Diagrams.Solve.Tridiagonal:solveTriDiagonal from diagrams-solve-0.1, A

Percentage Accurate: 85.1% → 93.0%

Time: 12.8s

Alternatives: 10

Speedup: 0.3×

Specification

Math

\[\begin{array}{l} \\ \frac{x - y \cdot z}{t - a \cdot z} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t, a)
    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
    code = (x - (y * z)) / (t - (a * z))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 85.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x - y \cdot z}{t - a \cdot z} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t, a)
    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
    code = (x - (y * z)) / (t - (a * z))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 93.0% accurate, 0.0× speedup

Math

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

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma z (- a) t)))
   (if (<= (/ (- x (* y z)) (- t (* z a))) INFINITY)
     (fma -1.0 (* y (/ z t_1)) (/ x t_1))
     (/ y a))))

C

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

Julia

function code(x, y, z, t, a)
	t_1 = fma(z, Float64(-a), t)
	tmp = 0.0
	if (Float64(Float64(x - Float64(y * z)) / Float64(t - Float64(z * a))) <= Inf)
		tmp = fma(-1.0, Float64(y * Float64(z / t_1)), Float64(x / t_1));
	else
		tmp = Float64(y / a);
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < +inf.0

   1. Initial program 90.5%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Step-by-step derivation

      1. *-commutative 90.5%

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

   3. Simplified 90.5%

      \[\leadsto \color{blue}{\frac{x - y \cdot z}{t - z \cdot a}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 90.5%

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

      1. fma-define 90.5%

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

      2. associate-/l* 94.5%

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

      3. sub-neg 94.5%

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

      4. mul-1-neg 94.5%

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

      5. +-commutative 94.5%

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

      6. mul-1-neg 94.5%

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

      7. distribute-rgt-neg-in 94.5%

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

      8. fma-undefine 94.5%

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

      9. fma-undefine 94.5%

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

      10. distribute-rgt-neg-in 94.5%

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

      11. mul-1-neg 94.5%

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

      12. associate-*r* 94.5%

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

      13. neg-mul-1 94.5%

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

      14. *-commutative 94.5%

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

      15. fma-define 94.5%

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

      16. cancel-sign-sub-inv 94.5%

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

      17. *-commutative 94.5%

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

      18. +-commutative 94.5%

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

      19. fma-define 94.5%

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

   7. Simplified 94.5%

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

   if +inf.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z)))

   1. Initial program 0.0%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Step-by-step derivation

      1. *-commutative 0.0%

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

   3. Simplified 0.0%

      \[\leadsto \color{blue}{\frac{x - y \cdot z}{t - z \cdot a}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 100.0%

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

4. Final simplification 94.7%

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

Alternative 2: 90.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t - z \cdot a\\ t_2 := \frac{x - y \cdot z}{t\_1}\\ \mathbf{if}\;t\_2 \leq -\infty \lor \neg \left(t\_2 \leq 5 \cdot 10^{+300}\right):\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot z}{z \cdot a - t} + \frac{x}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- t (* z a))) (t_2 (/ (- x (* y z)) t_1)))
   (if (or (<= t_2 (- INFINITY)) (not (<= t_2 5e+300)))
     (/ (- y (/ x z)) a)
     (+ (/ (* y z) (- (* z a) t)) (/ x t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = t - (z * a);
	double t_2 = (x - (y * z)) / t_1;
	double tmp;
	if ((t_2 <= -((double) INFINITY)) || !(t_2 <= 5e+300)) {
		tmp = (y - (x / z)) / a;
	} else {
		tmp = ((y * z) / ((z * a) - t)) + (x / t_1);
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = t - (z * a);
	double t_2 = (x - (y * z)) / t_1;
	double tmp;
	if ((t_2 <= -Double.POSITIVE_INFINITY) || !(t_2 <= 5e+300)) {
		tmp = (y - (x / z)) / a;
	} else {
		tmp = ((y * z) / ((z * a) - t)) + (x / t_1);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = t - (z * a)
	t_2 = (x - (y * z)) / t_1
	tmp = 0
	if (t_2 <= -math.inf) or not (t_2 <= 5e+300):
		tmp = (y - (x / z)) / a
	else:
		tmp = ((y * z) / ((z * a) - t)) + (x / t_1)
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(t - Float64(z * a))
	t_2 = Float64(Float64(x - Float64(y * z)) / t_1)
	tmp = 0.0
	if ((t_2 <= Float64(-Inf)) || !(t_2 <= 5e+300))
		tmp = Float64(Float64(y - Float64(x / z)) / a);
	else
		tmp = Float64(Float64(Float64(y * z) / Float64(Float64(z * a) - t)) + Float64(x / t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = t - (z * a);
	t_2 = (x - (y * z)) / t_1;
	tmp = 0.0;
	if ((t_2 <= -Inf) || ~((t_2 <= 5e+300)))
		tmp = (y - (x / z)) / a;
	else
		tmp = ((y * z) / ((z * a) - t)) + (x / t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, If[Or[LessEqual[t$95$2, (-Infinity)], N[Not[LessEqual[t$95$2, 5e+300]], $MachinePrecision]], N[(N[(y - N[(x / z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(N[(y * z), $MachinePrecision] / N[(N[(z * a), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] + N[(x / t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -inf.0 or 5.00000000000000026e300 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z)))

