0 + a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.0], t$95$3, If[LessEqual[t$95$2, 2e+288], N[(N[(N[(1.0 / t), $MachinePrecision] * N[(z * y), $MachinePrecision] + x), $MachinePrecision] / N[(1.0 + a), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 27.4%

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

   3. Taylor expanded in z around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 93.7%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -2.00000000000000007e62

   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 t around 0

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

      1. lower-/.f64 11.8

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

   5. Applied rewrites 11.8%

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

   6. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. associate-/l* N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 82.2

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

   8. Applied rewrites 82.2%

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

   if -2.00000000000000007e62 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -3.99999999999979e-311

   1. Initial program 99.7%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 88.5

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

   5. Applied rewrites 88.5%

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

   if -3.99999999999979e-311 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 0.0 or 2e288 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t)))

   1. Initial program 20.7%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. distribute-lft-in N/A

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

      10. *-rgt-identity N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 69.8

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

   5. Applied rewrites 69.8%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 81.1%

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

   if 0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 2e288

   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 b around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 82.2

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

   5. Applied rewrites 82.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. associate-/r/ N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 82.2

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

   7. Applied rewrites 82.2%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{t}, y \cdot z, x\right)}}{a + 1} \]
3. Recombined 5 regimes into one program.

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq -\infty:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{b}{t}, y, a\right), t, t\right)} \cdot z\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq -2 \cdot 10^{+62}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(b, \frac{y}{t}, 1 + a\right)}\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq -4 \cdot 10^{-311}:\\ \;\;\;\;\frac{\frac{z \cdot y}{t} + x}{1 + a}\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq 2 \cdot 10^{+288}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{t}, z \cdot y, x\right)}{1 + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot y}{t} + x\\ t_2 := \frac{t\_1}{\frac{b \cdot y}{t} + \left(1 + a\right)}\\ t_3 := \frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\frac{z}{\mathsf{fma}\left(b, y, t\right)} \cdot y\\ \mathbf{elif}\;t\_2 \leq -2 \cdot 10^{+62}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(b, \frac{y}{t}, 1 + a\right)}\\ \mathbf{elif}\;t\_2 \leq -4 \cdot 10^{-311}:\\ \;\;\;\;\frac{t\_1}{1 + a}\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+288}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{t}, z \cdot y, x\right)}{1 + a}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (/ (* z y) t) x))
        (t_2 (/ t_1 (+ (/ (* b y) t) (+ 1.0 a))))
        (t_3 (/ (fma t (/ x y) z) b)))
   (if (<= t_2 (- INFINITY))
     (* (/ z (fma b y t)) y)
     (if (<= t_2 -2e+62)
       (/ x (fma b (/ y t) (+ 1.0 a)))
       (if (<= t_2 -4e-311)
         (/ t_1 (+ 1.0 a))
         (if (<= t_2 0.0)
           t_3
           (if (<= t_2 2e+288)
             (/ (fma (/ 1.0 t) (* z y) x) (+ 1.0 a))
             t_3)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((z * y) / t) + x;
	double t_2 = t_1 / (((b * y) / t) + (1.0 + a));
	double t_3 = fma(t, (x / y), z) / b;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = (z / fma(b, y, t)) * y;
	} else if (t_2 <= -2e+62) {
		tmp = x / fma(b, (y / t), (1.0 + a));
	} else if (t_2 <= -4e-311) {
		tmp = t_1 / (1.0 + a);
	} else if (t_2 <= 0.0) {
		tmp = t_3;
	} else if (t_2 <= 2e+288) {
		tmp = fma((1.0 / t), (z * y), x) / (1.0 + a);
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(z * y) / t) + x)
	t_2 = Float64(t_1 / Float64(Float64(Float64(b * y) / t) + Float64(1.0 + a)))
	t_3 = Float64(fma(t, Float64(x / y), z) / b)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(Float64(z / fma(b, y, t)) * y);
	elseif (t_2 <= -2e+62)
		tmp = Float64(x / fma(b, Float64(y / t), Float64(1.0 + a)));
	elseif (t_2 <= -4e-311)
		tmp = Float64(t_1 / Float64(1.0 + a));
	elseif (t_2 <= 0.0)
		tmp = t_3;
	elseif (t_2 <= 2e+288)
		tmp = Float64(fma(Float64(1.0 / t), Float64(z * y), x) / Float64(1.0 + a));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(z * y), $MachinePrecision] / t), $MachinePrecision] + x), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / N[(N[(N[(b * y), $MachinePrecision] / t), $MachinePrecision] + N[(1.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(t * N[(x / y), $MachinePrecision] + z), $MachinePrecision] / b), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(N[(z / N[(b * y + t), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t$95$2, -2e+62], N[(x / N[(b * N[(y / t), $MachinePrecision] + N[(1.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -4e-311], N[(t$95$1 / N[(1.0 + a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.0], t$95$3, If[LessEqual[t$95$2, 2e+288], N[(N[(N[(1.0 / t), $MachinePrecision] * N[(z * y), $MachinePrecision] + x), $MachinePrecision] / N[(1.0 + a), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 27.4%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 64.1

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

   5. Applied rewrites 64.1%

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

   6. Taylor expanded in z around inf

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

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. *-lft-identity N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-/l* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 17.7

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

   8. Applied rewrites 17.7%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 79.4%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -2.00000000000000007e62

   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 t around 0

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

      1. lower-/.f64 11.8

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

   5. Applied rewrites 11.8%

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

   6. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. associate-/l* N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 82.2

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

   8. Applied rewrites 82.2%

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

   if -2.00000000000000007e62 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -3.99999999999979e-311

   1. Initial program 99.7%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 88.5

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

   5. Applied rewrites 88.5%

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

   if -3.99999999999979e-311 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 0.0 or 2e288 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t)))

   1. Initial program 20.7%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. distribute-lft-in N/A

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

      10. *-rgt-identity N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 69.8

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

   5. Applied rewrites 69.8%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 81.1%

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

   if 0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 2e288

   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 b around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 82.2

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

   5. Applied rewrites 82.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. associate-/r/ N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 82.2

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

   7. Applied rewrites 82.2%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{t}, y \cdot z, x\right)}}{a + 1} \]
3. Recombined 5 regimes into one program.

