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

Percentage Accurate: 84.8% → 97.3%

Time: 12.2s

Alternatives: 12

Speedup: 0.2×

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: 84.8% 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: 97.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - y \cdot z\\ t_2 := t - z \cdot a\\ t_3 := \frac{t\_1}{t\_2}\\ \mathbf{if}\;t\_3 \leq -5 \cdot 10^{-321}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;\frac{\frac{t\_1}{\frac{t}{a} - z}}{a}\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{-47}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{\mathsf{fma}\left(z, a, 0 - t\right)}, \frac{x}{t\_2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- x (* y z))) (t_2 (- t (* z a))) (t_3 (/ t_1 t_2)))
   (if (<= t_3 -5e-321)
     t_3
     (if (<= t_3 0.0)
       (/ (/ t_1 (- (/ t a) z)) a)
       (if (<= t_3 5e-47)
         t_3
         (if (<= t_3 INFINITY)
           (fma y (/ z (fma z a (- 0.0 t))) (/ x t_2))
           (/ y a)))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x - (y * z);
	double t_2 = t - (z * a);
	double t_3 = t_1 / t_2;
	double tmp;
	if (t_3 <= -5e-321) {
		tmp = t_3;
	} else if (t_3 <= 0.0) {
		tmp = (t_1 / ((t / a) - z)) / a;
	} else if (t_3 <= 5e-47) {
		tmp = t_3;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = fma(y, (z / fma(z, a, (0.0 - t))), (x / t_2));
	} else {
		tmp = y / a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x - Float64(y * z))
	t_2 = Float64(t - Float64(z * a))
	t_3 = Float64(t_1 / t_2)
	tmp = 0.0
	if (t_3 <= -5e-321)
		tmp = t_3;
	elseif (t_3 <= 0.0)
		tmp = Float64(Float64(t_1 / Float64(Float64(t / a) - z)) / a);
	elseif (t_3 <= 5e-47)
		tmp = t_3;
	elseif (t_3 <= Inf)
		tmp = fma(y, Float64(z / fma(z, a, Float64(0.0 - t))), Float64(x / t_2));
	else
		tmp = Float64(y / a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 / t$95$2), $MachinePrecision]}, If[LessEqual[t$95$3, -5e-321], t$95$3, If[LessEqual[t$95$3, 0.0], N[(N[(t$95$1 / N[(N[(t / a), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[t$95$3, 5e-47], t$95$3, If[LessEqual[t$95$3, Infinity], N[(y * N[(z / N[(z * a + N[(0.0 - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x / t$95$2), $MachinePrecision]), $MachinePrecision], N[(y / a), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -4.99994e-321 or -0.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 5.00000000000000011e-47

   1. Initial program 97.7%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Add Preprocessing

   if -4.99994e-321 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -0.0

   1. Initial program 49.8%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

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

      1. *-lowering-*.f64 N/A

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

      2. --lowering--.f64 N/A

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

      3. /-lowering-/.f64 49.8

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

   5. Simplified 49.8%

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

      1. *-commutative N/A

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. --lowering--.f64 N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 N/A

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

      8. --lowering--.f64 N/A

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

      9. /-lowering-/.f64 99.7

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

   7. Applied egg-rr 99.7%

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

   if 5.00000000000000011e-47 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < +inf.0

   1. Initial program 91.6%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

      3. distribute-rgt-neg-in N/A

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

      4. mul-1-neg N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{z}{\mathsf{fma}\left(z, a, 0 - t\right)}, \frac{x}{t - z \cdot a}\right)} \]

   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. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. /-lowering-/.f64 100.0

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

   5. Simplified 100.0%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y \cdot z}{t - z \cdot a} \leq -5 \cdot 10^{-321}:\\ \;\;\;\;\frac{x - y \cdot z}{t - z \cdot a}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq 0:\\ \;\;\;\;\frac{\frac{x - y \cdot z}{\frac{t}{a} - z}}{a}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq 5 \cdot 10^{-47}:\\ \;\;\;\;\frac{x - y \cdot z}{t - z \cdot a}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{\mathsf{fma}\left(z, a, 0 - t\right)}, \frac{x}{t - z \cdot a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - y \cdot z\\ t_2 := \frac{t\_1}{t - z \cdot a}\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{-321}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;\frac{\frac{t\_1}{\frac{t}{a} - z}}{a}\\ \mathbf{elif}\;t\_2 \leq 10^{+307}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y}{\mathsf{fma}\left(z, a, 0 - t\right)}, \frac{x}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- x (* y z))) (t_2 (/ t_1 (- t (* z a)))))
   (if (<= t_2 -5e-321)
     t_2
     (if (<= t_2 0.0)
       (/ (/ t_1 (- (/ t a) z)) a)
       (if (<= t_2 1e+307)
         t_2
         (if (<= t_2 INFINITY)
           (fma z (/ y (fma z a (- 0.0 t))) (/ x t))
           (/ y a)))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x - (y * z);
	double t_2 = t_1 / (t - (z * a));
	double tmp;
	if (t_2 <= -5e-321) {
		tmp = t_2;
	} else if (t_2 <= 0.0) {
		tmp = (t_1 / ((t / a) - z)) / a;
	} else if (t_2 <= 1e+307) {
		tmp = t_2;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = fma(z, (y / fma(z, a, (0.0 - t))), (x / t));
	} else {
		tmp = y / a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x - Float64(y * z))
	t_2 = Float64(t_1 / Float64(t - Float64(z * a)))
	tmp = 0.0
	if (t_2 <= -5e-321)
		tmp = t_2;
	elseif (t_2 <= 0.0)
		tmp = Float64(Float64(t_1 / Float64(Float64(t / a) - z)) / a);
	elseif (t_2 <= 1e+307)
		tmp = t_2;
	elseif (t_2 <= Inf)
		tmp = fma(z, Float64(y / fma(z, a, Float64(0.0 - t))), Float64(x / t));
	else
		tmp = Float64(y / a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e-321], t$95$2, If[LessEqual[t$95$2, 0.0], N[(N[(t$95$1 / N[(N[(t / a), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[t$95$2, 1e+307], t$95$2, If[LessEqual[t$95$2, Infinity], N[(z * N[(y / N[(z * a + N[(0.0 - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x / t), $MachinePrecision]), $MachinePrecision], N[(y / a), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -4.99994e-321 or -0.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 9.99999999999999986e306