   1. Initial program 41.6%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Step-by-step derivation

      1. *-commutative 41.6%

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

   3. Simplified 41.6%

      \[\leadsto \color{blue}{\frac{x - y \cdot z}{t - z \cdot a}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 41.6%

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

      1. fma-define 41.6%

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

      2. associate-/l* 78.3%

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

      3. sub-neg 78.3%

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

      4. mul-1-neg 78.3%

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

      5. +-commutative 78.3%

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

      6. mul-1-neg 78.3%

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

      7. distribute-rgt-neg-in 78.3%

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

      8. fma-undefine 78.3%

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

      9. fma-undefine 78.3%

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

      10. distribute-rgt-neg-in 78.3%

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

      11. mul-1-neg 78.3%

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

      12. associate-*r* 78.3%

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

      13. neg-mul-1 78.3%

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

      14. *-commutative 78.3%

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

      15. fma-define 78.3%

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

      16. cancel-sign-sub-inv 78.3%

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

      17. *-commutative 78.3%

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

      18. +-commutative 78.3%

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

      19. fma-define 78.3%

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

   7. Simplified 78.3%

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

   8. Taylor expanded in a around inf 83.7%

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

      1. mul-1-neg 83.7%

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

      2. unsub-neg 83.7%

         \[\leadsto \frac{\color{blue}{y - \frac{x}{z}}}{a} \]

   10. Simplified 83.7%

       \[\leadsto \color{blue}{\frac{y - \frac{x}{z}}{a}} \]

   if -inf.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 5.00000000000000026e300

   1. Initial program 95.3%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Step-by-step derivation

      1. *-commutative 95.3%

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

   3. Simplified 95.3%

      \[\leadsto \color{blue}{\frac{x - y \cdot z}{t - z \cdot a}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 95.3%

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

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y \cdot z}{t - z \cdot a} \leq -\infty \lor \neg \left(\frac{x - y \cdot z}{t - z \cdot a} \leq 5 \cdot 10^{+300}\right):\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot z}{z \cdot a - t} + \frac{x}{t - z \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - y \cdot z}{t - z \cdot a}\\ \mathbf{if}\;t\_1 \leq -\infty \lor \neg \left(t\_1 \leq 5 \cdot 10^{+300}\right):\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- x (* y z)) (- t (* z a)))))
   (if (or (<= t_1 (- INFINITY)) (not (<= t_1 5e+300)))
     (/ (- y (/ x z)) a)
     t_1)))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (x - (y * z)) / (t - (z * a));
	double tmp;
	if ((t_1 <= -((double) INFINITY)) || !(t_1 <= 5e+300)) {
		tmp = (y - (x / z)) / a;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (x - (y * z)) / (t - (z * a));
	double tmp;
	if ((t_1 <= -Double.POSITIVE_INFINITY) || !(t_1 <= 5e+300)) {
		tmp = (y - (x / z)) / a;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (x - (y * z)) / (t - (z * a))
	tmp = 0
	if (t_1 <= -math.inf) or not (t_1 <= 5e+300):
		tmp = (y - (x / z)) / a
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(x - Float64(y * z)) / Float64(t - Float64(z * a)))
	tmp = 0.0
	if ((t_1 <= Float64(-Inf)) || !(t_1 <= 5e+300))
		tmp = Float64(Float64(y - Float64(x / z)) / a);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (x - (y * z)) / (t - (z * a));
	tmp = 0.0;
	if ((t_1 <= -Inf) || ~((t_1 <= 5e+300)))
		tmp = (y - (x / z)) / a;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, (-Infinity)], N[Not[LessEqual[t$95$1, 5e+300]], $MachinePrecision]], N[(N[(y - N[(x / z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], t$95$1]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -inf.0 or 5.00000000000000026e300 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z)))

   1. Initial program 41.6%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Step-by-step derivation

      1. *-commutative 41.6%

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

   3. Simplified 41.6%

      \[\leadsto \color{blue}{\frac{x - y \cdot z}{t - z \cdot a}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 41.6%

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

      1. fma-define 41.6%

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

      2. associate-/l* 78.3%

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

      3. sub-neg 78.3%

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

      4. mul-1-neg 78.3%

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

      5. +-commutative 78.3%

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

      6. mul-1-neg 78.3%

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

      7. distribute-rgt-neg-in 78.3%

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

      8. fma-undefine 78.3%

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

      9. fma-undefine 78.3%

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

      10. distribute-rgt-neg-in 78.3%

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

      11. mul-1-neg 78.3%

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

      12. associate-*r* 78.3%

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

      13. neg-mul-1 78.3%

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

      14. *-commutative 78.3%

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

      15. fma-define 78.3%

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

      16. cancel-sign-sub-inv 78.3%

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

      17. *-commutative 78.3%

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

      18. +-commutative 78.3%

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

      19. fma-define 78.3%

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

   7. Simplified 78.3%

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

   8. Taylor expanded in a around inf 83.7%

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

      1. mul-1-neg 83.7%

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

      2. unsub-neg 83.7%

         \[\leadsto \frac{\color{blue}{y - \frac{x}{z}}}{a} \]

