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

Percentage Accurate: 85.1% → 92.8%

Time: 11.0s

Alternatives: 11

Speedup: 0.4×

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: 92.8% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, y, z, t, a)
	t_1 = Float64(t - Float64(z * a))
	tmp = 0.0
	if (Float64(Float64(x - Float64(y * z)) / t_1) <= Inf)
		tmp = fma(y, Float64(z / fma(z, a, Float64(0.0 - t))), 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[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], Infinity], N[(y * N[(z / N[(z * a + N[(0.0 - t), $MachinePrecision]), $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 87.2%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   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 93.4%

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

   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 2 regimes into one program.

4. Final simplification 93.7%

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

Alternative 2: 91.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - y \cdot z}{t - z \cdot a}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{\mathsf{fma}\left(z, a, 0 - t\right)}, \frac{x}{t}\right)\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+305}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;z \cdot \frac{y}{\mathsf{fma}\left(z, a, 0\right) - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- x (* y z)) (- t (* z a)))))
   (if (<= t_1 (- INFINITY))
     (fma y (/ z (fma z a (- 0.0 t))) (/ x t))
     (if (<= t_1 4e+305)
       t_1
       (if (<= t_1 INFINITY) (* z (/ y (- (fma z a 0.0) 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 <= -((double) INFINITY)) {
		tmp = fma(y, (z / fma(z, a, (0.0 - t))), (x / t));
	} else if (t_1 <= 4e+305) {
		tmp = t_1;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = z * (y / (fma(z, a, 0.0) - 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 <= Float64(-Inf))
		tmp = fma(y, Float64(z / fma(z, a, Float64(0.0 - t))), Float64(x / t));
	elseif (t_1 <= 4e+305)
		tmp = t_1;
	elseif (t_1 <= Inf)
		tmp = Float64(z * Float64(y / Float64(fma(z, a, 0.0) - 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, (-Infinity)], N[(y * N[(z / N[(z * a + N[(0.0 - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 4e+305], t$95$1, If[LessEqual[t$95$1, Infinity], N[(z * N[(y / N[(N[(z * a + 0.0), $MachinePrecision] - t), $MachinePrecision]), $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))) < -inf.0

   1. Initial program 37.5%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   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 99.5%

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

   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 93.3

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

   8. Simplified 93.3%

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

   if -inf.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 3.9999999999999998e305

   1. Initial program 94.0%

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

   if 3.9999999999999998e305 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < +inf.0

   1. Initial program 60.3%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   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 95.7%

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

   6. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{y \cdot z}{a \cdot z - t}} \]
   7. 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. *-lowering-*.f64 79.9

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

   8. Simplified 79.9%

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

      1. associate-*r/ N/A

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

      2. associate-*l/ N/A

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

      3. *-commutative N/A

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

      4. +-rgt-identity N/A

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

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

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

      6. +-rgt-identity N/A

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

      7. *-commutative N/A

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

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

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

      9. *-commutative N/A

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

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

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

      11. +-rgt-identity N/A

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

      12. accelerator-lowering-fma.f64 87.6

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

   10. Applied egg-rr 87.6%

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

   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 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y \cdot z}{t - z \cdot a} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{\mathsf{fma}\left(z, a, 0 - t\right)}, \frac{x}{t}\right)\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq 4 \cdot 10^{+305}:\\ \;\;\;\;\frac{x - y \cdot z}{t - z \cdot a}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq \infty:\\ \;\;\;\;z \cdot \frac{y}{\mathsf{fma}\left(z, a, 0\right) - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - y \cdot z}{t - z \cdot a}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+305}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;z \cdot \frac{y}{\mathsf{fma}\left(z, a, 0\right) - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- x (* y z)) (- t (* z a)))))
   (if (<= t_1 (- INFINITY))
     (/ (- y (/ x z)) a)
     (if (<= t_1 4e+305)
       t_1
       (if (<= t_1 INFINITY) (* z (/ y (- (fma z a 0.0) 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 <= -((double) INFINITY)) {
		tmp = (y - (x / z)) / a;
	} else if (t_1 <= 4e+305) {
		tmp = t_1;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = z * (y / (fma(z, a, 0.0) - 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 <= Float64(-Inf))
		tmp = Float64(Float64(y - Float64(x / z)) / a);
	elseif (t_1 <= 4e+305)
		tmp = t_1;
	elseif (t_1 <= Inf)
		tmp = Float64(z * Float64(y / Float64(fma(z, a, 0.0) - 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, (-Infinity)], N[(N[(y - N[(x / z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[t$95$1, 4e+305], t$95$1, If[LessEqual[t$95$1, Infinity], N[(z * N[(y / N[(N[(z * a + 0.0), $MachinePrecision] - t), $MachinePrecision]), $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))) < -inf.0