   1. Initial program 98.2%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Add Preprocessing

   if -4.99994e-321 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -0.0

   1. Initial program 49.8%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

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

      1. *-lowering-*.f64 N/A

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

      2. --lowering--.f64 N/A

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

      3. /-lowering-/.f64 49.8

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

   5. Simplified 49.8%

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

      1. *-commutative N/A

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. --lowering--.f64 N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 N/A

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

      8. --lowering--.f64 N/A

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

      9. /-lowering-/.f64 99.7

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

   7. Applied egg-rr 99.7%

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

   if 9.99999999999999986e306 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < +inf.0

   1. Initial program 73.6%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

      3. distribute-rgt-neg-in N/A

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

      4. mul-1-neg N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{z}{\mathsf{fma}\left(z, a, 0 - t\right)}, \frac{x}{t - z \cdot a}\right)} \]

   6. Taylor expanded in t around inf

      \[\leadsto \mathsf{fma}\left(y, \frac{z}{\mathsf{fma}\left(z, a, 0 - t\right)}, \color{blue}{\frac{x}{t}}\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 96.5

         \[\leadsto \mathsf{fma}\left(y, \frac{z}{\mathsf{fma}\left(z, a, 0 - t\right)}, \color{blue}{\frac{x}{t}}\right) \]

   8. Simplified 96.5%

      \[\leadsto \mathsf{fma}\left(y, \frac{z}{\mathsf{fma}\left(z, a, 0 - t\right)}, \color{blue}{\frac{x}{t}}\right) \]
   9. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{y \cdot z}{z \cdot a + \left(0 - t\right)}} + \frac{x}{t} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{z \cdot y}}{z \cdot a + \left(0 - t\right)} + \frac{x}{t} \]

      3. sub0-neg N/A

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

      4. sub-neg N/A

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

      5. associate-/l* N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. sub-neg N/A

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

      9. sub0-neg N/A

         \[\leadsto \mathsf{fma}\left(z, \frac{y}{z \cdot a + \color{blue}{\left(0 - t\right)}}, \frac{x}{t}\right) \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(z, \frac{y}{\color{blue}{\mathsf{fma}\left(z, a, 0 - t\right)}}, \frac{x}{t}\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma}\left(z, \frac{y}{\mathsf{fma}\left(z, a, \color{blue}{0 - t}\right)}, \frac{x}{t}\right) \]

      12. /-lowering-/.f64 96.6

          \[\leadsto \mathsf{fma}\left(z, \frac{y}{\mathsf{fma}\left(z, a, 0 - t\right)}, \color{blue}{\frac{x}{t}}\right) \]

   10. Applied egg-rr 96.6%

       \[\leadsto \color{blue}{\mathsf{fma}\left(z, \frac{y}{\mathsf{fma}\left(z, a, 0 - t\right)}, \frac{x}{t}\right)} \]

   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. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. /-lowering-/.f64 100.0

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

   5. Simplified 100.0%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y \cdot z}{t - z \cdot a} \leq -5 \cdot 10^{-321}:\\ \;\;\;\;\frac{x - y \cdot z}{t - z \cdot a}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq 0:\\ \;\;\;\;\frac{\frac{x - y \cdot z}{\frac{t}{a} - z}}{a}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq 10^{+307}:\\ \;\;\;\;\frac{x - y \cdot z}{t - z \cdot a}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y}{\mathsf{fma}\left(z, a, 0 - t\right)}, \frac{x}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - y \cdot z}{t - z \cdot a}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-321}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{elif}\;t\_1 \leq 10^{+307}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y}{\mathsf{fma}\left(z, a, 0 - t\right)}, \frac{x}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- x (* y z)) (- t (* z a)))))
   (if (<= t_1 -5e-321)
     t_1
     (if (<= t_1 0.0)
       (/ (- y (/ x z)) a)
       (if (<= t_1 1e+307)
         t_1
         (if (<= t_1 INFINITY)
           (fma z (/ y (fma z a (- 0.0 t))) (/ x t))
           (/ y a)))))))

