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

Percentage Accurate: 75.2% → 89.4%

Time: 10.5s

Alternatives: 17

Speedup: 0.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 75.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 89.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{z \cdot y}{t} + x}{\left(1 + a\right) + \frac{b \cdot y}{t}}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{y}{t}, b, a\right), t, t\right)} \cdot z\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{-163}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-306}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{z}{t}, y, x\right)}{\mathsf{fma}\left(\frac{b}{t}, y, 1 + a\right)}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+273}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\frac{t}{y} + b}{z}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (+ (/ (* z y) t) x) (+ (+ 1.0 a) (/ (* b y) t)))))
   (if (<= t_1 (- INFINITY))
     (* (/ y (fma (fma (/ y t) b a) t t)) z)
     (if (<= t_1 -1e-163)
       t_1
       (if (<= t_1 2e-306)
         (/ (fma (/ z t) y x) (fma (/ b t) y (+ 1.0 a)))
         (if (<= t_1 2e+273) t_1 (/ 1.0 (/ (+ (/ t y) b) z))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (((z * y) / t) + x) / ((1.0 + a) + ((b * y) / t));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (y / fma(fma((y / t), b, a), t, t)) * z;
	} else if (t_1 <= -1e-163) {
		tmp = t_1;
	} else if (t_1 <= 2e-306) {
		tmp = fma((z / t), y, x) / fma((b / t), y, (1.0 + a));
	} else if (t_1 <= 2e+273) {
		tmp = t_1;
	} else {
		tmp = 1.0 / (((t / y) + b) / z);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(Float64(z * y) / t) + x) / Float64(Float64(1.0 + a) + Float64(Float64(b * y) / t)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(y / fma(fma(Float64(y / t), b, a), t, t)) * z);
	elseif (t_1 <= -1e-163)
		tmp = t_1;
	elseif (t_1 <= 2e-306)
		tmp = Float64(fma(Float64(z / t), y, x) / fma(Float64(b / t), y, Float64(1.0 + a)));
	elseif (t_1 <= 2e+273)
		tmp = t_1;
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(t / y) + b) / z));
	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[(1.0 + a), $MachinePrecision] + N[(N[(b * y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(y / N[(N[(N[(y / t), $MachinePrecision] * b + a), $MachinePrecision] * t + t), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[t$95$1, -1e-163], t$95$1, If[LessEqual[t$95$1, 2e-306], N[(N[(N[(z / t), $MachinePrecision] * y + x), $MachinePrecision] / N[(N[(b / t), $MachinePrecision] * y + N[(1.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+273], t$95$1, N[(1.0 / N[(N[(N[(t / y), $MachinePrecision] + b), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 12.3%

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

   3. Taylor expanded in x around 0

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

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. *-lft-identity N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 92.2

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

   5. Applied rewrites 92.2%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -9.99999999999999923e-164 or 2.00000000000000006e-306 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 1.99999999999999989e273

   1. Initial program 99.7%

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

   if -9.99999999999999923e-164 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 2.00000000000000006e-306

   1. Initial program 58.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 60.3

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 76.7

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 76.7

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

   4. Applied rewrites 76.7%

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

   if 1.99999999999999989e273 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t)))

   1. Initial program 9.3%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 9.3

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 9.4

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 9.4

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lift-/.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. associate-/l* N/A

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

      21. *-commutative N/A

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

      22. lower-fma.f64 N/A

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

      23. lower-/.f64 12.7

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

   4. Applied rewrites 12.7%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 22.8

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

   7. Applied rewrites 22.8%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 47.9%

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

   11. Taylor expanded in y around inf

       \[\leadsto \frac{1}{\frac{b + \frac{t}{y}}{z}} \]
   12. Step-by-step derivation

   13. Applied rewrites 89.1%

       \[\leadsto \frac{1}{\frac{\frac{t}{y} + b}{z}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{z \cdot y}{t} + x}{\left(1 + a\right) + \frac{b \cdot y}{t}} \leq -\infty:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{y}{t}, b, a\right), t, t\right)} \cdot z\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\left(1 + a\right) + \frac{b \cdot y}{t}} \leq -1 \cdot 10^{-163}:\\ \;\;\;\;\frac{\frac{z \cdot y}{t} + x}{\left(1 + a\right) + \frac{b \cdot y}{t}}\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\left(1 + a\right) + \frac{b \cdot y}{t}} \leq 2 \cdot 10^{-306}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{z}{t}, y, x\right)}{\mathsf{fma}\left(\frac{b}{t}, y, 1 + a\right)}\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\left(1 + a\right) + \frac{b \cdot y}{t}} \leq 2 \cdot 10^{+273}:\\ \;\;\;\;\frac{\frac{z \cdot y}{t} + x}{\left(1 + a\right) + \frac{b \cdot y}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\frac{t}{y} + b}{z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 74.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot y}{t} + x\\ t_2 := \frac{t\_1}{1 + a}\\ t_3 := \frac{t\_1}{\left(1 + a\right) + \frac{b \cdot y}{t}}\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{y}{t}, b, a\right), t, t\right)} \cdot z\\ \mathbf{elif}\;t\_3 \leq -5 \cdot 10^{-317}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{-293}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{+273}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\frac{t}{y} + b}{z}}\\ \end{array} \end{array} \]

