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

Percentage Accurate: 74.8% → 91.5%

Time: 20.5s

Alternatives: 17

Speedup: 0.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 74.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 91.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)}\\ t_2 := \frac{y}{t} \cdot \frac{z}{1 + \mathsf{fma}\left(y, \frac{b}{t}, a\right)}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq -5 \cdot 10^{-317}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;\frac{z}{b} + \left(\frac{\frac{t}{\frac{b}{x}}}{y} - \frac{t}{\frac{y \cdot {b}^{2}}{z \cdot \left(a + 1\right)}}\right)\\ \mathbf{elif}\;t_1 \leq 10^{+263}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (* y z) t)) (+ (/ (* y b) t) (+ a 1.0))))
        (t_2 (* (/ y t) (/ z (+ 1.0 (fma y (/ b t) a))))))
   (if (<= t_1 (- INFINITY))
     t_2
     (if (<= t_1 -5e-317)
       t_1
       (if (<= t_1 0.0)
         (+
          (/ z b)
          (- (/ (/ t (/ b x)) y) (/ t (/ (* y (pow b 2.0)) (* z (+ a 1.0))))))
         (if (<= t_1 1e+263) t_1 (if (<= t_1 INFINITY) t_2 (/ z b))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x + ((y * z) / t)) / (((y * b) / t) + (a + 1.0));
	double t_2 = (y / t) * (z / (1.0 + fma(y, (b / t), a)));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_2;
	} else if (t_1 <= -5e-317) {
		tmp = t_1;
	} else if (t_1 <= 0.0) {
		tmp = (z / b) + (((t / (b / x)) / y) - (t / ((y * pow(b, 2.0)) / (z * (a + 1.0)))));
	} else if (t_1 <= 1e+263) {
		tmp = t_1;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_2;
	} else {
		tmp = z / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x + Float64(Float64(y * z) / t)) / Float64(Float64(Float64(y * b) / t) + Float64(a + 1.0)))
	t_2 = Float64(Float64(y / t) * Float64(z / Float64(1.0 + fma(y, Float64(b / t), a))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = t_2;
	elseif (t_1 <= -5e-317)
		tmp = t_1;
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(z / b) + Float64(Float64(Float64(t / Float64(b / x)) / y) - Float64(t / Float64(Float64(y * (b ^ 2.0)) / Float64(z * Float64(a + 1.0))))));
	elseif (t_1 <= 1e+263)
		tmp = t_1;
	elseif (t_1 <= Inf)
		tmp = t_2;
	else
		tmp = Float64(z / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(y * b), $MachinePrecision] / t), $MachinePrecision] + N[(a + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y / t), $MachinePrecision] * N[(z / N[(1.0 + N[(y * N[(b / t), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], t$95$2, If[LessEqual[t$95$1, -5e-317], t$95$1, If[LessEqual[t$95$1, 0.0], N[(N[(z / b), $MachinePrecision] + N[(N[(N[(t / N[(b / x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] - N[(t / N[(N[(y * N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] / N[(z * N[(a + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+263], t$95$1, If[LessEqual[t$95$1, Infinity], t$95$2, N[(z / b), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t))) < -inf.0 or 1.00000000000000002e263 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t))) < +inf.0

   1. Initial program 28.7%

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

      1. associate-/l* 57.0%

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

      2. associate-+l+ 57.0%

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

      3. associate-/l* 57.0%

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

   3. Simplified 57.0%

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

   4. Taylor expanded in x around 0 56.2%

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

      1. times-frac 94.5%

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

      2. +-commutative 94.5%

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

      3. *-commutative 94.5%

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

      4. associate-*r/ 94.3%

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

      5. fma-udef 94.3%

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

   6. Simplified 94.3%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t))) < -5.00000017e-317 or 0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t))) < 1.00000000000000002e263

   1. Initial program 99.4%

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

   if -5.00000017e-317 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t))) < 0.0

   1. Initial program 49.8%

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

      1. associate-/l* 49.6%

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

      2. associate-+l+ 49.6%

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

      3. associate-/l* 71.9%

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

   3. Simplified 71.9%

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

      1. clear-num 71.9%

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

      2. associate-+r+ 71.9%

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

      3. associate-/l* 49.6%

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

      4. associate-/l* 49.8%

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

      5. inv-pow 49.8%

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

      6. associate-+l+ 49.8%

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

      7. +-commutative 49.8%

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

      8. *-commutative 49.8%

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

      9. associate-*l/ 74.9%

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

      10. *-commutative 74.9%

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

      11. fma-def 74.9%

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

      12. +-commutative 74.9%

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

      13. associate-/l* 71.9%

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

      14. associate-/r/ 67.0%

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

      15. fma-def 67.0%

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

   5. Applied egg-rr 67.0%

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

      1. unpow-1 67.0%

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

      2. fma-udef 67.0%

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

      3. associate-*l/ 74.9%

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

      4. associate-*r/ 74.7%

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

      5. fma-udef 74.7%

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

   7. Simplified 74.7%

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

   8. Taylor expanded in y around inf 63.0%

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

      1. associate--l+ 63.0%

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

      2. associate-/r* 64.2%

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

      3. associate-/l* 77.2%

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

      4. associate-/l* 77.2%

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

      5. *-commutative 77.2%

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

   10. Simplified 77.2%

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

   if +inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.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. Step-by-step derivation

      1. associate-/l* 1.1%

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

      2. associate-+l+ 1.1%

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

      3. associate-/l* 13.0%

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

   3. Simplified 13.0%

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

   4. Taylor expanded in y around inf 94.2%

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

4. Final simplification 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq -\infty:\\ \;\;\;\;\frac{y}{t} \cdot \frac{z}{1 + \mathsf{fma}\left(y, \frac{b}{t}, a\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq -5 \cdot 10^{-317}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq 0:\\ \;\;\;\;\frac{z}{b} + \left(\frac{\frac{t}{\frac{b}{x}}}{y} - \frac{t}{\frac{y \cdot {b}^{2}}{z \cdot \left(a + 1\right)}}\right)\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq 10^{+263}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq \infty:\\ \;\;\;\;\frac{y}{t} \cdot \frac{z}{1 + \mathsf{fma}\left(y, \frac{b}{t}, a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]

Alternative 2: 91.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)}\\ t_2 := \frac{y}{t} \cdot \frac{z}{1 + \mathsf{fma}\left(y, \frac{b}{t}, a\right)}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq -5 \cdot 10^{-282}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{-103}:\\ \;\;\;\;\frac{x + \frac{y}{\frac{t}{z}}}{a + \left(1 + b \cdot \frac{y}{t}\right)}\\ \mathbf{elif}\;t_1 \leq 10^{+263}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (* y z) t)) (+ (/ (* y b) t) (+ a 1.0))))
        (t_2 (* (/ y t) (/ z (+ 1.0 (fma y (/ b t) a))))))
   (if (<= t_1 (- INFINITY))
     t_2
     (if (<= t_1 -5e-282)
       t_1
       (if (<= t_1 2e-103)
         (/ (+ x (/ y (/ t z))) (+ a (+ 1.0 (* b (/ y t)))))
         (if (<= t_1 1e+263) t_1 (if (<= t_1 INFINITY) t_2 (/ z b))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x + ((y * z) / t)) / (((y * b) / t) + (a + 1.0));
	double t_2 = (y / t) * (z / (1.0 + fma(y, (b / t), a)));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_2;
	} else if (t_1 <= -5e-282) {
		tmp = t_1;
	} else if (t_1 <= 2e-103) {
		tmp = (x + (y / (t / z))) / (a + (1.0 + (b * (y / t))));
	} else if (t_1 <= 1e+263) {
		tmp = t_1;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_2;
	} else {
		tmp = z / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x + Float64(Float64(y * z) / t)) / Float64(Float64(Float64(y * b) / t) + Float64(a + 1.0)))
	t_2 = Float64(Float64(y / t) * Float64(z / Float64(1.0 + fma(y, Float64(b / t), a))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = t_2;
	elseif (t_1 <= -5e-282)
		tmp = t_1;
	elseif (t_1 <= 2e-103)
		tmp = Float64(Float64(x + Float64(y / Float64(t / z))) / Float64(a + Float64(1.0 + Float64(b * Float64(y / t)))));
	elseif (t_1 <= 1e+263)
		tmp = t_1;
	elseif (t_1 <= Inf)
		tmp = t_2;
	else
		tmp = Float64(z / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(y * b), $MachinePrecision] / t), $MachinePrecision] + N[(a + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y / t), $MachinePrecision] * N[(z / N[(1.0 + N[(y * N[(b / t), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], t$95$2, If[LessEqual[t$95$1, -5e-282], t$95$1, If[LessEqual[t$95$1, 2e-103], N[(N[(x + N[(y / N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a + N[(1.0 + N[(b * N[(y / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+263], t$95$1, If[LessEqual[t$95$1, Infinity], t$95$2, N[(z / b), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t))) < -inf.0 or 1.00000000000000002e263 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t))) < +inf.0

   1. Initial program 28.7%

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

      1. associate-/l* 57.0%

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

      2. associate-+l+ 57.0%

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

      3. associate-/l* 57.0%

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

   3. Simplified 57.0%

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

   4. Taylor expanded in x around 0 56.2%

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

      1. times-frac 94.5%

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

      2. +-commutative 94.5%

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

      3. *-commutative 94.5%

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

      4. associate-*r/ 94.3%

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

      5. fma-udef 94.3%

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

   6. Simplified 94.3%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t))) < -5.0000000000000001e-282 or 1.99999999999999992e-103 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t))) < 1.00000000000000002e263

   1. Initial program 99.7%

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

   if -5.0000000000000001e-282 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t))) < 1.99999999999999992e-103

   1. Initial program 72.6%

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

      1. associate-/l* 73.4%

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

      2. associate-+l+ 73.4%

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

      3. associate-/l* 81.8%

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

   3. Simplified 81.8%

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

      1. associate-/r/ 71.2%

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

   5. Applied egg-rr 85.1%

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

   if +inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.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. Step-by-step derivation

      1. associate-/l* 1.1%

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

      2. associate-+l+ 1.1%

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

      3. associate-/l* 13.0%

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

   3. Simplified 13.0%

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

   4. Taylor expanded in y around inf 94.2%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq -\infty:\\ \;\;\;\;\frac{y}{t} \cdot \frac{z}{1 + \mathsf{fma}\left(y, \frac{b}{t}, a\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq -5 \cdot 10^{-282}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq 2 \cdot 10^{-103}:\\ \;\;\;\;\frac{x + \frac{y}{\frac{t}{z}}}{a + \left(1 + b \cdot \frac{y}{t}\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq 10^{+263}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq \infty:\\ \;\;\;\;\frac{y}{t} \cdot \frac{z}{1 + \mathsf{fma}\left(y, \frac{b}{t}, a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]