   1. Initial program 37.5%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   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 99.5%

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

   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 86.7

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

   8. Simplified 86.7%

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

   if -inf.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 3.9999999999999998e305

   1. Initial program 94.0%

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

   if 3.9999999999999998e305 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < +inf.0

   1. Initial program 60.3%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   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 95.7%

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

   6. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{y \cdot z}{a \cdot z - t}} \]
   7. 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. *-lowering-*.f64 79.9

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

   8. Simplified 79.9%

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

      1. associate-*r/ N/A

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

      2. associate-*l/ N/A

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

      3. *-commutative N/A

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

      4. +-rgt-identity N/A

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

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

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

      6. +-rgt-identity N/A

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

      7. *-commutative N/A

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

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

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

      9. *-commutative N/A

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

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

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

      11. +-rgt-identity N/A

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

      12. accelerator-lowering-fma.f64 87.6

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

   10. Applied egg-rr 87.6%

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

   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 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y \cdot z}{t - z \cdot a} \leq -\infty:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq 4 \cdot 10^{+305}:\\ \;\;\;\;\frac{x - y \cdot z}{t - z \cdot a}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq \infty:\\ \;\;\;\;z \cdot \frac{y}{\mathsf{fma}\left(z, a, 0\right) - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 65.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{+47}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{-168}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{elif}\;z \leq 1.95 \cdot 10^{+97}:\\ \;\;\;\;\frac{x - y \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -5e+47)
   (/ y a)
   (if (<= z 2.1e-168)
     (/ x (- t (* z a)))
     (if (<= z 1.95e+97) (/ (- x (* y z)) t) (/ y a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -5e+47) {
		tmp = y / a;
	} else if (z <= 2.1e-168) {
		tmp = x / (t - (z * a));
	} else if (z <= 1.95e+97) {
		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 <= (-5d+47)) then
        tmp = y / a
    else if (z <= 2.1d-168) then
        tmp = x / (t - (z * a))
    else if (z <= 1.95d+97) 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 <= -5e+47) {
		tmp = y / a;
	} else if (z <= 2.1e-168) {
		tmp = x / (t - (z * a));
	} else if (z <= 1.95e+97) {
		tmp = (x - (y * z)) / t;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -5e+47:
		tmp = y / a
	elif z <= 2.1e-168:
		tmp = x / (t - (z * a))
	elif z <= 1.95e+97:
		tmp = (x - (y * z)) / t
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -5e+47)
		tmp = Float64(y / a);
	elseif (z <= 2.1e-168)
		tmp = Float64(x / Float64(t - Float64(z * a)));
	elseif (z <= 1.95e+97)
		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 <= -5e+47)
		tmp = y / a;
	elseif (z <= 2.1e-168)
		tmp = x / (t - (z * a));
	elseif (z <= 1.95e+97)
		tmp = (x - (y * z)) / t;
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -5e+47], N[(y / a), $MachinePrecision], If[LessEqual[z, 2.1e-168], N[(x / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.95e+97], N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], N[(y / a), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.00000000000000022e47 or 1.95e97 < z

   1. Initial program 61.7%

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

   3. Taylor expanded in z around inf

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

      1. /-lowering-/.f64 68.5

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

   5. Simplified 68.5%

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

   if -5.00000000000000022e47 < z < 2.09999999999999994e-168

   1. Initial program 99.0%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   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 81.9