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 (+ (+ 1.0 a) (/ (* b y) t)))))
   (if (<= t_3 (- INFINITY))
     (* (/ y (fma (fma (/ y t) b a) t t)) z)
     (if (<= t_3 -5e-317)
       t_2
       (if (<= t_3 5e-293)
         (/ (fma t (/ x y) z) b)
         (if (<= t_3 2e+273) t_2 (/ 1.0 (/ (+ (/ t y) b) z))))))))

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 / ((1.0 + a) + ((b * y) / t));
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = (y / fma(fma((y / t), b, a), t, t)) * z;
	} else if (t_3 <= -5e-317) {
		tmp = t_2;
	} else if (t_3 <= 5e-293) {
		tmp = fma(t, (x / y), z) / b;
	} else if (t_3 <= 2e+273) {
		tmp = t_2;
	} else {
		tmp = 1.0 / (((t / y) + b) / z);
	}
	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(1.0 + a) + Float64(Float64(b * y) / t)))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(Float64(y / fma(fma(Float64(y / t), b, a), t, t)) * z);
	elseif (t_3 <= -5e-317)
		tmp = t_2;
	elseif (t_3 <= 5e-293)
		tmp = Float64(fma(t, Float64(x / y), z) / b);
	elseif (t_3 <= 2e+273)
		tmp = t_2;
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(t / y) + b) / z));
	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[(1.0 + a), $MachinePrecision] + N[(N[(b * y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], N[(N[(y / N[(N[(N[(y / t), $MachinePrecision] * b + a), $MachinePrecision] * t + t), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[t$95$3, -5e-317], t$95$2, If[LessEqual[t$95$3, 5e-293], N[(N[(t * N[(x / y), $MachinePrecision] + z), $MachinePrecision] / b), $MachinePrecision], If[LessEqual[t$95$3, 2e+273], t$95$2, N[(1.0 / N[(N[(N[(t / y), $MachinePrecision] + b), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 12.3%

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

   3. Taylor expanded in x around 0

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

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. *-lft-identity N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 92.2

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

   5. Applied rewrites 92.2%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -5.00000017e-317 or 5.0000000000000003e-293 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 1.99999999999999989e273

   1. Initial program 98.8%

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

   3. Taylor expanded in y around 0

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

      1. lower-+.f64 76.0

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

   5. Applied rewrites 76.0%

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

   if -5.00000017e-317 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 5.0000000000000003e-293

   1. Initial program 51.4%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-rgt-in N/A

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

      8. *-lft-identity N/A

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

      9. lower-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. associate-/l* N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 N/A

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

   5. Applied rewrites 63.1%

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

   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 70.1%

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

   if 1.99999999999999989e273 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t)))

   1. Initial program 9.3%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 9.3

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 9.4

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 9.4

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lift-/.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. associate-/l* N/A

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

      21. *-commutative N/A

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

      22. lower-fma.f64 N/A

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

      23. lower-/.f64 12.7

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

   4. Applied rewrites 12.7%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 22.8

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

   7. Applied rewrites 22.8%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 47.9%

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

   11. Taylor expanded in y around inf

       \[\leadsto \frac{1}{\frac{b + \frac{t}{y}}{z}} \]
   12. Step-by-step derivation

   13. Applied rewrites 89.1%

       \[\leadsto \frac{1}{\frac{\frac{t}{y} + b}{z}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{z \cdot y}{t} + x}{\left(1 + a\right) + \frac{b \cdot y}{t}} \leq -\infty:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{y}{t}, b, a\right), t, t\right)} \cdot z\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\left(1 + a\right) + \frac{b \cdot y}{t}} \leq -5 \cdot 10^{-317}:\\ \;\;\;\;\frac{\frac{z \cdot y}{t} + x}{1 + a}\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\left(1 + a\right) + \frac{b \cdot y}{t}} \leq 5 \cdot 10^{-293}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\left(1 + a\right) + \frac{b \cdot y}{t}} \leq 2 \cdot 10^{+273}:\\ \;\;\;\;\frac{\frac{z \cdot y}{t} + x}{1 + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\frac{t}{y} + b}{z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot y}{t} + x\\ t_2 := \frac{t\_1}{1 + a}\\ t_3 := \frac{t\_1}{\left(1 + a\right) + \frac{b \cdot y}{t}}\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;\frac{z}{\mathsf{fma}\left(\mathsf{fma}\left(b, \frac{y}{t}, a\right), t, t\right)} \cdot y\\ \mathbf{elif}\;t\_3 \leq -5 \cdot 10^{-317}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{-293}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{+273}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\frac{t}{y} + b}{z}}\\ \end{array} \end{array} \]