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq -\infty:\\ \;\;\;\;\frac{z}{\mathsf{fma}\left(b, y, t\right)} \cdot y\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq -2 \cdot 10^{+62}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(b, \frac{y}{t}, 1 + a\right)}\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq -4 \cdot 10^{-311}:\\ \;\;\;\;\frac{\frac{z \cdot y}{t} + x}{1 + a}\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq 2 \cdot 10^{+288}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{t}, z \cdot y, x\right)}{1 + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot y}{t} + x\\ t_2 := \frac{t\_1}{1 + a}\\ t_3 := \frac{t\_1}{\frac{b \cdot y}{t} + \left(1 + a\right)}\\ t_4 := \frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;\frac{z}{\mathsf{fma}\left(b, y, t\right)} \cdot y\\ \mathbf{elif}\;t\_3 \leq -2 \cdot 10^{+62}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(b, \frac{y}{t}, 1 + a\right)}\\ \mathbf{elif}\;t\_3 \leq -4 \cdot 10^{-311}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{+288}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_4\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (/ (* z y) t) x))
        (t_2 (/ t_1 (+ 1.0 a)))
        (t_3 (/ t_1 (+ (/ (* b y) t) (+ 1.0 a))))
        (t_4 (/ (fma t (/ x y) z) b)))
   (if (<= t_3 (- INFINITY))
     (* (/ z (fma b y t)) y)
     (if (<= t_3 -2e+62)
       (/ x (fma b (/ y t) (+ 1.0 a)))
       (if (<= t_3 -4e-311)
         t_2
         (if (<= t_3 0.0) t_4 (if (<= t_3 2e+288) t_2 t_4)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((z * y) / t) + x;
	double t_2 = t_1 / (1.0 + a);
	double t_3 = t_1 / (((b * y) / t) + (1.0 + a));
	double t_4 = fma(t, (x / y), z) / b;
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = (z / fma(b, y, t)) * y;
	} else if (t_3 <= -2e+62) {
		tmp = x / fma(b, (y / t), (1.0 + a));
	} else if (t_3 <= -4e-311) {
		tmp = t_2;
	} else if (t_3 <= 0.0) {
		tmp = t_4;
	} else if (t_3 <= 2e+288) {
		tmp = t_2;
	} else {
		tmp = t_4;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(z * y) / t) + x)
	t_2 = Float64(t_1 / Float64(1.0 + a))
	t_3 = Float64(t_1 / Float64(Float64(Float64(b * y) / t) + Float64(1.0 + a)))
	t_4 = Float64(fma(t, Float64(x / y), z) / b)
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(Float64(z / fma(b, y, t)) * y);
	elseif (t_3 <= -2e+62)
		tmp = Float64(x / fma(b, Float64(y / t), Float64(1.0 + a)));
	elseif (t_3 <= -4e-311)
		tmp = t_2;
	elseif (t_3 <= 0.0)
		tmp = t_4;
	elseif (t_3 <= 2e+288)
		tmp = t_2;
	else
		tmp = t_4;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(z * y), $MachinePrecision] / t), $MachinePrecision] + x), $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[(N[(b * y), $MachinePrecision] / t), $MachinePrecision] + N[(1.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(t * N[(x / y), $MachinePrecision] + z), $MachinePrecision] / b), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], N[(N[(z / N[(b * y + t), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t$95$3, -2e+62], N[(x / N[(b * N[(y / t), $MachinePrecision] + N[(1.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -4e-311], t$95$2, If[LessEqual[t$95$3, 0.0], t$95$4, If[LessEqual[t$95$3, 2e+288], t$95$2, t$95$4]]]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 27.4%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 64.1

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

   5. Applied rewrites 64.1%

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

   6. Taylor expanded in z around inf

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

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. *-lft-identity N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-/l* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 17.7

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

   8. Applied rewrites 17.7%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 79.4%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -2.00000000000000007e62

   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 t around 0

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

      1. lower-/.f64 11.8

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

   5. Applied rewrites 11.8%

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

   6. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. associate-/l* N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 82.2

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

   8. Applied rewrites 82.2%

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

   if -2.00000000000000007e62 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -3.99999999999979e-311 or 0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 2e288

   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 b around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 84.4

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

   5. Applied rewrites 84.4%

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

   if -3.99999999999979e-311 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 0.0 or 2e288 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t)))

