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

Percentage Accurate: 75.5% → 87.5%

Time: 11.5s

Alternatives: 12

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: 75.5% 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: 87.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a + 1\right) + \frac{y \cdot b}{t}\\ t_2 := x + \frac{y \cdot z}{t}\\ t_3 := \frac{t_2}{t_1}\\ t_4 := \frac{x + \frac{z}{\frac{t}{y}}}{t_1}\\ \mathbf{if}\;t_3 \leq -1 \cdot 10^{-317}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;t_3 \leq 2 \cdot 10^{-95}:\\ \;\;\;\;\frac{t_2}{\left(a + 1\right) + \frac{y}{\frac{t}{b}}}\\ \mathbf{elif}\;t_3 \leq \infty:\\ \;\;\;\;t_4\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (+ a 1.0) (/ (* y b) t)))
        (t_2 (+ x (/ (* y z) t)))
        (t_3 (/ t_2 t_1))
        (t_4 (/ (+ x (/ z (/ t y))) t_1)))
   (if (<= t_3 -1e-317)
     t_4
     (if (<= t_3 2e-95)
       (/ t_2 (+ (+ a 1.0) (/ y (/ t b))))
       (if (<= t_3 INFINITY) t_4 (/ z b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a + 1.0) + ((y * b) / t);
	double t_2 = x + ((y * z) / t);
	double t_3 = t_2 / t_1;
	double t_4 = (x + (z / (t / y))) / t_1;
	double tmp;
	if (t_3 <= -1e-317) {
		tmp = t_4;
	} else if (t_3 <= 2e-95) {
		tmp = t_2 / ((a + 1.0) + (y / (t / b)));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = t_4;
	} else {
		tmp = z / b;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a + 1.0) + ((y * b) / t);
	double t_2 = x + ((y * z) / t);
	double t_3 = t_2 / t_1;
	double t_4 = (x + (z / (t / y))) / t_1;
	double tmp;
	if (t_3 <= -1e-317) {
		tmp = t_4;
	} else if (t_3 <= 2e-95) {
		tmp = t_2 / ((a + 1.0) + (y / (t / b)));
	} else if (t_3 <= Double.POSITIVE_INFINITY) {
		tmp = t_4;
	} else {
		tmp = z / b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (a + 1.0) + ((y * b) / t)
	t_2 = x + ((y * z) / t)
	t_3 = t_2 / t_1
	t_4 = (x + (z / (t / y))) / t_1
	tmp = 0
	if t_3 <= -1e-317:
		tmp = t_4
	elif t_3 <= 2e-95:
		tmp = t_2 / ((a + 1.0) + (y / (t / b)))
	elif t_3 <= math.inf:
		tmp = t_4
	else:
		tmp = z / b
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(a + 1.0) + Float64(Float64(y * b) / t))
	t_2 = Float64(x + Float64(Float64(y * z) / t))
	t_3 = Float64(t_2 / t_1)
	t_4 = Float64(Float64(x + Float64(z / Float64(t / y))) / t_1)
	tmp = 0.0
	if (t_3 <= -1e-317)
		tmp = t_4;
	elseif (t_3 <= 2e-95)
		tmp = Float64(t_2 / Float64(Float64(a + 1.0) + Float64(y / Float64(t / b))));
	elseif (t_3 <= Inf)
		tmp = t_4;
	else
		tmp = Float64(z / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (a + 1.0) + ((y * b) / t);
	t_2 = x + ((y * z) / t);
	t_3 = t_2 / t_1;
	t_4 = (x + (z / (t / y))) / t_1;
	tmp = 0.0;
	if (t_3 <= -1e-317)
		tmp = t_4;
	elseif (t_3 <= 2e-95)
		tmp = t_2 / ((a + 1.0) + (y / (t / b)));
	elseif (t_3 <= Inf)
		tmp = t_4;
	else
		tmp = z / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(a + 1.0), $MachinePrecision] + N[(N[(y * b), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 / t$95$1), $MachinePrecision]}, Block[{t$95$4 = N[(N[(x + N[(z / N[(t / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, If[LessEqual[t$95$3, -1e-317], t$95$4, If[LessEqual[t$95$3, 2e-95], N[(t$95$2 / N[(N[(a + 1.0), $MachinePrecision] + N[(y / N[(t / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], t$95$4, N[(z / b), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t))) < -1.00000023e-317 or 1.99999999999999998e-95 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t))) < +inf.0

   1. Initial program 89.3%

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

      1. *-commutative 89.3%

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

      2. associate-/l* 94.0%

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

      3. associate-*l/ 91.6%

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

   3. Simplified 91.6%

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

   4. Taylor expanded in y around 0 94.0%

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

   if -1.00000023e-317 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t))) < 1.99999999999999998e-95

   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* 80.7%

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

   3. Simplified 80.7%

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

   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. *-commutative 0.0%

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

      2. associate-/l* 0.0%

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

      3. associate-*l/ 0.3%

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

   3. Simplified 0.3%

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

   4. Taylor expanded in t around 0 100.0%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq -1 \cdot 10^{-317}:\\ \;\;\;\;\frac{x + \frac{z}{\frac{t}{y}}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq 2 \cdot 10^{-95}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y}{\frac{t}{b}}}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq \infty:\\ \;\;\;\;\frac{x + \frac{z}{\frac{t}{y}}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]