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 <= -5e-321) {
		tmp = t_1;
	} else if (t_1 <= 0.0) {
		tmp = (y - (x / z)) / a;
	} else if (t_1 <= 1e+307) {
		tmp = t_1;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = fma(z, (y / fma(z, a, (0.0 - t))), (x / t));
	} else {
		tmp = y / a;
	}
	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 <= -5e-321)
		tmp = t_1;
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(y - Float64(x / z)) / a);
	elseif (t_1 <= 1e+307)
		tmp = t_1;
	elseif (t_1 <= Inf)
		tmp = fma(z, Float64(y / fma(z, a, Float64(0.0 - t))), Float64(x / t));
	else
		tmp = Float64(y / a);
	end
	return 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[LessEqual[t$95$1, -5e-321], t$95$1, If[LessEqual[t$95$1, 0.0], N[(N[(y - N[(x / z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[t$95$1, 1e+307], t$95$1, If[LessEqual[t$95$1, Infinity], N[(z * N[(y / N[(z * a + N[(0.0 - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x / t), $MachinePrecision]), $MachinePrecision], N[(y / a), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -4.99994e-321 or -0.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 9.99999999999999986e306

   1. Initial program 98.2%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Add Preprocessing

   if -4.99994e-321 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -0.0

   1. Initial program 49.8%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

      3. distribute-rgt-neg-in N/A

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

      4. mul-1-neg N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 49.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{z}{\mathsf{fma}\left(z, a, 0 - t\right)}, \frac{x}{t - z \cdot a}\right)} \]

   6. Taylor expanded in a around inf

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

      1. /-lowering-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. --lowering--.f64 N/A

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

      5. /-lowering-/.f64 96.6

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

   8. Simplified 96.6%

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

   if 9.99999999999999986e306 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < +inf.0

   1. Initial program 73.6%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

      3. distribute-rgt-neg-in N/A

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

      4. mul-1-neg N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{z}{\mathsf{fma}\left(z, a, 0 - t\right)}, \frac{x}{t - z \cdot a}\right)} \]

   6. Taylor expanded in t around inf

      \[\leadsto \mathsf{fma}\left(y, \frac{z}{\mathsf{fma}\left(z, a, 0 - t\right)}, \color{blue}{\frac{x}{t}}\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 96.5

         \[\leadsto \mathsf{fma}\left(y, \frac{z}{\mathsf{fma}\left(z, a, 0 - t\right)}, \color{blue}{\frac{x}{t}}\right) \]

   8. Simplified 96.5%

      \[\leadsto \mathsf{fma}\left(y, \frac{z}{\mathsf{fma}\left(z, a, 0 - t\right)}, \color{blue}{\frac{x}{t}}\right) \]
   9. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{y \cdot z}{z \cdot a + \left(0 - t\right)}} + \frac{x}{t} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{z \cdot y}}{z \cdot a + \left(0 - t\right)} + \frac{x}{t} \]

      3. sub0-neg N/A

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

      4. sub-neg N/A

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

      5. associate-/l* N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. sub-neg N/A

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

      9. sub0-neg N/A

         \[\leadsto \mathsf{fma}\left(z, \frac{y}{z \cdot a + \color{blue}{\left(0 - t\right)}}, \frac{x}{t}\right) \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(z, \frac{y}{\color{blue}{\mathsf{fma}\left(z, a, 0 - t\right)}}, \frac{x}{t}\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma}\left(z, \frac{y}{\mathsf{fma}\left(z, a, \color{blue}{0 - t}\right)}, \frac{x}{t}\right) \]

      12. /-lowering-/.f64 96.6

          \[\leadsto \mathsf{fma}\left(z, \frac{y}{\mathsf{fma}\left(z, a, 0 - t\right)}, \color{blue}{\frac{x}{t}}\right) \]

   10. Applied egg-rr 96.6%

       \[\leadsto \color{blue}{\mathsf{fma}\left(z, \frac{y}{\mathsf{fma}\left(z, a, 0 - t\right)}, \frac{x}{t}\right)} \]

   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. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. /-lowering-/.f64 100.0

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

   5. Simplified 100.0%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y \cdot z}{t - z \cdot a} \leq -5 \cdot 10^{-321}:\\ \;\;\;\;\frac{x - y \cdot z}{t - z \cdot a}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq 0:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq 10^{+307}:\\ \;\;\;\;\frac{x - y \cdot z}{t - z \cdot a}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y}{\mathsf{fma}\left(z, a, 0 - t\right)}, \frac{x}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - y \cdot z}{t - z \cdot a}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-321}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{elif}\;t\_1 \leq 10^{+307}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{\mathsf{fma}\left(z, a, 0 - t\right)}, \frac{x}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- x (* y z)) (- t (* z a)))))
   (if (<= t_1 -5e-321)
     t_1
     (if (<= t_1 0.0)
       (/ (- y (/ x z)) a)
       (if (<= t_1 1e+307)
         t_1
         (if (<= t_1 INFINITY)
           (fma y (/ z (fma z a (- 0.0 t))) (/ x t))
           (/ y a)))))))

