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

Percentage Accurate: 85.3% → 93.2%

Time: 10.8s

Alternatives: 12

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: 85.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 93.2% accurate, 0.4× speedup

Math

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

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 (- 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, -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(-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 + (-t)), $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 87.5%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   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. lower-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. Applied rewrites 93.4%

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

   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. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 93.6%

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

Alternative 2: 90.9% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- t (* z a))) (t_2 (/ (- x (* y z)) t_1)))
   (if (<= t_2 (- INFINITY))
     (* z (/ y (- (* z a) t)))
     (if (<= t_2 INFINITY) (/ (fma (- z) y 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 t_2 = (x - (y * z)) / t_1;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = z * (y / ((z * a) - t));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = fma(-z, y, x) / t_1;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(t - Float64(z * a))
	t_2 = Float64(Float64(x - Float64(y * z)) / t_1)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(z * Float64(y / Float64(Float64(z * a) - t)));
	elseif (t_2 <= Inf)
		tmp = Float64(fma(Float64(-z), y, 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]}, Block[{t$95$2 = N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(z * N[(y / N[(N[(z * a), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(N[((-z) * y + x), $MachinePrecision] / t$95$1), $MachinePrecision], N[(y / a), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 42.0%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-neg-in N/A

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

      9. associate-*r* N/A

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

      10. distribute-lft-neg-out N/A

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

      11. mul-1-neg N/A

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

      12. remove-double-neg N/A

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

      13. *-commutative N/A

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

      14. mul-1-neg N/A

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

      15. lower-fma.f64 N/A

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

      16. mul-1-neg N/A

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

      17. lower-neg.f64 29.1

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

   5. Applied rewrites 29.1%

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

   7. Applied rewrites 86.9%

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

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

   1. Initial program 92.2%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-neg-in N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 92.2

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

   4. Applied rewrites 92.2%

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

   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. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 92.0%

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

Alternative 3: 90.9% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- x (* y z)) (- t (* z a)))))
   (if (<= t_1 (- INFINITY))
     (* z (/ y (- (* z a) t)))
     (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 <= -((double) INFINITY)) {
		tmp = z * (y / ((z * a) - t));
	} 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 <= -Double.POSITIVE_INFINITY) {
		tmp = z * (y / ((z * a) - t));
	} 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 <= -math.inf:
		tmp = z * (y / ((z * a) - t))
	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 <= Float64(-Inf))
		tmp = Float64(z * Float64(y / Float64(Float64(z * a) - t)));
	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 <= -Inf)
		tmp = z * (y / ((z * a) - t));
	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, (-Infinity)], N[(z * N[(y / N[(N[(z * a), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]), $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))) < -inf.0

   1. Initial program 42.0%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-neg-in N/A

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

      9. associate-*r* N/A

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

      10. distribute-lft-neg-out N/A

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

      11. mul-1-neg N/A

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

      12. remove-double-neg N/A

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

      13. *-commutative N/A

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

      14. mul-1-neg N/A

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

      15. lower-fma.f64 N/A

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

      16. mul-1-neg N/A

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

      17. lower-neg.f64 29.1

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

   5. Applied rewrites 29.1%

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

   7. Applied rewrites 86.9%

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

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

   1. Initial program 92.2%

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

   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. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 92.0%

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

Alternative 4: 66.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \frac{z}{z \cdot a - t}\\ \mathbf{if}\;z \leq -1.6 \cdot 10^{+124}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -2.4 \cdot 10^{-80}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -2.7 \cdot 10^{-153}:\\ \;\;\;\;\frac{x - y \cdot z}{t}\\ \mathbf{elif}\;z \leq 1.28 \cdot 10^{-36}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* y (/ z (- (* z a) t)))))
   (if (<= z -1.6e+124)
     (/ y a)
     (if (<= z -2.4e-80)
       t_1
       (if (<= z -2.7e-153)
         (/ (- x (* y z)) t)
         (if (<= z 1.28e-36) (/ x (- t (* z a))) t_1))))))