Alternative 2: 62.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + z \cdot \frac{y}{t}}{a + 1}\\ \mathbf{if}\;t \leq -3600:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -2.0865 \cdot 10^{-138}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;t \leq -9.2 \cdot 10^{-268}:\\ \;\;\;\;\frac{y \cdot z}{t \cdot \left(1 + \left(a + \frac{y \cdot b}{t}\right)\right)}\\ \mathbf{elif}\;t \leq 6.2 \cdot 10^{-60}:\\ \;\;\;\;\frac{y \cdot z}{t + y \cdot b}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (+ x (* z (/ y t))) (+ a 1.0))))
   (if (<= t -3600.0)
     t_1
     (if (<= t -2.0865e-138)
       (/ z b)
       (if (<= t -9.2e-268)
         (/ (* y z) (* t (+ 1.0 (+ a (/ (* y b) t)))))
         (if (<= t 6.2e-60) (/ (* y z) (+ t (* y b))) t_1))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x + (z * (y / t))) / (a + 1.0);
	double tmp;
	if (t <= -3600.0) {
		tmp = t_1;
	} else if (t <= -2.0865e-138) {
		tmp = z / b;
	} else if (t <= -9.2e-268) {
		tmp = (y * z) / (t * (1.0 + (a + ((y * b) / t))));
	} else if (t <= 6.2e-60) {
		tmp = (y * z) / (t + (y * b));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x + (z * (y / t))) / (a + 1.0d0)
    if (t <= (-3600.0d0)) then
        tmp = t_1
    else if (t <= (-2.0865d-138)) then
        tmp = z / b
    else if (t <= (-9.2d-268)) then
        tmp = (y * z) / (t * (1.0d0 + (a + ((y * b) / t))))
    else if (t <= 6.2d-60) then
        tmp = (y * z) / (t + (y * b))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x + (z * (y / t))) / (a + 1.0);
	double tmp;
	if (t <= -3600.0) {
		tmp = t_1;
	} else if (t <= -2.0865e-138) {
		tmp = z / b;
	} else if (t <= -9.2e-268) {
		tmp = (y * z) / (t * (1.0 + (a + ((y * b) / t))));
	} else if (t <= 6.2e-60) {
		tmp = (y * z) / (t + (y * b));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (x + (z * (y / t))) / (a + 1.0)
	tmp = 0
	if t <= -3600.0:
		tmp = t_1
	elif t <= -2.0865e-138:
		tmp = z / b
	elif t <= -9.2e-268:
		tmp = (y * z) / (t * (1.0 + (a + ((y * b) / t))))
	elif t <= 6.2e-60:
		tmp = (y * z) / (t + (y * b))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x + Float64(z * Float64(y / t))) / Float64(a + 1.0))
	tmp = 0.0
	if (t <= -3600.0)
		tmp = t_1;
	elseif (t <= -2.0865e-138)
		tmp = Float64(z / b);
	elseif (t <= -9.2e-268)
		tmp = Float64(Float64(y * z) / Float64(t * Float64(1.0 + Float64(a + Float64(Float64(y * b) / t)))));
	elseif (t <= 6.2e-60)
		tmp = Float64(Float64(y * z) / Float64(t + Float64(y * b)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (x + (z * (y / t))) / (a + 1.0);
	tmp = 0.0;
	if (t <= -3600.0)
		tmp = t_1;
	elseif (t <= -2.0865e-138)
		tmp = z / b;
	elseif (t <= -9.2e-268)
		tmp = (y * z) / (t * (1.0 + (a + ((y * b) / t))));
	elseif (t <= 6.2e-60)
		tmp = (y * z) / (t + (y * b));
	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[(z * N[(y / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -3600.0], t$95$1, If[LessEqual[t, -2.0865e-138], N[(z / b), $MachinePrecision], If[LessEqual[t, -9.2e-268], N[(N[(y * z), $MachinePrecision] / N[(t * N[(1.0 + N[(a + N[(N[(y * b), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 6.2e-60], N[(N[(y * z), $MachinePrecision] / N[(t + N[(y * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -3600 or 6.19999999999999976e-60 < t

   1. Initial program 86.6%

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

      1. *-commutative 86.6%

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

      2. associate-/l* 93.0%

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

      3. associate-*l/ 96.0%

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

   3. Simplified 96.0%

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

   4. Taylor expanded in y around 0 93.0%

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

   5. Taylor expanded in b around 0 77.4%

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

      1. +-commutative 77.4%

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

      2. associate-/l* 81.5%

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

      3. associate-/r/ 83.2%

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

   7. Simplified 83.2%

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

   if -3600 < t < -2.08649999999999994e-138

   1. Initial program 61.8%

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

      1. *-commutative 61.8%

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

      2. associate-/l* 64.5%

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

      3. associate-*l/ 64.4%

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

   3. Simplified 64.4%

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

   4. Taylor expanded in t around 0 54.0%

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

   if -2.08649999999999994e-138 < t < -9.20000000000000042e-268

   1. Initial program 71.0%

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

      1. *-commutative 71.0%

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

      2. associate-/l* 60.1%

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

      3. associate-*l/ 56.0%

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

   3. Simplified 56.0%

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

   4. Taylor expanded in x around 0 71.2%

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

   if -9.20000000000000042e-268 < t < 6.19999999999999976e-60

   1. Initial program 67.8%

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

      1. +-commutative 67.8%

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

      2. associate-*l/ 56.7%

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

      3. fma-def 56.7%

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

      4. associate-+l+ 56.7%

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

      5. +-commutative 56.7%

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

      6. associate-*l/ 49.8%

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

      7. fma-def 49.8%

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

   3. Simplified 49.8%

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

   4. Taylor expanded in a around 0 61.3%

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

   5. Taylor expanded in z around inf 57.4%

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

   6. Taylor expanded in y around 0 72.5%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3600:\\ \;\;\;\;\frac{x + z \cdot \frac{y}{t}}{a + 1}\\ \mathbf{elif}\;t \leq -2.0865 \cdot 10^{-138}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;t \leq -9.2 \cdot 10^{-268}:\\ \;\;\;\;\frac{y \cdot z}{t \cdot \left(1 + \left(a + \frac{y \cdot b}{t}\right)\right)}\\ \mathbf{elif}\;t \leq 6.2 \cdot 10^{-60}:\\ \;\;\;\;\frac{y \cdot z}{t + y \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + z \cdot \frac{y}{t}}{a + 1}\\ \end{array} \]