   1. Initial program 20.7%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. distribute-lft-in N/A

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

      10. *-rgt-identity N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 69.8

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

   5. Applied rewrites 69.8%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 81.1%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq -\infty:\\ \;\;\;\;\frac{z}{\mathsf{fma}\left(b, y, t\right)} \cdot y\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq -2 \cdot 10^{+62}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(b, \frac{y}{t}, 1 + a\right)}\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq -4 \cdot 10^{-311}:\\ \;\;\;\;\frac{\frac{z \cdot y}{t} + x}{1 + a}\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq 2 \cdot 10^{+288}:\\ \;\;\;\;\frac{\frac{z \cdot y}{t} + x}{1 + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)}\\ t_2 := \frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\frac{z}{\mathsf{fma}\left(b, y, t\right)} \cdot y\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{+62}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(b, \frac{y}{t}, 1 + a\right)}\\ \mathbf{elif}\;t\_1 \leq -4 \cdot 10^{-311}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{z}{t}, y, x\right)}{1 + a}\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+288}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}{1 + a}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (+ (/ (* z y) t) x) (+ (/ (* b y) t) (+ 1.0 a))))
        (t_2 (/ (fma t (/ x y) z) b)))
   (if (<= t_1 (- INFINITY))
     (* (/ z (fma b y t)) y)
     (if (<= t_1 -2e+62)
       (/ x (fma b (/ y t) (+ 1.0 a)))
       (if (<= t_1 -4e-311)
         (/ (fma (/ z t) y x) (+ 1.0 a))
         (if (<= t_1 0.0)
           t_2
           (if (<= t_1 2e+288) (/ (fma z (/ y t) x) (+ 1.0 a)) t_2)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (((z * y) / t) + x) / (((b * y) / t) + (1.0 + a));
	double t_2 = fma(t, (x / y), z) / b;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (z / fma(b, y, t)) * y;
	} else if (t_1 <= -2e+62) {
		tmp = x / fma(b, (y / t), (1.0 + a));
	} else if (t_1 <= -4e-311) {
		tmp = fma((z / t), y, x) / (1.0 + a);
	} else if (t_1 <= 0.0) {
		tmp = t_2;
	} else if (t_1 <= 2e+288) {
		tmp = fma(z, (y / t), x) / (1.0 + a);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(Float64(z * y) / t) + x) / Float64(Float64(Float64(b * y) / t) + Float64(1.0 + a)))
	t_2 = Float64(fma(t, Float64(x / y), z) / b)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(z / fma(b, y, t)) * y);
	elseif (t_1 <= -2e+62)
		tmp = Float64(x / fma(b, Float64(y / t), Float64(1.0 + a)));
	elseif (t_1 <= -4e-311)
		tmp = Float64(fma(Float64(z / t), y, x) / Float64(1.0 + a));
	elseif (t_1 <= 0.0)
		tmp = t_2;
	elseif (t_1 <= 2e+288)
		tmp = Float64(fma(z, Float64(y / t), x) / Float64(1.0 + a));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(N[(z * y), $MachinePrecision] / t), $MachinePrecision] + x), $MachinePrecision] / N[(N[(N[(b * y), $MachinePrecision] / t), $MachinePrecision] + N[(1.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t * N[(x / y), $MachinePrecision] + z), $MachinePrecision] / b), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(z / N[(b * y + t), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t$95$1, -2e+62], N[(x / N[(b * N[(y / t), $MachinePrecision] + N[(1.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -4e-311], N[(N[(N[(z / t), $MachinePrecision] * y + x), $MachinePrecision] / N[(1.0 + a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], t$95$2, If[LessEqual[t$95$1, 2e+288], N[(N[(z * N[(y / t), $MachinePrecision] + x), $MachinePrecision] / N[(1.0 + a), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 27.4%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 64.1

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

   5. Applied rewrites 64.1%

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

   6. Taylor expanded in z around inf

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

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. *-lft-identity N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-/l* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 17.7

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

   8. Applied rewrites 17.7%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 79.4%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -2.00000000000000007e62

   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 t around 0

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

      1. lower-/.f64 11.8

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

   5. Applied rewrites 11.8%

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

   6. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. associate-/l* N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 82.2

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

   8. Applied rewrites 82.2%

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

   if -2.00000000000000007e62 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -3.99999999999979e-311

   1. Initial program 99.7%

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

   3. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*l/ N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 86.4

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

   5. Applied rewrites 86.4%

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

   if -3.99999999999979e-311 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 0.0 or 2e288 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t)))

   1. Initial program 20.7%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. distribute-lft-in N/A

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

      10. *-rgt-identity N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 69.8

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

   5. Applied rewrites 69.8%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 81.1%

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

   if 0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 2e288

   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 b around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 82.2

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

   5. Applied rewrites 82.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 82.2

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

   7. Applied rewrites 82.2%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}}{a + 1} \]
3. Recombined 5 regimes into one program.