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 <= -5e-321) {
		tmp = t_1;
	} else if (t_1 <= 0.0) {
		tmp = (y - (x / z)) / a;
	} else if (t_1 <= 1e+307) {
		tmp = t_1;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = fma(y, (z / fma(z, a, (0.0 - t))), (x / t));
	} else {
		tmp = y / a;
	}
	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 <= -5e-321)
		tmp = t_1;
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(y - Float64(x / z)) / a);
	elseif (t_1 <= 1e+307)
		tmp = t_1;
	elseif (t_1 <= Inf)
		tmp = fma(y, Float64(z / fma(z, a, Float64(0.0 - t))), Float64(x / t));
	else
		tmp = Float64(y / a);
	end
	return 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[LessEqual[t$95$1, -5e-321], t$95$1, If[LessEqual[t$95$1, 0.0], N[(N[(y - N[(x / z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[t$95$1, 1e+307], t$95$1, If[LessEqual[t$95$1, Infinity], N[(y * N[(z / N[(z * a + N[(0.0 - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x / t), $MachinePrecision]), $MachinePrecision], N[(y / a), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -4.99994e-321 or -0.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 9.99999999999999986e306

   1. Initial program 98.2%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Add Preprocessing

   if -4.99994e-321 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -0.0

   1. Initial program 49.8%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

      3. distribute-rgt-neg-in N/A

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

      4. mul-1-neg N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 49.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{z}{\mathsf{fma}\left(z, a, 0 - t\right)}, \frac{x}{t - z \cdot a}\right)} \]

   6. Taylor expanded in a around inf

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

      1. /-lowering-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. --lowering--.f64 N/A

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

      5. /-lowering-/.f64 96.6

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

   8. Simplified 96.6%

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

   if 9.99999999999999986e306 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < +inf.0

   1. Initial program 73.6%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

      3. distribute-rgt-neg-in N/A

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

      4. mul-1-neg N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{z}{\mathsf{fma}\left(z, a, 0 - t\right)}, \frac{x}{t - z \cdot a}\right)} \]

   6. Taylor expanded in t around inf

      \[\leadsto \mathsf{fma}\left(y, \frac{z}{\mathsf{fma}\left(z, a, 0 - t\right)}, \color{blue}{\frac{x}{t}}\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 96.5

         \[\leadsto \mathsf{fma}\left(y, \frac{z}{\mathsf{fma}\left(z, a, 0 - t\right)}, \color{blue}{\frac{x}{t}}\right) \]

   8. Simplified 96.5%

      \[\leadsto \mathsf{fma}\left(y, \frac{z}{\mathsf{fma}\left(z, a, 0 - t\right)}, \color{blue}{\frac{x}{t}}\right) \]

   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. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. /-lowering-/.f64 100.0

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

   5. Simplified 100.0%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y \cdot z}{t - z \cdot a} \leq -5 \cdot 10^{-321}:\\ \;\;\;\;\frac{x - y \cdot z}{t - z \cdot a}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq 0:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq 10^{+307}:\\ \;\;\;\;\frac{x - y \cdot z}{t - z \cdot a}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{\mathsf{fma}\left(z, a, 0 - t\right)}, \frac{x}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - y \cdot z}{t - z \cdot a}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-321}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- x (* y z)) (- t (* z a)))))
   (if (<= t_1 -5e-321)
     t_1
     (if (<= t_1 0.0)
       (/ (- y (/ x z)) a)
       (if (<= t_1 INFINITY) t_1 (/ y a))))))

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 <= -5e-321) {
		tmp = t_1;
	} else if (t_1 <= 0.0) {
		tmp = (y - (x / z)) / a;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = y / a;
	}
	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 <= -5e-321) {
		tmp = t_1;
	} else if (t_1 <= 0.0) {
		tmp = (y - (x / z)) / a;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (x - (y * z)) / (t - (z * a))
	tmp = 0
	if t_1 <= -5e-321:
		tmp = t_1
	elif t_1 <= 0.0:
		tmp = (y - (x / z)) / a
	elif t_1 <= math.inf:
		tmp = t_1
	else:
		tmp = y / a
	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 <= -5e-321)
		tmp = t_1;
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(y - Float64(x / z)) / a);
	elseif (t_1 <= Inf)
		tmp = t_1;
	else
		tmp = Float64(y / a);
	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 <= -5e-321)
		tmp = t_1;
	elseif (t_1 <= 0.0)
		tmp = (y - (x / z)) / a;
	elseif (t_1 <= Inf)
		tmp = t_1;
	else
		tmp = y / a;
	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[LessEqual[t$95$1, -5e-321], t$95$1, If[LessEqual[t$95$1, 0.0], N[(N[(y - N[(x / z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[t$95$1, Infinity], t$95$1, N[(y / a), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -4.99994e-321 or -0.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < +inf.0

   1. Initial program 95.8%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Add Preprocessing

   if -4.99994e-321 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -0.0

   1. Initial program 49.8%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

      3. distribute-rgt-neg-in N/A

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

      4. mul-1-neg N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 49.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{z}{\mathsf{fma}\left(z, a, 0 - t\right)}, \frac{x}{t - z \cdot a}\right)} \]

   6. Taylor expanded in a around inf

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

      1. /-lowering-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. --lowering--.f64 N/A

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

      5. /-lowering-/.f64 96.6

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

   8. Simplified 96.6%

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

   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. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. /-lowering-/.f64 100.0