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 <= -1.6e+124) {
		tmp = y / a;
	} else if (z <= -2.4e-80) {
		tmp = t_1;
	} else if (z <= -2.7e-153) {
		tmp = (x - (y * z)) / t;
	} else if (z <= 1.28e-36) {
		tmp = x / (t - (z * a));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = y * (z / ((z * a) - t))
    if (z <= (-1.6d+124)) then
        tmp = y / a
    else if (z <= (-2.4d-80)) then
        tmp = t_1
    else if (z <= (-2.7d-153)) then
        tmp = (x - (y * z)) / t
    else if (z <= 1.28d-36) then
        tmp = x / (t - (z * a))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = y * (z / ((z * a) - t));
	double tmp;
	if (z <= -1.6e+124) {
		tmp = y / a;
	} else if (z <= -2.4e-80) {
		tmp = t_1;
	} else if (z <= -2.7e-153) {
		tmp = (x - (y * z)) / t;
	} else if (z <= 1.28e-36) {
		tmp = x / (t - (z * a));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = y * (z / ((z * a) - t))
	tmp = 0
	if z <= -1.6e+124:
		tmp = y / a
	elif z <= -2.4e-80:
		tmp = t_1
	elif z <= -2.7e-153:
		tmp = (x - (y * z)) / t
	elif z <= 1.28e-36:
		tmp = x / (t - (z * a))
	else:
		tmp = t_1
	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 <= -1.6e+124)
		tmp = Float64(y / a);
	elseif (z <= -2.4e-80)
		tmp = t_1;
	elseif (z <= -2.7e-153)
		tmp = Float64(Float64(x - Float64(y * z)) / t);
	elseif (z <= 1.28e-36)
		tmp = Float64(x / Float64(t - Float64(z * a)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = y * (z / ((z * a) - t));
	tmp = 0.0;
	if (z <= -1.6e+124)
		tmp = y / a;
	elseif (z <= -2.4e-80)
		tmp = t_1;
	elseif (z <= -2.7e-153)
		tmp = (x - (y * z)) / t;
	elseif (z <= 1.28e-36)
		tmp = x / (t - (z * a));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(y * N[(z / N[(N[(z * a), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.6e+124], N[(y / a), $MachinePrecision], If[LessEqual[z, -2.4e-80], t$95$1, If[LessEqual[z, -2.7e-153], N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[z, 1.28e-36], N[(x / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -1.59999999999999996e124

   1. Initial program 53.0%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 69.8

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

   5. Applied rewrites 69.8%

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

   if -1.59999999999999996e124 < z < -2.3999999999999999e-80 or 1.28e-36 < z

   1. Initial program 80.8%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-neg-in N/A

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

      9. associate-*r* N/A

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

      10. distribute-lft-neg-out N/A

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

      11. mul-1-neg N/A

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

      12. remove-double-neg N/A

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

      13. *-commutative N/A

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

      14. mul-1-neg N/A

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

      15. lower-fma.f64 N/A

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

      16. mul-1-neg N/A

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

      17. lower-neg.f64 57.7

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

   5. Applied rewrites 57.7%

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

   7. Applied rewrites 69.4%

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

   if -2.3999999999999999e-80 < z < -2.70000000000000009e-153

   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. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 84.4

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

   5. Applied rewrites 84.4%

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

   if -2.70000000000000009e-153 < z < 1.28e-36

   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. lower-/.f64 N/A

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

      2. lower--.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. lower-*.f64 82.8

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

   5. Applied rewrites 82.8%

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

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.6 \cdot 10^{+124}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -2.4 \cdot 10^{-80}:\\ \;\;\;\;y \cdot \frac{z}{z \cdot a - t}\\ \mathbf{elif}\;z \leq -2.7 \cdot 10^{-153}:\\ \;\;\;\;\frac{x - y \cdot z}{t}\\ \mathbf{elif}\;z \leq 1.28 \cdot 10^{-36}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z}{z \cdot a - t}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.6 \cdot 10^{+66}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{elif}\;z \leq -3 \cdot 10^{-130}:\\ \;\;\;\;\frac{y \cdot z}{\mathsf{fma}\left(z, a, -t\right)}\\ \mathbf{elif}\;z \leq 1.28 \cdot 10^{-36}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z}{z \cdot a - t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -5.6e+66)
   (/ (- y (/ x z)) a)
   (if (<= z -3e-130)
     (/ (* y z) (fma z a (- t)))
     (if (<= z 1.28e-36) (/ 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 <= -5.6e+66) {
		tmp = (y - (x / z)) / a;
	} else if (z <= -3e-130) {
		tmp = (y * z) / fma(z, a, -t);
	} else if (z <= 1.28e-36) {
		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 <= -5.6e+66)
		tmp = Float64(Float64(y - Float64(x / z)) / a);
	elseif (z <= -3e-130)
		tmp = Float64(Float64(y * z) / fma(z, a, Float64(-t)));
	elseif (z <= 1.28e-36)
		tmp = Float64(x / Float64(t - Float64(z * a)));
	else
		tmp = Float64(y * Float64(z / Float64(Float64(z * a) - t)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -5.6e+66], N[(N[(y - N[(x / z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[z, -3e-130], N[(N[(y * z), $MachinePrecision] / N[(z * a + (-t)), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.28e-36], 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 4 regimes