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq -\infty:\\ \;\;\;\;\frac{z}{\mathsf{fma}\left(b, y, t\right)} \cdot y\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq -2 \cdot 10^{+62}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(b, \frac{y}{t}, 1 + a\right)}\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq -4 \cdot 10^{-311}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{z}{t}, y, x\right)}{1 + a}\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq 2 \cdot 10^{+288}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}{1 + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 72.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\mathsf{fma}\left(\frac{z}{t}, y, x\right)}{1 + a}\\ t_2 := \frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)}\\ t_3 := \frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\frac{z}{\mathsf{fma}\left(b, y, t\right)} \cdot y\\ \mathbf{elif}\;t\_2 \leq -2 \cdot 10^{+62}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(b, \frac{y}{t}, 1 + a\right)}\\ \mathbf{elif}\;t\_2 \leq -4 \cdot 10^{-311}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+288}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (fma (/ z t) y x) (+ 1.0 a)))
        (t_2 (/ (+ (/ (* z y) t) x) (+ (/ (* b y) t) (+ 1.0 a))))
        (t_3 (/ (fma t (/ x y) z) b)))
   (if (<= t_2 (- INFINITY))
     (* (/ z (fma b y t)) y)
     (if (<= t_2 -2e+62)
       (/ x (fma b (/ y t) (+ 1.0 a)))
       (if (<= t_2 -4e-311)
         t_1
         (if (<= t_2 0.0) t_3 (if (<= t_2 2e+288) t_1 t_3)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma((z / t), y, x) / (1.0 + a);
	double t_2 = (((z * y) / t) + x) / (((b * y) / t) + (1.0 + a));
	double t_3 = fma(t, (x / y), z) / b;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = (z / fma(b, y, t)) * y;
	} else if (t_2 <= -2e+62) {
		tmp = x / fma(b, (y / t), (1.0 + a));
	} else if (t_2 <= -4e-311) {
		tmp = t_1;
	} else if (t_2 <= 0.0) {
		tmp = t_3;
	} else if (t_2 <= 2e+288) {
		tmp = t_1;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(fma(Float64(z / t), y, x) / Float64(1.0 + a))
	t_2 = Float64(Float64(Float64(Float64(z * y) / t) + x) / Float64(Float64(Float64(b * y) / t) + Float64(1.0 + a)))
	t_3 = Float64(fma(t, Float64(x / y), z) / b)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(Float64(z / fma(b, y, t)) * y);
	elseif (t_2 <= -2e+62)
		tmp = Float64(x / fma(b, Float64(y / t), Float64(1.0 + a)));
	elseif (t_2 <= -4e-311)
		tmp = t_1;
	elseif (t_2 <= 0.0)
		tmp = t_3;
	elseif (t_2 <= 2e+288)
		tmp = t_1;
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(z / t), $MachinePrecision] * y + x), $MachinePrecision] / N[(1.0 + a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(z * y), $MachinePrecision] / t), $MachinePrecision] + x), $MachinePrecision] / N[(N[(N[(b * y), $MachinePrecision] / t), $MachinePrecision] + N[(1.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(t * N[(x / y), $MachinePrecision] + z), $MachinePrecision] / b), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(N[(z / N[(b * y + t), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t$95$2, -2e+62], N[(x / N[(b * N[(y / t), $MachinePrecision] + N[(1.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -4e-311], t$95$1, If[LessEqual[t$95$2, 0.0], t$95$3, If[LessEqual[t$95$2, 2e+288], t$95$1, t$95$3]]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 27.4%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 64.1

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

   5. Applied rewrites 64.1%

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

   6. Taylor expanded in z around inf

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

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. *-lft-identity N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-/l* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 17.7

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

   8. Applied rewrites 17.7%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 79.4%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -2.00000000000000007e62

   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 t around 0

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

      1. lower-/.f64 11.8

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

   5. Applied rewrites 11.8%

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

   6. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. associate-/l* N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 82.2

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

   8. Applied rewrites 82.2%

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

   if -2.00000000000000007e62 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -3.99999999999979e-311 or 0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 2e288

   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 b around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*l/ N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 82.9

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

   5. Applied rewrites 82.9%

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

   if -3.99999999999979e-311 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 0.0 or 2e288 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t)))

   1. Initial program 20.7%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. distribute-lft-in N/A

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

      10. *-rgt-identity N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 69.8

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

   5. Applied rewrites 69.8%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 81.1%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq -\infty:\\ \;\;\;\;\frac{z}{\mathsf{fma}\left(b, y, t\right)} \cdot y\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq -2 \cdot 10^{+62}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(b, \frac{y}{t}, 1 + a\right)}\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq -4 \cdot 10^{-311}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{z}{t}, y, x\right)}{1 + a}\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq 2 \cdot 10^{+288}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{z}{t}, y, x\right)}{1 + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 90.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{b \cdot y}{t} + \left(1 + a\right)\\ t_2 := \frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ t_3 := \frac{\frac{z \cdot y}{t} + x}{t\_1}\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{b}{t}, y, a\right), t, t\right)} \cdot z\\ \mathbf{elif}\;t\_3 \leq -4 \cdot 10^{-311}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{+288}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{t}, z \cdot y, x\right)}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (/ (* b y) t) (+ 1.0 a)))
        (t_2 (/ (fma t (/ x y) z) b))
        (t_3 (/ (+ (/ (* z y) t) x) t_1)))
   (if (<= t_3 (- INFINITY))
     (* (/ y (fma (fma (/ b t) y a) t t)) z)
     (if (<= t_3 -4e-311)
       t_3
       (if (<= t_3 0.0)
         t_2
         (if (<= t_3 2e+288) (/ (fma (/ 1.0 t) (* z y) x) t_1) t_2))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((b * y) / t) + (1.0 + a);
	double t_2 = fma(t, (x / y), z) / b;
	double t_3 = (((z * y) / t) + x) / t_1;
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = (y / fma(fma((b / t), y, a), t, t)) * z;
	} else if (t_3 <= -4e-311) {
		tmp = t_3;
	} else if (t_3 <= 0.0) {
		tmp = t_2;
	} else if (t_3 <= 2e+288) {
		tmp = fma((1.0 / t), (z * y), x) / t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(b * y) / t) + Float64(1.0 + a))
	t_2 = Float64(fma(t, Float64(x / y), z) / b)
	t_3 = Float64(Float64(Float64(Float64(z * y) / t) + x) / t_1)
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(Float64(y / fma(fma(Float64(b / t), y, a), t, t)) * z);
	elseif (t_3 <= -4e-311)
		tmp = t_3;
	elseif (t_3 <= 0.0)
		tmp = t_2;
	elseif (t_3 <= 2e+288)
		tmp = Float64(fma(Float64(1.0 / t), Float64(z * y), x) / t_1);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(b * y), $MachinePrecision] / t), $MachinePrecision] + N[(1.0 + a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t * N[(x / y), $MachinePrecision] + z), $MachinePrecision] / b), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[(z * y), $MachinePrecision] / t), $MachinePrecision] + x), $MachinePrecision] / t$95$1), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], N[(N[(y / N[(N[(N[(b / t), $MachinePrecision] * y + a), $MachinePrecision] * t + t), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[t$95$3, -4e-311], t$95$3, If[LessEqual[t$95$3, 0.0], t$95$2, If[LessEqual[t$95$3, 2e+288], N[(N[(N[(1.0 / t), $MachinePrecision] * N[(z * y), $MachinePrecision] + x), $MachinePrecision] / t$95$1), $MachinePrecision], t$95$2]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 27.4%