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

   5. Simplified 100.0%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y \cdot z}{t - z \cdot a} \leq -5 \cdot 10^{-321}:\\ \;\;\;\;\frac{x - y \cdot z}{t - z \cdot a}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq 0:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq \infty:\\ \;\;\;\;\frac{x - y \cdot z}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \frac{z}{z \cdot a - t}\\ \mathbf{if}\;z \leq -4.5 \cdot 10^{+168}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -7.2 \cdot 10^{-32}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 5 \cdot 10^{+25}:\\ \;\;\;\;\frac{x - y \cdot z}{t}\\ \mathbf{elif}\;z \leq 1.65 \cdot 10^{+206}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* y (/ z (- (* z a) t)))))
   (if (<= z -4.5e+168)
     (/ y a)
     (if (<= z -7.2e-32)
       t_1
       (if (<= z 5e+25)
         (/ (- x (* y z)) t)
         (if (<= z 1.65e+206) t_1 (/ y a)))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = y * (z / ((z * a) - t));
	double tmp;
	if (z <= -4.5e+168) {
		tmp = y / a;
	} else if (z <= -7.2e-32) {
		tmp = t_1;
	} else if (z <= 5e+25) {
		tmp = (x - (y * z)) / t;
	} else if (z <= 1.65e+206) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_1 = y * (z / ((z * a) - t))
    if (z <= (-4.5d+168)) then
        tmp = y / a
    else if (z <= (-7.2d-32)) then
        tmp = t_1
    else if (z <= 5d+25) then
        tmp = (x - (y * z)) / t
    else if (z <= 1.65d+206) then
        tmp = t_1
    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 t_1 = y * (z / ((z * a) - t));
	double tmp;
	if (z <= -4.5e+168) {
		tmp = y / a;
	} else if (z <= -7.2e-32) {
		tmp = t_1;
	} else if (z <= 5e+25) {
		tmp = (x - (y * z)) / t;
	} else if (z <= 1.65e+206) {
		tmp = t_1;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = y * (z / ((z * a) - t))
	tmp = 0
	if z <= -4.5e+168:
		tmp = y / a
	elif z <= -7.2e-32:
		tmp = t_1
	elif z <= 5e+25:
		tmp = (x - (y * z)) / t
	elif z <= 1.65e+206:
		tmp = t_1
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(y * Float64(z / Float64(Float64(z * a) - t)))
	tmp = 0.0
	if (z <= -4.5e+168)
		tmp = Float64(y / a);
	elseif (z <= -7.2e-32)
		tmp = t_1;
	elseif (z <= 5e+25)
		tmp = Float64(Float64(x - Float64(y * z)) / t);
	elseif (z <= 1.65e+206)
		tmp = t_1;
	else
		tmp = Float64(y / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = y * (z / ((z * a) - t));
	tmp = 0.0;
	if (z <= -4.5e+168)
		tmp = y / a;
	elseif (z <= -7.2e-32)
		tmp = t_1;
	elseif (z <= 5e+25)
		tmp = (x - (y * z)) / t;
	elseif (z <= 1.65e+206)
		tmp = t_1;
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(y * N[(z / N[(N[(z * a), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -4.5e+168], N[(y / a), $MachinePrecision], If[LessEqual[z, -7.2e-32], t$95$1, If[LessEqual[z, 5e+25], N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[z, 1.65e+206], t$95$1, N[(y / a), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.50000000000000012e168 or 1.64999999999999992e206 < z

   1. Initial program 58.2%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. /-lowering-/.f64 74.8

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

   5. Simplified 74.8%

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

   if -4.50000000000000012e168 < z < -7.19999999999999986e-32 or 5.00000000000000024e25 < z < 1.64999999999999992e206

   1. Initial program 80.8%

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

      1. div-inv N/A

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

      2. flip-- N/A

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

      3. clear-num N/A

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

      4. frac-2neg N/A

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

      5. metadata-eval N/A

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

      6. frac-times N/A

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

      7. metadata-eval N/A

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

      8. /-lowering-/.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. clear-num N/A

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

      11. flip-- N/A

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

      12. /-lowering-/.f64 N/A

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

      13. --lowering--.f64 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. neg-sub0 N/A

          \[\leadsto \frac{-1}{\frac{1}{x - y \cdot z} \cdot \color{blue}{\left(0 - \left(t - a \cdot z\right)\right)}} \]

   4. Applied egg-rr 79.6%

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

   5. Taylor expanded in x around 0

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

      1. associate-/l* N/A

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

      2. *-lowering-*.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. --lowering--.f64 N/A