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 (+ (+ 1.0 a) (/ (* b y) t)))))
   (if (<= t_3 (- INFINITY))
     (* (/ z (fma (fma b (/ y t) a) t t)) y)
     (if (<= t_3 -5e-317)
       t_2
       (if (<= t_3 5e-293)
         (/ (fma t (/ x y) z) b)
         (if (<= t_3 2e+273) t_2 (/ 1.0 (/ (+ (/ t y) b) z))))))))

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 / ((1.0 + a) + ((b * y) / t));
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = (z / fma(fma(b, (y / t), a), t, t)) * y;
	} else if (t_3 <= -5e-317) {
		tmp = t_2;
	} else if (t_3 <= 5e-293) {
		tmp = fma(t, (x / y), z) / b;
	} else if (t_3 <= 2e+273) {
		tmp = t_2;
	} else {
		tmp = 1.0 / (((t / y) + b) / z);
	}
	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(1.0 + a) + Float64(Float64(b * y) / t)))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(Float64(z / fma(fma(b, Float64(y / t), a), t, t)) * y);
	elseif (t_3 <= -5e-317)
		tmp = t_2;
	elseif (t_3 <= 5e-293)
		tmp = Float64(fma(t, Float64(x / y), z) / b);
	elseif (t_3 <= 2e+273)
		tmp = t_2;
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(t / y) + b) / z));
	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[(1.0 + a), $MachinePrecision] + N[(N[(b * y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], N[(N[(z / N[(N[(b * N[(y / t), $MachinePrecision] + a), $MachinePrecision] * t + t), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t$95$3, -5e-317], t$95$2, If[LessEqual[t$95$3, 5e-293], N[(N[(t * N[(x / y), $MachinePrecision] + z), $MachinePrecision] / b), $MachinePrecision], If[LessEqual[t$95$3, 2e+273], t$95$2, N[(1.0 / N[(N[(N[(t / y), $MachinePrecision] + b), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 12.3%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-rgt-in N/A

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

      8. *-lft-identity N/A

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

      9. lower-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. associate-/l* N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 N/A

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

   5. Applied rewrites 91.8%

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

   6. Taylor expanded in x around 0

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

      1. associate-/l* N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-*.f64 N/A

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

      5. *-rgt-identity N/A

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

      6. distribute-lft-in N/A

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

      7. lower-/.f64 N/A

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

      8. distribute-lft-in N/A

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

      9. *-rgt-identity N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. +-commutative N/A

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

      14. associate-/l* N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 91.9

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

   8. Applied rewrites 91.9%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -5.00000017e-317 or 5.0000000000000003e-293 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 1.99999999999999989e273

   1. Initial program 98.8%

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

   3. Taylor expanded in y around 0

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

      1. lower-+.f64 76.0

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

   5. Applied rewrites 76.0%

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

   if -5.00000017e-317 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 5.0000000000000003e-293

   1. Initial program 51.4%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-rgt-in N/A

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

      8. *-lft-identity N/A

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

      9. lower-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. associate-/l* N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 N/A

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

   5. Applied rewrites 63.1%

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

   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 70.1%

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

   if 1.99999999999999989e273 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t)))

   1. Initial program 9.3%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 9.3

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 9.4

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 9.4

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lift-/.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. associate-/l* N/A