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

   3. Taylor expanded in z around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 93.7%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -3.99999999999979e-311

   1. Initial program 99.6%

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

   if -3.99999999999979e-311 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 0.0 or 2e288 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t)))

   1. Initial program 20.7%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. distribute-lft-in N/A

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

      10. *-rgt-identity N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 69.8

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

   5. Applied rewrites 69.8%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 81.1%

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

   if 0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 2e288

   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. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. associate-/r/ N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 99.6

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 99.6

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

   4. Applied rewrites 99.6%

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

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq -\infty:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{b}{t}, y, a\right), t, t\right)} \cdot z\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq -4 \cdot 10^{-311}:\\ \;\;\;\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)}\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq 2 \cdot 10^{+288}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{t}, z \cdot y, x\right)}{\frac{b \cdot y}{t} + \left(1 + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 90.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ t_2 := \frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{b}{t}, y, a\right), t, t\right)} \cdot z\\ \mathbf{elif}\;t\_2 \leq -4 \cdot 10^{-311}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+288}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (fma t (/ x y) z) b))
        (t_2 (/ (+ (/ (* z y) t) x) (+ (/ (* b y) t) (+ 1.0 a)))))
   (if (<= t_2 (- INFINITY))
     (* (/ y (fma (fma (/ b t) y a) t t)) z)
     (if (<= t_2 -4e-311)
       t_2
       (if (<= t_2 0.0) t_1 (if (<= t_2 2e+288) t_2 t_1))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(t, (x / y), z) / b;
	double t_2 = (((z * y) / t) + x) / (((b * y) / t) + (1.0 + a));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = (y / fma(fma((b / t), y, a), t, t)) * z;
	} else if (t_2 <= -4e-311) {
		tmp = t_2;
	} else if (t_2 <= 0.0) {
		tmp = t_1;
	} else if (t_2 <= 2e+288) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(fma(t, Float64(x / y), z) / b)
	t_2 = Float64(Float64(Float64(Float64(z * y) / t) + x) / Float64(Float64(Float64(b * y) / t) + Float64(1.0 + a)))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(Float64(y / fma(fma(Float64(b / t), y, a), t, t)) * z);
	elseif (t_2 <= -4e-311)
		tmp = t_2;
	elseif (t_2 <= 0.0)
		tmp = t_1;
	elseif (t_2 <= 2e+288)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t * N[(x / y), $MachinePrecision] + z), $MachinePrecision] / b), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(z * y), $MachinePrecision] / t), $MachinePrecision] + x), $MachinePrecision] / N[(N[(N[(b * y), $MachinePrecision] / t), $MachinePrecision] + N[(1.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(N[(y / N[(N[(N[(b / t), $MachinePrecision] * y + a), $MachinePrecision] * t + t), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[t$95$2, -4e-311], t$95$2, If[LessEqual[t$95$2, 0.0], t$95$1, If[LessEqual[t$95$2, 2e+288], t$95$2, t$95$1]]]]]]

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 27.4%

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

   3. Taylor expanded in z around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 93.7%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -3.99999999999979e-311 or 0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 2e288

   1. Initial program 99.6%

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

   if -3.99999999999979e-311 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 0.0 or 2e288 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t)))

   1. Initial program 20.7%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. distribute-lft-in N/A

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

      10. *-rgt-identity N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 69.8

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

   5. Applied rewrites 69.8%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 81.1%

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

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq -\infty:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{b}{t}, y, a\right), t, t\right)} \cdot z\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq -4 \cdot 10^{-311}:\\ \;\;\;\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)}\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq 2 \cdot 10^{+288}:\\ \;\;\;\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 87.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{b \cdot y}{t} + \left(1 + a\right)\\ t_2 := \frac{\frac{z}{t} \cdot y + x}{t\_1}\\ t_3 := \frac{\frac{z \cdot y}{t} + x}{t\_1}\\ t_4 := \frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{b}{t}, y, a\right), t, t\right)} \cdot z\\ \mathbf{elif}\;t\_3 \leq -4 \cdot 10^{-311}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{+288}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_4\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (/ (* b y) t) (+ 1.0 a)))
        (t_2 (/ (+ (* (/ z t) y) x) t_1))
        (t_3 (/ (+ (/ (* z y) t) x) t_1))
        (t_4 (/ (fma t (/ x y) z) b)))
   (if (<= t_3 (- INFINITY))
     (* (/ y (fma (fma (/ b t) y a) t t)) z)
     (if (<= t_3 -4e-311)
       t_2
       (if (<= t_3 0.0) t_4 (if (<= t_3 2e+288) t_2 t_4))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((b * y) / t) + (1.0 + a);
	double t_2 = (((z / t) * y) + x) / t_1;
	double t_3 = (((z * y) / t) + x) / t_1;
	double t_4 = fma(t, (x / y), z) / b;
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = (y / fma(fma((b / t), y, a), t, t)) * z;
	} else if (t_3 <= -4e-311) {
		tmp = t_2;
	} else if (t_3 <= 0.0) {
		tmp = t_4;
	} else if (t_3 <= 2e+288) {
		tmp = t_2;
	} else {
		tmp = t_4;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(b * y) / t) + Float64(1.0 + a))
	t_2 = Float64(Float64(Float64(Float64(z / t) * y) + x) / t_1)
	t_3 = Float64(Float64(Float64(Float64(z * y) / t) + x) / t_1)
	t_4 = Float64(fma(t, Float64(x / y), z) / b)
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(Float64(y / fma(fma(Float64(b / t), y, a), t, t)) * z);
	elseif (t_3 <= -4e-311)
		tmp = t_2;
	elseif (t_3 <= 0.0)
		tmp = t_4;
	elseif (t_3 <= 2e+288)
		tmp = t_2;
	else
		tmp = t_4;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(b * y), $MachinePrecision] / t), $MachinePrecision] + N[(1.0 + a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(z / t), $MachinePrecision] * y), $MachinePrecision] + x), $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[(z * y), $MachinePrecision] / t), $MachinePrecision] + x), $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$4 = N[(N[(t * N[(x / y), $MachinePrecision] + z), $MachinePrecision] / b), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], N[(N[(y / N[(N[(N[(b / t), $MachinePrecision] * y + a), $MachinePrecision] * t + t), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[t$95$3, -4e-311], t$95$2, If[LessEqual[t$95$3, 0.0], t$95$4, If[LessEqual[t$95$3, 2e+288], t$95$2, t$95$4]]]]]]]]