2. if z < -5.6000000000000001e66

   1. Initial program 60.9%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   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. lower-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. Applied rewrites 79.4%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 74.2%

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

   if -5.6000000000000001e66 < z < -2.99999999999999986e-130

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-neg-in N/A

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

      9. associate-*r* N/A

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

      10. distribute-lft-neg-out N/A

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

      11. mul-1-neg N/A

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

      12. remove-double-neg N/A

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

      13. *-commutative N/A

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

      14. mul-1-neg N/A

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

      15. lower-fma.f64 N/A

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

      16. mul-1-neg N/A

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

      17. lower-neg.f64 68.5

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

   5. Applied rewrites 68.5%

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

   if -2.99999999999999986e-130 < z < 1.28e-36

   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. lower-/.f64 N/A

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

      2. lower--.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. lower-*.f64 82.5

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

   5. Applied rewrites 82.5%

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

   if 1.28e-36 < z

   1. Initial program 75.4%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-neg-in N/A

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

      9. associate-*r* N/A

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

      10. distribute-lft-neg-out N/A

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

      11. mul-1-neg N/A

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

      12. remove-double-neg N/A

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

      13. *-commutative N/A

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

      14. mul-1-neg N/A

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

      15. lower-fma.f64 N/A

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

      16. mul-1-neg N/A

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

      17. lower-neg.f64 58.1

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

   5. Applied rewrites 58.1%

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

   7. Applied rewrites 71.4%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.6 \cdot 10^{+66}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{elif}\;z \leq -3 \cdot 10^{-130}:\\ \;\;\;\;\frac{y \cdot z}{\mathsf{fma}\left(z, a, -t\right)}\\ \mathbf{elif}\;z \leq 1.28 \cdot 10^{-36}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z}{z \cdot a - t}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 63.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.7 \cdot 10^{+107}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -5.6 \cdot 10^{-130}:\\ \;\;\;\;z \cdot \frac{y}{z \cdot a - t}\\ \mathbf{elif}\;z \leq 1.28 \cdot 10^{-36}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{y}{\mathsf{fma}\left(z, a, -t\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -4.7e+107)
   (/ y a)
   (if (<= z -5.6e-130)
     (* z (/ y (- (* z a) t)))
     (if (<= z 1.28e-36) (/ x (- t (* z a))) (* z (/ y (fma z a (- t))))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -4.7e+107) {
		tmp = y / a;
	} else if (z <= -5.6e-130) {
		tmp = z * (y / ((z * a) - t));
	} else if (z <= 1.28e-36) {
		tmp = x / (t - (z * a));
	} else {
		tmp = z * (y / fma(z, a, -t));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -4.7e+107)
		tmp = Float64(y / a);
	elseif (z <= -5.6e-130)
		tmp = Float64(z * Float64(y / Float64(Float64(z * a) - t)));
	elseif (z <= 1.28e-36)
		tmp = Float64(x / Float64(t - Float64(z * a)));
	else
		tmp = Float64(z * Float64(y / fma(z, a, Float64(-t))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -4.7e+107], N[(y / a), $MachinePrecision], If[LessEqual[z, -5.6e-130], N[(z * N[(y / N[(N[(z * a), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.28e-36], N[(x / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z * N[(y / N[(z * a + (-t)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if z < -4.7000000000000001e107