Alternative 3: 67.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z}{t}}{a + 1}\\ t_2 := \frac{z}{b} + \frac{t}{b} \cdot \frac{x}{y}\\ \mathbf{if}\;y \leq -3.1 \cdot 10^{+31}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 1.02 \cdot 10^{-112}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 3.35 \cdot 10^{-47}:\\ \;\;\;\;\frac{x}{1 + \left(a + \frac{y \cdot b}{t}\right)}\\ \mathbf{elif}\;y \leq 5.6 \cdot 10^{+58}:\\ \;\;\;\;\frac{y}{\frac{t \cdot a}{z}} + \frac{x}{a + 1}\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{+118} \lor \neg \left(y \leq 7.8 \cdot 10^{+128}\right):\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (* y z) t)) (+ a 1.0)))
        (t_2 (+ (/ z b) (* (/ t b) (/ x y)))))
   (if (<= y -3.1e+31)
     t_2
     (if (<= y 1.02e-112)
       t_1
       (if (<= y 3.35e-47)
         (/ x (+ 1.0 (+ a (/ (* y b) t))))
         (if (<= y 5.6e+58)
           (+ (/ y (/ (* t a) z)) (/ x (+ a 1.0)))
           (if (or (<= y 3.1e+118) (not (<= y 7.8e+128))) t_2 t_1)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x + ((y * z) / t)) / (a + 1.0);
	double t_2 = (z / b) + ((t / b) * (x / y));
	double tmp;
	if (y <= -3.1e+31) {
		tmp = t_2;
	} else if (y <= 1.02e-112) {
		tmp = t_1;
	} else if (y <= 3.35e-47) {
		tmp = x / (1.0 + (a + ((y * b) / t)));
	} else if (y <= 5.6e+58) {
		tmp = (y / ((t * a) / z)) + (x / (a + 1.0));
	} else if ((y <= 3.1e+118) || !(y <= 7.8e+128)) {
		tmp = t_2;
	} 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) :: t_2
    real(8) :: tmp
    t_1 = (x + ((y * z) / t)) / (a + 1.0d0)
    t_2 = (z / b) + ((t / b) * (x / y))
    if (y <= (-3.1d+31)) then
        tmp = t_2
    else if (y <= 1.02d-112) then
        tmp = t_1
    else if (y <= 3.35d-47) then
        tmp = x / (1.0d0 + (a + ((y * b) / t)))
    else if (y <= 5.6d+58) then
        tmp = (y / ((t * a) / z)) + (x / (a + 1.0d0))
    else if ((y <= 3.1d+118) .or. (.not. (y <= 7.8d+128))) then
        tmp = t_2
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x + ((y * z) / t)) / (a + 1.0);
	double t_2 = (z / b) + ((t / b) * (x / y));
	double tmp;
	if (y <= -3.1e+31) {
		tmp = t_2;
	} else if (y <= 1.02e-112) {
		tmp = t_1;
	} else if (y <= 3.35e-47) {
		tmp = x / (1.0 + (a + ((y * b) / t)));
	} else if (y <= 5.6e+58) {
		tmp = (y / ((t * a) / z)) + (x / (a + 1.0));
	} else if ((y <= 3.1e+118) || !(y <= 7.8e+128)) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (x + ((y * z) / t)) / (a + 1.0)
	t_2 = (z / b) + ((t / b) * (x / y))
	tmp = 0
	if y <= -3.1e+31:
		tmp = t_2
	elif y <= 1.02e-112:
		tmp = t_1
	elif y <= 3.35e-47:
		tmp = x / (1.0 + (a + ((y * b) / t)))
	elif y <= 5.6e+58:
		tmp = (y / ((t * a) / z)) + (x / (a + 1.0))
	elif (y <= 3.1e+118) or not (y <= 7.8e+128):
		tmp = t_2
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x + Float64(Float64(y * z) / t)) / Float64(a + 1.0))
	t_2 = Float64(Float64(z / b) + Float64(Float64(t / b) * Float64(x / y)))
	tmp = 0.0
	if (y <= -3.1e+31)
		tmp = t_2;
	elseif (y <= 1.02e-112)
		tmp = t_1;
	elseif (y <= 3.35e-47)
		tmp = Float64(x / Float64(1.0 + Float64(a + Float64(Float64(y * b) / t))));
	elseif (y <= 5.6e+58)
		tmp = Float64(Float64(y / Float64(Float64(t * a) / z)) + Float64(x / Float64(a + 1.0)));
	elseif ((y <= 3.1e+118) || !(y <= 7.8e+128))
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (x + ((y * z) / t)) / (a + 1.0);
	t_2 = (z / b) + ((t / b) * (x / y));
	tmp = 0.0;
	if (y <= -3.1e+31)
		tmp = t_2;
	elseif (y <= 1.02e-112)
		tmp = t_1;
	elseif (y <= 3.35e-47)
		tmp = x / (1.0 + (a + ((y * b) / t)));
	elseif (y <= 5.6e+58)
		tmp = (y / ((t * a) / z)) + (x / (a + 1.0));
	elseif ((y <= 3.1e+118) || ~((y <= 7.8e+128)))
		tmp = t_2;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[(a + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(z / b), $MachinePrecision] + N[(N[(t / b), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -3.1e+31], t$95$2, If[LessEqual[y, 1.02e-112], t$95$1, If[LessEqual[y, 3.35e-47], N[(x / N[(1.0 + N[(a + N[(N[(y * b), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5.6e+58], N[(N[(y / N[(N[(t * a), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] + N[(x / N[(a + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[y, 3.1e+118], N[Not[LessEqual[y, 7.8e+128]], $MachinePrecision]], t$95$2, t$95$1]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -3.1000000000000002e31 or 5.5999999999999996e58 < y < 3.09999999999999986e118 or 7.7999999999999994e128 < y

   1. Initial program 51.5%

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

      1. associate-/l* 61.5%

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

      2. associate-+l+ 61.5%

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

      3. associate-/l* 70.3%

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

   3. Simplified 70.3%

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

      1. clear-num 70.3%

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

      2. associate-+r+ 70.3%

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

      3. associate-/l* 61.4%

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

      4. associate-/l* 51.5%

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

      5. inv-pow 51.5%

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

      6. associate-+l+ 51.5%

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

      7. +-commutative 51.5%

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

      8. *-commutative 51.5%

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

      9. associate-*l/ 58.6%

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

      10. *-commutative 58.6%

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

      11. fma-def 58.6%

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

      12. +-commutative 58.6%

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

      13. associate-/l* 70.3%

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

      14. associate-/r/ 60.9%

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

      15. fma-def 60.9%

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

   5. Applied egg-rr 60.9%

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

      1. unpow-1 60.9%

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

      2. fma-udef 60.9%

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

      3. associate-*l/ 58.6%

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

      4. associate-*r/ 71.0%

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

      5. fma-udef 71.0%

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

   7. Simplified 71.0%

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

   8. Taylor expanded in b around inf 32.3%

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

      1. times-frac 40.9%

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

      2. associate-/l* 47.3%

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

   10. Simplified 47.3%

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

   11. Taylor expanded in t around 0 61.0%

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

       1. +-commutative 61.0%

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

       2. times-frac 68.4%

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

   13. Simplified 68.4%

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

   if -3.1000000000000002e31 < y < 1.01999999999999996e-112 or 3.09999999999999986e118 < y < 7.7999999999999994e128

   1. Initial program 93.6%

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

      1. associate-/l* 88.1%

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

      2. associate-+l+ 88.1%

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

      3. associate-/l* 86.5%

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

   3. Simplified 86.5%

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

   4. Taylor expanded in b around 0 80.2%

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

   if 1.01999999999999996e-112 < y < 3.3499999999999999e-47

   1. Initial program 88.4%

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

      1. associate-/l* 88.4%

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

      2. associate-+l+ 88.4%

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

      3. associate-/l* 88.4%

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

   3. Simplified 88.4%

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

   4. Taylor expanded in x around inf 82.9%

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

   if 3.3499999999999999e-47 < y < 5.5999999999999996e58

   1. Initial program 70.8%

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

      1. associate-/l* 70.8%

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

      2. associate-+l+ 70.8%

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

      3. associate-/l* 64.8%

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

   3. Simplified 64.8%

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

   4. Taylor expanded in y around 0 57.1%

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

   5. Taylor expanded in a around inf 63.7%

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

      1. associate-/l* 63.7%

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

      2. *-commutative 63.7%

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

   7. Simplified 63.7%

      \[\leadsto \color{blue}{\frac{y}{\frac{t \cdot a}{z}}} + \frac{x}{1 + a} \]
3. Recombined 4 regimes into one program.

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.1 \cdot 10^{+31}:\\ \;\;\;\;\frac{z}{b} + \frac{t}{b} \cdot \frac{x}{y}\\ \mathbf{elif}\;y \leq 1.02 \cdot 10^{-112}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{a + 1}\\ \mathbf{elif}\;y \leq 3.35 \cdot 10^{-47}:\\ \;\;\;\;\frac{x}{1 + \left(a + \frac{y \cdot b}{t}\right)}\\ \mathbf{elif}\;y \leq 5.6 \cdot 10^{+58}:\\ \;\;\;\;\frac{y}{\frac{t \cdot a}{z}} + \frac{x}{a + 1}\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{+118} \lor \neg \left(y \leq 7.8 \cdot 10^{+128}\right):\\ \;\;\;\;\frac{z}{b} + \frac{t}{b} \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{a + 1}\\ \end{array} \]

Alternative 4: 68.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y \cdot z}{y \cdot b + t \cdot \left(a + 1\right)}\\ t_2 := \frac{z}{b} + \frac{t}{b} \cdot \frac{x}{y}\\ \mathbf{if}\;y \leq -4.7 \cdot 10^{+30}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{-52}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{a + 1}\\ \mathbf{elif}\;y \leq 96000000000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 6.6 \cdot 10^{+78}:\\ \;\;\;\;\frac{x + y \cdot \frac{z}{t}}{a + 1}\\ \mathbf{elif}\;y \leq 6.6 \cdot 10^{+128}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (* y z) (+ (* y b) (* t (+ a 1.0)))))
        (t_2 (+ (/ z b) (* (/ t b) (/ x y)))))
   (if (<= y -4.7e+30)
     t_2
     (if (<= y 3.5e-52)
       (/ (+ x (/ (* y z) t)) (+ a 1.0))
       (if (<= y 96000000000000.0)
         t_1
         (if (<= y 6.6e+78)
           (/ (+ x (* y (/ z t))) (+ a 1.0))
           (if (<= y 6.6e+128) t_1 t_2)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (y * z) / ((y * b) + (t * (a + 1.0)));
	double t_2 = (z / b) + ((t / b) * (x / y));
	double tmp;
	if (y <= -4.7e+30) {
		tmp = t_2;
	} else if (y <= 3.5e-52) {
		tmp = (x + ((y * z) / t)) / (a + 1.0);
	} else if (y <= 96000000000000.0) {
		tmp = t_1;
	} else if (y <= 6.6e+78) {
		tmp = (x + (y * (z / t))) / (a + 1.0);
	} else if (y <= 6.6e+128) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	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) :: t_2
    real(8) :: tmp
    t_1 = (y * z) / ((y * b) + (t * (a + 1.0d0)))
    t_2 = (z / b) + ((t / b) * (x / y))
    if (y <= (-4.7d+30)) then
        tmp = t_2
    else if (y <= 3.5d-52) then
        tmp = (x + ((y * z) / t)) / (a + 1.0d0)
    else if (y <= 96000000000000.0d0) then
        tmp = t_1
    else if (y <= 6.6d+78) then
        tmp = (x + (y * (z / t))) / (a + 1.0d0)
    else if (y <= 6.6d+128) then
        tmp = t_1
    else
        tmp = t_2
    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 = (y * z) / ((y * b) + (t * (a + 1.0)));
	double t_2 = (z / b) + ((t / b) * (x / y));
	double tmp;
	if (y <= -4.7e+30) {
		tmp = t_2;
	} else if (y <= 3.5e-52) {
		tmp = (x + ((y * z) / t)) / (a + 1.0);
	} else if (y <= 96000000000000.0) {
		tmp = t_1;
	} else if (y <= 6.6e+78) {
		tmp = (x + (y * (z / t))) / (a + 1.0);
	} else if (y <= 6.6e+128) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (y * z) / ((y * b) + (t * (a + 1.0)))
	t_2 = (z / b) + ((t / b) * (x / y))
	tmp = 0
	if y <= -4.7e+30:
		tmp = t_2
	elif y <= 3.5e-52:
		tmp = (x + ((y * z) / t)) / (a + 1.0)
	elif y <= 96000000000000.0:
		tmp = t_1
	elif y <= 6.6e+78:
		tmp = (x + (y * (z / t))) / (a + 1.0)
	elif y <= 6.6e+128:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(y * z) / Float64(Float64(y * b) + Float64(t * Float64(a + 1.0))))
	t_2 = Float64(Float64(z / b) + Float64(Float64(t / b) * Float64(x / y)))
	tmp = 0.0
	if (y <= -4.7e+30)
		tmp = t_2;
	elseif (y <= 3.5e-52)
		tmp = Float64(Float64(x + Float64(Float64(y * z) / t)) / Float64(a + 1.0));
	elseif (y <= 96000000000000.0)
		tmp = t_1;
	elseif (y <= 6.6e+78)
		tmp = Float64(Float64(x + Float64(y * Float64(z / t))) / Float64(a + 1.0));
	elseif (y <= 6.6e+128)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (y * z) / ((y * b) + (t * (a + 1.0)));
	t_2 = (z / b) + ((t / b) * (x / y));
	tmp = 0.0;
	if (y <= -4.7e+30)
		tmp = t_2;
	elseif (y <= 3.5e-52)
		tmp = (x + ((y * z) / t)) / (a + 1.0);
	elseif (y <= 96000000000000.0)
		tmp = t_1;
	elseif (y <= 6.6e+78)
		tmp = (x + (y * (z / t))) / (a + 1.0);
	elseif (y <= 6.6e+128)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(y * z), $MachinePrecision] / N[(N[(y * b), $MachinePrecision] + N[(t * N[(a + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(z / b), $MachinePrecision] + N[(N[(t / b), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -4.7e+30], t$95$2, If[LessEqual[y, 3.5e-52], N[(N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[(a + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 96000000000000.0], t$95$1, If[LessEqual[y, 6.6e+78], N[(N[(x + N[(y * N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6.6e+128], t$95$1, t$95$2]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -4.6999999999999999e30 or 6.6000000000000001e128 < y