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

      21. *-commutative N/A

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

      22. lower-fma.f64 N/A

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

      23. lower-/.f64 12.7

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

   4. Applied rewrites 12.7%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 22.8

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

   7. Applied rewrites 22.8%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 47.9%

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

   11. Taylor expanded in y around inf

       \[\leadsto \frac{1}{\frac{b + \frac{t}{y}}{z}} \]
   12. Step-by-step derivation

   13. Applied rewrites 89.1%

       \[\leadsto \frac{1}{\frac{\frac{t}{y} + b}{z}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{z \cdot y}{t} + x}{\left(1 + a\right) + \frac{b \cdot y}{t}} \leq -\infty:\\ \;\;\;\;\frac{z}{\mathsf{fma}\left(\mathsf{fma}\left(b, \frac{y}{t}, a\right), t, t\right)} \cdot y\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\left(1 + a\right) + \frac{b \cdot y}{t}} \leq -5 \cdot 10^{-317}:\\ \;\;\;\;\frac{\frac{z \cdot y}{t} + x}{1 + a}\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\left(1 + a\right) + \frac{b \cdot y}{t}} \leq 5 \cdot 10^{-293}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\left(1 + a\right) + \frac{b \cdot y}{t}} \leq 2 \cdot 10^{+273}:\\ \;\;\;\;\frac{\frac{z \cdot y}{t} + x}{1 + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\frac{t}{y} + b}{z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{z \cdot y}{t} + x}{\left(1 + a\right) + \frac{b \cdot y}{t}}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+217}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{y}{t}, b, a\right), t, t\right)} \cdot z\\ \mathbf{elif}\;t\_1 \leq 10^{+243}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{z}{t}, y, x\right)}{\mathsf{fma}\left(\frac{b}{t}, y, 1 + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\frac{t}{y} + b}{z}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (+ (/ (* z y) t) x) (+ (+ 1.0 a) (/ (* b y) t)))))
   (if (<= t_1 -5e+217)
     (* (/ y (fma (fma (/ y t) b a) t t)) z)
     (if (<= t_1 1e+243)
       (/ (fma (/ z t) y x) (fma (/ b t) y (+ 1.0 a)))
       (/ 1.0 (/ (+ (/ t y) b) z))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (((z * y) / t) + x) / ((1.0 + a) + ((b * y) / t));
	double tmp;
	if (t_1 <= -5e+217) {
		tmp = (y / fma(fma((y / t), b, a), t, t)) * z;
	} else if (t_1 <= 1e+243) {
		tmp = fma((z / t), y, x) / fma((b / t), y, (1.0 + a));
	} else {
		tmp = 1.0 / (((t / y) + b) / z);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(Float64(z * y) / t) + x) / Float64(Float64(1.0 + a) + Float64(Float64(b * y) / t)))
	tmp = 0.0
	if (t_1 <= -5e+217)
		tmp = Float64(Float64(y / fma(fma(Float64(y / t), b, a), t, t)) * z);
	elseif (t_1 <= 1e+243)
		tmp = Float64(fma(Float64(z / t), y, x) / fma(Float64(b / t), y, Float64(1.0 + a)));
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(t / y) + b) / z));
	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[(1.0 + a), $MachinePrecision] + N[(N[(b * y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+217], N[(N[(y / N[(N[(N[(y / t), $MachinePrecision] * b + a), $MachinePrecision] * t + t), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[t$95$1, 1e+243], N[(N[(N[(z / t), $MachinePrecision] * y + x), $MachinePrecision] / N[(N[(b / t), $MachinePrecision] * y + N[(1.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(t / y), $MachinePrecision] + b), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -5.00000000000000041e217

   1. Initial program 28.7%

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

   3. Taylor expanded in x around 0

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

      1. 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

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. *-lft-identity N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 87.6

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

   5. Applied rewrites 87.6%

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

   if -5.00000000000000041e217 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 1.0000000000000001e243

   1. Initial program 87.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 83.2

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 85.5

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 85.5

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

   4. Applied rewrites 85.5%

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

   if 1.0000000000000001e243 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t)))

   1. Initial program 12.4%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 12.4

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 12.5

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 12.5

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lift-/.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. associate-/l* N/A

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

      21. *-commutative N/A

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

      22. lower-fma.f64 N/A

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

      23. lower-/.f64 12.3

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

   4. Applied rewrites 12.3%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 22.0

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

   7. Applied rewrites 22.0%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 46.3%

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

   11. Taylor expanded in y around inf

       \[\leadsto \frac{1}{\frac{b + \frac{t}{y}}{z}} \]
   12. Step-by-step derivation

   13. Applied rewrites 86.0%

       \[\leadsto \frac{1}{\frac{\frac{t}{y} + b}{z}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{z \cdot y}{t} + x}{\left(1 + a\right) + \frac{b \cdot y}{t}} \leq -5 \cdot 10^{+217}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{y}{t}, b, a\right), t, t\right)} \cdot z\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\left(1 + a\right) + \frac{b \cdot y}{t}} \leq 10^{+243}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{z}{t}, y, x\right)}{\mathsf{fma}\left(\frac{b}{t}, y, 1 + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\frac{t}{y} + b}{z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 90.2% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 80.3%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-rgt-in N/A

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

      8. *-lft-identity N/A

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

      9. lower-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. associate-/l* N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 N/A

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

   5. Applied rewrites 85.7%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 90.2%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 90.8%

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

Alternative 6: 74.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{if}\;b \leq -3.7 \cdot 10^{+146}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(\frac{y}{t}, b, 1 + a\right)}\\ \mathbf{elif}\;b \leq -1.6 \cdot 10^{+94}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 4.4 \cdot 10^{+145}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{\mathsf{fma}\left(b, y, \mathsf{fma}\left(a, t, t\right)\right)}, y, \frac{x}{1 + a}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (fma t (/ x y) z) b)))
   (if (<= b -3.7e+146)
     (/ x (fma (/ y t) b (+ 1.0 a)))
     (if (<= b -1.6e+94)
       t_1
       (if (<= b 4.4e+145)
         (fma (/ z (fma b y (fma a t t))) y (/ 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 (b <= -3.7e+146) {
		tmp = x / fma((y / t), b, (1.0 + a));
	} else if (b <= -1.6e+94) {
		tmp = t_1;
	} else if (b <= 4.4e+145) {
		tmp = fma((z / fma(b, y, fma(a, t, t))), y, (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 (b <= -3.7e+146)
		tmp = Float64(x / fma(Float64(y / t), b, Float64(1.0 + a)));
	elseif (b <= -1.6e+94)
		tmp = t_1;
	elseif (b <= 4.4e+145)
		tmp = fma(Float64(z / fma(b, y, fma(a, t, t))), y, 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[b, -3.7e+146], N[(x / N[(N[(y / t), $MachinePrecision] * b + N[(1.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, -1.6e+94], t$95$1, If[LessEqual[b, 4.4e+145], N[(N[(z / N[(b * y + N[(a * t + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y + N[(x / N[(1.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.70000000000000004e146