   10. Simplified 83.7%

       \[\leadsto \color{blue}{\frac{y - \frac{x}{z}}{a}} \]

   if -inf.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 5.00000000000000026e300

   1. Initial program 95.3%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y \cdot z}{t - z \cdot a} \leq -\infty \lor \neg \left(\frac{x - y \cdot z}{t - z \cdot a} \leq 5 \cdot 10^{+300}\right):\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y \cdot z}{t - z \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 65.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -7.4 \cdot 10^{+143}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -7.5 \cdot 10^{-110}:\\ \;\;\;\;\frac{x - y \cdot z}{t}\\ \mathbf{elif}\;z \leq 1800000:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -7.4e+143)
   (/ y a)
   (if (<= z -7.5e-110)
     (/ (- x (* y z)) t)
     (if (<= z 1800000.0) (/ x (- t (* z a))) (/ y a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -7.4e+143) {
		tmp = y / a;
	} else if (z <= -7.5e-110) {
		tmp = (x - (y * z)) / t;
	} else if (z <= 1800000.0) {
		tmp = x / (t - (z * a));
	} else {
		tmp = y / a;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    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) :: tmp
    if (z <= (-7.4d+143)) then
        tmp = y / a
    else if (z <= (-7.5d-110)) then
        tmp = (x - (y * z)) / t
    else if (z <= 1800000.0d0) then
        tmp = x / (t - (z * a))
    else
        tmp = y / a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -7.4e+143) {
		tmp = y / a;
	} else if (z <= -7.5e-110) {
		tmp = (x - (y * z)) / t;
	} else if (z <= 1800000.0) {
		tmp = x / (t - (z * a));
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -7.4e+143:
		tmp = y / a
	elif z <= -7.5e-110:
		tmp = (x - (y * z)) / t
	elif z <= 1800000.0:
		tmp = x / (t - (z * a))
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -7.4e+143)
		tmp = Float64(y / a);
	elseif (z <= -7.5e-110)
		tmp = Float64(Float64(x - Float64(y * z)) / t);
	elseif (z <= 1800000.0)
		tmp = Float64(x / Float64(t - Float64(z * a)));
	else
		tmp = Float64(y / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -7.4e+143)
		tmp = y / a;
	elseif (z <= -7.5e-110)
		tmp = (x - (y * z)) / t;
	elseif (z <= 1800000.0)
		tmp = x / (t - (z * a));
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -7.4e+143], N[(y / a), $MachinePrecision], If[LessEqual[z, -7.5e-110], N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[z, 1800000.0], N[(x / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y / a), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -7.4000000000000003e143 or 1.8e6 < z

   1. Initial program 65.8%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Step-by-step derivation

      1. *-commutative 65.8%

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

   3. Simplified 65.8%

      \[\leadsto \color{blue}{\frac{x - y \cdot z}{t - z \cdot a}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 69.0%

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

   if -7.4000000000000003e143 < z < -7.50000000000000053e-110

   1. Initial program 91.5%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Step-by-step derivation

      1. *-commutative 91.5%

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

   3. Simplified 91.5%

      \[\leadsto \color{blue}{\frac{x - y \cdot z}{t - z \cdot a}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf 64.0%

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

   if -7.50000000000000053e-110 < z < 1.8e6

   1. Initial program 99.8%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Step-by-step derivation

      1. *-commutative 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{x - y \cdot z}{t - z \cdot a}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 75.0%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.4 \cdot 10^{+143}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -7.5 \cdot 10^{-110}:\\ \;\;\;\;\frac{x - y \cdot z}{t}\\ \mathbf{elif}\;z \leq 1800000:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y - \frac{x}{z}}{a}\\ \mathbf{if}\;z \leq -2.95 \cdot 10^{+35}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -1.22 \cdot 10^{-107}:\\ \;\;\;\;\frac{x - y \cdot z}{t}\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{-30}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- y (/ x z)) a)))
   (if (<= z -2.95e+35)
     t_1
     (if (<= z -1.22e-107)
       (/ (- x (* y z)) t)
       (if (<= z 2.4e-30) (/ x (- t (* z a))) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (y - (x / z)) / a;
	double tmp;
	if (z <= -2.95e+35) {
		tmp = t_1;
	} else if (z <= -1.22e-107) {
		tmp = (x - (y * z)) / t;
	} else if (z <= 2.4e-30) {
		tmp = x / (t - (z * a));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    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) :: t_1
    real(8) :: tmp
    t_1 = (y - (x / z)) / a
    if (z <= (-2.95d+35)) then
        tmp = t_1
    else if (z <= (-1.22d-107)) then
        tmp = (x - (y * z)) / t
    else if (z <= 2.4d-30) then
        tmp = x / (t - (z * 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 t_1 = (y - (x / z)) / a;
	double tmp;
	if (z <= -2.95e+35) {
		tmp = t_1;
	} else if (z <= -1.22e-107) {
		tmp = (x - (y * z)) / t;
	} else if (z <= 2.4e-30) {
		tmp = x / (t - (z * a));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (y - (x / z)) / a
	tmp = 0
	if z <= -2.95e+35:
		tmp = t_1
	elif z <= -1.22e-107:
		tmp = (x - (y * z)) / t
	elif z <= 2.4e-30:
		tmp = x / (t - (z * a))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(y - Float64(x / z)) / a)
	tmp = 0.0
	if (z <= -2.95e+35)
		tmp = t_1;
	elseif (z <= -1.22e-107)
		tmp = Float64(Float64(x - Float64(y * z)) / t);
	elseif (z <= 2.4e-30)
		tmp = Float64(x / Float64(t - Float64(z * a)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (y - (x / z)) / a;
	tmp = 0.0;
	if (z <= -2.95e+35)
		tmp = t_1;
	elseif (z <= -1.22e-107)
		tmp = (x - (y * z)) / t;
	elseif (z <= 2.4e-30)
		tmp = x / (t - (z * a));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y - N[(x / z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]}, If[LessEqual[z, -2.95e+35], t$95$1, If[LessEqual[z, -1.22e-107], N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[z, 2.4e-30], N[(x / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.94999999999999993e35 or 2.39999999999999985e-30 < z