Alternative 3: 80.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8.4 \cdot 10^{+173} \lor \neg \left(y \leq 1.75 \cdot 10^{+214}\right):\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + z \cdot \frac{y}{t}}{\left(a + 1\right) + b \cdot \frac{y}{t}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -8.4e+173) (not (<= y 1.75e+214)))
   (/ z b)
   (/ (+ x (* z (/ y t))) (+ (+ 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 <= -8.4e+173) || !(y <= 1.75e+214)) {
		tmp = z / b;
	} else {
		tmp = (x + (z * (y / t))) / ((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 <= (-8.4d+173)) .or. (.not. (y <= 1.75d+214))) then
        tmp = z / b
    else
        tmp = (x + (z * (y / t))) / ((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 <= -8.4e+173) || !(y <= 1.75e+214)) {
		tmp = z / b;
	} else {
		tmp = (x + (z * (y / t))) / ((a + 1.0) + (b * (y / t)));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -8.4e+173) or not (y <= 1.75e+214):
		tmp = z / b
	else:
		tmp = (x + (z * (y / t))) / ((a + 1.0) + (b * (y / t)))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -8.4e+173) || !(y <= 1.75e+214))
		tmp = Float64(z / b);
	else
		tmp = Float64(Float64(x + Float64(z * Float64(y / t))) / Float64(Float64(a + 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 <= -8.4e+173) || ~((y <= 1.75e+214)))
		tmp = z / b;
	else
		tmp = (x + (z * (y / t))) / ((a + 1.0) + (b * (y / t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -8.4e+173], N[Not[LessEqual[y, 1.75e+214]], $MachinePrecision]], N[(z / b), $MachinePrecision], N[(N[(x + N[(z * N[(y / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(a + 1.0), $MachinePrecision] + N[(b * N[(y / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -8.40000000000000001e173 or 1.75e214 < y

   1. Initial program 49.1%

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

      1. *-commutative 49.1%

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

      2. associate-/l* 47.1%

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

      3. associate-*l/ 45.9%

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

   3. Simplified 45.9%

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

   4. Taylor expanded in t around 0 72.3%

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

   if -8.40000000000000001e173 < y < 1.75e214

   1. Initial program 86.4%

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

      1. *-commutative 86.4%

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

      2. associate-/l* 88.2%

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

      3. associate-*l/ 88.5%

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

   3. Simplified 88.5%

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

      1. div-inv 88.5%

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

      2. clear-num 88.5%

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

   5. Applied egg-rr 88.5%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.4 \cdot 10^{+173} \lor \neg \left(y \leq 1.75 \cdot 10^{+214}\right):\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + z \cdot \frac{y}{t}}{\left(a + 1\right) + b \cdot \frac{y}{t}}\\ \end{array} \]

Alternative 4: 60.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.3 \cdot 10^{+156}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;y \leq 3.7 \cdot 10^{+76}:\\ \;\;\;\;\frac{x}{1 + \left(a + \frac{y \cdot b}{t}\right)}\\ \mathbf{elif}\;y \leq 3.3 \cdot 10^{+166}:\\ \;\;\;\;\frac{y \cdot z}{t + y \cdot b}\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{+209}:\\ \;\;\;\;\left(x + \frac{y}{\frac{t}{z}}\right) \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -2.3e+156)
   (/ z b)
   (if (<= y 3.7e+76)
     (/ x (+ 1.0 (+ a (/ (* y b) t))))
     (if (<= y 3.3e+166)
       (/ (* y z) (+ t (* y b)))
       (if (<= y 2.6e+209) (* (+ x (/ y (/ t z))) (/ 1.0 a)) (/ z b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -2.3e+156) {
		tmp = z / b;
	} else if (y <= 3.7e+76) {
		tmp = x / (1.0 + (a + ((y * b) / t)));
	} else if (y <= 3.3e+166) {
		tmp = (y * z) / (t + (y * b));
	} else if (y <= 2.6e+209) {
		tmp = (x + (y / (t / z))) * (1.0 / a);
	} else {
		tmp = z / b;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (y <= (-2.3d+156)) then
        tmp = z / b
    else if (y <= 3.7d+76) then
        tmp = x / (1.0d0 + (a + ((y * b) / t)))
    else if (y <= 3.3d+166) then
        tmp = (y * z) / (t + (y * b))
    else if (y <= 2.6d+209) then
        tmp = (x + (y / (t / z))) * (1.0d0 / a)
    else
        tmp = z / b
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -2.3e+156) {
		tmp = z / b;
	} else if (y <= 3.7e+76) {
		tmp = x / (1.0 + (a + ((y * b) / t)));
	} else if (y <= 3.3e+166) {
		tmp = (y * z) / (t + (y * b));
	} else if (y <= 2.6e+209) {
		tmp = (x + (y / (t / z))) * (1.0 / a);
	} else {
		tmp = z / b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -2.3e+156:
		tmp = z / b
	elif y <= 3.7e+76:
		tmp = x / (1.0 + (a + ((y * b) / t)))
	elif y <= 3.3e+166:
		tmp = (y * z) / (t + (y * b))
	elif y <= 2.6e+209:
		tmp = (x + (y / (t / z))) * (1.0 / a)
	else:
		tmp = z / b
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -2.3e+156)
		tmp = Float64(z / b);
	elseif (y <= 3.7e+76)
		tmp = Float64(x / Float64(1.0 + Float64(a + Float64(Float64(y * b) / t))));
	elseif (y <= 3.3e+166)
		tmp = Float64(Float64(y * z) / Float64(t + Float64(y * b)));
	elseif (y <= 2.6e+209)
		tmp = Float64(Float64(x + Float64(y / Float64(t / z))) * Float64(1.0 / a));
	else
		tmp = Float64(z / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -2.3e+156)
		tmp = z / b;
	elseif (y <= 3.7e+76)
		tmp = x / (1.0 + (a + ((y * b) / t)));
	elseif (y <= 3.3e+166)
		tmp = (y * z) / (t + (y * b));
	elseif (y <= 2.6e+209)
		tmp = (x + (y / (t / z))) * (1.0 / a);
	else
		tmp = z / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -2.3e+156], N[(z / b), $MachinePrecision], If[LessEqual[y, 3.7e+76], N[(x / N[(1.0 + N[(a + N[(N[(y * b), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.3e+166], N[(N[(y * z), $MachinePrecision] / N[(t + N[(y * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.6e+209], N[(N[(x + N[(y / N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / a), $MachinePrecision]), $MachinePrecision], N[(z / b), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -2.2999999999999999e156 or 2.6e209 < y