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 67.8

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

   7. Simplified 67.8%

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

   if -7.19999999999999986e-32 < z < 5.00000000000000024e25

   1. Initial program 99.8%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

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

      1. /-lowering-/.f64 N/A

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

      2. --lowering--.f64 N/A

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

      3. *-lowering-*.f64 78.0

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

   5. Simplified 78.0%

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

Alternative 7: 70.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y - \frac{x}{z}}{a}\\ \mathbf{if}\;z \leq -3.5 \cdot 10^{+30}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{+28}:\\ \;\;\;\;\frac{x - y \cdot z}{t}\\ \mathbf{elif}\;z \leq 5.7 \cdot 10^{+203}:\\ \;\;\;\;y \cdot \frac{z}{z \cdot a - t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- y (/ x z)) a)))
   (if (<= z -3.5e+30)
     t_1
     (if (<= z 5.5e+28)
       (/ (- x (* y z)) t)
       (if (<= z 5.7e+203) (* y (/ z (- (* z a) t))) 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 <= -3.5e+30) {
		tmp = t_1;
	} else if (z <= 5.5e+28) {
		tmp = (x - (y * z)) / t;
	} else if (z <= 5.7e+203) {
		tmp = y * (z / ((z * a) - t));
	} 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 <= (-3.5d+30)) then
        tmp = t_1
    else if (z <= 5.5d+28) then
        tmp = (x - (y * z)) / t
    else if (z <= 5.7d+203) then
        tmp = y * (z / ((z * a) - t))
    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 <= -3.5e+30) {
		tmp = t_1;
	} else if (z <= 5.5e+28) {
		tmp = (x - (y * z)) / t;
	} else if (z <= 5.7e+203) {
		tmp = y * (z / ((z * a) - t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (y - (x / z)) / a
	tmp = 0
	if z <= -3.5e+30:
		tmp = t_1
	elif z <= 5.5e+28:
		tmp = (x - (y * z)) / t
	elif z <= 5.7e+203:
		tmp = y * (z / ((z * a) - t))
	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 <= -3.5e+30)
		tmp = t_1;
	elseif (z <= 5.5e+28)
		tmp = Float64(Float64(x - Float64(y * z)) / t);
	elseif (z <= 5.7e+203)
		tmp = Float64(y * Float64(z / Float64(Float64(z * a) - t)));
	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 <= -3.5e+30)
		tmp = t_1;
	elseif (z <= 5.5e+28)
		tmp = (x - (y * z)) / t;
	elseif (z <= 5.7e+203)
		tmp = y * (z / ((z * a) - t));
	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, -3.5e+30], t$95$1, If[LessEqual[z, 5.5e+28], N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[z, 5.7e+203], N[(y * N[(z / N[(N[(z * a), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.50000000000000021e30 or 5.7e203 < z

   1. Initial program 66.2%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

      3. distribute-rgt-neg-in N/A

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

      4. mul-1-neg N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 72.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{z}{\mathsf{fma}\left(z, a, 0 - t\right)}, \frac{x}{t - z \cdot a}\right)} \]

   6. Taylor expanded in a around inf

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

      1. /-lowering-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. --lowering--.f64 N/A

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

      5. /-lowering-/.f64 85.0

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

   8. Simplified 85.0%

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

   if -3.50000000000000021e30 < z < 5.5000000000000003e28

   1. Initial program 99.8%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

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

      1. /-lowering-/.f64 N/A

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

      2. --lowering--.f64 N/A

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

      3. *-lowering-*.f64 76.1

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

   5. Simplified 76.1%

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

   if 5.5000000000000003e28 < z < 5.7e203

   1. Initial program 69.6%

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

      1. div-inv N/A

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

      2. flip-- N/A

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

      3. clear-num N/A

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

      4. frac-2neg N/A

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

      5. metadata-eval N/A

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

      6. frac-times N/A

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

      7. metadata-eval N/A

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

      8. /-lowering-/.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. clear-num N/A

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

      11. flip-- N/A

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

      12. /-lowering-/.f64 N/A

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

      13. --lowering--.f64 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. neg-sub0 N/A

          \[\leadsto \frac{-1}{\frac{1}{x - y \cdot z} \cdot \color{blue}{\left(0 - \left(t - a \cdot z\right)\right)}} \]

   4. Applied egg-rr 69.4%

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

   5. Taylor expanded in x around 0

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

      1. associate-/l* N/A

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

      2. *-lowering-*.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. --lowering--.f64 N/A

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 74.5

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

   7. Simplified 74.5%

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

Alternative 8: 61.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.3 \cdot 10^{+28}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{+20}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{elif}\;z \leq 1.45 \cdot 10^{+201}:\\ \;\;\;\;y \cdot \frac{0 - z}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -3.3e+28)
   (/ y a)
   (if (<= z 1.6e+20)
     (/ x (- t (* z a)))
     (if (<= z 1.45e+201) (* y (/ (- 0.0 z) t)) (/ y a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -3.3e+28) {
		tmp = y / a;
	} else if (z <= 1.6e+20) {
		tmp = x / (t - (z * a));
	} else if (z <= 1.45e+201) {
		tmp = y * ((0.0 - z) / t);
	} 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 <= (-3.3d+28)) then
        tmp = y / a
    else if (z <= 1.6d+20) then
        tmp = x / (t - (z * a))
    else if (z <= 1.45d+201) then
        tmp = y * ((0.0d0 - z) / t)
    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 <= -3.3e+28) {
		tmp = y / a;
	} else if (z <= 1.6e+20) {
		tmp = x / (t - (z * a));
	} else if (z <= 1.45e+201) {
		tmp = y * ((0.0 - z) / t);
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -3.3e+28:
		tmp = y / a
	elif z <= 1.6e+20:
		tmp = x / (t - (z * a))
	elif z <= 1.45e+201:
		tmp = y * ((0.0 - z) / t)
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -3.3e+28)
		tmp = Float64(y / a);
	elseif (z <= 1.6e+20)
		tmp = Float64(x / Float64(t - Float64(z * a)));
	elseif (z <= 1.45e+201)
		tmp = Float64(y * Float64(Float64(0.0 - z) / t));
	else
		tmp = Float64(y / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -3.3e+28)
		tmp = y / a;
	elseif (z <= 1.6e+20)
		tmp = x / (t - (z * a));
	elseif (z <= 1.45e+201)
		tmp = y * ((0.0 - z) / t);
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -3.3e+28], N[(y / a), $MachinePrecision], If[LessEqual[z, 1.6e+20], N[(x / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.45e+201], N[(y * N[(N[(0.0 - z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(y / a), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.3e28 or 1.4500000000000001e201 < z