   1. Initial program 49.9%

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

      1. associate-/l* 59.4%

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

      2. associate-+l+ 59.4%

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

      3. associate-/l* 69.6%

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

   3. Simplified 69.6%

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

      1. clear-num 69.6%

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

      2. associate-+r+ 69.6%

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

      3. associate-/l* 59.3%

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

      4. associate-/l* 49.8%

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

      5. inv-pow 49.8%

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

      6. associate-+l+ 49.8%

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

      7. +-commutative 49.8%

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

      8. *-commutative 49.8%

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

      9. associate-*l/ 58.0%

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

      10. *-commutative 58.0%

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

      11. fma-def 58.0%

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

      12. +-commutative 58.0%

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

      13. associate-/l* 69.6%

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

      14. associate-/r/ 60.8%

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

      15. fma-def 60.8%

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

   5. Applied egg-rr 60.8%

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

      1. unpow-1 60.8%

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

      2. fma-udef 60.8%

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

      3. associate-*l/ 58.0%

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

      4. associate-*r/ 70.5%

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

      5. fma-udef 70.5%

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

   7. Simplified 70.5%

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

   8. Taylor expanded in b around inf 30.6%

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

      1. times-frac 40.5%

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

      2. associate-/l* 47.0%

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

   10. Simplified 47.0%

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

   11. Taylor expanded in t around 0 60.7%

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

       1. +-commutative 60.7%

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

       2. times-frac 68.3%

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

   13. Simplified 68.3%

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

   if -4.6999999999999999e30 < y < 3.5e-52

   1. Initial program 92.6%

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

      1. associate-/l* 88.3%

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

      2. associate-+l+ 88.3%

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

      3. associate-/l* 86.9%

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

   3. Simplified 86.9%

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

   4. Taylor expanded in b around 0 76.6%

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

   if 3.5e-52 < y < 9.6e13 or 6.6e78 < y < 6.6000000000000001e128

   1. Initial program 73.6%

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

      1. associate-/l* 70.1%

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

      2. associate-+l+ 70.1%

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

      3. associate-/l* 66.4%

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

   3. Simplified 66.4%

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

      1. clear-num 66.2%

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

      2. associate-+r+ 66.2%

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

      3. associate-/l* 70.1%

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

      4. associate-/l* 73.7%

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

      5. inv-pow 73.7%

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

      6. associate-+l+ 73.7%

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

      7. +-commutative 73.7%

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

      8. *-commutative 73.7%

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

      9. associate-*l/ 70.0%

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

      10. *-commutative 70.0%

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

      11. fma-def 70.0%

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

      12. +-commutative 70.0%

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

      13. associate-/l* 66.2%

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

      14. associate-/r/ 62.1%

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

      15. fma-def 62.1%

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

   5. Applied egg-rr 62.1%

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

      1. unpow-1 62.1%

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

      2. fma-udef 62.1%

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

      3. associate-*l/ 70.0%

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

      4. associate-*r/ 66.2%

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

      5. fma-udef 66.2%

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

   7. Simplified 66.2%

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

   8. Taylor expanded in z around inf 66.7%

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

      1. *-commutative 66.7%

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

      2. associate-/l* 65.9%

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

      3. associate-/r/ 62.8%

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

   10. Simplified 62.8%

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

   11. Taylor expanded in t around 0 74.2%

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

   if 9.6e13 < y < 6.6e78

   1. Initial program 68.1%

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

      1. associate-/l* 83.8%

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

      2. associate-+l+ 83.8%

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

      3. associate-/l* 83.8%

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

   3. Simplified 83.8%

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

      1. clear-num 83.6%

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

      2. associate-+r+ 83.6%

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

      3. associate-/l* 83.6%

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

      4. associate-/l* 68.0%

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

      5. inv-pow 68.0%

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

      6. associate-+l+ 68.0%

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

      7. +-commutative 68.0%

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

      8. *-commutative 68.0%

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

      9. associate-*l/ 68.0%

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

      10. *-commutative 68.0%

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

      11. fma-def 68.0%

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

      12. +-commutative 68.0%

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

      13. associate-/l* 83.4%

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

      14. associate-/r/ 83.4%

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

      15. fma-def 83.4%

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

   5. Applied egg-rr 83.4%

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

      1. unpow-1 83.4%

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

      2. fma-udef 83.4%

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

      3. associate-*l/ 68.0%

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

      4. associate-*r/ 83.6%

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

      5. fma-udef 83.6%

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

   7. Simplified 83.6%

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

   8. Taylor expanded in b around 0 67.8%

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

      1. associate-/l* 75.6%

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

   10. Simplified 75.6%

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

   11. Taylor expanded in y around 0 67.8%

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

       1. associate-*r/ 75.7%

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

   13. Simplified 75.7%

       \[\leadsto \frac{x + \color{blue}{y \cdot \frac{z}{t}}}{1 + a} \]
3. Recombined 4 regimes into one program.

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.7 \cdot 10^{+30}:\\ \;\;\;\;\frac{z}{b} + \frac{t}{b} \cdot \frac{x}{y}\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{-52}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{a + 1}\\ \mathbf{elif}\;y \leq 96000000000000:\\ \;\;\;\;\frac{y \cdot z}{y \cdot b + t \cdot \left(a + 1\right)}\\ \mathbf{elif}\;y \leq 6.6 \cdot 10^{+78}:\\ \;\;\;\;\frac{x + y \cdot \frac{z}{t}}{a + 1}\\ \mathbf{elif}\;y \leq 6.6 \cdot 10^{+128}:\\ \;\;\;\;\frac{y \cdot z}{y \cdot b + t \cdot \left(a + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b} + \frac{t}{b} \cdot \frac{x}{y}\\ \end{array} \]

Alternative 5: 80.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -5.5 \cdot 10^{-145} \lor \neg \left(t \leq 2.56 \cdot 10^{-182}\right):\\ \;\;\;\;\frac{x + \frac{y}{\frac{t}{z}}}{a + \left(1 + b \cdot \frac{y}{t}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot z}{y \cdot b + t \cdot \left(a + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= t -5.5e-145) (not (<= t 2.56e-182)))
   (/ (+ x (/ y (/ t z))) (+ a (+ 1.0 (* b (/ y t)))))
   (/ (* y z) (+ (* y b) (* t (+ a 1.0))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((t <= -5.5e-145) || !(t <= 2.56e-182)) {
		tmp = (x + (y / (t / z))) / (a + (1.0 + (b * (y / t))));
	} else {
		tmp = (y * z) / ((y * b) + (t * (a + 1.0)));
	}
	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 <= (-5.5d-145)) .or. (.not. (t <= 2.56d-182))) then
        tmp = (x + (y / (t / z))) / (a + (1.0d0 + (b * (y / t))))
    else
        tmp = (y * z) / ((y * b) + (t * (a + 1.0d0)))
    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 <= -5.5e-145) || !(t <= 2.56e-182)) {
		tmp = (x + (y / (t / z))) / (a + (1.0 + (b * (y / t))));
	} else {
		tmp = (y * z) / ((y * b) + (t * (a + 1.0)));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (t <= -5.5e-145) or not (t <= 2.56e-182):
		tmp = (x + (y / (t / z))) / (a + (1.0 + (b * (y / t))))
	else:
		tmp = (y * z) / ((y * b) + (t * (a + 1.0)))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((t <= -5.5e-145) || !(t <= 2.56e-182))
		tmp = Float64(Float64(x + Float64(y / Float64(t / z))) / Float64(a + Float64(1.0 + Float64(b * Float64(y / t)))));
	else
		tmp = Float64(Float64(y * z) / Float64(Float64(y * b) + Float64(t * Float64(a + 1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((t <= -5.5e-145) || ~((t <= 2.56e-182)))
		tmp = (x + (y / (t / z))) / (a + (1.0 + (b * (y / t))));
	else
		tmp = (y * z) / ((y * b) + (t * (a + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[t, -5.5e-145], N[Not[LessEqual[t, 2.56e-182]], $MachinePrecision]], N[(N[(x + N[(y / N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a + N[(1.0 + N[(b * N[(y / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * z), $MachinePrecision] / N[(N[(y * b), $MachinePrecision] + N[(t * N[(a + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -5.50000000000000015e-145 or 2.5600000000000001e-182 < t

   1. Initial program 80.7%

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

      1. associate-/l* 85.4%

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

      2. associate-+l+ 85.4%

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

      3. associate-/l* 89.7%

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

   3. Simplified 89.7%

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

      1. associate-/r/ 58.6%

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

   5. Applied egg-rr 88.2%

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

   if -5.50000000000000015e-145 < t < 2.5600000000000001e-182

   1. Initial program 51.0%

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

      1. associate-/l* 42.4%

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

      2. associate-+l+ 42.4%

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

      3. associate-/l* 39.2%

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

   3. Simplified 39.2%

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

      1. clear-num 39.1%

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

      2. associate-+r+ 39.1%

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

      3. associate-/l* 42.4%

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

      4. associate-/l* 50.9%

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

      5. inv-pow 50.9%

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

      6. associate-+l+ 50.9%

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

      7. +-commutative 50.9%

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

      8. *-commutative 50.9%

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

      9. associate-*l/ 47.6%

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

      10. *-commutative 47.6%

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

      11. fma-def 47.6%

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

      12. +-commutative 47.6%

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

      13. associate-/l* 39.1%

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

      14. associate-/r/ 35.1%

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

      15. fma-def 35.1%

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

   5. Applied egg-rr 35.1%

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

      1. unpow-1 35.1%

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

      2. fma-udef 35.1%

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

      3. associate-*l/ 47.6%

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

      4. associate-*r/ 39.1%

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

      5. fma-udef 39.1%

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

   7. Simplified 39.1%

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

   8. Taylor expanded in z around inf 48.9%

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

      1. *-commutative 48.9%

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

      2. associate-/l* 48.5%

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

      3. associate-/r/ 40.2%

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

   10. Simplified 40.2%

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

   11. Taylor expanded in t around 0 73.7%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.5 \cdot 10^{-145} \lor \neg \left(t \leq 2.56 \cdot 10^{-182}\right):\\ \;\;\;\;\frac{x + \frac{y}{\frac{t}{z}}}{a + \left(1 + b \cdot \frac{y}{t}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot z}{y \cdot b + t \cdot \left(a + 1\right)}\\ \end{array} \]

Alternative 6: 82.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -6.5 \cdot 10^{-172} \lor \neg \left(t \leq 2.56 \cdot 10^{-182}\right):\\ \;\;\;\;\frac{x + \frac{y}{\frac{t}{z}}}{a + \left(1 + \frac{y}{\frac{t}{b}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b} + \frac{x \cdot t}{y \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= t -6.5e-172) (not (<= t 2.56e-182)))
   (/ (+ x (/ y (/ t z))) (+ a (+ 1.0 (/ y (/ t b)))))
   (+ (/ z b) (/ (* x t) (* y b)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((t <= -6.5e-172) || !(t <= 2.56e-182)) {
		tmp = (x + (y / (t / z))) / (a + (1.0 + (y / (t / b))));
	} else {
		tmp = (z / b) + ((x * t) / (y * 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 ((t <= (-6.5d-172)) .or. (.not. (t <= 2.56d-182))) then
        tmp = (x + (y / (t / z))) / (a + (1.0d0 + (y / (t / b))))
    else
        tmp = (z / b) + ((x * t) / (y * 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 ((t <= -6.5e-172) || !(t <= 2.56e-182)) {
		tmp = (x + (y / (t / z))) / (a + (1.0 + (y / (t / b))));
	} else {
		tmp = (z / b) + ((x * t) / (y * b));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (t <= -6.5e-172) or not (t <= 2.56e-182):
		tmp = (x + (y / (t / z))) / (a + (1.0 + (y / (t / b))))
	else:
		tmp = (z / b) + ((x * t) / (y * b))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((t <= -6.5e-172) || !(t <= 2.56e-182))
		tmp = Float64(Float64(x + Float64(y / Float64(t / z))) / Float64(a + Float64(1.0 + Float64(y / Float64(t / b)))));
	else
		tmp = Float64(Float64(z / b) + Float64(Float64(x * t) / Float64(y * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((t <= -6.5e-172) || ~((t <= 2.56e-182)))
		tmp = (x + (y / (t / z))) / (a + (1.0 + (y / (t / b))));
	else
		tmp = (z / b) + ((x * t) / (y * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[t, -6.5e-172], N[Not[LessEqual[t, 2.56e-182]], $MachinePrecision]], N[(N[(x + N[(y / N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a + N[(1.0 + N[(y / N[(t / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(z / b), $MachinePrecision] + N[(N[(x * t), $MachinePrecision] / N[(y * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -6.50000000000000012e-172 or 2.5600000000000001e-182 < t