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 27.4%

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

   3. Taylor expanded in z around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 93.7%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -3.99999999999979e-311 or 0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 2e288

   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. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 95.0

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

   4. Applied rewrites 95.0%

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

   if -3.99999999999979e-311 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 0.0 or 2e288 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t)))

   1. Initial program 20.7%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. distribute-lft-in N/A

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

      10. *-rgt-identity N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 69.8

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

   5. Applied rewrites 69.8%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 81.1%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq -\infty:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{b}{t}, y, a\right), t, t\right)} \cdot z\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq -4 \cdot 10^{-311}:\\ \;\;\;\;\frac{\frac{z}{t} \cdot y + x}{\frac{b \cdot y}{t} + \left(1 + a\right)}\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq 2 \cdot 10^{+288}:\\ \;\;\;\;\frac{\frac{z}{t} \cdot y + x}{\frac{b \cdot y}{t} + \left(1 + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 58.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{if}\;y \leq -2.4 \cdot 10^{+16}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 3 \cdot 10^{-147}:\\ \;\;\;\;\frac{x}{1 + a}\\ \mathbf{elif}\;y \leq 2.45 \cdot 10^{+82}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{z}{t}, y, x\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (fma t (/ x y) z) b)))
   (if (<= y -2.4e+16)
     t_1
     (if (<= y 3e-147)
       (/ x (+ 1.0 a))
       (if (<= y 2.45e+82) (/ (fma (/ z t) y x) a) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(t, (x / y), z) / b;
	double tmp;
	if (y <= -2.4e+16) {
		tmp = t_1;
	} else if (y <= 3e-147) {
		tmp = x / (1.0 + a);
	} else if (y <= 2.45e+82) {
		tmp = fma((z / t), y, x) / a;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(fma(t, Float64(x / y), z) / b)
	tmp = 0.0
	if (y <= -2.4e+16)
		tmp = t_1;
	elseif (y <= 3e-147)
		tmp = Float64(x / Float64(1.0 + a));
	elseif (y <= 2.45e+82)
		tmp = Float64(fma(Float64(z / t), y, x) / a);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t * N[(x / y), $MachinePrecision] + z), $MachinePrecision] / b), $MachinePrecision]}, If[LessEqual[y, -2.4e+16], t$95$1, If[LessEqual[y, 3e-147], N[(x / N[(1.0 + a), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.45e+82], N[(N[(N[(z / t), $MachinePrecision] * y + x), $MachinePrecision] / a), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.4e16 or 2.45e82 < y

   1. Initial program 46.9%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. distribute-lft-in N/A

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

      10. *-rgt-identity N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 51.9

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

   5. Applied rewrites 51.9%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 65.4%

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

   if -2.4e16 < y < 3.0000000000000002e-147

   1. Initial program 97.1%

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

   3. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 76.3

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

   5. Applied rewrites 76.3%

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

   if 3.0000000000000002e-147 < y < 2.45e82

   1. Initial program 80.8%

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

   3. Taylor expanded in a around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*l/ N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 52.4

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

   5. Applied rewrites 52.4%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.4 \cdot 10^{+16}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{elif}\;y \leq 3 \cdot 10^{-147}:\\ \;\;\;\;\frac{x}{1 + a}\\ \mathbf{elif}\;y \leq 2.45 \cdot 10^{+82}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{z}{t}, y, x\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 65.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{if}\;y \leq -6.2 \cdot 10^{+16}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 2.45 \cdot 10^{+82}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(b, \frac{y}{t}, 1 + a\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (fma t (/ x y) z) b)))
   (if (<= y -6.2e+16)
     t_1
     (if (<= y 2.45e+82) (/ x (fma b (/ y t) (+ 1.0 a))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(t, (x / y), z) / b;
	double tmp;
	if (y <= -6.2e+16) {
		tmp = t_1;
	} else if (y <= 2.45e+82) {
		tmp = x / fma(b, (y / t), (1.0 + a));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(fma(t, Float64(x / y), z) / b)
	tmp = 0.0
	if (y <= -6.2e+16)
		tmp = t_1;
	elseif (y <= 2.45e+82)
		tmp = Float64(x / fma(b, Float64(y / t), Float64(1.0 + a)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t * N[(x / y), $MachinePrecision] + z), $MachinePrecision] / b), $MachinePrecision]}, If[LessEqual[y, -6.2e+16], t$95$1, If[LessEqual[y, 2.45e+82], N[(x / N[(b * N[(y / t), $MachinePrecision] + N[(1.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -6.2e16 or 2.45e82 < y