   1. Initial program 48.5%

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

      1. *-commutative 48.5%

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

      2. associate-/l* 46.5%

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

      3. associate-*l/ 47.1%

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

   3. Simplified 47.1%

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

   4. Taylor expanded in t around 0 70.5%

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

   if -2.2999999999999999e156 < y < 3.6999999999999999e76

   1. Initial program 90.8%

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

      1. *-commutative 90.8%

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

      2. associate-/l* 90.0%

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

      3. associate-*l/ 89.9%

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

   3. Simplified 89.9%

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

   4. Taylor expanded in x around inf 68.3%

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

   if 3.6999999999999999e76 < y < 3.3000000000000002e166

   1. Initial program 73.0%

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

      1. +-commutative 73.0%

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

      2. associate-*l/ 78.1%

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

      3. fma-def 78.1%

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

      4. associate-+l+ 78.1%

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

      5. +-commutative 78.1%

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

      6. associate-*l/ 78.0%

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

      7. fma-def 78.0%

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

   3. Simplified 78.0%

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

   4. Taylor expanded in a around 0 62.6%

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

   5. Taylor expanded in z around inf 62.9%

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

   6. Taylor expanded in y around 0 62.9%

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

   if 3.3000000000000002e166 < y < 2.6e209

   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. *-commutative 55.2%

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

      2. associate-/l* 90.2%

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

      3. associate-*l/ 90.3%

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

   3. Simplified 90.3%

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

   4. Taylor expanded in a around inf 29.3%

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

      1. div-inv 29.1%

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

      2. +-commutative 29.1%

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

      3. associate-/l* 55.5%

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

   6. Applied egg-rr 55.5%

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

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.3 \cdot 10^{+156}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;y \leq 3.7 \cdot 10^{+76}:\\ \;\;\;\;\frac{x}{1 + \left(a + \frac{y \cdot b}{t}\right)}\\ \mathbf{elif}\;y \leq 3.3 \cdot 10^{+166}:\\ \;\;\;\;\frac{y \cdot z}{t + y \cdot b}\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{+209}:\\ \;\;\;\;\left(x + \frac{y}{\frac{t}{z}}\right) \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]

Alternative 5: 60.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.55 \cdot 10^{+175}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;y \leq 1.65 \cdot 10^{+75}:\\ \;\;\;\;\frac{x}{\left(a + 1\right) + b \cdot \frac{y}{t}}\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{+166}:\\ \;\;\;\;\frac{y \cdot z}{t + y \cdot b}\\ \mathbf{elif}\;y \leq 2.55 \cdot 10^{+209}:\\ \;\;\;\;\left(x + \frac{y}{\frac{t}{z}}\right) \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -2.55e+175)
   (/ z b)
   (if (<= y 1.65e+75)
     (/ x (+ (+ a 1.0) (* b (/ y t))))
     (if (<= y 9.5e+166)
       (/ (* y z) (+ t (* y b)))
       (if (<= y 2.55e+209) (* (+ x (/ y (/ t z))) (/ 1.0 a)) (/ z b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -2.55e+175) {
		tmp = z / b;
	} else if (y <= 1.65e+75) {
		tmp = x / ((a + 1.0) + (b * (y / t)));
	} else if (y <= 9.5e+166) {
		tmp = (y * z) / (t + (y * b));
	} else if (y <= 2.55e+209) {
		tmp = (x + (y / (t / z))) * (1.0 / a);
	} else {
		tmp = z / b;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (y <= (-2.55d+175)) then
        tmp = z / b
    else if (y <= 1.65d+75) then
        tmp = x / ((a + 1.0d0) + (b * (y / t)))
    else if (y <= 9.5d+166) then
        tmp = (y * z) / (t + (y * b))
    else if (y <= 2.55d+209) then
        tmp = (x + (y / (t / z))) * (1.0d0 / a)
    else
        tmp = z / b
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -2.55e+175) {
		tmp = z / b;
	} else if (y <= 1.65e+75) {
		tmp = x / ((a + 1.0) + (b * (y / t)));
	} else if (y <= 9.5e+166) {
		tmp = (y * z) / (t + (y * b));
	} else if (y <= 2.55e+209) {
		tmp = (x + (y / (t / z))) * (1.0 / a);
	} else {
		tmp = z / b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -2.55e+175:
		tmp = z / b
	elif y <= 1.65e+75:
		tmp = x / ((a + 1.0) + (b * (y / t)))
	elif y <= 9.5e+166:
		tmp = (y * z) / (t + (y * b))
	elif y <= 2.55e+209:
		tmp = (x + (y / (t / z))) * (1.0 / a)
	else:
		tmp = z / b
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -2.55e+175)
		tmp = Float64(z / b);
	elseif (y <= 1.65e+75)
		tmp = Float64(x / Float64(Float64(a + 1.0) + Float64(b * Float64(y / t))));
	elseif (y <= 9.5e+166)
		tmp = Float64(Float64(y * z) / Float64(t + Float64(y * b)));
	elseif (y <= 2.55e+209)
		tmp = Float64(Float64(x + Float64(y / Float64(t / z))) * Float64(1.0 / a));
	else
		tmp = Float64(z / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -2.55e+175)
		tmp = z / b;
	elseif (y <= 1.65e+75)
		tmp = x / ((a + 1.0) + (b * (y / t)));
	elseif (y <= 9.5e+166)
		tmp = (y * z) / (t + (y * b));
	elseif (y <= 2.55e+209)
		tmp = (x + (y / (t / z))) * (1.0 / a);
	else
		tmp = z / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -2.55e+175], N[(z / b), $MachinePrecision], If[LessEqual[y, 1.65e+75], N[(x / N[(N[(a + 1.0), $MachinePrecision] + N[(b * N[(y / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 9.5e+166], N[(N[(y * z), $MachinePrecision] / N[(t + N[(y * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.55e+209], N[(N[(x + N[(y / N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / a), $MachinePrecision]), $MachinePrecision], N[(z / b), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -2.55000000000000003e175 or 2.55000000000000011e209 < y