   1. Initial program 80.2%

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

      1. associate-/l* 84.2%

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

      2. associate-+l+ 84.2%

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

      3. associate-/l* 88.4%

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

   3. Simplified 88.4%

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

   if -6.50000000000000012e-172 < t < 2.5600000000000001e-182

   1. Initial program 46.8%

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

      1. associate-/l* 38.5%

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

      2. associate-+l+ 38.5%

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

      3. associate-/l* 34.5%

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

   3. Simplified 34.5%

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

      1. clear-num 34.5%

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

      2. associate-+r+ 34.5%

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

      3. associate-/l* 38.5%

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

      4. associate-/l* 46.7%

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

      5. inv-pow 46.7%

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

      6. associate-+l+ 46.7%

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

      7. +-commutative 46.7%

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

      8. *-commutative 46.7%

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

      9. associate-*l/ 42.7%

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

      10. *-commutative 42.7%

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

      11. fma-def 42.7%

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

      12. +-commutative 42.7%

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

      13. associate-/l* 34.5%

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

      14. associate-/r/ 29.6%

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

      15. fma-def 29.6%

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

   5. Applied egg-rr 29.6%

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

      1. unpow-1 29.6%

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

      2. fma-udef 29.6%

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

      3. associate-*l/ 42.7%

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

      4. associate-*r/ 34.5%

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

      5. fma-udef 34.5%

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

   7. Simplified 34.5%

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

   8. Taylor expanded in b around inf 45.9%

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

      1. times-frac 32.2%

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

      2. associate-/l* 32.2%

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

   10. Simplified 32.2%

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

   11. Taylor expanded in t around 0 77.8%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -6.5 \cdot 10^{-172} \lor \neg \left(t \leq 2.56 \cdot 10^{-182}\right):\\ \;\;\;\;\frac{x + \frac{y}{\frac{t}{z}}}{a + \left(1 + \frac{y}{\frac{t}{b}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b} + \frac{x \cdot t}{y \cdot b}\\ \end{array} \]

Alternative 7: 82.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.3 \cdot 10^{-172}:\\ \;\;\;\;\frac{x + \frac{y}{\frac{t}{z}}}{a + \left(1 + \frac{y}{\frac{t}{b}}\right)}\\ \mathbf{elif}\;t \leq 2.56 \cdot 10^{-182}:\\ \;\;\;\;\frac{z}{b} + \frac{x \cdot t}{y \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{z}{\frac{t}{y}}}{1 + \left(a + y \cdot \frac{b}{t}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= t -2.3e-172)
   (/ (+ x (/ y (/ t z))) (+ a (+ 1.0 (/ y (/ t b)))))
   (if (<= t 2.56e-182)
     (+ (/ z b) (/ (* x t) (* y b)))
     (/ (+ x (/ z (/ t y))) (+ 1.0 (+ a (* y (/ b t))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= -2.3e-172) {
		tmp = (x + (y / (t / z))) / (a + (1.0 + (y / (t / b))));
	} else if (t <= 2.56e-182) {
		tmp = (z / b) + ((x * t) / (y * b));
	} else {
		tmp = (x + (z / (t / y))) / (1.0 + (a + (y * (b / t))));
	}
	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 <= (-2.3d-172)) then
        tmp = (x + (y / (t / z))) / (a + (1.0d0 + (y / (t / b))))
    else if (t <= 2.56d-182) then
        tmp = (z / b) + ((x * t) / (y * b))
    else
        tmp = (x + (z / (t / y))) / (1.0d0 + (a + (y * (b / t))))
    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 <= -2.3e-172) {
		tmp = (x + (y / (t / z))) / (a + (1.0 + (y / (t / b))));
	} else if (t <= 2.56e-182) {
		tmp = (z / b) + ((x * t) / (y * b));
	} else {
		tmp = (x + (z / (t / y))) / (1.0 + (a + (y * (b / t))));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if t <= -2.3e-172:
		tmp = (x + (y / (t / z))) / (a + (1.0 + (y / (t / b))))
	elif t <= 2.56e-182:
		tmp = (z / b) + ((x * t) / (y * b))
	else:
		tmp = (x + (z / (t / y))) / (1.0 + (a + (y * (b / t))))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (t <= -2.3e-172)
		tmp = Float64(Float64(x + Float64(y / Float64(t / z))) / Float64(a + Float64(1.0 + Float64(y / Float64(t / b)))));
	elseif (t <= 2.56e-182)
		tmp = Float64(Float64(z / b) + Float64(Float64(x * t) / Float64(y * b)));
	else
		tmp = Float64(Float64(x + Float64(z / Float64(t / y))) / Float64(1.0 + Float64(a + Float64(y * Float64(b / t)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (t <= -2.3e-172)
		tmp = (x + (y / (t / z))) / (a + (1.0 + (y / (t / b))));
	elseif (t <= 2.56e-182)
		tmp = (z / b) + ((x * t) / (y * b));
	else
		tmp = (x + (z / (t / y))) / (1.0 + (a + (y * (b / t))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[t, -2.3e-172], N[(N[(x + N[(y / N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a + N[(1.0 + N[(y / N[(t / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.56e-182], N[(N[(z / b), $MachinePrecision] + N[(N[(x * t), $MachinePrecision] / N[(y * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x + N[(z / N[(t / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(a + N[(y * N[(b / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -2.29999999999999995e-172

   1. Initial program 85.4%

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

      1. associate-/l* 84.6%

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

      2. associate-+l+ 84.6%

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

      3. associate-/l* 88.2%

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

   3. Simplified 88.2%

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

   if -2.29999999999999995e-172 < t < 2.5600000000000001e-182

   1. Initial program 46.8%

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

      1. associate-/l* 38.5%

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

      2. associate-+l+ 38.5%

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

      3. associate-/l* 34.5%

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

   3. Simplified 34.5%

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

      1. clear-num 34.5%

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

      2. associate-+r+ 34.5%

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

      3. associate-/l* 38.5%

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

      4. associate-/l* 46.7%

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

      5. inv-pow 46.7%

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

      6. associate-+l+ 46.7%

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

      7. +-commutative 46.7%

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

      8. *-commutative 46.7%

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

      9. associate-*l/ 42.7%

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

      10. *-commutative 42.7%

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

      11. fma-def 42.7%

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

      12. +-commutative 42.7%

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

      13. associate-/l* 34.5%

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

      14. associate-/r/ 29.6%

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

      15. fma-def 29.6%

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

   5. Applied egg-rr 29.6%

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

      1. unpow-1 29.6%

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

      2. fma-udef 29.6%

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

      3. associate-*l/ 42.7%

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

      4. associate-*r/ 34.5%

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

      5. fma-udef 34.5%

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

   7. Simplified 34.5%

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

   8. Taylor expanded in b around inf 45.9%

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

      1. times-frac 32.2%

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

      2. associate-/l* 32.2%

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

   10. Simplified 32.2%

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

   11. Taylor expanded in t around 0 77.8%

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

   if 2.5600000000000001e-182 < t

   1. Initial program 74.7%

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

      1. *-commutative 74.7%

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

      2. associate-/l* 83.7%

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

      3. +-commutative 83.7%

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

      4. associate-+l+ 83.7%

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

      5. associate-*r/ 89.4%

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

      6. *-commutative 89.4%

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

   3. Simplified 89.4%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.3 \cdot 10^{-172}:\\ \;\;\;\;\frac{x + \frac{y}{\frac{t}{z}}}{a + \left(1 + \frac{y}{\frac{t}{b}}\right)}\\ \mathbf{elif}\;t \leq 2.56 \cdot 10^{-182}:\\ \;\;\;\;\frac{z}{b} + \frac{x \cdot t}{y \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{z}{\frac{t}{y}}}{1 + \left(a + y \cdot \frac{b}{t}\right)}\\ \end{array} \]

Alternative 8: 59.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z}{b} + \frac{t}{b} \cdot \frac{x}{y}\\ \mathbf{if}\;y \leq -46000000000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.14 \cdot 10^{-73}:\\ \;\;\;\;\frac{x}{a + 1}\\ \mathbf{elif}\;y \leq 3 \cdot 10^{+15}:\\ \;\;\;\;\frac{z}{b} + \frac{x \cdot t}{y \cdot b}\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{+60}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (/ z b) (* (/ t b) (/ x y)))))
   (if (<= y -46000000000000.0)
     t_1
     (if (<= y 1.14e-73)
       (/ x (+ a 1.0))
       (if (<= y 3e+15)
         (+ (/ z b) (/ (* x t) (* y b)))
         (if (<= y 1.5e+60) (/ (+ x (/ (* y z) t)) a) t_1))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (z / b) + ((t / b) * (x / y));
	double tmp;
	if (y <= -46000000000000.0) {
		tmp = t_1;
	} else if (y <= 1.14e-73) {
		tmp = x / (a + 1.0);
	} else if (y <= 3e+15) {
		tmp = (z / b) + ((x * t) / (y * b));
	} else if (y <= 1.5e+60) {
		tmp = (x + ((y * z) / t)) / a;
	} 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 = (z / b) + ((t / b) * (x / y))
    if (y <= (-46000000000000.0d0)) then
        tmp = t_1
    else if (y <= 1.14d-73) then
        tmp = x / (a + 1.0d0)
    else if (y <= 3d+15) then
        tmp = (z / b) + ((x * t) / (y * b))
    else if (y <= 1.5d+60) then
        tmp = (x + ((y * z) / t)) / a
    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 = (z / b) + ((t / b) * (x / y));
	double tmp;
	if (y <= -46000000000000.0) {
		tmp = t_1;
	} else if (y <= 1.14e-73) {
		tmp = x / (a + 1.0);
	} else if (y <= 3e+15) {
		tmp = (z / b) + ((x * t) / (y * b));
	} else if (y <= 1.5e+60) {
		tmp = (x + ((y * z) / t)) / a;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (z / b) + ((t / b) * (x / y))
	tmp = 0
	if y <= -46000000000000.0:
		tmp = t_1
	elif y <= 1.14e-73:
		tmp = x / (a + 1.0)
	elif y <= 3e+15:
		tmp = (z / b) + ((x * t) / (y * b))
	elif y <= 1.5e+60:
		tmp = (x + ((y * z) / t)) / a
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(z / b) + Float64(Float64(t / b) * Float64(x / y)))
	tmp = 0.0
	if (y <= -46000000000000.0)
		tmp = t_1;
	elseif (y <= 1.14e-73)
		tmp = Float64(x / Float64(a + 1.0));
	elseif (y <= 3e+15)
		tmp = Float64(Float64(z / b) + Float64(Float64(x * t) / Float64(y * b)));
	elseif (y <= 1.5e+60)
		tmp = Float64(Float64(x + Float64(Float64(y * z) / t)) / a);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (z / b) + ((t / b) * (x / y));
	tmp = 0.0;
	if (y <= -46000000000000.0)
		tmp = t_1;
	elseif (y <= 1.14e-73)
		tmp = x / (a + 1.0);
	elseif (y <= 3e+15)
		tmp = (z / b) + ((x * t) / (y * b));
	elseif (y <= 1.5e+60)
		tmp = (x + ((y * z) / t)) / a;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(z / b), $MachinePrecision] + N[(N[(t / b), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -46000000000000.0], t$95$1, If[LessEqual[y, 1.14e-73], N[(x / N[(a + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3e+15], N[(N[(z / b), $MachinePrecision] + N[(N[(x * t), $MachinePrecision] / N[(y * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.5e+60], N[(N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -4.6e13 or 1.4999999999999999e60 < y