   1. Initial program 73.8%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Step-by-step derivation

      1. *-commutative 73.8%

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

   3. Simplified 73.8%

      \[\leadsto \color{blue}{\frac{x - y \cdot z}{t - z \cdot a}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 73.8%

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

      1. fma-define 73.8%

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

      2. associate-/l* 83.9%

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

      3. sub-neg 83.9%

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

      4. mul-1-neg 83.9%

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

      5. +-commutative 83.9%

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

      6. mul-1-neg 83.9%

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

      7. distribute-rgt-neg-in 83.9%

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

      8. fma-undefine 83.9%

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

      9. fma-undefine 83.9%

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

      10. distribute-rgt-neg-in 83.9%

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

      11. mul-1-neg 83.9%

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

      12. associate-*r* 83.9%

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

      13. neg-mul-1 83.9%

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

      14. *-commutative 83.9%

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

      15. fma-define 83.9%

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

      16. cancel-sign-sub-inv 83.9%

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

      17. *-commutative 83.9%

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

      18. +-commutative 83.9%

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

      19. fma-define 83.9%

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

   7. Simplified 83.9%

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

   8. Taylor expanded in a around inf 73.4%

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

      1. mul-1-neg 73.4%

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

      2. unsub-neg 73.4%

         \[\leadsto \frac{\color{blue}{y - \frac{x}{z}}}{a} \]

   10. Simplified 73.4%

       \[\leadsto \color{blue}{\frac{y - \frac{x}{z}}{a}} \]

   if -2.94999999999999993e35 < z < -1.22000000000000001e-107

   1. Initial program 94.4%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Step-by-step derivation

      1. *-commutative 94.4%

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

   3. Simplified 94.4%

      \[\leadsto \color{blue}{\frac{x - y \cdot z}{t - z \cdot a}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf 73.9%

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

   if -1.22000000000000001e-107 < z < 2.39999999999999985e-30

   1. Initial program 99.8%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Step-by-step derivation

      1. *-commutative 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{x - y \cdot z}{t - z \cdot a}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 78.5%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.95 \cdot 10^{+35}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{elif}\;z \leq -1.22 \cdot 10^{-107}:\\ \;\;\;\;\frac{x - y \cdot z}{t}\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{-30}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 72.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y - \frac{x}{z}}{a}\\ \mathbf{if}\;z \leq -4.2 \cdot 10^{+34}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -1.05 \cdot 10^{-105}:\\ \;\;\;\;\frac{x}{t} - y \cdot \frac{z}{t}\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{-31}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- y (/ x z)) a)))
   (if (<= z -4.2e+34)
     t_1
     (if (<= z -1.05e-105)
       (- (/ x t) (* y (/ z t)))
       (if (<= z 2.1e-31) (/ x (- t (* z a))) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (y - (x / z)) / a;
	double tmp;
	if (z <= -4.2e+34) {
		tmp = t_1;
	} else if (z <= -1.05e-105) {
		tmp = (x / t) - (y * (z / t));
	} else if (z <= 2.1e-31) {
		tmp = x / (t - (z * a));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    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) :: t_1
    real(8) :: tmp
    t_1 = (y - (x / z)) / a
    if (z <= (-4.2d+34)) then
        tmp = t_1
    else if (z <= (-1.05d-105)) then
        tmp = (x / t) - (y * (z / t))
    else if (z <= 2.1d-31) then
        tmp = x / (t - (z * 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 t_1 = (y - (x / z)) / a;
	double tmp;
	if (z <= -4.2e+34) {
		tmp = t_1;
	} else if (z <= -1.05e-105) {
		tmp = (x / t) - (y * (z / t));
	} else if (z <= 2.1e-31) {
		tmp = x / (t - (z * a));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (y - (x / z)) / a
	tmp = 0
	if z <= -4.2e+34:
		tmp = t_1
	elif z <= -1.05e-105:
		tmp = (x / t) - (y * (z / t))
	elif z <= 2.1e-31:
		tmp = x / (t - (z * a))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(y - Float64(x / z)) / a)
	tmp = 0.0
	if (z <= -4.2e+34)
		tmp = t_1;
	elseif (z <= -1.05e-105)
		tmp = Float64(Float64(x / t) - Float64(y * Float64(z / t)));
	elseif (z <= 2.1e-31)
		tmp = Float64(x / Float64(t - Float64(z * a)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (y - (x / z)) / a;
	tmp = 0.0;
	if (z <= -4.2e+34)
		tmp = t_1;
	elseif (z <= -1.05e-105)
		tmp = (x / t) - (y * (z / t));
	elseif (z <= 2.1e-31)
		tmp = x / (t - (z * a));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y - N[(x / z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]}, If[LessEqual[z, -4.2e+34], t$95$1, If[LessEqual[z, -1.05e-105], N[(N[(x / t), $MachinePrecision] - N[(y * N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.1e-31], N[(x / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.20000000000000035e34 or 2.09999999999999991e-31 < z