   1. Initial program 58.5%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 67.9

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

   5. Applied rewrites 67.9%

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

   if -4.7000000000000001e107 < z < -5.60000000000000032e-130

   1. Initial program 93.6%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-neg-in N/A

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

      9. associate-*r* N/A

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

      10. distribute-lft-neg-out N/A

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

      11. mul-1-neg N/A

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

      12. remove-double-neg N/A

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

      13. *-commutative N/A

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

      14. mul-1-neg N/A

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

      15. lower-fma.f64 N/A

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

      16. mul-1-neg N/A

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

      17. lower-neg.f64 59.2

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

   5. Applied rewrites 59.2%

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

   7. Applied rewrites 63.2%

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

   if -5.60000000000000032e-130 < z < 1.28e-36

   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. lower-/.f64 N/A

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

      2. lower--.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. lower-*.f64 82.5

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

   5. Applied rewrites 82.5%

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

   if 1.28e-36 < z

   1. Initial program 75.4%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-neg-in N/A

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

      9. associate-*r* N/A

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

      10. distribute-lft-neg-out N/A

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

      11. mul-1-neg N/A

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

      12. remove-double-neg N/A

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

      13. *-commutative N/A

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

      14. mul-1-neg N/A

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

      15. lower-fma.f64 N/A

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

      16. mul-1-neg N/A

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

      17. lower-neg.f64 58.1

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

   5. Applied rewrites 58.1%

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

   7. Applied rewrites 71.4%

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

   9. Applied rewrites 66.7%

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

Alternative 7: 63.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot \frac{y}{z \cdot a - t}\\ \mathbf{if}\;z \leq -4.7 \cdot 10^{+107}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -5.6 \cdot 10^{-130}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.28 \cdot 10^{-36}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* z (/ y (- (* z a) t)))))
   (if (<= z -4.7e+107)
     (/ y a)
     (if (<= z -5.6e-130) t_1 (if (<= z 1.28e-36) (/ x (- t (* z a))) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = z * (y / ((z * a) - t));
	double tmp;
	if (z <= -4.7e+107) {
		tmp = y / a;
	} else if (z <= -5.6e-130) {
		tmp = t_1;
	} else if (z <= 1.28e-36) {
		tmp = x / (t - (z * a));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = z * (y / ((z * a) - t))
    if (z <= (-4.7d+107)) then
        tmp = y / a
    else if (z <= (-5.6d-130)) then
        tmp = t_1
    else if (z <= 1.28d-36) then
        tmp = x / (t - (z * a))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = z * (y / ((z * a) - t));
	double tmp;
	if (z <= -4.7e+107) {
		tmp = y / a;
	} else if (z <= -5.6e-130) {
		tmp = t_1;
	} else if (z <= 1.28e-36) {
		tmp = x / (t - (z * a));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = z * (y / ((z * a) - t))
	tmp = 0
	if z <= -4.7e+107:
		tmp = y / a
	elif z <= -5.6e-130:
		tmp = t_1
	elif z <= 1.28e-36:
		tmp = x / (t - (z * a))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(z * Float64(y / Float64(Float64(z * a) - t)))
	tmp = 0.0
	if (z <= -4.7e+107)
		tmp = Float64(y / a);
	elseif (z <= -5.6e-130)
		tmp = t_1;
	elseif (z <= 1.28e-36)
		tmp = Float64(x / Float64(t - Float64(z * a)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = z * (y / ((z * a) - t));
	tmp = 0.0;
	if (z <= -4.7e+107)
		tmp = y / a;
	elseif (z <= -5.6e-130)
		tmp = t_1;
	elseif (z <= 1.28e-36)
		tmp = x / (t - (z * a));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(z * N[(y / N[(N[(z * a), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -4.7e+107], N[(y / a), $MachinePrecision], If[LessEqual[z, -5.6e-130], t$95$1, If[LessEqual[z, 1.28e-36], N[(x / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.7000000000000001e107

   1. Initial program 58.5%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 67.9

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

   5. Applied rewrites 67.9%

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

   if -4.7000000000000001e107 < z < -5.60000000000000032e-130 or 1.28e-36 < z

   1. Initial program 82.4%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-neg-in N/A

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

      9. associate-*r* N/A

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

      10. distribute-lft-neg-out N/A

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

      11. mul-1-neg N/A

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

      12. remove-double-neg N/A

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

      13. *-commutative N/A

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

      14. mul-1-neg N/A

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

      15. lower-fma.f64 N/A

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

      16. mul-1-neg N/A

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

      17. lower-neg.f64 58.5

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

   5. Applied rewrites 58.5%

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

   7. Applied rewrites 65.3%

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

   if -5.60000000000000032e-130 < z < 1.28e-36

   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. lower-/.f64 N/A

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

      2. lower--.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. lower-*.f64 82.5