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

   5. Simplified 81.9%

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

   if 2.09999999999999994e-168 < z < 1.95e97

   1. Initial program 91.8%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   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 70.7

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

   5. Simplified 70.7%

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

Alternative 5: 55.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.3 \cdot 10^{+40}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 7.5 \cdot 10^{-87}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{+65}:\\ \;\;\;\;y \cdot \frac{z}{0 - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -4.3e+40)
   (/ y a)
   (if (<= z 7.5e-87)
     (/ x t)
     (if (<= z 1.1e+65) (* y (/ z (- 0.0 t))) (/ y a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -4.3e+40) {
		tmp = y / a;
	} else if (z <= 7.5e-87) {
		tmp = x / t;
	} else if (z <= 1.1e+65) {
		tmp = y * (z / (0.0 - 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 <= (-4.3d+40)) then
        tmp = y / a
    else if (z <= 7.5d-87) then
        tmp = x / t
    else if (z <= 1.1d+65) then
        tmp = y * (z / (0.0d0 - 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 <= -4.3e+40) {
		tmp = y / a;
	} else if (z <= 7.5e-87) {
		tmp = x / t;
	} else if (z <= 1.1e+65) {
		tmp = y * (z / (0.0 - t));
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -4.3e+40:
		tmp = y / a
	elif z <= 7.5e-87:
		tmp = x / t
	elif z <= 1.1e+65:
		tmp = y * (z / (0.0 - t))
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -4.3e+40)
		tmp = Float64(y / a);
	elseif (z <= 7.5e-87)
		tmp = Float64(x / t);
	elseif (z <= 1.1e+65)
		tmp = Float64(y * Float64(z / Float64(0.0 - 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 <= -4.3e+40)
		tmp = y / a;
	elseif (z <= 7.5e-87)
		tmp = x / t;
	elseif (z <= 1.1e+65)
		tmp = y * (z / (0.0 - t));
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -4.3e+40], N[(y / a), $MachinePrecision], If[LessEqual[z, 7.5e-87], N[(x / t), $MachinePrecision], If[LessEqual[z, 1.1e+65], N[(y * N[(z / N[(0.0 - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y / a), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.3000000000000002e40 or 1.0999999999999999e65 < z

   1. Initial program 63.1%

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

   3. Taylor expanded in z around inf

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

      1. /-lowering-/.f64 66.5

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

   5. Simplified 66.5%

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

   if -4.3000000000000002e40 < z < 7.5000000000000002e-87

   1. Initial program 99.1%

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

   3. Taylor expanded in z around 0

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

      1. /-lowering-/.f64 68.3

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

   5. Simplified 68.3%

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

   if 7.5000000000000002e-87 < z < 1.0999999999999999e65

   1. Initial program 93.0%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   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 96.3%

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

   6. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{y \cdot z}{a \cdot z - t}} \]
   7. 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. *-lowering-*.f64 68.9

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

   8. Simplified 68.9%

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

   9. Taylor expanded in a around 0

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

       1. mul-1-neg N/A

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

       2. neg-sub0 N/A

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

       3. --lowering--.f64 47.7

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

   11. Simplified 47.7%

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

Alternative 6: 72.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y - \frac{x}{z}}{a}\\ \mathbf{if}\;z \leq -1.75 \cdot 10^{-16}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{+14}:\\ \;\;\;\;\frac{x - y \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- y (/ x z)) a)))
   (if (<= z -1.75e-16) t_1 (if (<= z 1.4e+14) (/ (- x (* y z)) 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 <= -1.75e-16) {
		tmp = t_1;
	} else if (z <= 1.4e+14) {
		tmp = (x - (y * z)) / 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 <= (-1.75d-16)) then
        tmp = t_1
    else if (z <= 1.4d+14) then
        tmp = (x - (y * z)) / 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 <= -1.75e-16) {
		tmp = t_1;
	} else if (z <= 1.4e+14) {
		tmp = (x - (y * z)) / 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 <= -1.75e-16:
		tmp = t_1
	elif z <= 1.4e+14:
		tmp = (x - (y * z)) / 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 <= -1.75e-16)
		tmp = t_1;
	elseif (z <= 1.4e+14)
		tmp = Float64(Float64(x - Float64(y * z)) / 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 <= -1.75e-16)
		tmp = t_1;
	elseif (z <= 1.4e+14)
		tmp = (x - (y * z)) / 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, -1.75e-16], t$95$1, If[LessEqual[z, 1.4e+14], N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.75000000000000009e-16 or 1.4e14 < z