   1. Initial program 73.8%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Step-by-step derivation

      1. *-commutative 73.8%

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

   3. Simplified 73.8%

      \[\leadsto \color{blue}{\frac{x - y \cdot z}{t - z \cdot a}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 73.8%

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

      1. fma-define 73.8%

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

      2. associate-/l* 83.9%

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

      3. sub-neg 83.9%

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

      4. mul-1-neg 83.9%

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

      5. +-commutative 83.9%

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

      6. mul-1-neg 83.9%

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

      7. distribute-rgt-neg-in 83.9%

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

      8. fma-undefine 83.9%

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

      9. fma-undefine 83.9%

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

      10. distribute-rgt-neg-in 83.9%

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

      11. mul-1-neg 83.9%

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

      12. associate-*r* 83.9%

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

      13. neg-mul-1 83.9%

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

      14. *-commutative 83.9%

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

      15. fma-define 83.9%

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

      16. cancel-sign-sub-inv 83.9%

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

      17. *-commutative 83.9%

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

      18. +-commutative 83.9%

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

      19. fma-define 83.9%

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

   7. Simplified 83.9%

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

   8. Taylor expanded in a around inf 73.4%

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

      1. mul-1-neg 73.4%

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

      2. unsub-neg 73.4%

         \[\leadsto \frac{\color{blue}{y - \frac{x}{z}}}{a} \]

   10. Simplified 73.4%

       \[\leadsto \color{blue}{\frac{y - \frac{x}{z}}{a}} \]

   if -4.20000000000000035e34 < z < -1.05e-105

   1. Initial program 94.4%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Step-by-step derivation

      1. *-commutative 94.4%

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

   3. Simplified 94.4%

      \[\leadsto \color{blue}{\frac{x - y \cdot z}{t - z \cdot a}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 94.4%

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

      1. fma-define 94.4%

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

      2. associate-/l* 99.6%

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

      3. sub-neg 99.6%

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

      4. mul-1-neg 99.6%

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

      5. +-commutative 99.6%

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

      6. mul-1-neg 99.6%

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

      7. distribute-rgt-neg-in 99.6%

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

      8. fma-undefine 99.6%

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

      9. fma-undefine 99.6%

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

      10. distribute-rgt-neg-in 99.6%

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

      11. mul-1-neg 99.6%

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

      12. associate-*r* 99.6%

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

      13. neg-mul-1 99.6%

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

      14. *-commutative 99.6%

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

      15. fma-define 99.6%

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

      16. cancel-sign-sub-inv 99.6%

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

      17. *-commutative 99.6%

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

      18. +-commutative 99.6%

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

      19. fma-define 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in a around 0 73.9%