   1. Initial program 67.0%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. /-lowering-/.f64 66.7

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

   5. Simplified 66.7%

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

   if -3.3e28 < z < 1.6e20

   1. Initial program 99.8%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

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

      2. --lowering--.f64 N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 72.4

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

   5. Simplified 72.4%

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

   if 1.6e20 < z < 1.4500000000000001e201

   1. Initial program 71.6%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

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

      1. *-lowering-*.f64 N/A

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

      2. --lowering--.f64 N/A

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

      3. /-lowering-/.f64 58.3

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

   5. Simplified 58.3%

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

   6. Taylor expanded in a around 0

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

      1. /-lowering-/.f64 N/A

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

      2. --lowering--.f64 N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 52.1

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

   8. Simplified 52.1%

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

   9. Taylor expanded in x around 0

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

       1. mul-1-neg N/A

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

       2. associate-/l* N/A

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

       3. distribute-rgt-neg-in N/A

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

       4. mul-1-neg N/A

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

       5. *-lowering-*.f64 N/A

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

       6. mul-1-neg N/A

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

       7. distribute-neg-frac2 N/A

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

       8. mul-1-neg N/A

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

       9. /-lowering-/.f64 N/A

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

       10. mul-1-neg N/A

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

       11. neg-sub0 N/A

           \[\leadsto y \cdot \frac{z}{\color{blue}{0 - t}} \]

       12. --lowering--.f64 53.7

           \[\leadsto y \cdot \frac{z}{\color{blue}{0 - t}} \]

   11. Simplified 53.7%

       \[\leadsto \color{blue}{y \cdot \frac{z}{0 - t}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.3 \cdot 10^{+28}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{+20}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{elif}\;z \leq 1.45 \cdot 10^{+201}:\\ \;\;\;\;y \cdot \frac{0 - z}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 51.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.05 \cdot 10^{-35}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 9.2 \cdot 10^{+18}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;z \leq 1.45 \cdot 10^{+201}:\\ \;\;\;\;y \cdot \frac{0 - z}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.05e-35)
   (/ y a)
   (if (<= z 9.2e+18)
     (/ x t)
     (if (<= z 1.45e+201) (* y (/ (- 0.0 z) t)) (/ y a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.05e-35) {
		tmp = y / a;
	} else if (z <= 9.2e+18) {
		tmp = x / t;
	} else if (z <= 1.45e+201) {
		tmp = y * ((0.0 - z) / t);
	} 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 <= (-1.05d-35)) then
        tmp = y / a
    else if (z <= 9.2d+18) then
        tmp = x / t
    else if (z <= 1.45d+201) then
        tmp = y * ((0.0d0 - z) / t)
    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 <= -1.05e-35) {
		tmp = y / a;
	} else if (z <= 9.2e+18) {
		tmp = x / t;
	} else if (z <= 1.45e+201) {
		tmp = y * ((0.0 - z) / t);
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.05e-35:
		tmp = y / a
	elif z <= 9.2e+18:
		tmp = x / t
	elif z <= 1.45e+201:
		tmp = y * ((0.0 - z) / t)
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.05e-35)
		tmp = Float64(y / a);
	elseif (z <= 9.2e+18)
		tmp = Float64(x / t);
	elseif (z <= 1.45e+201)
		tmp = Float64(y * Float64(Float64(0.0 - z) / t));
	else
		tmp = Float64(y / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -1.05e-35)
		tmp = y / a;
	elseif (z <= 9.2e+18)
		tmp = x / t;
	elseif (z <= 1.45e+201)
		tmp = y * ((0.0 - z) / t);
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.05e-35], N[(y / a), $MachinePrecision], If[LessEqual[z, 9.2e+18], N[(x / t), $MachinePrecision], If[LessEqual[z, 1.45e+201], N[(y * N[(N[(0.0 - z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(y / a), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.05e-35 or 1.4500000000000001e201 < z

   1. Initial program 71.0%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. /-lowering-/.f64 61.9

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

   5. Simplified 61.9%

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

   if -1.05e-35 < z < 9.2e18

   1. Initial program 99.8%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. /-lowering-/.f64 61.4

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

   5. Simplified 61.4%

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

   if 9.2e18 < z < 1.4500000000000001e201

   1. Initial program 71.6%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

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

      1. *-lowering-*.f64 N/A

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

      2. --lowering--.f64 N/A

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

      3. /-lowering-/.f64 58.3

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

   5. Simplified 58.3%

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

   6. Taylor expanded in a around 0

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

      1. /-lowering-/.f64 N/A

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

      2. --lowering--.f64 N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 52.1

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

   8. Simplified 52.1%

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

   9. Taylor expanded in x around 0

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

       1. mul-1-neg N/A

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

       2. associate-/l* N/A

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

       3. distribute-rgt-neg-in N/A

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

       4. mul-1-neg N/A

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

       5. *-lowering-*.f64 N/A

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

       6. mul-1-neg N/A

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

       7. distribute-neg-frac2 N/A

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

       8. mul-1-neg N/A

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

       9. /-lowering-/.f64 N/A

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

       10. mul-1-neg N/A

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

       11. neg-sub0 N/A

           \[\leadsto y \cdot \frac{z}{\color{blue}{0 - t}} \]