   1. Initial program 55.2%

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

      1. associate-/l* 63.5%

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

      2. associate-+l+ 63.5%

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

      3. associate-/l* 71.6%

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

   3. Simplified 71.6%

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

      1. clear-num 71.5%

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

      2. associate-+r+ 71.5%

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

      3. associate-/l* 63.4%

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

      4. associate-/l* 55.2%

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

      5. inv-pow 55.2%

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

      6. associate-+l+ 55.2%

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

      7. +-commutative 55.2%

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

      8. *-commutative 55.2%

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

      9. associate-*l/ 61.6%

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

      10. *-commutative 61.6%

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

      11. fma-def 61.6%

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

      12. +-commutative 61.6%

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

      13. associate-/l* 71.5%

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

      14. associate-/r/ 63.7%

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

      15. fma-def 63.7%

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

   5. Applied egg-rr 63.7%

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

      1. unpow-1 63.7%

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

      2. fma-udef 63.7%

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

      3. associate-*l/ 61.6%

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

      4. associate-*r/ 72.1%

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

      5. fma-udef 72.2%

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

   7. Simplified 72.2%

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

   8. Taylor expanded in b around inf 31.2%

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

      1. times-frac 38.9%

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

      2. associate-/l* 44.8%

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

   10. Simplified 44.8%

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

   11. Taylor expanded in t around 0 57.2%

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

       1. +-commutative 57.2%

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

       2. times-frac 63.9%

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

   13. Simplified 63.9%

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

   if -4.6e13 < y < 1.14000000000000005e-73

   1. Initial program 93.4%

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

      1. associate-/l* 88.7%

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

      2. associate-+l+ 88.7%

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

      3. associate-/l* 87.1%

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

   3. Simplified 87.1%

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

   4. Taylor expanded in y around 0 64.8%

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

   if 1.14000000000000005e-73 < y < 3e15

   1. Initial program 72.2%

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

      1. associate-/l* 72.2%

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

      2. associate-+l+ 72.2%

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

      3. associate-/l* 66.6%

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

   3. Simplified 66.6%

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

      1. clear-num 66.5%

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

      2. associate-+r+ 66.5%

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

      3. associate-/l* 72.2%

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

      4. associate-/l* 72.1%

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

      5. inv-pow 72.1%

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

      6. associate-+l+ 72.1%

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

      7. +-commutative 72.1%

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

      8. *-commutative 72.1%

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

      9. associate-*l/ 66.6%

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

      10. *-commutative 66.6%

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

      11. fma-def 66.6%

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

      12. +-commutative 66.6%

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

      13. associate-/l* 66.5%

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

      14. associate-/r/ 66.6%

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

      15. fma-def 66.6%

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

   5. Applied egg-rr 66.6%

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

      1. unpow-1 66.6%

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

      2. fma-udef 66.6%

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

      3. associate-*l/ 66.6%

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

      4. associate-*r/ 66.5%

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

      5. fma-udef 66.5%

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

   7. Simplified 66.5%

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

   8. Taylor expanded in b around inf 54.0%

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

      1. times-frac 37.3%

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

      2. associate-/l* 37.3%

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

   10. Simplified 37.3%

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

   11. Taylor expanded in t around 0 62.1%

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

   if 3e15 < y < 1.4999999999999999e60

   1. Initial program 86.3%

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

      1. associate-/l* 86.1%

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

      2. associate-+l+ 86.1%

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

      3. associate-/l* 86.1%

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

   3. Simplified 86.1%

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

   4. Taylor expanded in a around inf 72.0%

      \[\leadsto \color{blue}{\frac{x + \frac{y \cdot z}{t}}{a}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -46000000000000:\\ \;\;\;\;\frac{z}{b} + \frac{t}{b} \cdot \frac{x}{y}\\ \mathbf{elif}\;y \leq 1.14 \cdot 10^{-73}:\\ \;\;\;\;\frac{x}{a + 1}\\ \mathbf{elif}\;y \leq 3 \cdot 10^{+15}:\\ \;\;\;\;\frac{z}{b} + \frac{x \cdot t}{y \cdot b}\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{+60}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b} + \frac{t}{b} \cdot \frac{x}{y}\\ \end{array} \]

Alternative 9: 65.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + y \cdot \frac{z}{t}}{a + 1}\\ t_2 := \frac{z}{b} + \frac{t}{b} \cdot \frac{x}{y}\\ \mathbf{if}\;y \leq -7.2 \cdot 10^{+31}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{-73}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{+18}:\\ \;\;\;\;\frac{z}{b} + \frac{x \cdot t}{y \cdot b}\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{+75}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (+ x (* y (/ z t))) (+ a 1.0)))
        (t_2 (+ (/ z b) (* (/ t b) (/ x y)))))
   (if (<= y -7.2e+31)
     t_2
     (if (<= y 1.8e-73)
       t_1
       (if (<= y 1.1e+18)
         (+ (/ z b) (/ (* x t) (* y b)))
         (if (<= y 1.7e+75) t_1 t_2))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x + (y * (z / t))) / (a + 1.0);
	double t_2 = (z / b) + ((t / b) * (x / y));
	double tmp;
	if (y <= -7.2e+31) {
		tmp = t_2;
	} else if (y <= 1.8e-73) {
		tmp = t_1;
	} else if (y <= 1.1e+18) {
		tmp = (z / b) + ((x * t) / (y * b));
	} else if (y <= 1.7e+75) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	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) :: t_2
    real(8) :: tmp
    t_1 = (x + (y * (z / t))) / (a + 1.0d0)
    t_2 = (z / b) + ((t / b) * (x / y))
    if (y <= (-7.2d+31)) then
        tmp = t_2
    else if (y <= 1.8d-73) then
        tmp = t_1
    else if (y <= 1.1d+18) then
        tmp = (z / b) + ((x * t) / (y * b))
    else if (y <= 1.7d+75) then
        tmp = t_1
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x + (y * (z / t))) / (a + 1.0);
	double t_2 = (z / b) + ((t / b) * (x / y));
	double tmp;
	if (y <= -7.2e+31) {
		tmp = t_2;
	} else if (y <= 1.8e-73) {
		tmp = t_1;
	} else if (y <= 1.1e+18) {
		tmp = (z / b) + ((x * t) / (y * b));
	} else if (y <= 1.7e+75) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (x + (y * (z / t))) / (a + 1.0)
	t_2 = (z / b) + ((t / b) * (x / y))
	tmp = 0
	if y <= -7.2e+31:
		tmp = t_2
	elif y <= 1.8e-73:
		tmp = t_1
	elif y <= 1.1e+18:
		tmp = (z / b) + ((x * t) / (y * b))
	elif y <= 1.7e+75:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x + Float64(y * Float64(z / t))) / Float64(a + 1.0))
	t_2 = Float64(Float64(z / b) + Float64(Float64(t / b) * Float64(x / y)))
	tmp = 0.0
	if (y <= -7.2e+31)
		tmp = t_2;
	elseif (y <= 1.8e-73)
		tmp = t_1;
	elseif (y <= 1.1e+18)
		tmp = Float64(Float64(z / b) + Float64(Float64(x * t) / Float64(y * b)));
	elseif (y <= 1.7e+75)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (x + (y * (z / t))) / (a + 1.0);
	t_2 = (z / b) + ((t / b) * (x / y));
	tmp = 0.0;
	if (y <= -7.2e+31)
		tmp = t_2;
	elseif (y <= 1.8e-73)
		tmp = t_1;
	elseif (y <= 1.1e+18)
		tmp = (z / b) + ((x * t) / (y * b));
	elseif (y <= 1.7e+75)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x + N[(y * N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(z / b), $MachinePrecision] + N[(N[(t / b), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -7.2e+31], t$95$2, If[LessEqual[y, 1.8e-73], t$95$1, If[LessEqual[y, 1.1e+18], N[(N[(z / b), $MachinePrecision] + N[(N[(x * t), $MachinePrecision] / N[(y * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.7e+75], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -7.19999999999999992e31 or 1.70000000000000006e75 < y

   1. Initial program 54.1%

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

      1. associate-/l* 62.2%

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

      2. associate-+l+ 62.2%

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

      3. associate-/l* 70.9%

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

   3. Simplified 70.9%

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

      1. clear-num 70.8%

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

      2. associate-+r+ 70.8%

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

      3. associate-/l* 62.2%

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

      4. associate-/l* 54.1%

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

      5. inv-pow 54.1%

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

      6. associate-+l+ 54.1%

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

      7. +-commutative 54.1%

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

      8. *-commutative 54.1%

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

      9. associate-*l/ 61.0%

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

      10. *-commutative 61.0%

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

      11. fma-def 61.0%

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

      12. +-commutative 61.0%

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

      13. associate-/l* 70.8%

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

      14. associate-/r/ 62.5%

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

      15. fma-def 62.5%

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

   5. Applied egg-rr 62.5%

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

      1. unpow-1 62.5%

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

      2. fma-udef 62.5%

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

      3. associate-*l/ 61.0%

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

      4. associate-*r/ 71.6%

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

      5. fma-udef 71.6%

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

   7. Simplified 71.6%

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

   8. Taylor expanded in b around inf 31.7%

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

      1. times-frac 40.1%

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

      2. associate-/l* 46.4%

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

   10. Simplified 46.4%

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

   11. Taylor expanded in t around 0 58.9%

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

       1. +-commutative 58.9%

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

       2. times-frac 66.2%

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

   13. Simplified 66.2%

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

   if -7.19999999999999992e31 < y < 1.8e-73 or 1.1e18 < y < 1.70000000000000006e75

   1. Initial program 91.3%

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

      1. associate-/l* 87.9%

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

      2. associate-+l+ 87.9%

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

      3. associate-/l* 86.4%

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

   3. Simplified 86.4%

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

      1. clear-num 85.9%

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

      2. associate-+r+ 85.9%

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

      3. associate-/l* 87.3%

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

      4. associate-/l* 90.7%

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

      5. inv-pow 90.7%

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

      6. associate-+l+ 90.7%

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

      7. +-commutative 90.7%

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

      8. *-commutative 90.7%

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

      9. associate-*l/ 89.3%

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

      10. *-commutative 89.3%

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

      11. fma-def 89.3%

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

      12. +-commutative 89.3%

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

      13. associate-/l* 86.0%

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

      14. associate-/r/ 90.4%

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

      15. fma-def 90.4%

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

   5. Applied egg-rr 90.4%

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

      1. unpow-1 90.4%

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

      2. fma-udef 90.4%

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

      3. associate-*l/ 89.3%

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

      4. associate-*r/ 85.9%

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

      5. fma-udef 85.9%

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

   7. Simplified 85.9%

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

   8. Taylor expanded in b around 0 77.9%

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

      1. associate-/l* 75.2%

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

   10. Simplified 75.2%

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

   11. Taylor expanded in y around 0 77.9%

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

       1. associate-*r/ 75.2%

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

   13. Simplified 75.2%

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

   if 1.8e-73 < y < 1.1e18

   1. Initial program 72.2%

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

      1. associate-/l* 72.2%

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

      2. associate-+l+ 72.2%

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

      3. associate-/l* 66.6%

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

   3. Simplified 66.6%

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

      1. clear-num 66.5%

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

      2. associate-+r+ 66.5%

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

      3. associate-/l* 72.2%

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

      4. associate-/l* 72.1%

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

      5. inv-pow 72.1%

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

      6. associate-+l+ 72.1%

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

      7. +-commutative 72.1%

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

      8. *-commutative 72.1%

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

      9. associate-*l/ 66.6%

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

      10. *-commutative 66.6%

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

      11. fma-def 66.6%

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

      12. +-commutative 66.6%

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

      13. associate-/l* 66.5%

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

      14. associate-/r/ 66.6%

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

      15. fma-def 66.6%

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

   5. Applied egg-rr 66.6%

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

      1. unpow-1 66.6%

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

      2. fma-udef 66.6%

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

      3. associate-*l/ 66.6%

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

      4. associate-*r/ 66.5%

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

      5. fma-udef 66.5%

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

   7. Simplified 66.5%

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

   8. Taylor expanded in b around inf 54.0%

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

      1. times-frac 37.3%

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

      2. associate-/l* 37.3%

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

   10. Simplified 37.3%

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

   11. Taylor expanded in t around 0 62.1%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{+31}:\\ \;\;\;\;\frac{z}{b} + \frac{t}{b} \cdot \frac{x}{y}\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{-73}:\\ \;\;\;\;\frac{x + y \cdot \frac{z}{t}}{a + 1}\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{+18}:\\ \;\;\;\;\frac{z}{b} + \frac{x \cdot t}{y \cdot b}\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{+75}:\\ \;\;\;\;\frac{x + y \cdot \frac{z}{t}}{a + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b} + \frac{t}{b} \cdot \frac{x}{y}\\ \end{array} \]