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

      1. +-commutative 73.9%

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

      2. mul-1-neg 73.9%

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

      3. sub-neg 73.9%

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

      4. *-commutative 73.9%

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

      5. div-sub 73.9%

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

   10. Simplified 73.9%

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

       1. div-sub 73.9%

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

       2. *-commutative 73.9%

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

       3. associate-/l* 76.5%

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

   12. Applied egg-rr 76.5%

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

   if -1.05e-105 < z < 2.09999999999999991e-31

   1. Initial program 99.8%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Step-by-step derivation

      1. *-commutative 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{x - y \cdot z}{t - z \cdot a}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 78.5%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.2 \cdot 10^{+34}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{elif}\;z \leq -1.05 \cdot 10^{-105}:\\ \;\;\;\;\frac{x}{t} - y \cdot \frac{z}{t}\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{-31}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 71.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.4 \cdot 10^{+35}:\\ \;\;\;\;\frac{y}{a} - \frac{\frac{x}{a}}{z}\\ \mathbf{elif}\;z \leq -3.95 \cdot 10^{-106}:\\ \;\;\;\;\frac{x}{t} - y \cdot \frac{z}{t}\\ \mathbf{elif}\;z \leq 6.3 \cdot 10^{-31}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -2.4e+35)
   (- (/ y a) (/ (/ x a) z))
   (if (<= z -3.95e-106)
     (- (/ x t) (* y (/ z t)))
     (if (<= z 6.3e-31) (/ x (- t (* z a))) (/ (- y (/ x z)) a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.4e+35) {
		tmp = (y / a) - ((x / a) / z);
	} else if (z <= -3.95e-106) {
		tmp = (x / t) - (y * (z / t));
	} else if (z <= 6.3e-31) {
		tmp = x / (t - (z * a));
	} else {
		tmp = (y - (x / z)) / a;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    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) :: tmp
    if (z <= (-2.4d+35)) then
        tmp = (y / a) - ((x / a) / z)
    else if (z <= (-3.95d-106)) then
        tmp = (x / t) - (y * (z / t))
    else if (z <= 6.3d-31) then
        tmp = x / (t - (z * a))
    else
        tmp = (y - (x / z)) / a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.4e+35) {
		tmp = (y / a) - ((x / a) / z);
	} else if (z <= -3.95e-106) {
		tmp = (x / t) - (y * (z / t));
	} else if (z <= 6.3e-31) {
		tmp = x / (t - (z * a));
	} else {
		tmp = (y - (x / z)) / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -2.4e+35:
		tmp = (y / a) - ((x / a) / z)
	elif z <= -3.95e-106:
		tmp = (x / t) - (y * (z / t))
	elif z <= 6.3e-31:
		tmp = x / (t - (z * a))
	else:
		tmp = (y - (x / z)) / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -2.4e+35)
		tmp = Float64(Float64(y / a) - Float64(Float64(x / a) / z));
	elseif (z <= -3.95e-106)
		tmp = Float64(Float64(x / t) - Float64(y * Float64(z / t)));
	elseif (z <= 6.3e-31)
		tmp = Float64(x / Float64(t - Float64(z * a)));
	else
		tmp = Float64(Float64(y - Float64(x / z)) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -2.4e+35)
		tmp = (y / a) - ((x / a) / z);
	elseif (z <= -3.95e-106)
		tmp = (x / t) - (y * (z / t));
	elseif (z <= 6.3e-31)
		tmp = x / (t - (z * a));
	else
		tmp = (y - (x / z)) / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -2.4e+35], N[(N[(y / a), $MachinePrecision] - N[(N[(x / a), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, -3.95e-106], N[(N[(x / t), $MachinePrecision] - N[(y * N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 6.3e-31], N[(x / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y - N[(x / z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if z < -2.40000000000000015e35

   1. Initial program 77.9%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Step-by-step derivation

      1. *-commutative 77.9%

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

   3. Simplified 77.9%

      \[\leadsto \color{blue}{\frac{x - y \cdot z}{t - z \cdot a}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 57.4%

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

      1. +-commutative 57.4%

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

      2. associate--l+ 57.4%

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

      3. associate-/r* 59.6%

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

      4. associate-*r/ 59.6%

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

      5. associate-/r* 59.6%

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

      6. associate-*r/ 59.6%

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

      7. div-sub 59.6%

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

      8. distribute-lft-out-- 59.6%

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

      9. associate-*r/ 59.6%

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

      10. mul-1-neg 59.6%

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

      11. unsub-neg 59.6%

          \[\leadsto \color{blue}{\frac{y}{a} - \frac{\frac{x}{a} - \frac{t \cdot y}{{a}^{2}}}{z}} \]

   7. Simplified 59.6%

      \[\leadsto \color{blue}{\frac{y}{a} - \frac{\frac{x}{a} - \frac{y \cdot t}{{a}^{2}}}{z}} \]

   8. Taylor expanded in x around inf 70.7%

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

      1. associate-/r* 72.9%

         \[\leadsto \frac{y}{a} - \color{blue}{\frac{\frac{x}{a}}{z}} \]

   10. Simplified 72.9%

       \[\leadsto \frac{y}{a} - \color{blue}{\frac{\frac{x}{a}}{z}} \]

   if -2.40000000000000015e35 < z < -3.94999999999999993e-106

   1. Initial program 94.4%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Step-by-step derivation

      1. *-commutative 94.4%

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

   3. Simplified 94.4%

      \[\leadsto \color{blue}{\frac{x - y \cdot z}{t - z \cdot a}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 94.4%

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

      1. fma-define 94.4%

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

      2. associate-/l* 99.6%

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

      3. sub-neg 99.6%

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

      4. mul-1-neg 99.6%

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

      5. +-commutative 99.6%

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

      6. mul-1-neg 99.6%

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

      7. distribute-rgt-neg-in 99.6%

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

      8. fma-undefine 99.6%

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

      9. fma-undefine 99.6%

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

      10. distribute-rgt-neg-in 99.6%

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

      11. mul-1-neg 99.6%

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

      12. associate-*r* 99.6%

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

      13. neg-mul-1 99.6%

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

      14. *-commutative 99.6%

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

      15. fma-define 99.6%

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

      16. cancel-sign-sub-inv 99.6%

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

      17. *-commutative 99.6%

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

      18. +-commutative 99.6%

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

      19. fma-define 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in a around 0 73.9%