   1. Initial program 66.0%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   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 80.3%

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

   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 75.9

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

   8. Simplified 75.9%

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

   if -1.75000000000000009e-16 < z < 1.4e14

   1. Initial program 99.9%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   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 81.8

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

   5. Simplified 81.8%

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

Alternative 7: 65.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{+48}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{-83}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z}{\mathsf{fma}\left(z, a, 0 - t\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1e+48)
   (/ y a)
   (if (<= z 1.7e-83) (/ x (- t (* z a))) (* y (/ z (fma z a (- 0.0 t)))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1e+48) {
		tmp = y / a;
	} else if (z <= 1.7e-83) {
		tmp = x / (t - (z * a));
	} else {
		tmp = y * (z / fma(z, a, (0.0 - t)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1e+48)
		tmp = Float64(y / a);
	elseif (z <= 1.7e-83)
		tmp = Float64(x / Float64(t - Float64(z * a)));
	else
		tmp = Float64(y * Float64(z / fma(z, a, Float64(0.0 - t))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1e+48], N[(y / a), $MachinePrecision], If[LessEqual[z, 1.7e-83], N[(x / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(z / N[(z * a + N[(0.0 - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.00000000000000004e48

   1. Initial program 59.2%

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

   3. Taylor expanded in z around inf

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

      1. /-lowering-/.f64 70.7

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

   5. Simplified 70.7%

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

   if -1.00000000000000004e48 < z < 1.6999999999999999e-83

   1. Initial program 99.1%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   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 81.9

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

   5. Simplified 81.9%

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

   if 1.6999999999999999e-83 < z

   1. Initial program 75.9%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   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 88.3%

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

   6. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{y \cdot z}{a \cdot z - t}} \]
   7. 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. *-lowering-*.f64 68.3

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

   8. Simplified 68.3%

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. neg-sub0 N/A

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

      5. --lowering--.f64 68.3

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

   10. Applied egg-rr 68.3%

       \[\leadsto y \cdot \frac{z}{\color{blue}{\mathsf{fma}\left(z, a, 0 - t\right)}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 65.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -7 \cdot 10^{+47}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 1.62 \cdot 10^{-83}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z}{z \cdot a - t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -7e+47)
   (/ y a)
   (if (<= z 1.62e-83) (/ x (- t (* z a))) (* y (/ z (- (* z a) t))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -7e+47) {
		tmp = y / a;
	} else if (z <= 1.62e-83) {
		tmp = x / (t - (z * a));
	} else {
		tmp = y * (z / ((z * a) - 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+47)) then
        tmp = y / a
    else if (z <= 1.62d-83) then
        tmp = x / (t - (z * a))
    else
        tmp = y * (z / ((z * a) - 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+47) {
		tmp = y / a;
	} else if (z <= 1.62e-83) {
		tmp = x / (t - (z * a));
	} else {
		tmp = y * (z / ((z * a) - t));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -7e+47:
		tmp = y / a
	elif z <= 1.62e-83:
		tmp = x / (t - (z * a))
	else:
		tmp = y * (z / ((z * a) - t))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -7e+47)
		tmp = Float64(y / a);
	elseif (z <= 1.62e-83)
		tmp = Float64(x / Float64(t - Float64(z * a)));
	else
		tmp = Float64(y * Float64(z / Float64(Float64(z * a) - t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -7e+47)
		tmp = y / a;
	elseif (z <= 1.62e-83)
		tmp = x / (t - (z * a));
	else
		tmp = y * (z / ((z * a) - t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -7e+47], N[(y / a), $MachinePrecision], If[LessEqual[z, 1.62e-83], N[(x / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(z / N[(N[(z * a), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -7.00000000000000031e47