       12. --lowering--.f64 53.7

           \[\leadsto y \cdot \frac{z}{\color{blue}{0 - t}} \]

   11. Simplified 53.7%

       \[\leadsto \color{blue}{y \cdot \frac{z}{0 - t}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 60.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.05 \cdot 10^{-35}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 9.2 \cdot 10^{+18}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;z \leq 1.45 \cdot 10^{+201}:\\ \;\;\;\;y \cdot \frac{0 - z}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 62.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.8 \cdot 10^{+33}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 1.45 \cdot 10^{+201}:\\ \;\;\;\;\frac{x - y \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -5.8e+33)
   (/ y a)
   (if (<= z 1.45e+201) (/ (- x (* y z)) t) (/ y a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -5.8e+33) {
		tmp = y / a;
	} else if (z <= 1.45e+201) {
		tmp = (x - (y * z)) / t;
	} 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 <= (-5.8d+33)) then
        tmp = y / a
    else if (z <= 1.45d+201) then
        tmp = (x - (y * z)) / t
    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 <= -5.8e+33) {
		tmp = y / a;
	} else if (z <= 1.45e+201) {
		tmp = (x - (y * z)) / t;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -5.8e+33:
		tmp = y / a
	elif z <= 1.45e+201:
		tmp = (x - (y * z)) / t
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -5.8e+33)
		tmp = Float64(y / a);
	elseif (z <= 1.45e+201)
		tmp = Float64(Float64(x - Float64(y * z)) / t);
	else
		tmp = Float64(y / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -5.8e+33)
		tmp = y / a;
	elseif (z <= 1.45e+201)
		tmp = (x - (y * z)) / t;
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -5.8e+33], N[(y / a), $MachinePrecision], If[LessEqual[z, 1.45e+201], N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], N[(y / a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -5.80000000000000049e33 or 1.4500000000000001e201 < z

   1. Initial program 66.2%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. /-lowering-/.f64 67.0

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

   5. Simplified 67.0%

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

   if -5.80000000000000049e33 < z < 1.4500000000000001e201

   1. Initial program 94.8%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

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

      1. /-lowering-/.f64 N/A

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

      2. --lowering--.f64 N/A

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

      3. *-lowering-*.f64 71.6

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

   5. Simplified 71.6%

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

Alternative 11: 54.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.35 \cdot 10^{-35}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{+30}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.35e-35) (/ y a) (if (<= z 4.2e+30) (/ x t) (/ y a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.35e-35) {
		tmp = y / a;
	} else if (z <= 4.2e+30) {
		tmp = x / t;
	} 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 <= (-1.35d-35)) then
        tmp = y / a
    else if (z <= 4.2d+30) then
        tmp = x / t
    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 <= -1.35e-35) {
		tmp = y / a;
	} else if (z <= 4.2e+30) {
		tmp = x / t;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.35e-35:
		tmp = y / a
	elif z <= 4.2e+30:
		tmp = x / t
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.35e-35)
		tmp = Float64(y / a);
	elseif (z <= 4.2e+30)
		tmp = Float64(x / t);
	else
		tmp = Float64(y / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -1.35e-35)
		tmp = y / a;
	elseif (z <= 4.2e+30)
		tmp = x / t;
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.35e-35], N[(y / a), $MachinePrecision], If[LessEqual[z, 4.2e+30], N[(x / t), $MachinePrecision], N[(y / a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.3499999999999999e-35 or 4.2e30 < z

   1. Initial program 70.7%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. /-lowering-/.f64 56.7

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

   5. Simplified 56.7%

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

   if -1.3499999999999999e-35 < z < 4.2e30

   1. Initial program 99.8%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. /-lowering-/.f64 61.2

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

   5. Simplified 61.2%

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

Alternative 12: 35.0% accurate, 2.3× 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 85.3%

   \[\frac{x - y \cdot z}{t - a \cdot z} \]
2. Add Preprocessing

3. Taylor expanded in z around 0

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

   1. /-lowering-/.f64 35.1

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

5. Simplified 35.1%

   \[\leadsto \color{blue}{\frac{x}{t}} \]
6. Add Preprocessing

Developer Target 1: 97.4% accurate, 0.5× 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 2024196 
(FPCore (x y z t a)
  :name "Diagrams.Solve.Tridiagonal:solveTriDiagonal from diagrams-solve-0.1, A"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< z -32113435955957344) (- (/ x (- t (* a z))) (/ y (- (/ t z) a))) (if (< z 4392440296622287/125000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (* (- x (* y z)) (/ 1 (- t (* a z)))) (- (/ x (- t (* a z))) (/ y (- (/ t z) a))))))

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