   1. Initial program 71.5%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-+.f64 72.0

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

   5. Applied rewrites 72.0%

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

   if -3.70000000000000004e146 < b < -1.60000000000000007e94 or 4.40000000000000017e145 < b

   1. Initial program 47.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-rgt-in N/A

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

      8. *-lft-identity N/A

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

      9. lower-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. associate-/l* N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 N/A

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

   5. Applied rewrites 61.2%

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

   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 75.7%

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

   if -1.60000000000000007e94 < b < 4.40000000000000017e145

   1. Initial program 84.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-rgt-in N/A

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

      8. *-lft-identity N/A

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

      9. lower-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. associate-/l* N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 N/A

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

   5. Applied rewrites 87.2%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 91.8%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 83.0%

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

Alternative 7: 73.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{z}{\mathsf{fma}\left(b, y, \mathsf{fma}\left(a, t, t\right)\right)}, y, \frac{x}{1 + a}\right)\\ \mathbf{if}\;a \leq -6.2 \cdot 10^{-15}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 2.2 \cdot 10^{+42}:\\ \;\;\;\;\frac{\frac{z \cdot y}{t} + x}{\mathsf{fma}\left(\frac{y}{t}, b, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma (/ z (fma b y (fma a t t))) y (/ x (+ 1.0 a)))))
   (if (<= a -6.2e-15)
     t_1
     (if (<= a 2.2e+42) (/ (+ (/ (* z y) t) x) (fma (/ y t) b 1.0)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma((z / fma(b, y, fma(a, t, t))), y, (x / (1.0 + a)));
	double tmp;
	if (a <= -6.2e-15) {
		tmp = t_1;
	} else if (a <= 2.2e+42) {
		tmp = (((z * y) / t) + x) / fma((y / t), b, 1.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(Float64(z / fma(b, y, fma(a, t, t))), y, Float64(x / Float64(1.0 + a)))
	tmp = 0.0
	if (a <= -6.2e-15)
		tmp = t_1;
	elseif (a <= 2.2e+42)
		tmp = Float64(Float64(Float64(Float64(z * y) / t) + x) / fma(Float64(y / t), b, 1.0));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(z / N[(b * y + N[(a * t + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y + N[(x / N[(1.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -6.2e-15], t$95$1, If[LessEqual[a, 2.2e+42], N[(N[(N[(N[(z * y), $MachinePrecision] / t), $MachinePrecision] + x), $MachinePrecision] / N[(N[(y / t), $MachinePrecision] * b + 1.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -6.1999999999999998e-15 or 2.2000000000000001e42 < a

   1. Initial program 72.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-rgt-in N/A

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

      8. *-lft-identity N/A

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

      9. lower-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. associate-/l* N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 N/A

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

   5. Applied rewrites 77.2%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 85.6%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 77.4%

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

   if -6.1999999999999998e-15 < a < 2.2000000000000001e42

   1. Initial program 77.5%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 81.5

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

   5. Applied rewrites 81.5%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -6.2 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{\mathsf{fma}\left(b, y, \mathsf{fma}\left(a, t, t\right)\right)}, y, \frac{x}{1 + a}\right)\\ \mathbf{elif}\;a \leq 2.2 \cdot 10^{+42}:\\ \;\;\;\;\frac{\frac{z \cdot y}{t} + x}{\mathsf{fma}\left(\frac{y}{t}, b, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{\mathsf{fma}\left(b, y, \mathsf{fma}\left(a, t, t\right)\right)}, y, \frac{x}{1 + a}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 67.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{if}\;b \leq -3.7 \cdot 10^{+146}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(\frac{y}{t}, b, 1 + a\right)}\\ \mathbf{elif}\;b \leq -1.2 \cdot 10^{+56}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 4.4 \cdot 10^{+145}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{1 + a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (fma t (/ x y) z) b)))
   (if (<= b -3.7e+146)
     (/ x (fma (/ y t) b (+ 1.0 a)))
     (if (<= b -1.2e+56)
       t_1
       (if (<= b 4.4e+145) (/ (fma (/ y t) z 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 (b <= -3.7e+146) {
		tmp = x / fma((y / t), b, (1.0 + a));
	} else if (b <= -1.2e+56) {
		tmp = t_1;
	} else if (b <= 4.4e+145) {
		tmp = fma((y / t), z, 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 (b <= -3.7e+146)
		tmp = Float64(x / fma(Float64(y / t), b, Float64(1.0 + a)));
	elseif (b <= -1.2e+56)
		tmp = t_1;
	elseif (b <= 4.4e+145)
		tmp = Float64(fma(Float64(y / t), z, 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[b, -3.7e+146], N[(x / N[(N[(y / t), $MachinePrecision] * b + N[(1.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, -1.2e+56], t$95$1, If[LessEqual[b, 4.4e+145], N[(N[(N[(y / t), $MachinePrecision] * z + x), $MachinePrecision] / N[(1.0 + a), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.70000000000000004e146