   1. Initial program 59.2%

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

   3. Taylor expanded in z around inf

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

      1. /-lowering-/.f64 70.7

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

   5. Simplified 70.7%

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

   if -7.00000000000000031e47 < z < 1.6199999999999999e-83

   1. Initial program 99.1%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   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 81.9

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

   5. Simplified 81.9%

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

   if 1.6199999999999999e-83 < z

   1. Initial program 75.9%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   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 88.3%

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

   6. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{y \cdot z}{a \cdot z - t}} \]
   7. 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. *-lowering-*.f64 68.3

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

   8. Simplified 68.3%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7 \cdot 10^{+47}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 1.62 \cdot 10^{-83}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z}{z \cdot a - t}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 66.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.2 \cdot 10^{+47}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{+121}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -5.2e+47) (/ y a) (if (<= z 1.8e+121) (/ x (- t (* z a))) (/ y a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -5.2e+47) {
		tmp = y / a;
	} else if (z <= 1.8e+121) {
		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 <= (-5.2d+47)) then
        tmp = y / a
    else if (z <= 1.8d+121) 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 <= -5.2e+47) {
		tmp = y / a;
	} else if (z <= 1.8e+121) {
		tmp = x / (t - (z * a));
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -5.2e+47:
		tmp = y / a
	elif z <= 1.8e+121:
		tmp = x / (t - (z * a))
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -5.2e+47)
		tmp = Float64(y / a);
	elseif (z <= 1.8e+121)
		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 <= -5.2e+47)
		tmp = y / a;
	elseif (z <= 1.8e+121)
		tmp = x / (t - (z * a));
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -5.2e+47], N[(y / a), $MachinePrecision], If[LessEqual[z, 1.8e+121], N[(x / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y / a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -5.20000000000000007e47 or 1.79999999999999991e121 < z

   1. Initial program 60.1%

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

   3. Taylor expanded in z around inf

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

      1. /-lowering-/.f64 70.2

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

   5. Simplified 70.2%

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

   if -5.20000000000000007e47 < z < 1.79999999999999991e121

   1. Initial program 96.9%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   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 71.3

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

   5. Simplified 71.3%

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

Alternative 10: 55.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.1 \cdot 10^{+40}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 1.55 \cdot 10^{+14}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -2.1e+40) (/ y a) (if (<= z 1.55e+14) (/ x t) (/ y a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.1e+40) {
		tmp = y / a;
	} else if (z <= 1.55e+14) {
		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 <= (-2.1d+40)) then
        tmp = y / a
    else if (z <= 1.55d+14) 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 <= -2.1e+40) {
		tmp = y / a;
	} else if (z <= 1.55e+14) {
		tmp = x / t;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -2.1e+40:
		tmp = y / a
	elif z <= 1.55e+14:
		tmp = x / t
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -2.1e+40)
		tmp = Float64(y / a);
	elseif (z <= 1.55e+14)
		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 <= -2.1e+40)
		tmp = y / a;
	elseif (z <= 1.55e+14)
		tmp = x / t;
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -2.1e+40], N[(y / a), $MachinePrecision], If[LessEqual[z, 1.55e+14], N[(x / t), $MachinePrecision], N[(y / a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.1000000000000001e40 or 1.55e14 < z

   1. Initial program 64.0%

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

   3. Taylor expanded in z around inf

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

      1. /-lowering-/.f64 63.8

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

   5. Simplified 63.8%

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

   if -2.1000000000000001e40 < z < 1.55e14

   1. Initial program 99.2%

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

   3. Taylor expanded in z around 0

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

      1. /-lowering-/.f64 62.5

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

   5. Simplified 62.5%

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

Alternative 11: 36.5% 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 83.1%

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

3. Taylor expanded in z around 0

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

   1. /-lowering-/.f64 39.6

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

5. Simplified 39.6%

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

Developer Target 1: 97.3% 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 2024195 
(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))))