   1. Initial program 49.1%

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

      1. *-commutative 49.1%

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

      2. associate-/l* 47.1%

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

      3. associate-*l/ 45.9%

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

   3. Simplified 45.9%

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

   4. Taylor expanded in t around 0 72.3%

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

   if -2.55000000000000003e175 < y < 1.64999999999999999e75

   1. Initial program 89.8%

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

      1. *-commutative 89.8%

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

      2. associate-/l* 89.1%

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

      3. associate-*l/ 89.5%

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

   3. Simplified 89.5%

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

   4. Taylor expanded in x around inf 67.7%

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

   if 1.64999999999999999e75 < y < 9.49999999999999984e166

   1. Initial program 73.0%

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

      1. +-commutative 73.0%

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

      2. associate-*l/ 78.1%

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

      3. fma-def 78.1%

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

      4. associate-+l+ 78.1%

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

      5. +-commutative 78.1%

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

      6. associate-*l/ 78.0%

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

      7. fma-def 78.0%

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

   3. Simplified 78.0%

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

   4. Taylor expanded in a around 0 62.6%

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

   5. Taylor expanded in z around inf 62.9%

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

   6. Taylor expanded in y around 0 62.9%

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

   if 9.49999999999999984e166 < y < 2.55000000000000011e209

   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. *-commutative 55.2%

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

      2. associate-/l* 90.2%

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

      3. associate-*l/ 90.3%

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

   3. Simplified 90.3%

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

   4. Taylor expanded in a around inf 29.3%

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

      1. div-inv 29.1%

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

      2. +-commutative 29.1%

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

      3. associate-/l* 55.5%

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

   6. Applied egg-rr 55.5%

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

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.55 \cdot 10^{+175}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;y \leq 1.65 \cdot 10^{+75}:\\ \;\;\;\;\frac{x}{\left(a + 1\right) + b \cdot \frac{y}{t}}\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{+166}:\\ \;\;\;\;\frac{y \cdot z}{t + y \cdot b}\\ \mathbf{elif}\;y \leq 2.55 \cdot 10^{+209}:\\ \;\;\;\;\left(x + \frac{y}{\frac{t}{z}}\right) \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]

Alternative 6: 69.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -130000 \lor \neg \left(a \leq 2.8 \cdot 10^{+25}\right):\\ \;\;\;\;\frac{x + z \cdot \frac{y}{t}}{a + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{1 + y \cdot \frac{b}{t}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= a -130000.0) (not (<= a 2.8e+25)))
   (/ (+ x (* z (/ y t))) (+ a 1.0))
   (/ (+ x (/ (* y z) t)) (+ 1.0 (* y (/ b t))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((a <= -130000.0) || !(a <= 2.8e+25)) {
		tmp = (x + (z * (y / t))) / (a + 1.0);
	} else {
		tmp = (x + ((y * z) / t)) / (1.0 + (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 ((a <= (-130000.0d0)) .or. (.not. (a <= 2.8d+25))) then
        tmp = (x + (z * (y / t))) / (a + 1.0d0)
    else
        tmp = (x + ((y * z) / t)) / (1.0d0 + (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 ((a <= -130000.0) || !(a <= 2.8e+25)) {
		tmp = (x + (z * (y / t))) / (a + 1.0);
	} else {
		tmp = (x + ((y * z) / t)) / (1.0 + (y * (b / t)));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (a <= -130000.0) or not (a <= 2.8e+25):
		tmp = (x + (z * (y / t))) / (a + 1.0)
	else:
		tmp = (x + ((y * z) / t)) / (1.0 + (y * (b / t)))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((a <= -130000.0) || !(a <= 2.8e+25))
		tmp = Float64(Float64(x + Float64(z * Float64(y / t))) / Float64(a + 1.0));
	else
		tmp = Float64(Float64(x + Float64(Float64(y * z) / t)) / Float64(1.0 + Float64(y * Float64(b / t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((a <= -130000.0) || ~((a <= 2.8e+25)))
		tmp = (x + (z * (y / t))) / (a + 1.0);
	else
		tmp = (x + ((y * z) / t)) / (1.0 + (y * (b / t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[a, -130000.0], N[Not[LessEqual[a, 2.8e+25]], $MachinePrecision]], N[(N[(x + N[(z * N[(y / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(y * N[(b / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.3e5 or 2.8000000000000002e25 < a

   1. Initial program 77.9%

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

      1. *-commutative 77.9%

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

      2. associate-/l* 79.5%

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

      3. associate-*l/ 80.2%

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

   3. Simplified 80.2%

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

   4. Taylor expanded in y around 0 79.5%

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

   5. Taylor expanded in b around 0 65.4%

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

      1. +-commutative 65.4%

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

      2. associate-/l* 67.4%

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

      3. associate-/r/ 69.6%

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

   7. Simplified 69.6%

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

   if -1.3e5 < a < 2.8000000000000002e25

   1. Initial program 78.8%

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

      1. +-commutative 78.8%

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

      2. associate-*l/ 79.1%

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

      3. fma-def 79.1%

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

      4. associate-+l+ 79.1%

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

      5. +-commutative 79.1%

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

      6. associate-*l/ 78.7%

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

      7. fma-def 78.7%

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

   3. Simplified 78.7%

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

   4. Taylor expanded in a around 0 77.3%

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

      1. associate-/l* 77.1%

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

      2. div-inv 77.1%

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

   6. Applied egg-rr 77.1%

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

      1. associate-*r/ 77.1%

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

      2. *-rgt-identity 77.1%

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

      3. associate-/l* 77.3%

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

      4. associate-*r/ 77.1%

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

   8. Simplified 77.1%

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

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -130000 \lor \neg \left(a \leq 2.8 \cdot 10^{+25}\right):\\ \;\;\;\;\frac{x + z \cdot \frac{y}{t}}{a + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{1 + y \cdot \frac{b}{t}}\\ \end{array} \]

Alternative 7: 69.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -130000 \lor \neg \left(a \leq 2.95 \cdot 10^{+25}\right):\\ \;\;\;\;\frac{x + z \cdot \frac{y}{t}}{a + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{1 + \frac{y \cdot b}{t}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= a -130000.0) (not (<= a 2.95e+25)))
   (/ (+ x (* z (/ y t))) (+ a 1.0))
   (/ (+ x (/ (* y z) t)) (+ 1.0 (/ (* y b) t)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((a <= -130000.0) || !(a <= 2.95e+25)) {
		tmp = (x + (z * (y / t))) / (a + 1.0);
	} else {
		tmp = (x + ((y * z) / t)) / (1.0 + ((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 ((a <= (-130000.0d0)) .or. (.not. (a <= 2.95d+25))) then
        tmp = (x + (z * (y / t))) / (a + 1.0d0)
    else
        tmp = (x + ((y * z) / t)) / (1.0d0 + ((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 ((a <= -130000.0) || !(a <= 2.95e+25)) {
		tmp = (x + (z * (y / t))) / (a + 1.0);
	} else {
		tmp = (x + ((y * z) / t)) / (1.0 + ((y * b) / t));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (a <= -130000.0) or not (a <= 2.95e+25):
		tmp = (x + (z * (y / t))) / (a + 1.0)
	else:
		tmp = (x + ((y * z) / t)) / (1.0 + ((y * b) / t))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((a <= -130000.0) || !(a <= 2.95e+25))
		tmp = Float64(Float64(x + Float64(z * Float64(y / t))) / Float64(a + 1.0));
	else
		tmp = Float64(Float64(x + Float64(Float64(y * z) / t)) / Float64(1.0 + Float64(Float64(y * b) / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((a <= -130000.0) || ~((a <= 2.95e+25)))
		tmp = (x + (z * (y / t))) / (a + 1.0);
	else
		tmp = (x + ((y * z) / t)) / (1.0 + ((y * b) / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[a, -130000.0], N[Not[LessEqual[a, 2.95e+25]], $MachinePrecision]], N[(N[(x + N[(z * N[(y / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[(y * b), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.3e5 or 2.95e25 < a