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

      1. +-commutative 73.9%

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

      2. mul-1-neg 73.9%

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

      3. sub-neg 73.9%

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

      4. *-commutative 73.9%

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

      5. div-sub 73.9%

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

   10. Simplified 73.9%

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

       1. div-sub 73.9%

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

       2. *-commutative 73.9%

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

       3. associate-/l* 76.5%

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

   12. Applied egg-rr 76.5%

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

   if -3.94999999999999993e-106 < z < 6.3000000000000002e-31

   1. Initial program 99.8%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Step-by-step derivation

      1. *-commutative 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{x - y \cdot z}{t - z \cdot a}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 78.5%

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

   if 6.3000000000000002e-31 < z

   1. Initial program 71.2%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Step-by-step derivation

      1. *-commutative 71.2%

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

   3. Simplified 71.2%

      \[\leadsto \color{blue}{\frac{x - y \cdot z}{t - z \cdot a}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 71.2%

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

      1. fma-define 71.2%

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

      2. associate-/l* 82.1%

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

      3. sub-neg 82.1%

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

      4. mul-1-neg 82.1%

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

      5. +-commutative 82.1%

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

      6. mul-1-neg 82.1%

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

      7. distribute-rgt-neg-in 82.1%

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

      8. fma-undefine 82.1%

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

      9. fma-undefine 82.1%

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

      10. distribute-rgt-neg-in 82.1%

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

      11. mul-1-neg 82.1%

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

      12. associate-*r* 82.1%

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

      13. neg-mul-1 82.1%

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

      14. *-commutative 82.1%

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

      15. fma-define 82.1%

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

      16. cancel-sign-sub-inv 82.1%

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

      17. *-commutative 82.1%

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

      18. +-commutative 82.1%

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

      19. fma-define 82.1%

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

   7. Simplified 82.1%

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

   8. Taylor expanded in a around inf 73.9%

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

      1. mul-1-neg 73.9%

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

      2. unsub-neg 73.9%

         \[\leadsto \frac{\color{blue}{y - \frac{x}{z}}}{a} \]

   10. Simplified 73.9%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.4 \cdot 10^{+35}:\\ \;\;\;\;\frac{y}{a} - \frac{\frac{x}{a}}{z}\\ \mathbf{elif}\;z \leq -3.95 \cdot 10^{-106}:\\ \;\;\;\;\frac{x}{t} - y \cdot \frac{z}{t}\\ \mathbf{elif}\;z \leq 6.3 \cdot 10^{-31}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 66.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.2 \cdot 10^{+90} \lor \neg \left(z \leq 11500000\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -2.2e+90) (not (<= z 11500000.0))) (/ y a) (/ x (- t (* z a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -2.2e+90) || !(z <= 11500000.0)) {
		tmp = y / a;
	} else {
		tmp = x / (t - (z * a));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    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) :: tmp
    if ((z <= (-2.2d+90)) .or. (.not. (z <= 11500000.0d0))) then
        tmp = y / a
    else
        tmp = x / (t - (z * a))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -2.2e+90) || !(z <= 11500000.0)) {
		tmp = y / a;
	} else {
		tmp = x / (t - (z * a));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (z <= -2.2e+90) or not (z <= 11500000.0):
		tmp = y / a
	else:
		tmp = x / (t - (z * a))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -2.2e+90) || !(z <= 11500000.0))
		tmp = Float64(y / a);
	else
		tmp = Float64(x / Float64(t - Float64(z * a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((z <= -2.2e+90) || ~((z <= 11500000.0)))
		tmp = y / a;
	else
		tmp = x / (t - (z * a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -2.2e+90], N[Not[LessEqual[z, 11500000.0]], $MachinePrecision]], N[(y / a), $MachinePrecision], N[(x / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.1999999999999999e90 or 1.15e7 < z

   1. Initial program 69.0%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Step-by-step derivation

      1. *-commutative 69.0%

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

   3. Simplified 69.0%

      \[\leadsto \color{blue}{\frac{x - y \cdot z}{t - z \cdot a}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 66.2%

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

   if -2.1999999999999999e90 < z < 1.15e7

   1. Initial program 97.0%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Step-by-step derivation

      1. *-commutative 97.0%

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

   3. Simplified 97.0%

      \[\leadsto \color{blue}{\frac{x - y \cdot z}{t - z \cdot a}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 65.2%