   1. Initial program 46.9%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. distribute-lft-in N/A

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

      10. *-rgt-identity N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 51.9

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

   5. Applied rewrites 51.9%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 65.4%

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

   if -6.2e16 < y < 2.45e82

   1. Initial program 92.7%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 11.4

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

   5. Applied rewrites 11.4%

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

   6. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. associate-/l* N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 73.9

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

   8. Applied rewrites 73.9%

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

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.2 \cdot 10^{+16}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{elif}\;y \leq 2.45 \cdot 10^{+82}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(b, \frac{y}{t}, 1 + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 59.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{if}\;y \leq -2.4 \cdot 10^{+16}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{+82}:\\ \;\;\;\;\frac{x}{1 + a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (fma t (/ x y) z) b)))
   (if (<= y -2.4e+16) t_1 (if (<= y 2.2e+82) (/ x (+ 1.0 a)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(t, (x / y), z) / b;
	double tmp;
	if (y <= -2.4e+16) {
		tmp = t_1;
	} else if (y <= 2.2e+82) {
		tmp = x / (1.0 + a);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(fma(t, Float64(x / y), z) / b)
	tmp = 0.0
	if (y <= -2.4e+16)
		tmp = t_1;
	elseif (y <= 2.2e+82)
		tmp = Float64(x / Float64(1.0 + a));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t * N[(x / y), $MachinePrecision] + z), $MachinePrecision] / b), $MachinePrecision]}, If[LessEqual[y, -2.4e+16], t$95$1, If[LessEqual[y, 2.2e+82], N[(x / N[(1.0 + a), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.4e16 or 2.2000000000000001e82 < y

   1. Initial program 46.9%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. distribute-lft-in N/A

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

      10. *-rgt-identity N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 51.9

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

   5. Applied rewrites 51.9%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 65.4%

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

   if -2.4e16 < y < 2.2000000000000001e82

   1. Initial program 92.7%

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

   3. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 65.2

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

   5. Applied rewrites 65.2%

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

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.4 \cdot 10^{+16}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{+82}:\\ \;\;\;\;\frac{x}{1 + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 39.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{+16}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{-170}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot x - x, a, x\right)\\ \mathbf{elif}\;y \leq 4.1 \cdot 10^{+91}:\\ \;\;\;\;\frac{x}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -2e+16)
   (/ z b)
   (if (<= y 1.75e-170)
     (fma (- (* a x) x) a x)
     (if (<= y 4.1e+91) (/ x a) (/ z b)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -2e+16) {
		tmp = z / b;
	} else if (y <= 1.75e-170) {
		tmp = fma(((a * x) - x), a, x);
	} else if (y <= 4.1e+91) {
		tmp = x / a;
	} else {
		tmp = z / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -2e+16)
		tmp = Float64(z / b);
	elseif (y <= 1.75e-170)
		tmp = fma(Float64(Float64(a * x) - x), a, x);
	elseif (y <= 4.1e+91)
		tmp = Float64(x / a);
	else
		tmp = Float64(z / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -2e+16], N[(z / b), $MachinePrecision], If[LessEqual[y, 1.75e-170], N[(N[(N[(a * x), $MachinePrecision] - x), $MachinePrecision] * a + x), $MachinePrecision], If[LessEqual[y, 4.1e+91], N[(x / a), $MachinePrecision], N[(z / b), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2e16 or 4.1000000000000002e91 < y

   1. Initial program 46.5%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 56.6

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

   5. Applied rewrites 56.6%

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

   if -2e16 < y < 1.74999999999999992e-170

   1. Initial program 97.0%

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

   3. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 75.3

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

   5. Applied rewrites 75.3%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 48.0%

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

   if 1.74999999999999992e-170 < y < 4.1000000000000002e91

   1. Initial program 83.1%

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

   3. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 41.0

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

   5. Applied rewrites 41.0%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 35.7%

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

Alternative 14: 39.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{+16}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{-170}:\\ \;\;\;\;\frac{x}{1}\\ \mathbf{elif}\;y \leq 4.1 \cdot 10^{+91}:\\ \;\;\;\;\frac{x}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -2e+16)
   (/ z b)
   (if (<= y 1.75e-170) (/ x 1.0) (if (<= y 4.1e+91) (/ x a) (/ z b)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -2e+16) {
		tmp = z / b;
	} else if (y <= 1.75e-170) {
		tmp = x / 1.0;
	} else if (y <= 4.1e+91) {
		tmp = x / a;
	} else {
		tmp = z / b;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (y <= (-2d+16)) then
        tmp = z / b
    else if (y <= 1.75d-170) then
        tmp = x / 1.0d0
    else if (y <= 4.1d+91) then
        tmp = x / a
    else
        tmp = z / b
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -2e+16) {
		tmp = z / b;
	} else if (y <= 1.75e-170) {
		tmp = x / 1.0;
	} else if (y <= 4.1e+91) {
		tmp = x / a;
	} else {
		tmp = z / b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -2e+16:
		tmp = z / b
	elif y <= 1.75e-170:
		tmp = x / 1.0
	elif y <= 4.1e+91:
		tmp = x / a
	else:
		tmp = z / b
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -2e+16)
		tmp = Float64(z / b);
	elseif (y <= 1.75e-170)
		tmp = Float64(x / 1.0);
	elseif (y <= 4.1e+91)
		tmp = Float64(x / a);
	else
		tmp = Float64(z / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -2e+16)
		tmp = z / b;
	elseif (y <= 1.75e-170)
		tmp = x / 1.0;
	elseif (y <= 4.1e+91)
		tmp = x / a;
	else
		tmp = z / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -2e+16], N[(z / b), $MachinePrecision], If[LessEqual[y, 1.75e-170], N[(x / 1.0), $MachinePrecision], If[LessEqual[y, 4.1e+91], N[(x / a), $MachinePrecision], N[(z / b), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2e16 or 4.1000000000000002e91 < y