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

   5. Applied rewrites 82.5%

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

Alternative 8: 65.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.9 \cdot 10^{+91}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 1.85 \cdot 10^{+34}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{elif}\;z \leq 1.22 \cdot 10^{+101}:\\ \;\;\;\;z \cdot \frac{y}{-t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -2.9e+91)
   (/ y a)
   (if (<= z 1.85e+34)
     (/ x (- t (* z a)))
     (if (<= z 1.22e+101) (* z (/ y (- t))) (/ y a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.9e+91) {
		tmp = y / a;
	} else if (z <= 1.85e+34) {
		tmp = x / (t - (z * a));
	} else if (z <= 1.22e+101) {
		tmp = z * (y / -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 <= (-2.9d+91)) then
        tmp = y / a
    else if (z <= 1.85d+34) then
        tmp = x / (t - (z * a))
    else if (z <= 1.22d+101) then
        tmp = z * (y / -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 <= -2.9e+91) {
		tmp = y / a;
	} else if (z <= 1.85e+34) {
		tmp = x / (t - (z * a));
	} else if (z <= 1.22e+101) {
		tmp = z * (y / -t);
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -2.9e+91:
		tmp = y / a
	elif z <= 1.85e+34:
		tmp = x / (t - (z * a))
	elif z <= 1.22e+101:
		tmp = z * (y / -t)
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -2.9e+91)
		tmp = Float64(y / a);
	elseif (z <= 1.85e+34)
		tmp = Float64(x / Float64(t - Float64(z * a)));
	elseif (z <= 1.22e+101)
		tmp = Float64(z * Float64(y / Float64(-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 <= -2.9e+91)
		tmp = y / a;
	elseif (z <= 1.85e+34)
		tmp = x / (t - (z * a));
	elseif (z <= 1.22e+101)
		tmp = z * (y / -t);
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -2.9e+91], N[(y / a), $MachinePrecision], If[LessEqual[z, 1.85e+34], N[(x / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.22e+101], N[(z * N[(y / (-t)), $MachinePrecision]), $MachinePrecision], N[(y / a), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.90000000000000014e91 or 1.22e101 < z

   1. Initial program 59.4%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 68.1

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

   5. Applied rewrites 68.1%

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

   if -2.90000000000000014e91 < z < 1.85000000000000004e34

   1. Initial program 98.5%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   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. lower-/.f64 N/A

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

      2. lower--.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. lower-*.f64 66.7

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

   5. Applied rewrites 66.7%

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

   if 1.85000000000000004e34 < z < 1.22e101

   1. Initial program 83.6%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   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. lower-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. Applied rewrites 99.6%

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

   6. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.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. lower-*.f64 49.5

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

   8. Applied rewrites 49.5%

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

   10. Applied rewrites 65.4%

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

   11. Taylor expanded in z around inf

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

   13. Applied rewrites 57.1%

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

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.9 \cdot 10^{+91}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 1.85 \cdot 10^{+34}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{elif}\;z \leq 1.22 \cdot 10^{+101}:\\ \;\;\;\;z \cdot \frac{y}{-t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 61.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.1 \cdot 10^{+55}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;a \leq 4.3 \cdot 10^{+18}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-z, y, x\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -1.1e+55)
   (/ y a)
   (if (<= a 4.3e+18) (/ (fma (- z) y x) t) (/ x (- t (* z a))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -1.1e+55) {
		tmp = y / a;
	} else if (a <= 4.3e+18) {
		tmp = fma(-z, y, x) / t;
	} else {
		tmp = x / (t - (z * a));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -1.1e+55)
		tmp = Float64(y / a);
	elseif (a <= 4.3e+18)
		tmp = Float64(fma(Float64(-z), y, x) / t);
	else
		tmp = Float64(x / Float64(t - Float64(z * a)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -1.1e+55], N[(y / a), $MachinePrecision], If[LessEqual[a, 4.3e+18], N[(N[((-z) * y + x), $MachinePrecision] / t), $MachinePrecision], N[(x / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.10000000000000005e55