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

4. Final simplification 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.2 \cdot 10^{+90} \lor \neg \left(z \leq 11500000\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 56.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -7 \cdot 10^{+15} \lor \neg \left(z \leq 1.4 \cdot 10^{-30}\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -7e+15) (not (<= z 1.4e-30))) (/ y a) (/ x t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -7e+15) || !(z <= 1.4e-30)) {
		tmp = y / a;
	} else {
		tmp = x / t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    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) :: tmp
    if ((z <= (-7d+15)) .or. (.not. (z <= 1.4d-30))) then
        tmp = y / a
    else
        tmp = x / t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -7e+15) || !(z <= 1.4e-30)) {
		tmp = y / a;
	} else {
		tmp = x / t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (z <= -7e+15) or not (z <= 1.4e-30):
		tmp = y / a
	else:
		tmp = x / t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -7e+15) || !(z <= 1.4e-30))
		tmp = Float64(y / a);
	else
		tmp = Float64(x / t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((z <= -7e+15) || ~((z <= 1.4e-30)))
		tmp = y / a;
	else
		tmp = x / t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -7e+15], N[Not[LessEqual[z, 1.4e-30]], $MachinePrecision]], N[(y / a), $MachinePrecision], N[(x / t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -7e15 or 1.39999999999999994e-30 < z

   1. Initial program 73.1%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Step-by-step derivation

      1. *-commutative 73.1%

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

   3. Simplified 73.1%

      \[\leadsto \color{blue}{\frac{x - y \cdot z}{t - z \cdot a}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 57.3%

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

   if -7e15 < z < 1.39999999999999994e-30

   1. Initial program 99.8%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Step-by-step derivation

      1. *-commutative 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{x - y \cdot z}{t - z \cdot a}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 54.3%

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

4. Final simplification 55.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7 \cdot 10^{+15} \lor \neg \left(z \leq 1.4 \cdot 10^{-30}\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 36.0% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t, a)
    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
    code = x / t
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.7%

   \[\frac{x - y \cdot z}{t - a \cdot z} \]
2. Step-by-step derivation

   1. *-commutative 87.7%

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

3. Simplified 87.7%

   \[\leadsto \color{blue}{\frac{x - y \cdot z}{t - z \cdot a}} \]
4. Add Preprocessing

5. Taylor expanded in z around 0 36.7%

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

6. Final simplification 36.7%

   \[\leadsto \frac{x}{t} \]
7. Add Preprocessing

Developer target: 97.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t - a \cdot z\\ t_2 := \frac{x}{t\_1} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{if}\;z < -32113435955957344:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z < 3.5139522372978296 \cdot 10^{-86}:\\ \;\;\;\;\left(x - y \cdot z\right) \cdot \frac{1}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- t (* a z))) (t_2 (- (/ x t_1) (/ y (- (/ t z) a)))))
   (if (< z -32113435955957344.0)
     t_2
     (if (< z 3.5139522372978296e-86) (* (- x (* y z)) (/ 1.0 t_1)) t_2))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = t - (a * z);
	double t_2 = (x / t_1) - (y / ((t / z) - a));
	double tmp;
	if (z < -32113435955957344.0) {
		tmp = t_2;
	} else if (z < 3.5139522372978296e-86) {
		tmp = (x - (y * z)) * (1.0 / t_1);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = t - (a * z)
    t_2 = (x / t_1) - (y / ((t / z) - a))
    if (z < (-32113435955957344.0d0)) then
        tmp = t_2
    else if (z < 3.5139522372978296d-86) then
        tmp = (x - (y * z)) * (1.0d0 / 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 t_1 = t - (a * z);
	double t_2 = (x / t_1) - (y / ((t / z) - a));
	double tmp;
	if (z < -32113435955957344.0) {
		tmp = t_2;
	} else if (z < 3.5139522372978296e-86) {
		tmp = (x - (y * z)) * (1.0 / t_1);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = t - (a * z)
	t_2 = (x / t_1) - (y / ((t / z) - a))
	tmp = 0
	if z < -32113435955957344.0:
		tmp = t_2
	elif z < 3.5139522372978296e-86:
		tmp = (x - (y * z)) * (1.0 / t_1)
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(t - Float64(a * z))
	t_2 = Float64(Float64(x / t_1) - Float64(y / Float64(Float64(t / z) - a)))
	tmp = 0.0
	if (z < -32113435955957344.0)
		tmp = t_2;
	elseif (z < 3.5139522372978296e-86)
		tmp = Float64(Float64(x - Float64(y * z)) * Float64(1.0 / t_1));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = t - (a * z);
	t_2 = (x / t_1) - (y / ((t / z) - a));
	tmp = 0.0;
	if (z < -32113435955957344.0)
		tmp = t_2;
	elseif (z < 3.5139522372978296e-86)
		tmp = (x - (y * z)) * (1.0 / t_1);
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(t - N[(a * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x / t$95$1), $MachinePrecision] - N[(y / N[(N[(t / z), $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Less[z, -32113435955957344.0], t$95$2, If[Less[z, 3.5139522372978296e-86], N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] * N[(1.0 / t$95$1), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Reproduce

herbie shell --seed 2024044 
(FPCore (x y z t a)
  :name "Diagrams.Solve.Tridiagonal:solveTriDiagonal from diagrams-solve-0.1, A"
  :precision binary64

  :herbie-target
  (if (< z -32113435955957344.0) (- (/ x (- t (* a z))) (/ y (- (/ t z) a))) (if (< z 3.5139522372978296e-86) (* (- x (* y z)) (/ 1.0 (- t (* a z)))) (- (/ x (- t (* a z))) (/ y (- (/ t z) a)))))

  (/ (- x (* y z)) (- t (* a z))))