Alternative 10: 65.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -7.2 \cdot 10^{+73}:\\ \;\;\;\;\frac{x + y \cdot \frac{z}{t}}{a + 1}\\ \mathbf{elif}\;t \leq -2.3 \cdot 10^{-13}:\\ \;\;\;\;\frac{z}{b} + \frac{t}{b} \cdot \frac{x}{y}\\ \mathbf{elif}\;t \leq -7.5 \cdot 10^{-172}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{a + 1}\\ \mathbf{elif}\;t \leq 2.56 \cdot 10^{-182}:\\ \;\;\;\;\frac{z}{b} + \frac{x \cdot t}{y \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{\frac{t}{z}}}{a + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= t -7.2e+73)
   (/ (+ x (* y (/ z t))) (+ a 1.0))
   (if (<= t -2.3e-13)
     (+ (/ z b) (* (/ t b) (/ x y)))
     (if (<= t -7.5e-172)
       (/ (+ x (/ (* y z) t)) (+ a 1.0))
       (if (<= t 2.56e-182)
         (+ (/ z b) (/ (* x t) (* y b)))
         (/ (+ x (/ y (/ t z))) (+ a 1.0)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= -7.2e+73) {
		tmp = (x + (y * (z / t))) / (a + 1.0);
	} else if (t <= -2.3e-13) {
		tmp = (z / b) + ((t / b) * (x / y));
	} else if (t <= -7.5e-172) {
		tmp = (x + ((y * z) / t)) / (a + 1.0);
	} else if (t <= 2.56e-182) {
		tmp = (z / b) + ((x * t) / (y * b));
	} else {
		tmp = (x + (y / (t / z))) / (a + 1.0);
	}
	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 <= (-7.2d+73)) then
        tmp = (x + (y * (z / t))) / (a + 1.0d0)
    else if (t <= (-2.3d-13)) then
        tmp = (z / b) + ((t / b) * (x / y))
    else if (t <= (-7.5d-172)) then
        tmp = (x + ((y * z) / t)) / (a + 1.0d0)
    else if (t <= 2.56d-182) then
        tmp = (z / b) + ((x * t) / (y * b))
    else
        tmp = (x + (y / (t / z))) / (a + 1.0d0)
    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 <= -7.2e+73) {
		tmp = (x + (y * (z / t))) / (a + 1.0);
	} else if (t <= -2.3e-13) {
		tmp = (z / b) + ((t / b) * (x / y));
	} else if (t <= -7.5e-172) {
		tmp = (x + ((y * z) / t)) / (a + 1.0);
	} else if (t <= 2.56e-182) {
		tmp = (z / b) + ((x * t) / (y * b));
	} else {
		tmp = (x + (y / (t / z))) / (a + 1.0);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if t <= -7.2e+73:
		tmp = (x + (y * (z / t))) / (a + 1.0)
	elif t <= -2.3e-13:
		tmp = (z / b) + ((t / b) * (x / y))
	elif t <= -7.5e-172:
		tmp = (x + ((y * z) / t)) / (a + 1.0)
	elif t <= 2.56e-182:
		tmp = (z / b) + ((x * t) / (y * b))
	else:
		tmp = (x + (y / (t / z))) / (a + 1.0)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (t <= -7.2e+73)
		tmp = Float64(Float64(x + Float64(y * Float64(z / t))) / Float64(a + 1.0));
	elseif (t <= -2.3e-13)
		tmp = Float64(Float64(z / b) + Float64(Float64(t / b) * Float64(x / y)));
	elseif (t <= -7.5e-172)
		tmp = Float64(Float64(x + Float64(Float64(y * z) / t)) / Float64(a + 1.0));
	elseif (t <= 2.56e-182)
		tmp = Float64(Float64(z / b) + Float64(Float64(x * t) / Float64(y * b)));
	else
		tmp = Float64(Float64(x + Float64(y / Float64(t / z))) / Float64(a + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (t <= -7.2e+73)
		tmp = (x + (y * (z / t))) / (a + 1.0);
	elseif (t <= -2.3e-13)
		tmp = (z / b) + ((t / b) * (x / y));
	elseif (t <= -7.5e-172)
		tmp = (x + ((y * z) / t)) / (a + 1.0);
	elseif (t <= 2.56e-182)
		tmp = (z / b) + ((x * t) / (y * b));
	else
		tmp = (x + (y / (t / z))) / (a + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[t, -7.2e+73], N[(N[(x + N[(y * N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -2.3e-13], N[(N[(z / b), $MachinePrecision] + N[(N[(t / b), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -7.5e-172], N[(N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[(a + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.56e-182], N[(N[(z / b), $MachinePrecision] + N[(N[(x * t), $MachinePrecision] / N[(y * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x + N[(y / N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a + 1.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if t < -7.1999999999999998e73

   1. Initial program 86.7%

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

      1. associate-/l* 90.6%

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

      2. associate-+l+ 90.6%

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

      3. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

      1. clear-num 98.9%

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

      2. associate-+r+ 98.9%

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

      3. associate-/l* 89.6%

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

      4. associate-/l* 85.7%

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

      5. inv-pow 85.7%

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

      6. associate-+l+ 85.7%

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

      7. +-commutative 85.7%

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

      8. *-commutative 85.7%

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

      9. associate-*l/ 91.3%

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

      10. *-commutative 91.3%

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

      11. fma-def 91.3%

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

      12. +-commutative 91.3%

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

      13. associate-/l* 98.9%

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

      14. associate-/r/ 98.8%

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

      15. fma-def 98.8%

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

   5. Applied egg-rr 98.8%

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

      1. unpow-1 98.8%

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

      2. fma-udef 98.8%

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

      3. associate-*l/ 91.3%

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

      4. associate-*r/ 98.9%

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

      5. fma-udef 99.0%

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

   7. Simplified 99.0%

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

   8. Taylor expanded in b around 0 82.6%

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

      1. associate-/l* 84.5%

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

   10. Simplified 84.5%

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

   11. Taylor expanded in y around 0 82.6%

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

       1. associate-*r/ 84.5%

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

   13. Simplified 84.5%

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

   if -7.1999999999999998e73 < t < -2.29999999999999979e-13

   1. Initial program 74.8%

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

      1. associate-/l* 69.5%

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

      2. associate-+l+ 69.5%

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

      3. associate-/l* 69.7%

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

   3. Simplified 69.7%

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

      1. clear-num 69.7%

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

      2. associate-+r+ 69.7%

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

      3. associate-/l* 69.4%

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

      4. associate-/l* 74.9%

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

      5. inv-pow 74.9%

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

      6. associate-+l+ 74.9%

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

      7. +-commutative 74.9%

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

      8. *-commutative 74.9%

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

      9. associate-*l/ 74.8%

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

      10. *-commutative 74.8%

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

      11. fma-def 74.8%

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

      12. +-commutative 74.8%

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

      13. associate-/l* 69.5%

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

      14. associate-/r/ 75.1%

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

      15. fma-def 75.1%

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

   5. Applied egg-rr 75.1%

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

      1. unpow-1 75.1%

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

      2. fma-udef 75.1%

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

      3. associate-*l/ 74.8%

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

      4. associate-*r/ 69.5%

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

      5. fma-udef 69.5%

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

   7. Simplified 69.5%

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

   8. Taylor expanded in b around inf 47.7%

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

      1. times-frac 49.2%

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

      2. associate-/l* 55.2%

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

   10. Simplified 55.2%

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

   11. Taylor expanded in t around 0 71.4%

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

       1. +-commutative 71.4%

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

       2. times-frac 72.6%

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

   13. Simplified 72.6%

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

   if -2.29999999999999979e-13 < t < -7.4999999999999999e-172

   1. Initial program 87.8%

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

      1. associate-/l* 83.1%

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

      2. associate-+l+ 83.1%

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

      3. associate-/l* 80.8%

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

   3. Simplified 80.8%

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

   4. Taylor expanded in b around 0 60.9%

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

   if -7.4999999999999999e-172 < t < 2.5600000000000001e-182

   1. Initial program 46.8%

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

      1. associate-/l* 38.5%

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

      2. associate-+l+ 38.5%

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

      3. associate-/l* 34.5%

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

   3. Simplified 34.5%

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

      1. clear-num 34.5%

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

      2. associate-+r+ 34.5%

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

      3. associate-/l* 38.5%

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

      4. associate-/l* 46.7%

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

      5. inv-pow 46.7%

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

      6. associate-+l+ 46.7%

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

      7. +-commutative 46.7%

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

      8. *-commutative 46.7%

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

      9. associate-*l/ 42.7%

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

      10. *-commutative 42.7%

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

      11. fma-def 42.7%

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

      12. +-commutative 42.7%

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

      13. associate-/l* 34.5%

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

      14. associate-/r/ 29.6%

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

      15. fma-def 29.6%

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

   5. Applied egg-rr 29.6%

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

      1. unpow-1 29.6%

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

      2. fma-udef 29.6%

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

      3. associate-*l/ 42.7%

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

      4. associate-*r/ 34.5%

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

      5. fma-udef 34.5%

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

   7. Simplified 34.5%

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

   8. Taylor expanded in b around inf 45.9%

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

      1. times-frac 32.2%

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

      2. associate-/l* 32.2%

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

   10. Simplified 32.2%

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

   11. Taylor expanded in t around 0 77.8%

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

   if 2.5600000000000001e-182 < t

   1. Initial program 74.7%

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

      1. associate-/l* 83.7%

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

      2. associate-+l+ 83.7%

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

      3. associate-/l* 88.5%

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

   3. Simplified 88.5%

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

      1. clear-num 88.3%

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

      2. associate-+r+ 88.3%

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

      3. associate-/l* 83.5%

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

      4. associate-/l* 74.5%

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

      5. inv-pow 74.5%

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

      6. associate-+l+ 74.5%

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

      7. +-commutative 74.5%

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

      8. *-commutative 74.5%

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

      9. associate-*l/ 79.3%

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

      10. *-commutative 79.3%

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

      11. fma-def 79.3%

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

      12. +-commutative 79.3%

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

      13. associate-/l* 88.4%

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

      14. associate-/r/ 88.3%

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

      15. fma-def 88.3%

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

   5. Applied egg-rr 88.3%

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

      1. unpow-1 88.3%

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

      2. fma-udef 88.3%

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

      3. associate-*l/ 79.3%

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

      4. associate-*r/ 89.2%

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

      5. fma-udef 89.2%

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

   7. Simplified 89.2%

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

   8. Taylor expanded in b around 0 57.4%

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

      1. associate-/l* 65.3%

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

   10. Simplified 65.3%

       \[\leadsto \color{blue}{\frac{x + \frac{y}{\frac{t}{z}}}{1 + a}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -7.2 \cdot 10^{+73}:\\ \;\;\;\;\frac{x + y \cdot \frac{z}{t}}{a + 1}\\ \mathbf{elif}\;t \leq -2.3 \cdot 10^{-13}:\\ \;\;\;\;\frac{z}{b} + \frac{t}{b} \cdot \frac{x}{y}\\ \mathbf{elif}\;t \leq -7.5 \cdot 10^{-172}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{a + 1}\\ \mathbf{elif}\;t \leq 2.56 \cdot 10^{-182}:\\ \;\;\;\;\frac{z}{b} + \frac{x \cdot t}{y \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{\frac{t}{z}}}{a + 1}\\ \end{array} \]

Alternative 11: 60.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -49000000000000 \lor \neg \left(y \leq 2.7 \cdot 10^{+62}\right):\\ \;\;\;\;\frac{z}{b} + \frac{t}{b} \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -49000000000000.0) (not (<= y 2.7e+62)))
   (+ (/ z b) (* (/ t b) (/ x y)))
   (/ x (+ a 1.0))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -49000000000000.0) || !(y <= 2.7e+62)) {
		tmp = (z / b) + ((t / b) * (x / y));
	} else {
		tmp = x / (a + 1.0);
	}
	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 <= (-49000000000000.0d0)) .or. (.not. (y <= 2.7d+62))) then
        tmp = (z / b) + ((t / b) * (x / y))
    else
        tmp = x / (a + 1.0d0)
    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 <= -49000000000000.0) || !(y <= 2.7e+62)) {
		tmp = (z / b) + ((t / b) * (x / y));
	} else {
		tmp = x / (a + 1.0);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -49000000000000.0) or not (y <= 2.7e+62):
		tmp = (z / b) + ((t / b) * (x / y))
	else:
		tmp = x / (a + 1.0)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -49000000000000.0) || !(y <= 2.7e+62))
		tmp = Float64(Float64(z / b) + Float64(Float64(t / b) * Float64(x / y)));
	else
		tmp = Float64(x / Float64(a + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -49000000000000.0) || ~((y <= 2.7e+62)))
		tmp = (z / b) + ((t / b) * (x / y));
	else
		tmp = x / (a + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -49000000000000.0], N[Not[LessEqual[y, 2.7e+62]], $MachinePrecision]], N[(N[(z / b), $MachinePrecision] + N[(N[(t / b), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(a + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.9e13 or 2.7e62 < y

   1. Initial program 55.2%

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

      1. associate-/l* 63.5%

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

      2. associate-+l+ 63.5%

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

      3. associate-/l* 71.6%

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

   3. Simplified 71.6%

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

      1. clear-num 71.5%

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

      2. associate-+r+ 71.5%

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

      3. associate-/l* 63.4%

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

      4. associate-/l* 55.2%

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

      5. inv-pow 55.2%

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

      6. associate-+l+ 55.2%

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

      7. +-commutative 55.2%

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

      8. *-commutative 55.2%

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

      9. associate-*l/ 61.6%

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

      10. *-commutative 61.6%

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

      11. fma-def 61.6%

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

      12. +-commutative 61.6%

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

      13. associate-/l* 71.5%

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

      14. associate-/r/ 63.7%

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

      15. fma-def 63.7%

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

   5. Applied egg-rr 63.7%

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

      1. unpow-1 63.7%

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

      2. fma-udef 63.7%

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

      3. associate-*l/ 61.6%

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

      4. associate-*r/ 72.1%

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

      5. fma-udef 72.2%

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

   7. Simplified 72.2%

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

   8. Taylor expanded in b around inf 31.2%

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

      1. times-frac 38.9%

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

      2. associate-/l* 44.8%

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

   10. Simplified 44.8%

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

   11. Taylor expanded in t around 0 57.2%

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

       1. +-commutative 57.2%

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

       2. times-frac 63.9%

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

   13. Simplified 63.9%

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

   if -4.9e13 < y < 2.7e62

   1. Initial program 90.4%

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

      1. associate-/l* 86.5%

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

      2. associate-+l+ 86.5%

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

      3. associate-/l* 84.5%

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

   3. Simplified 84.5%

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

   4. Taylor expanded in y around 0 60.4%

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

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -49000000000000 \lor \neg \left(y \leq 2.7 \cdot 10^{+62}\right):\\ \;\;\;\;\frac{z}{b} + \frac{t}{b} \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a + 1}\\ \end{array} \]