   1. Initial program 71.5%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-+.f64 72.0

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

   5. Applied rewrites 72.0%

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

   if -3.70000000000000004e146 < b < -1.20000000000000007e56 or 4.40000000000000017e145 < b

   1. Initial program 52.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-rgt-in N/A

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

      8. *-lft-identity N/A

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

      9. lower-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. associate-/l* N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 N/A

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

   5. Applied rewrites 60.3%

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

   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 72.6%

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

   if -1.20000000000000007e56 < b < 4.40000000000000017e145

   1. Initial program 84.2%

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

   3. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-*l/ N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 76.0

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

   5. Applied rewrites 76.0%

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

Alternative 9: 70.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{if}\;y \leq -2.4 \cdot 10^{+81}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+14}:\\ \;\;\;\;\frac{\frac{z \cdot y}{t} + x}{1 + a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (fma t (/ x y) z) b)))
   (if (<= y -2.4e+81)
     t_1
     (if (<= y 2e+14) (/ (+ (/ (* z y) t) 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+81) {
		tmp = t_1;
	} else if (y <= 2e+14) {
		tmp = (((z * y) / t) + 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+81)
		tmp = t_1;
	elseif (y <= 2e+14)
		tmp = Float64(Float64(Float64(Float64(z * y) / t) + 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+81], t$95$1, If[LessEqual[y, 2e+14], N[(N[(N[(N[(z * y), $MachinePrecision] / t), $MachinePrecision] + x), $MachinePrecision] / N[(1.0 + a), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.3999999999999999e81 or 2e14 < y

   1. Initial program 47.7%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-rgt-in N/A

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

      8. *-lft-identity N/A

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

      9. lower-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. associate-/l* N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 N/A

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

   5. Applied rewrites 64.5%

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

   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 67.6%

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

   if -2.3999999999999999e81 < y < 2e14

   1. Initial program 92.2%

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

   3. Taylor expanded in y around 0

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

      1. lower-+.f64 77.4

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

   5. Applied rewrites 77.4%

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

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.4 \cdot 10^{+81}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+14}:\\ \;\;\;\;\frac{\frac{z \cdot y}{t} + x}{1 + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 65.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{if}\;y \leq -1.15 \cdot 10^{+39}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{-37}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(\frac{y}{t}, b, 1 + a\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (fma t (/ x y) z) b)))
   (if (<= y -1.15e+39)
     t_1
     (if (<= y 1.2e-37) (/ x (fma (/ y t) b (+ 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 <= -1.15e+39) {
		tmp = t_1;
	} else if (y <= 1.2e-37) {
		tmp = x / fma((y / t), b, (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 <= -1.15e+39)
		tmp = t_1;
	elseif (y <= 1.2e-37)
		tmp = Float64(x / fma(Float64(y / t), b, 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, -1.15e+39], t$95$1, If[LessEqual[y, 1.2e-37], N[(x / N[(N[(y / t), $MachinePrecision] * b + N[(1.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.15000000000000006e39 or 1.19999999999999995e-37 < y

   1. Initial program 52.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-rgt-in N/A

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

      8. *-lft-identity N/A

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

      9. lower-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. associate-/l* N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 N/A

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

   5. Applied rewrites 68.9%

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

   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 64.1%

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

   if -1.15000000000000006e39 < y < 1.19999999999999995e-37

   1. Initial program 94.5%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-+.f64 67.1

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

   5. Applied rewrites 67.1%

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

Alternative 11: 55.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.8 \cdot 10^{+33}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;y \leq 1.08 \cdot 10^{-72}:\\ \;\;\;\;\frac{x}{1 + a}\\ \mathbf{elif}\;y \leq 7 \cdot 10^{+23}:\\ \;\;\;\;\frac{z}{\mathsf{fma}\left(a, t, t\right)} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -3.8e+33)
   (/ z b)
   (if (<= y 1.08e-72)
     (/ x (+ 1.0 a))
     (if (<= y 7e+23) (* (/ z (fma a t t)) y) (/ z b)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -3.8e+33) {
		tmp = z / b;
	} else if (y <= 1.08e-72) {
		tmp = x / (1.0 + a);
	} else if (y <= 7e+23) {
		tmp = (z / fma(a, t, t)) * y;
	} else {
		tmp = z / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -3.8e+33)
		tmp = Float64(z / b);
	elseif (y <= 1.08e-72)
		tmp = Float64(x / Float64(1.0 + a));
	elseif (y <= 7e+23)
		tmp = Float64(Float64(z / fma(a, t, t)) * y);
	else
		tmp = Float64(z / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -3.8e+33], N[(z / b), $MachinePrecision], If[LessEqual[y, 1.08e-72], N[(x / N[(1.0 + a), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 7e+23], N[(N[(z / N[(a * t + t), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], N[(z / b), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.80000000000000002e33 or 7.0000000000000004e23 < y