   1. Initial program 70.2%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 62.9

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

   5. Applied rewrites 62.9%

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

   if -1.10000000000000005e55 < a < 4.3e18

   1. Initial program 90.2%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   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. lower-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. Applied rewrites 96.7%

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

   6. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.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. lower-*.f64 68.5

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

   8. Applied rewrites 68.5%

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

   10. Applied rewrites 68.5%

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

   if 4.3e18 < a

   1. Initial program 81.9%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   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. lower-/.f64 N/A

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

      2. lower--.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. lower-*.f64 61.1

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

   5. Applied rewrites 61.1%

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

Alternative 10: 61.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.1 \cdot 10^{+55}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;a \leq 4.3 \cdot 10^{+18}:\\ \;\;\;\;\frac{x - y \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -1.1e+55)
   (/ y a)
   (if (<= a 4.3e+18) (/ (- x (* y z)) t) (/ x (- t (* z a))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -1.1e+55) {
		tmp = y / a;
	} else if (a <= 4.3e+18) {
		tmp = (x - (y * z)) / t;
	} else {
		tmp = x / (t - (z * a));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (a <= (-1.1d+55)) then
        tmp = y / a
    else if (a <= 4.3d+18) then
        tmp = (x - (y * z)) / t
    else
        tmp = x / (t - (z * a))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -1.1e+55) {
		tmp = y / a;
	} else if (a <= 4.3e+18) {
		tmp = (x - (y * z)) / t;
	} else {
		tmp = x / (t - (z * a));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if a <= -1.1e+55:
		tmp = y / a
	elif a <= 4.3e+18:
		tmp = (x - (y * z)) / t
	else:
		tmp = x / (t - (z * a))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -1.1e+55)
		tmp = Float64(y / a);
	elseif (a <= 4.3e+18)
		tmp = Float64(Float64(x - Float64(y * z)) / t);
	else
		tmp = Float64(x / Float64(t - Float64(z * a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (a <= -1.1e+55)
		tmp = y / a;
	elseif (a <= 4.3e+18)
		tmp = (x - (y * z)) / t;
	else
		tmp = x / (t - (z * a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -1.1e+55], N[(y / a), $MachinePrecision], If[LessEqual[a, 4.3e+18], N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], N[(x / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.10000000000000005e55

   1. Initial program 70.2%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 62.9

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

   5. Applied rewrites 62.9%

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

   if -1.10000000000000005e55 < a < 4.3e18

   1. Initial program 90.2%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   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. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 68.5

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

   5. Applied rewrites 68.5%

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

   if 4.3e18 < a

   1. Initial program 81.9%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   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. lower-/.f64 N/A

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

      2. lower--.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. lower-*.f64 61.1

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

   5. Applied rewrites 61.1%

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

Alternative 11: 55.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.3 \cdot 10^{-47}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 4.6 \cdot 10^{-38}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -2.3e-47) (/ y a) (if (<= z 4.6e-38) (/ x t) (/ y a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.3e-47) {
		tmp = y / a;
	} else if (z <= 4.6e-38) {
		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 <= (-2.3d-47)) then
        tmp = y / a
    else if (z <= 4.6d-38) 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 <= -2.3e-47) {
		tmp = y / a;
	} else if (z <= 4.6e-38) {
		tmp = x / t;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -2.3e-47:
		tmp = y / a
	elif z <= 4.6e-38:
		tmp = x / t
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -2.3e-47)
		tmp = Float64(y / a);
	elseif (z <= 4.6e-38)
		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 <= -2.3e-47)
		tmp = y / a;
	elseif (z <= 4.6e-38)
		tmp = x / t;
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -2.3e-47], N[(y / a), $MachinePrecision], If[LessEqual[z, 4.6e-38], N[(x / t), $MachinePrecision], N[(y / a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.29999999999999982e-47 or 4.60000000000000003e-38 < z

   1. Initial program 73.6%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 54.7

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

   5. Applied rewrites 54.7%

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

   if -2.29999999999999982e-47 < z < 4.60000000000000003e-38

   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. lower-/.f64 60.7

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

   5. Applied rewrites 60.7%

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

Alternative 12: 35.9% 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 84.4%

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

3. Taylor expanded in z around 0

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

   1. lower-/.f64 32.6

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

5. Applied rewrites 32.6%

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

Developer Target 1: 97.6% 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 2024233 
(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))))