Alternative 12: 66.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9.5 \cdot 10^{+44} \lor \neg \left(y \leq 4.8 \cdot 10^{+58}\right):\\ \;\;\;\;\frac{z}{b} + \frac{t}{b} \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + \left(a + \frac{y \cdot b}{t}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -9.5e+44) (not (<= y 4.8e+58)))
   (+ (/ z b) (* (/ t b) (/ x y)))
   (/ x (+ 1.0 (+ a (/ (* y b) t))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -9.5e+44) || !(y <= 4.8e+58)) {
		tmp = (z / b) + ((t / b) * (x / y));
	} else {
		tmp = x / (1.0 + (a + ((y * b) / t)));
	}
	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 <= (-9.5d+44)) .or. (.not. (y <= 4.8d+58))) then
        tmp = (z / b) + ((t / b) * (x / y))
    else
        tmp = x / (1.0d0 + (a + ((y * b) / t)))
    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 <= -9.5e+44) || !(y <= 4.8e+58)) {
		tmp = (z / b) + ((t / b) * (x / y));
	} else {
		tmp = x / (1.0 + (a + ((y * b) / t)));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -9.5e+44) or not (y <= 4.8e+58):
		tmp = (z / b) + ((t / b) * (x / y))
	else:
		tmp = x / (1.0 + (a + ((y * b) / t)))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -9.5e+44) || !(y <= 4.8e+58))
		tmp = Float64(Float64(z / b) + Float64(Float64(t / b) * Float64(x / y)));
	else
		tmp = Float64(x / Float64(1.0 + Float64(a + Float64(Float64(y * b) / t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -9.5e+44) || ~((y <= 4.8e+58)))
		tmp = (z / b) + ((t / b) * (x / y));
	else
		tmp = x / (1.0 + (a + ((y * b) / t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -9.5e+44], N[Not[LessEqual[y, 4.8e+58]], $MachinePrecision]], N[(N[(z / b), $MachinePrecision] + N[(N[(t / b), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(1.0 + N[(a + N[(N[(y * b), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -9.5000000000000004e44 or 4.8e58 < y

   1. Initial program 52.4%

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

      1. associate-/l* 61.4%

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

      2. associate-+l+ 61.4%

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

      3. associate-/l* 70.1%

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

   3. Simplified 70.1%

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

      1. clear-num 70.0%

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

      2. associate-+r+ 70.0%

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

      3. associate-/l* 61.3%

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

      4. associate-/l* 52.4%

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

      5. inv-pow 52.4%

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

      6. associate-+l+ 52.4%

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

      7. +-commutative 52.4%

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

      8. *-commutative 52.4%

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

      9. associate-*l/ 59.3%

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

      10. *-commutative 59.3%

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

      11. fma-def 59.3%

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

      12. +-commutative 59.3%

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

      13. associate-/l* 70.0%

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

      14. associate-/r/ 61.6%

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

      15. fma-def 61.6%

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

   5. Applied egg-rr 61.6%

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

      1. unpow-1 61.6%

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

      2. fma-udef 61.6%

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

      3. associate-*l/ 59.3%

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

      4. associate-*r/ 70.7%

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

      5. fma-udef 70.7%

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

   7. Simplified 70.7%

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

   8. Taylor expanded in b around inf 30.3%

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

      1. times-frac 38.7%

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

      2. associate-/l* 45.1%

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

   10. Simplified 45.1%

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

   11. Taylor expanded in t around 0 58.5%

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

       1. +-commutative 58.5%

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

       2. times-frac 65.7%

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

   13. Simplified 65.7%

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

   if -9.5000000000000004e44 < y < 4.8e58

   1. Initial program 90.4%

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

      1. associate-/l* 86.7%

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

      2. associate-+l+ 86.7%

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

      3. associate-/l* 84.8%

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

   3. Simplified 84.8%

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

   4. Taylor expanded in x around inf 68.1%

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

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9.5 \cdot 10^{+44} \lor \neg \left(y \leq 4.8 \cdot 10^{+58}\right):\\ \;\;\;\;\frac{z}{b} + \frac{t}{b} \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + \left(a + \frac{y \cdot b}{t}\right)}\\ \end{array} \]

Alternative 13: 66.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.45 \cdot 10^{+48} \lor \neg \left(y \leq 5 \cdot 10^{+64}\right):\\ \;\;\;\;\frac{z}{b} + \frac{t}{b} \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a + \left(1 + b \cdot \frac{y}{t}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -1.45e+48) (not (<= y 5e+64)))
   (+ (/ z b) (* (/ t b) (/ x y)))
   (/ x (+ a (+ 1.0 (* b (/ y t)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -1.45e+48) || !(y <= 5e+64)) {
		tmp = (z / b) + ((t / b) * (x / y));
	} else {
		tmp = x / (a + (1.0 + (b * (y / t))));
	}
	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 <= (-1.45d+48)) .or. (.not. (y <= 5d+64))) then
        tmp = (z / b) + ((t / b) * (x / y))
    else
        tmp = x / (a + (1.0d0 + (b * (y / t))))
    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 <= -1.45e+48) || !(y <= 5e+64)) {
		tmp = (z / b) + ((t / b) * (x / y));
	} else {
		tmp = x / (a + (1.0 + (b * (y / t))));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -1.45e+48) or not (y <= 5e+64):
		tmp = (z / b) + ((t / b) * (x / y))
	else:
		tmp = x / (a + (1.0 + (b * (y / t))))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -1.45e+48) || !(y <= 5e+64))
		tmp = Float64(Float64(z / b) + Float64(Float64(t / b) * Float64(x / y)));
	else
		tmp = Float64(x / Float64(a + Float64(1.0 + Float64(b * Float64(y / t)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -1.45e+48) || ~((y <= 5e+64)))
		tmp = (z / b) + ((t / b) * (x / y));
	else
		tmp = x / (a + (1.0 + (b * (y / t))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -1.45e+48], N[Not[LessEqual[y, 5e+64]], $MachinePrecision]], N[(N[(z / b), $MachinePrecision] + N[(N[(t / b), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(a + N[(1.0 + N[(b * N[(y / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.4499999999999999e48 or 5e64 < y

   1. Initial program 52.4%

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

      1. associate-/l* 61.4%

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

      2. associate-+l+ 61.4%

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

      3. associate-/l* 70.1%

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

   3. Simplified 70.1%

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

      1. clear-num 70.0%

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

      2. associate-+r+ 70.0%

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

      3. associate-/l* 61.3%

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

      4. associate-/l* 52.4%

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

      5. inv-pow 52.4%

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

      6. associate-+l+ 52.4%

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

      7. +-commutative 52.4%

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

      8. *-commutative 52.4%

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

      9. associate-*l/ 59.3%

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

      10. *-commutative 59.3%

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

      11. fma-def 59.3%

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

      12. +-commutative 59.3%

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

      13. associate-/l* 70.0%

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

      14. associate-/r/ 61.6%

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

      15. fma-def 61.6%

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

   5. Applied egg-rr 61.6%

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

      1. unpow-1 61.6%

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

      2. fma-udef 61.6%

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

      3. associate-*l/ 59.3%

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

      4. associate-*r/ 70.7%

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

      5. fma-udef 70.7%

         \[\leadsto \frac{1}{\frac{a + \mathsf{fma}\left(y, \frac{b}{t}, 1\right)}{\color{blue}{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}}} \]

   7. Simplified 70.7%

      \[\leadsto \color{blue}{\frac{1}{\frac{a + \mathsf{fma}\left(y, \frac{b}{t}, 1\right)}{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}}} \]

   8. Taylor expanded in b around inf 30.3%

      \[\leadsto \color{blue}{\frac{t \cdot \left(x + \frac{y \cdot z}{t}\right)}{b \cdot y}} \]
   9. Step-by-step derivation

      1. times-frac 38.7%

         \[\leadsto \color{blue}{\frac{t}{b} \cdot \frac{x + \frac{y \cdot z}{t}}{y}} \]

      2. associate-/l* 45.1%

         \[\leadsto \frac{t}{b} \cdot \frac{x + \color{blue}{\frac{y}{\frac{t}{z}}}}{y} \]

   10. Simplified 45.1%

       \[\leadsto \color{blue}{\frac{t}{b} \cdot \frac{x + \frac{y}{\frac{t}{z}}}{y}} \]

   11. Taylor expanded in t around 0 58.5%

       \[\leadsto \color{blue}{\frac{z}{b} + \frac{t \cdot x}{b \cdot y}} \]
   12. Step-by-step derivation

       1. +-commutative 58.5%

          \[\leadsto \color{blue}{\frac{t \cdot x}{b \cdot y} + \frac{z}{b}} \]

       2. times-frac 65.7%

          \[\leadsto \color{blue}{\frac{t}{b} \cdot \frac{x}{y}} + \frac{z}{b} \]

   13. Simplified 65.7%

       \[\leadsto \color{blue}{\frac{t}{b} \cdot \frac{x}{y} + \frac{z}{b}} \]

   if -1.4499999999999999e48 < y < 5e64

   1. Initial program 90.4%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Step-by-step derivation

      1. associate-/l* 86.7%

         \[\leadsto \frac{x + \color{blue}{\frac{y}{\frac{t}{z}}}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

      2. associate-+l+ 86.7%

         \[\leadsto \frac{x + \frac{y}{\frac{t}{z}}}{\color{blue}{a + \left(1 + \frac{y \cdot b}{t}\right)}} \]

      3. associate-/l* 84.8%

         \[\leadsto \frac{x + \frac{y}{\frac{t}{z}}}{a + \left(1 + \color{blue}{\frac{y}{\frac{t}{b}}}\right)} \]

   3. Simplified 84.8%

      \[\leadsto \color{blue}{\frac{x + \frac{y}{\frac{t}{z}}}{a + \left(1 + \frac{y}{\frac{t}{b}}\right)}} \]

   4. Taylor expanded in x around inf 66.8%

      \[\leadsto \frac{\color{blue}{x}}{a + \left(1 + \frac{y}{\frac{t}{b}}\right)} \]
   5. Step-by-step derivation

      1. associate-/r/ 68.1%

         \[\leadsto \frac{x}{a + \left(1 + \color{blue}{\frac{y}{t} \cdot b}\right)} \]

   6. Applied egg-rr 68.1%

      \[\leadsto \frac{x}{a + \left(1 + \color{blue}{\frac{y}{t} \cdot b}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.45 \cdot 10^{+48} \lor \neg \left(y \leq 5 \cdot 10^{+64}\right):\\ \;\;\;\;\frac{z}{b} + \frac{t}{b} \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a + \left(1 + b \cdot \frac{y}{t}\right)}\\ \end{array} \]

Alternative 14: 41.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4900000000:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;y \leq -3.2 \cdot 10^{-173}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 3 \cdot 10^{+58}:\\ \;\;\;\;\frac{x}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -4900000000.0)
   (/ z b)
   (if (<= y -3.2e-173) x (if (<= y 3e+58) (/ x a) (/ z b)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -4900000000.0) {
		tmp = z / b;
	} else if (y <= -3.2e-173) {
		tmp = x;
	} else if (y <= 3e+58) {
		tmp = x / a;
	} else {
		tmp = z / b;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (y <= (-4900000000.0d0)) then
        tmp = z / b
    else if (y <= (-3.2d-173)) then
        tmp = x
    else if (y <= 3d+58) then
        tmp = x / a
    else
        tmp = z / b
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -4900000000.0) {
		tmp = z / b;
	} else if (y <= -3.2e-173) {
		tmp = x;
	} else if (y <= 3e+58) {
		tmp = x / a;
	} else {
		tmp = z / b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -4900000000.0:
		tmp = z / b
	elif y <= -3.2e-173:
		tmp = x
	elif y <= 3e+58:
		tmp = x / a
	else:
		tmp = z / b
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -4900000000.0)
		tmp = Float64(z / b);
	elseif (y <= -3.2e-173)
		tmp = x;
	elseif (y <= 3e+58)
		tmp = Float64(x / a);
	else
		tmp = Float64(z / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -4900000000.0)
		tmp = z / b;
	elseif (y <= -3.2e-173)
		tmp = x;
	elseif (y <= 3e+58)
		tmp = x / a;
	else
		tmp = z / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -4900000000.0], N[(z / b), $MachinePrecision], If[LessEqual[y, -3.2e-173], x, If[LessEqual[y, 3e+58], N[(x / a), $MachinePrecision], N[(z / b), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.9e9 or 3.0000000000000002e58 < y