   1. Initial program 50.7%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 53.5

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

   5. Applied rewrites 53.5%

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

   if -3.80000000000000002e33 < y < 1.07999999999999998e-72

   1. Initial program 94.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 61.8

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

   5. Applied rewrites 61.8%

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

   if 1.07999999999999998e-72 < y < 7.0000000000000004e23

   1. Initial program 79.1%

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

   3. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-*l/ N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 67.8

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

   5. Applied rewrites 67.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 49.7%

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

4. Final simplification 57.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.8 \cdot 10^{+33}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;y \leq 1.08 \cdot 10^{-72}:\\ \;\;\;\;\frac{x}{1 + a}\\ \mathbf{elif}\;y \leq 7 \cdot 10^{+23}:\\ \;\;\;\;\frac{z}{\mathsf{fma}\left(a, t, t\right)} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 60.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{if}\;y \leq -1.35 \cdot 10^{+36}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{-38}:\\ \;\;\;\;\frac{x}{1 + a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (fma t (/ x y) z) b)))
   (if (<= y -1.35e+36) t_1 (if (<= y 2.3e-38) (/ 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 <= -1.35e+36) {
		tmp = t_1;
	} else if (y <= 2.3e-38) {
		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 <= -1.35e+36)
		tmp = t_1;
	elseif (y <= 2.3e-38)
		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, -1.35e+36], t$95$1, If[LessEqual[y, 2.3e-38], N[(x / N[(1.0 + a), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.35e36 or 2.30000000000000002e-38 < y

   1. Initial program 52.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-rgt-in N/A

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

      8. *-lft-identity N/A

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

      9. lower-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. associate-/l* N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 N/A

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

   5. Applied rewrites 68.9%

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

   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 64.1%

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

   if -1.35e36 < y < 2.30000000000000002e-38

   1. Initial program 94.5%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 59.7

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

   5. Applied rewrites 59.7%

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

Alternative 13: 43.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.4 \cdot 10^{+226}:\\ \;\;\;\;\frac{x}{1}\\ \mathbf{elif}\;t \leq -6.3 \cdot 10^{+60}:\\ \;\;\;\;\frac{x}{a}\\ \mathbf{elif}\;t \leq 9 \cdot 10^{+22}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= t -1.4e+226)
   (/ x 1.0)
   (if (<= t -6.3e+60) (/ x a) (if (<= t 9e+22) (/ z b) (/ x a)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= -1.4e+226) {
		tmp = x / 1.0;
	} else if (t <= -6.3e+60) {
		tmp = x / a;
	} else if (t <= 9e+22) {
		tmp = z / b;
	} else {
		tmp = x / a;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (t <= (-1.4d+226)) then
        tmp = x / 1.0d0
    else if (t <= (-6.3d+60)) then
        tmp = x / a
    else if (t <= 9d+22) then
        tmp = z / b
    else
        tmp = x / a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= -1.4e+226) {
		tmp = x / 1.0;
	} else if (t <= -6.3e+60) {
		tmp = x / a;
	} else if (t <= 9e+22) {
		tmp = z / b;
	} else {
		tmp = x / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if t <= -1.4e+226:
		tmp = x / 1.0
	elif t <= -6.3e+60:
		tmp = x / a
	elif t <= 9e+22:
		tmp = z / b
	else:
		tmp = x / a
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (t <= -1.4e+226)
		tmp = Float64(x / 1.0);
	elseif (t <= -6.3e+60)
		tmp = Float64(x / a);
	elseif (t <= 9e+22)
		tmp = Float64(z / b);
	else
		tmp = Float64(x / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (t <= -1.4e+226)
		tmp = x / 1.0;
	elseif (t <= -6.3e+60)
		tmp = x / a;
	elseif (t <= 9e+22)
		tmp = z / b;
	else
		tmp = x / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[t, -1.4e+226], N[(x / 1.0), $MachinePrecision], If[LessEqual[t, -6.3e+60], N[(x / a), $MachinePrecision], If[LessEqual[t, 9e+22], N[(z / b), $MachinePrecision], N[(x / a), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.4000000000000001e226

   1. Initial program 67.6%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 60.7

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

   5. Applied rewrites 60.7%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 52.5%

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

   if -1.4000000000000001e226 < t < -6.3000000000000003e60 or 8.9999999999999996e22 < t

   1. Initial program 82.3%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-rgt-in N/A

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

      8. *-lft-identity N/A

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

      9. lower-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. associate-/l* N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 N/A

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

   5. Applied rewrites 91.5%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 49.8%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 39.3%