   1. Initial program 77.9%

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

      1. *-commutative 77.9%

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

      2. associate-/l* 79.5%

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

      3. associate-*l/ 80.2%

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

   3. Simplified 80.2%

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

   4. Taylor expanded in y around 0 79.5%

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

   5. Taylor expanded in b around 0 65.4%

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

      1. +-commutative 65.4%

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

      2. associate-/l* 67.4%

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

      3. associate-/r/ 69.6%

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

   7. Simplified 69.6%

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

   if -1.3e5 < a < 2.95e25

   1. Initial program 78.8%

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

      1. +-commutative 78.8%

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

      2. associate-*l/ 79.1%

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

      3. fma-def 79.1%

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

      4. associate-+l+ 79.1%

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

      5. +-commutative 79.1%

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

      6. associate-*l/ 78.7%

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

      7. fma-def 78.7%

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

   3. Simplified 78.7%

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

   4. Taylor expanded in a around 0 77.3%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -130000 \lor \neg \left(a \leq 2.95 \cdot 10^{+25}\right):\\ \;\;\;\;\frac{x + z \cdot \frac{y}{t}}{a + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{1 + \frac{y \cdot b}{t}}\\ \end{array} \]

Alternative 8: 42.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{1}{\frac{1}{x}}\\ \mathbf{if}\;a \leq -1.65 \cdot 10^{+98}:\\ \;\;\;\;\frac{x}{a}\\ \mathbf{elif}\;a \leq -9.5 \cdot 10^{-99}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;a \leq -2.75 \cdot 10^{-127}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 4.1 \cdot 10^{-277}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;a \leq 10^{-42}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 9.8 \cdot 10^{+25}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ 1.0 (/ 1.0 x))))
   (if (<= a -1.65e+98)
     (/ x a)
     (if (<= a -9.5e-99)
       (/ z b)
       (if (<= a -2.75e-127)
         t_1
         (if (<= a 4.1e-277)
           (/ z b)
           (if (<= a 1e-42) t_1 (if (<= a 9.8e+25) (/ z b) (/ x a)))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = 1.0 / (1.0 / x);
	double tmp;
	if (a <= -1.65e+98) {
		tmp = x / a;
	} else if (a <= -9.5e-99) {
		tmp = z / b;
	} else if (a <= -2.75e-127) {
		tmp = t_1;
	} else if (a <= 4.1e-277) {
		tmp = z / b;
	} else if (a <= 1e-42) {
		tmp = t_1;
	} else if (a <= 9.8e+25) {
		tmp = z / b;
	} else {
		tmp = x / a;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 1.0d0 / (1.0d0 / x)
    if (a <= (-1.65d+98)) then
        tmp = x / a
    else if (a <= (-9.5d-99)) then
        tmp = z / b
    else if (a <= (-2.75d-127)) then
        tmp = t_1
    else if (a <= 4.1d-277) then
        tmp = z / b
    else if (a <= 1d-42) then
        tmp = t_1
    else if (a <= 9.8d+25) then
        tmp = z / b
    else
        tmp = x / a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = 1.0 / (1.0 / x);
	double tmp;
	if (a <= -1.65e+98) {
		tmp = x / a;
	} else if (a <= -9.5e-99) {
		tmp = z / b;
	} else if (a <= -2.75e-127) {
		tmp = t_1;
	} else if (a <= 4.1e-277) {
		tmp = z / b;
	} else if (a <= 1e-42) {
		tmp = t_1;
	} else if (a <= 9.8e+25) {
		tmp = z / b;
	} else {
		tmp = x / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = 1.0 / (1.0 / x)
	tmp = 0
	if a <= -1.65e+98:
		tmp = x / a
	elif a <= -9.5e-99:
		tmp = z / b
	elif a <= -2.75e-127:
		tmp = t_1
	elif a <= 4.1e-277:
		tmp = z / b
	elif a <= 1e-42:
		tmp = t_1
	elif a <= 9.8e+25:
		tmp = z / b
	else:
		tmp = x / a
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(1.0 / Float64(1.0 / x))
	tmp = 0.0
	if (a <= -1.65e+98)
		tmp = Float64(x / a);
	elseif (a <= -9.5e-99)
		tmp = Float64(z / b);
	elseif (a <= -2.75e-127)
		tmp = t_1;
	elseif (a <= 4.1e-277)
		tmp = Float64(z / b);
	elseif (a <= 1e-42)
		tmp = t_1;
	elseif (a <= 9.8e+25)
		tmp = Float64(z / b);
	else
		tmp = Float64(x / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = 1.0 / (1.0 / x);
	tmp = 0.0;
	if (a <= -1.65e+98)
		tmp = x / a;
	elseif (a <= -9.5e-99)
		tmp = z / b;
	elseif (a <= -2.75e-127)
		tmp = t_1;
	elseif (a <= 4.1e-277)
		tmp = z / b;
	elseif (a <= 1e-42)
		tmp = t_1;
	elseif (a <= 9.8e+25)
		tmp = z / b;
	else
		tmp = x / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(1.0 / N[(1.0 / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -1.65e+98], N[(x / a), $MachinePrecision], If[LessEqual[a, -9.5e-99], N[(z / b), $MachinePrecision], If[LessEqual[a, -2.75e-127], t$95$1, If[LessEqual[a, 4.1e-277], N[(z / b), $MachinePrecision], If[LessEqual[a, 1e-42], t$95$1, If[LessEqual[a, 9.8e+25], N[(z / b), $MachinePrecision], N[(x / a), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.65000000000000014e98 or 9.8000000000000002e25 < a