   1. Initial program 46.5%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 56.6

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

   5. Applied rewrites 56.6%

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

   if -2e16 < y < 1.74999999999999992e-170

   1. Initial program 97.0%

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

   3. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 75.3

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

   5. Applied rewrites 75.3%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 47.8%

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

   if 1.74999999999999992e-170 < y < 4.1000000000000002e91

   1. Initial program 83.1%

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

   3. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 41.0

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

   5. Applied rewrites 41.0%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 35.7%

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

Alternative 15: 41.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 + a \leq -5 \cdot 10^{+19}:\\ \;\;\;\;\frac{x}{a}\\ \mathbf{elif}\;1 + a \leq 2:\\ \;\;\;\;\mathsf{fma}\left(-x, a, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (+ 1.0 a) -5e+19)
   (/ x a)
   (if (<= (+ 1.0 a) 2.0) (fma (- x) a x) (/ x a))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((1.0 + a) <= -5e+19) {
		tmp = x / a;
	} else if ((1.0 + a) <= 2.0) {
		tmp = fma(-x, a, x);
	} else {
		tmp = x / a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(1.0 + a) <= -5e+19)
		tmp = Float64(x / a);
	elseif (Float64(1.0 + a) <= 2.0)
		tmp = fma(Float64(-x), a, x);
	else
		tmp = Float64(x / a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(1.0 + a), $MachinePrecision], -5e+19], N[(x / a), $MachinePrecision], If[LessEqual[N[(1.0 + a), $MachinePrecision], 2.0], N[((-x) * a + x), $MachinePrecision], N[(x / a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 a #s(literal 1 binary64)) < -5e19 or 2 < (+.f64 a #s(literal 1 binary64))

   1. Initial program 74.9%

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

   3. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 44.6

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

   5. Applied rewrites 44.6%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 44.0%

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

   if -5e19 < (+.f64 a #s(literal 1 binary64)) < 2

   1. Initial program 67.2%

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

   3. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 37.3

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

   5. Applied rewrites 37.3%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 37.2%

      \[\leadsto \mathsf{fma}\left(-x, \color{blue}{a}, x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 40.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 + a \leq -5 \cdot 10^{+19}:\\ \;\;\;\;\frac{x}{a}\\ \mathbf{elif}\;1 + a \leq 2:\\ \;\;\;\;\mathsf{fma}\left(-x, a, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 55.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.2 \cdot 10^{+16}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;y \leq 2.45 \cdot 10^{+82}:\\ \;\;\;\;\frac{x}{1 + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -6.2e+16) (/ z b) (if (<= y 2.45e+82) (/ x (+ 1.0 a)) (/ z b))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -6.2e+16) {
		tmp = z / b;
	} else if (y <= 2.45e+82) {
		tmp = x / (1.0 + a);
	} else {
		tmp = z / b;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (y <= (-6.2d+16)) then
        tmp = z / b
    else if (y <= 2.45d+82) then
        tmp = x / (1.0d0 + a)
    else
        tmp = z / b
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -6.2e+16) {
		tmp = z / b;
	} else if (y <= 2.45e+82) {
		tmp = x / (1.0 + a);
	} else {
		tmp = z / b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -6.2e+16:
		tmp = z / b
	elif y <= 2.45e+82:
		tmp = x / (1.0 + a)
	else:
		tmp = z / b
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -6.2e+16)
		tmp = Float64(z / b);
	elseif (y <= 2.45e+82)
		tmp = Float64(x / Float64(1.0 + a));
	else
		tmp = Float64(z / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -6.2e+16)
		tmp = z / b;
	elseif (y <= 2.45e+82)
		tmp = x / (1.0 + a);
	else
		tmp = z / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -6.2e+16], N[(z / b), $MachinePrecision], If[LessEqual[y, 2.45e+82], N[(x / N[(1.0 + a), $MachinePrecision]), $MachinePrecision], N[(z / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -6.2e16 or 2.45e82 < y

   1. Initial program 46.9%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 56.2

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

   5. Applied rewrites 56.2%

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

   if -6.2e16 < y < 2.45e82

   1. Initial program 92.7%

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

   3. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 65.2

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

   5. Applied rewrites 65.2%

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

4. Final simplification 60.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.2 \cdot 10^{+16}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;y \leq 2.45 \cdot 10^{+82}:\\ \;\;\;\;\frac{x}{1 + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 19.6% accurate, 5.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 70.7%

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

3. Taylor expanded in t around inf

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

   1. lower-/.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f64 40.6

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

5. Applied rewrites 40.6%

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

6. Taylor expanded in a around 0

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

8. Applied rewrites 21.4%

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

Alternative 18: 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 70.7%

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

3. Taylor expanded in t around inf

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

   1. lower-/.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f64 40.6

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

5. Applied rewrites 40.6%

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

6. Taylor expanded in a around 0

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

8. Applied rewrites 21.4%

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

9. Taylor expanded in a around inf

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

11. Applied rewrites 3.7%

    \[\leadsto \left(-a\right) \cdot x \]
12. Add Preprocessing

Developer Target 1: 80.0% 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 2024237 
(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))))