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

   if -6.3000000000000003e60 < t < 8.9999999999999996e22

   1. Initial program 71.2%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 47.1

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

   5. Applied rewrites 47.1%

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

Alternative 14: 55.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.8 \cdot 10^{+33}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{-39}:\\ \;\;\;\;\frac{x}{1 + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -3.8e+33) (/ z b) (if (<= y 8.5e-39) (/ 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 <= -3.8e+33) {
		tmp = z / b;
	} else if (y <= 8.5e-39) {
		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 <= (-3.8d+33)) then
        tmp = z / b
    else if (y <= 8.5d-39) 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 <= -3.8e+33) {
		tmp = z / b;
	} else if (y <= 8.5e-39) {
		tmp = x / (1.0 + a);
	} else {
		tmp = z / b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -3.8e+33:
		tmp = z / b
	elif y <= 8.5e-39:
		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 <= -3.8e+33)
		tmp = Float64(z / b);
	elseif (y <= 8.5e-39)
		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 <= -3.8e+33)
		tmp = z / b;
	elseif (y <= 8.5e-39)
		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, -3.8e+33], N[(z / b), $MachinePrecision], If[LessEqual[y, 8.5e-39], N[(x / N[(1.0 + a), $MachinePrecision]), $MachinePrecision], N[(z / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.80000000000000002e33 or 8.5000000000000005e-39 < y

   1. Initial program 53.4%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 50.9

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

   5. Applied rewrites 50.9%

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

   if -3.80000000000000002e33 < y < 8.5000000000000005e-39

   1. Initial program 94.4%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 59.8

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

   5. Applied rewrites 59.8%

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

Alternative 15: 40.7% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -8.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{x}{a}\\ \mathbf{elif}\;a \leq 4.6 \cdot 10^{+21}:\\ \;\;\;\;\left(1 - a\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= a -8.8e-7) (/ x a) (if (<= a 4.6e+21) (* (- 1.0 a) x) (/ x a))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= -8.8e-7) {
		tmp = x / a;
	} else if (a <= 4.6e+21) {
		tmp = (1.0 - a) * x;
	} else {
		tmp = x / a;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-8.8d-7)) then
        tmp = x / a
    else if (a <= 4.6d+21) then
        tmp = (1.0d0 - a) * x
    else
        tmp = x / a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= -8.8e-7) {
		tmp = x / a;
	} else if (a <= 4.6e+21) {
		tmp = (1.0 - a) * x;
	} else {
		tmp = x / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if a <= -8.8e-7:
		tmp = x / a
	elif a <= 4.6e+21:
		tmp = (1.0 - a) * x
	else:
		tmp = x / a
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (a <= -8.8e-7)
		tmp = Float64(x / a);
	elseif (a <= 4.6e+21)
		tmp = Float64(Float64(1.0 - a) * x);
	else
		tmp = Float64(x / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (a <= -8.8e-7)
		tmp = x / a;
	elseif (a <= 4.6e+21)
		tmp = (1.0 - a) * x;
	else
		tmp = x / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[a, -8.8e-7], N[(x / a), $MachinePrecision], If[LessEqual[a, 4.6e+21], N[(N[(1.0 - a), $MachinePrecision] * x), $MachinePrecision], N[(x / a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -8.8000000000000004e-7 or 4.6e21 < a

   1. Initial program 71.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-rgt-in N/A

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

      8. *-lft-identity N/A

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

      9. lower-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. associate-/l* N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 N/A

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

   5. Applied rewrites 77.8%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 59.2%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 43.6%

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

   if -8.8000000000000004e-7 < a < 4.6e21

   1. Initial program 77.7%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 34.5

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

   5. Applied rewrites 34.5%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 34.6%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 34.6%

       \[\leadsto \left(1 - a\right) \cdot \color{blue}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 19.6% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \left(1 - a\right) \cdot x \end{array} \]

FPCore

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

C

double code(double x, double y, double z, double t, double a, double b) {
	return (1.0 - 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 = (1.0d0 - a) * x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(1.0 - a), $MachinePrecision] * x), $MachinePrecision]

Derivation

1. Initial program 75.0%

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

3. Taylor expanded in y around 0

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

   1. lower-/.f64 N/A

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

   2. lower-+.f64 39.0

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

5. Applied rewrites 39.0%

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

6. Taylor expanded in a around 0

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

8. Applied rewrites 19.9%

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

9. Taylor expanded in a around 0

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

11. Applied rewrites 19.9%

    \[\leadsto \left(1 - a\right) \cdot \color{blue}{x} \]
12. Add Preprocessing

Alternative 17: 4.1% 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 75.0%

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

3. Taylor expanded in y around 0

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

   1. lower-/.f64 N/A

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

   2. lower-+.f64 39.0

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

5. Applied rewrites 39.0%

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

6. Taylor expanded in a around 0

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

8. Applied rewrites 19.9%

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

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 4.6%

    \[\leadsto \left(-a\right) \cdot x \]
12. Add Preprocessing

Developer Target 1: 79.4% 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 2024295 
(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))))