   1. Initial program 79.4%

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

      1. *-commutative 79.4%

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

      2. associate-/l* 80.4%

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

      3. associate-*l/ 81.3%

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

   3. Simplified 81.3%

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

   4. Taylor expanded in x around inf 60.7%

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

   5. Taylor expanded in a around inf 54.8%

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

   if -1.65000000000000014e98 < a < -9.5000000000000008e-99 or -2.75000000000000018e-127 < a < 4.09999999999999989e-277 or 1.00000000000000004e-42 < a < 9.8000000000000002e25

   1. Initial program 76.6%

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

      1. *-commutative 76.6%

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

      2. associate-/l* 77.9%

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

      3. associate-*l/ 77.3%

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

   3. Simplified 77.3%

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

   4. Taylor expanded in t around 0 48.4%

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

   if -9.5000000000000008e-99 < a < -2.75000000000000018e-127 or 4.09999999999999989e-277 < a < 1.00000000000000004e-42

   1. Initial program 80.5%

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

      1. *-commutative 80.5%

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

      2. associate-/l* 80.4%

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

      3. associate-*l/ 80.4%

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

   3. Simplified 80.4%

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

   4. Taylor expanded in x around inf 59.4%

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

      1. clear-num 59.3%

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

      2. inv-pow 59.3%

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

      3. associate-+l+ 59.3%

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

      4. +-commutative 59.3%

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

      5. fma-def 59.3%

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

   6. Applied egg-rr 59.3%

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

      1. unpow-1 59.3%

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

   8. Simplified 59.3%

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

   9. Taylor expanded in a around 0 59.3%

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

   10. Taylor expanded in y around 0 49.7%

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

4. Final simplification 51.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.65 \cdot 10^{+98}:\\ \;\;\;\;\frac{x}{a}\\ \mathbf{elif}\;a \leq -9.5 \cdot 10^{-99}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;a \leq -2.75 \cdot 10^{-127}:\\ \;\;\;\;\frac{1}{\frac{1}{x}}\\ \mathbf{elif}\;a \leq 4.1 \cdot 10^{-277}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;a \leq 10^{-42}:\\ \;\;\;\;\frac{1}{\frac{1}{x}}\\ \mathbf{elif}\;a \leq 9.8 \cdot 10^{+25}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a}\\ \end{array} \]

Alternative 9: 64.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3600 \lor \neg \left(t \leq 6.2 \cdot 10^{-60}\right):\\ \;\;\;\;\frac{x + z \cdot \frac{y}{t}}{a + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= t -3600.0) (not (<= t 6.2e-60)))
   (/ (+ x (* z (/ y t))) (+ a 1.0))
   (/ z b)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((t <= -3600.0) || !(t <= 6.2e-60)) {
		tmp = (x + (z * (y / t))) / (a + 1.0);
	} 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 ((t <= (-3600.0d0)) .or. (.not. (t <= 6.2d-60))) then
        tmp = (x + (z * (y / t))) / (a + 1.0d0)
    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 ((t <= -3600.0) || !(t <= 6.2e-60)) {
		tmp = (x + (z * (y / t))) / (a + 1.0);
	} else {
		tmp = z / b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (t <= -3600.0) or not (t <= 6.2e-60):
		tmp = (x + (z * (y / t))) / (a + 1.0)
	else:
		tmp = z / b
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((t <= -3600.0) || !(t <= 6.2e-60))
		tmp = Float64(Float64(x + Float64(z * Float64(y / t))) / Float64(a + 1.0));
	else
		tmp = Float64(z / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((t <= -3600.0) || ~((t <= 6.2e-60)))
		tmp = (x + (z * (y / t))) / (a + 1.0);
	else
		tmp = z / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[t, -3600.0], N[Not[LessEqual[t, 6.2e-60]], $MachinePrecision]], N[(N[(x + N[(z * N[(y / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a + 1.0), $MachinePrecision]), $MachinePrecision], N[(z / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -3600 or 6.19999999999999976e-60 < t

   1. Initial program 86.6%

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

      1. *-commutative 86.6%

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

      2. associate-/l* 93.0%

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

      3. associate-*l/ 96.0%

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

   3. Simplified 96.0%

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

   4. Taylor expanded in y around 0 93.0%

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

   5. Taylor expanded in b around 0 77.4%

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

      1. +-commutative 77.4%

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

      2. associate-/l* 81.5%

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

      3. associate-/r/ 83.2%

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

   7. Simplified 83.2%

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

   if -3600 < t < 6.19999999999999976e-60

   1. Initial program 66.8%

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

      1. *-commutative 66.8%

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

      2. associate-/l* 59.9%

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

      3. associate-*l/ 55.9%

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

   3. Simplified 55.9%

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

   4. Taylor expanded in t around 0 60.5%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3600 \lor \neg \left(t \leq 6.2 \cdot 10^{-60}\right):\\ \;\;\;\;\frac{x + z \cdot \frac{y}{t}}{a + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]

Alternative 10: 55.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3.2 \cdot 10^{+18} \lor \neg \left(t \leq 6.8 \cdot 10^{-60}\right):\\ \;\;\;\;\frac{x}{a + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= t -3.2e+18) (not (<= t 6.8e-60))) (/ x (+ a 1.0)) (/ z b)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((t <= -3.2e+18) || !(t <= 6.8e-60)) {
		tmp = x / (a + 1.0);
	} 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 ((t <= (-3.2d+18)) .or. (.not. (t <= 6.8d-60))) then
        tmp = x / (a + 1.0d0)
    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 ((t <= -3.2e+18) || !(t <= 6.8e-60)) {
		tmp = x / (a + 1.0);
	} else {
		tmp = z / b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (t <= -3.2e+18) or not (t <= 6.8e-60):
		tmp = x / (a + 1.0)
	else:
		tmp = z / b
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((t <= -3.2e+18) || !(t <= 6.8e-60))
		tmp = Float64(x / Float64(a + 1.0));
	else
		tmp = Float64(z / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((t <= -3.2e+18) || ~((t <= 6.8e-60)))
		tmp = x / (a + 1.0);
	else
		tmp = z / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[t, -3.2e+18], N[Not[LessEqual[t, 6.8e-60]], $MachinePrecision]], N[(x / N[(a + 1.0), $MachinePrecision]), $MachinePrecision], N[(z / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -3.2e18 or 6.80000000000000013e-60 < t