   1. Initial program 55.2%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Step-by-step derivation

      1. associate-/l* 63.3%

         \[\leadsto \frac{x + \color{blue}{\frac{y}{\frac{t}{z}}}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

      2. associate-+l+ 63.3%

         \[\leadsto \frac{x + \frac{y}{\frac{t}{z}}}{\color{blue}{a + \left(1 + \frac{y \cdot b}{t}\right)}} \]

      3. associate-/l* 71.2%

         \[\leadsto \frac{x + \frac{y}{\frac{t}{z}}}{a + \left(1 + \color{blue}{\frac{y}{\frac{t}{b}}}\right)} \]

   3. Simplified 71.2%

      \[\leadsto \color{blue}{\frac{x + \frac{y}{\frac{t}{z}}}{a + \left(1 + \frac{y}{\frac{t}{b}}\right)}} \]

   4. Taylor expanded in y around inf 50.7%

      \[\leadsto \color{blue}{\frac{z}{b}} \]

   if -4.9e9 < y < -3.2e-173

   1. Initial program 89.6%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Step-by-step derivation

      1. associate-/l* 85.6%

         \[\leadsto \frac{x + \color{blue}{\frac{y}{\frac{t}{z}}}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

      2. associate-+l+ 85.6%

         \[\leadsto \frac{x + \frac{y}{\frac{t}{z}}}{\color{blue}{a + \left(1 + \frac{y \cdot b}{t}\right)}} \]

      3. associate-/l* 85.7%

         \[\leadsto \frac{x + \frac{y}{\frac{t}{z}}}{a + \left(1 + \color{blue}{\frac{y}{\frac{t}{b}}}\right)} \]

   3. Simplified 85.7%

      \[\leadsto \color{blue}{\frac{x + \frac{y}{\frac{t}{z}}}{a + \left(1 + \frac{y}{\frac{t}{b}}\right)}} \]

   4. Taylor expanded in x around inf 66.1%

      \[\leadsto \frac{\color{blue}{x}}{a + \left(1 + \frac{y}{\frac{t}{b}}\right)} \]

   5. Taylor expanded in a around 0 46.8%

      \[\leadsto \color{blue}{\frac{x}{1 + \frac{b \cdot y}{t}}} \]

   6. Taylor expanded in b around 0 36.2%

      \[\leadsto \color{blue}{x} \]

   if -3.2e-173 < y < 3.0000000000000002e58

   1. Initial program 91.7%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Step-by-step derivation

      1. associate-/l* 87.8%

         \[\leadsto \frac{x + \color{blue}{\frac{y}{\frac{t}{z}}}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

      2. associate-+l+ 87.8%

         \[\leadsto \frac{x + \frac{y}{\frac{t}{z}}}{\color{blue}{a + \left(1 + \frac{y \cdot b}{t}\right)}} \]

      3. associate-/l* 84.5%

         \[\leadsto \frac{x + \frac{y}{\frac{t}{z}}}{a + \left(1 + \color{blue}{\frac{y}{\frac{t}{b}}}\right)} \]

   3. Simplified 84.5%

      \[\leadsto \color{blue}{\frac{x + \frac{y}{\frac{t}{z}}}{a + \left(1 + \frac{y}{\frac{t}{b}}\right)}} \]

   4. Taylor expanded in x around inf 68.9%

      \[\leadsto \frac{\color{blue}{x}}{a + \left(1 + \frac{y}{\frac{t}{b}}\right)} \]

   5. Taylor expanded in a around inf 40.9%

      \[\leadsto \color{blue}{\frac{x}{a}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 44.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4900000000:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;y \leq -3.2 \cdot 10^{-173}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 3 \cdot 10^{+58}:\\ \;\;\;\;\frac{x}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]

Alternative 15: 56.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.2 \cdot 10^{+46} \lor \neg \left(y \leq 5.8 \cdot 10^{+59}\right):\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -6.2e+46) (not (<= y 5.8e+59))) (/ z b) (/ x (+ a 1.0))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -6.2e+46) || !(y <= 5.8e+59)) {
		tmp = z / b;
	} else {
		tmp = x / (a + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((y <= (-6.2d+46)) .or. (.not. (y <= 5.8d+59))) then
        tmp = z / b
    else
        tmp = x / (a + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -6.2e+46) || !(y <= 5.8e+59)) {
		tmp = z / b;
	} else {
		tmp = x / (a + 1.0);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -6.2e+46) or not (y <= 5.8e+59):
		tmp = z / b
	else:
		tmp = x / (a + 1.0)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -6.2e+46) || !(y <= 5.8e+59))
		tmp = Float64(z / b);
	else
		tmp = Float64(x / Float64(a + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -6.2e+46) || ~((y <= 5.8e+59)))
		tmp = z / b;
	else
		tmp = x / (a + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -6.2e+46], N[Not[LessEqual[y, 5.8e+59]], $MachinePrecision]], N[(z / b), $MachinePrecision], N[(x / N[(a + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -6.1999999999999995e46 or 5.79999999999999981e59 < y

   1. Initial program 52.4%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Step-by-step derivation

      1. associate-/l* 61.4%

         \[\leadsto \frac{x + \color{blue}{\frac{y}{\frac{t}{z}}}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

      2. associate-+l+ 61.4%

         \[\leadsto \frac{x + \frac{y}{\frac{t}{z}}}{\color{blue}{a + \left(1 + \frac{y \cdot b}{t}\right)}} \]

      3. associate-/l* 70.1%

         \[\leadsto \frac{x + \frac{y}{\frac{t}{z}}}{a + \left(1 + \color{blue}{\frac{y}{\frac{t}{b}}}\right)} \]

   3. Simplified 70.1%

      \[\leadsto \color{blue}{\frac{x + \frac{y}{\frac{t}{z}}}{a + \left(1 + \frac{y}{\frac{t}{b}}\right)}} \]

   4. Taylor expanded in y around inf 54.0%

      \[\leadsto \color{blue}{\frac{z}{b}} \]

   if -6.1999999999999995e46 < y < 5.79999999999999981e59

   1. Initial program 90.4%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Step-by-step derivation

      1. associate-/l* 86.7%

         \[\leadsto \frac{x + \color{blue}{\frac{y}{\frac{t}{z}}}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

      2. associate-+l+ 86.7%

         \[\leadsto \frac{x + \frac{y}{\frac{t}{z}}}{\color{blue}{a + \left(1 + \frac{y \cdot b}{t}\right)}} \]

      3. associate-/l* 84.8%

         \[\leadsto \frac{x + \frac{y}{\frac{t}{z}}}{a + \left(1 + \color{blue}{\frac{y}{\frac{t}{b}}}\right)} \]

   3. Simplified 84.8%

      \[\leadsto \color{blue}{\frac{x + \frac{y}{\frac{t}{z}}}{a + \left(1 + \frac{y}{\frac{t}{b}}\right)}} \]

   4. Taylor expanded in y around 0 57.9%

      \[\leadsto \color{blue}{\frac{x}{1 + a}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.2 \cdot 10^{+46} \lor \neg \left(y \leq 5.8 \cdot 10^{+59}\right):\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a + 1}\\ \end{array} \]

Alternative 16: 41.6% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.98 \lor \neg \left(a \leq 1\right):\\ \;\;\;\;\frac{x}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= a -0.98) (not (<= a 1.0))) (/ x a) x))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((a <= -0.98) || !(a <= 1.0)) {
		tmp = x / a;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((a <= (-0.98d0)) .or. (.not. (a <= 1.0d0))) then
        tmp = x / a
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((a <= -0.98) || !(a <= 1.0)) {
		tmp = x / a;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (a <= -0.98) or not (a <= 1.0):
		tmp = x / a
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((a <= -0.98) || !(a <= 1.0))
		tmp = Float64(x / a);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((a <= -0.98) || ~((a <= 1.0)))
		tmp = x / a;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[a, -0.98], N[Not[LessEqual[a, 1.0]], $MachinePrecision]], N[(x / a), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if a < -0.97999999999999998 or 1 < a

   1. Initial program 74.7%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Step-by-step derivation

      1. associate-/l* 74.8%

         \[\leadsto \frac{x + \color{blue}{\frac{y}{\frac{t}{z}}}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

      2. associate-+l+ 74.8%

         \[\leadsto \frac{x + \frac{y}{\frac{t}{z}}}{\color{blue}{a + \left(1 + \frac{y \cdot b}{t}\right)}} \]

      3. associate-/l* 77.9%

         \[\leadsto \frac{x + \frac{y}{\frac{t}{z}}}{a + \left(1 + \color{blue}{\frac{y}{\frac{t}{b}}}\right)} \]

   3. Simplified 77.9%

      \[\leadsto \color{blue}{\frac{x + \frac{y}{\frac{t}{z}}}{a + \left(1 + \frac{y}{\frac{t}{b}}\right)}} \]

   4. Taylor expanded in x around inf 51.2%

      \[\leadsto \frac{\color{blue}{x}}{a + \left(1 + \frac{y}{\frac{t}{b}}\right)} \]

   5. Taylor expanded in a around inf 44.2%

      \[\leadsto \color{blue}{\frac{x}{a}} \]

   if -0.97999999999999998 < a < 1

   1. Initial program 73.4%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Step-by-step derivation

      1. associate-/l* 76.8%

         \[\leadsto \frac{x + \color{blue}{\frac{y}{\frac{t}{z}}}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

      2. associate-+l+ 76.8%

         \[\leadsto \frac{x + \frac{y}{\frac{t}{z}}}{\color{blue}{a + \left(1 + \frac{y \cdot b}{t}\right)}} \]

      3. associate-/l* 79.0%

         \[\leadsto \frac{x + \frac{y}{\frac{t}{z}}}{a + \left(1 + \color{blue}{\frac{y}{\frac{t}{b}}}\right)} \]

   3. Simplified 79.0%

      \[\leadsto \color{blue}{\frac{x + \frac{y}{\frac{t}{z}}}{a + \left(1 + \frac{y}{\frac{t}{b}}\right)}} \]

   4. Taylor expanded in x around inf 51.0%

      \[\leadsto \frac{\color{blue}{x}}{a + \left(1 + \frac{y}{\frac{t}{b}}\right)} \]

   5. Taylor expanded in a around 0 48.0%

      \[\leadsto \color{blue}{\frac{x}{1 + \frac{b \cdot y}{t}}} \]

   6. Taylor expanded in b around 0 31.9%

      \[\leadsto \color{blue}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 38.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.98 \lor \neg \left(a \leq 1\right):\\ \;\;\;\;\frac{x}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 17: 19.9% accurate, 17.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

(FPCore (x y z t a b) :precision binary64 x)

C

double code(double x, double y, double z, double t, double a, double b) {
	return x;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = x
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return x;
}

Python

def code(x, y, z, t, a, b):
	return x

Julia

function code(x, y, z, t, a, b)
	return x
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = x;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := x

Derivation

1. Initial program 74.1%

   \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
2. Step-by-step derivation

   1. associate-/l* 75.8%

      \[\leadsto \frac{x + \color{blue}{\frac{y}{\frac{t}{z}}}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

   2. associate-+l+ 75.8%

      \[\leadsto \frac{x + \frac{y}{\frac{t}{z}}}{\color{blue}{a + \left(1 + \frac{y \cdot b}{t}\right)}} \]

   3. associate-/l* 78.5%

      \[\leadsto \frac{x + \frac{y}{\frac{t}{z}}}{a + \left(1 + \color{blue}{\frac{y}{\frac{t}{b}}}\right)} \]

3. Simplified 78.5%

   \[\leadsto \color{blue}{\frac{x + \frac{y}{\frac{t}{z}}}{a + \left(1 + \frac{y}{\frac{t}{b}}\right)}} \]

4. Taylor expanded in x around inf 51.1%

   \[\leadsto \frac{\color{blue}{x}}{a + \left(1 + \frac{y}{\frac{t}{b}}\right)} \]

5. Taylor expanded in a around 0 29.6%

   \[\leadsto \color{blue}{\frac{x}{1 + \frac{b \cdot y}{t}}} \]

6. Taylor expanded in b around 0 18.0%

   \[\leadsto \color{blue}{x} \]

7. Final simplification 18.0%

   \[\leadsto x \]

Developer target: 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 2023332 
(FPCore (x y z t a b)
  :name "Diagrams.Solve.Tridiagonal:solveCyclicTriDiagonal from diagrams-solve-0.1, B"
  :precision binary64

  :herbie-target
  (if (< t -1.3659085366310088e-271) (* 1.0 (* (+ x (* (/ y t) z)) (/ 1.0 (+ (+ a 1.0) (* (/ y t) b))))) (if (< t 3.036967103737246e-130) (/ z b) (* 1.0 (* (+ x (* (/ y t) z)) (/ 1.0 (+ (+ a 1.0) (* (/ y t) b)))))))

  (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t))))