   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. *-commutative 86.7%

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

      2. associate-/l* 93.4%

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

      3. associate-*l/ 96.5%

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

   3. Simplified 96.5%

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

   4. Taylor expanded in t around inf 62.7%

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

   if -3.2e18 < t < 6.80000000000000013e-60

   1. Initial program 67.7%

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

      1. *-commutative 67.7%

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

      2. associate-/l* 61.2%

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

      3. associate-*l/ 57.4%

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

   3. Simplified 57.4%

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

   4. Taylor expanded in t around 0 59.2%

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

4. Final simplification 61.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.2 \cdot 10^{+18} \lor \neg \left(t \leq 6.8 \cdot 10^{-60}\right):\\ \;\;\;\;\frac{x}{a + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]

Alternative 11: 42.6% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.06 \cdot 10^{+99}:\\ \;\;\;\;\frac{x}{a}\\ \mathbf{elif}\;a \leq 6.8 \cdot 10^{+26}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= a -1.06e+99) (/ x a) (if (<= a 6.8e+26) (/ z b) (/ x a))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= -1.06e+99) {
		tmp = x / a;
	} else if (a <= 6.8e+26) {
		tmp = z / b;
	} else {
		tmp = x / a;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-1.06d+99)) then
        tmp = x / a
    else if (a <= 6.8d+26) then
        tmp = z / b
    else
        tmp = x / a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= -1.06e+99) {
		tmp = x / a;
	} else if (a <= 6.8e+26) {
		tmp = z / b;
	} else {
		tmp = x / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if a <= -1.06e+99:
		tmp = x / a
	elif a <= 6.8e+26:
		tmp = z / b
	else:
		tmp = x / a
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (a <= -1.06e+99)
		tmp = Float64(x / a);
	elseif (a <= 6.8e+26)
		tmp = Float64(z / b);
	else
		tmp = Float64(x / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (a <= -1.06e+99)
		tmp = x / a;
	elseif (a <= 6.8e+26)
		tmp = z / b;
	else
		tmp = x / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[a, -1.06e+99], N[(x / a), $MachinePrecision], If[LessEqual[a, 6.8e+26], N[(z / b), $MachinePrecision], N[(x / a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.05999999999999999e99 or 6.8000000000000005e26 < a

   1. Initial program 79.4%

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

      1. *-commutative 79.4%

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

      2. associate-/l* 80.4%

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

      3. associate-*l/ 81.3%

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

   3. Simplified 81.3%

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

   4. Taylor expanded in x around inf 60.7%

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

   5. Taylor expanded in a around inf 54.8%

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

   if -1.05999999999999999e99 < a < 6.8000000000000005e26

   1. Initial program 77.8%

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

      1. *-commutative 77.8%

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

      2. associate-/l* 78.7%

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

      3. associate-*l/ 78.3%

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

   3. Simplified 78.3%

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

   4. Taylor expanded in t around 0 41.7%

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

4. Final simplification 46.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.06 \cdot 10^{+99}:\\ \;\;\;\;\frac{x}{a}\\ \mathbf{elif}\;a \leq 6.8 \cdot 10^{+26}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a}\\ \end{array} \]

Alternative 12: 26.0% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \frac{x}{a} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.4%

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

   1. *-commutative 78.4%

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

   2. associate-/l* 79.3%

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

   3. associate-*l/ 79.4%

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

3. Simplified 79.4%

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

4. Taylor expanded in x around inf 52.3%

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

5. Taylor expanded in a around inf 25.7%

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

6. Final simplification 25.7%

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

Developer target: 79.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \mathbf{if}\;t < -1.3659085366310088 \cdot 10^{-271}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t < 3.036967103737246 \cdot 10^{-130}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1
         (* 1.0 (* (+ x (* (/ y t) z)) (/ 1.0 (+ (+ a 1.0) (* (/ y t) b)))))))
   (if (< t -1.3659085366310088e-271)
     t_1
     (if (< t 3.036967103737246e-130) (/ z b) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = 1.0 * ((x + ((y / t) * z)) * (1.0 / ((a + 1.0) + ((y / t) * b))));
	double tmp;
	if (t < -1.3659085366310088e-271) {
		tmp = t_1;
	} else if (t < 3.036967103737246e-130) {
		tmp = z / b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 1.0d0 * ((x + ((y / t) * z)) * (1.0d0 / ((a + 1.0d0) + ((y / t) * b))))
    if (t < (-1.3659085366310088d-271)) then
        tmp = t_1
    else if (t < 3.036967103737246d-130) then
        tmp = z / b
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = 1.0 * ((x + ((y / t) * z)) * (1.0 / ((a + 1.0) + ((y / t) * b))));
	double tmp;
	if (t < -1.3659085366310088e-271) {
		tmp = t_1;
	} else if (t < 3.036967103737246e-130) {
		tmp = z / b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = 1.0 * ((x + ((y / t) * z)) * (1.0 / ((a + 1.0) + ((y / t) * b))))
	tmp = 0
	if t < -1.3659085366310088e-271:
		tmp = t_1
	elif t < 3.036967103737246e-130:
		tmp = z / b
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(1.0 * Float64(Float64(x + Float64(Float64(y / t) * z)) * Float64(1.0 / Float64(Float64(a + 1.0) + Float64(Float64(y / t) * b)))))
	tmp = 0.0
	if (t < -1.3659085366310088e-271)
		tmp = t_1;
	elseif (t < 3.036967103737246e-130)
		tmp = Float64(z / b);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = 1.0 * ((x + ((y / t) * z)) * (1.0 / ((a + 1.0) + ((y / t) * b))));
	tmp = 0.0;
	if (t < -1.3659085366310088e-271)
		tmp = t_1;
	elseif (t < 3.036967103737246e-130)
		tmp = z / b;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(1.0 * N[(N[(x + N[(N[(y / t), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(N[(a + 1.0), $MachinePrecision] + N[(N[(y / t), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Less[t, -1.3659085366310088e-271], t$95$1, If[Less[t, 3.036967103737246e-130], N[(z / b), $MachinePrecision], t$95$1]]]

Reproduce

herbie shell --seed 2023213 
(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))))