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

Percentage Accurate: 85.4% → 90.8%

Time: 4.5s

Alternatives: 8

Speedup: 0.7×

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: 85.4% 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: 90.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y - \frac{x}{z}}{a}\\ \mathbf{if}\;z \leq -8.5 \cdot 10^{+90}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{+122}:\\ \;\;\;\;\frac{x - y \cdot z}{\mathsf{fma}\left(-z, a, t\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- y (/ x z)) a)))
   (if (<= z -8.5e+90)
     t_1
     (if (<= z 1.2e+122) (/ (- x (* y z)) (fma (- z) a t)) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (y - (x / z)) / a;
	double tmp;
	if (z <= -8.5e+90) {
		tmp = t_1;
	} else if (z <= 1.2e+122) {
		tmp = (x - (y * z)) / fma(-z, a, t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(y - Float64(x / z)) / a)
	tmp = 0.0
	if (z <= -8.5e+90)
		tmp = t_1;
	elseif (z <= 1.2e+122)
		tmp = Float64(Float64(x - Float64(y * z)) / fma(Float64(-z), a, t));
	else
		tmp = t_1;
	end
	return 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, -8.5e+90], t$95$1, If[LessEqual[z, 1.2e+122], N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / N[((-z) * a + t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -8.5000000000000002e90 or 1.2000000000000001e122 < z

   1. Initial program 62.8%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. div-sub N/A

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

      7. sub-neg N/A

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

      8. distribute-lft-in N/A

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

      9. associate-/l* N/A

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

      10. *-inverses N/A

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

      11. *-rgt-identity N/A

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

      12. neg-mul-1 N/A

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

      13. remove-double-neg N/A

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

      14. +-commutative N/A

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

      15. mul-1-neg N/A

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

      16. unsub-neg N/A

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

      17. lower--.f64 N/A

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

      18. lower-/.f64 89.7

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

   5. Applied rewrites 89.7%

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

   if -8.5000000000000002e90 < z < 1.2000000000000001e122

   1. Initial program 98.7%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 98.7

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

   4. Applied rewrites 98.7%

      \[\leadsto \frac{x - y \cdot z}{\color{blue}{\mathsf{fma}\left(-z, a, t\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 90.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y - \frac{x}{z}}{a}\\ \mathbf{if}\;z \leq -8.5 \cdot 10^{+90}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{+122}:\\ \;\;\;\;\frac{x - y \cdot z}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- y (/ x z)) a)))
   (if (<= z -8.5e+90)
     t_1
     (if (<= z 1.2e+122) (/ (- x (* y z)) (- t (* a z))) 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 <= -8.5e+90) {
		tmp = t_1;
	} else if (z <= 1.2e+122) {
		tmp = (x - (y * z)) / (t - (a * z));
	} 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 <= (-8.5d+90)) then
        tmp = t_1
    else if (z <= 1.2d+122) then
        tmp = (x - (y * z)) / (t - (a * z))
    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 <= -8.5e+90) {
		tmp = t_1;
	} else if (z <= 1.2e+122) {
		tmp = (x - (y * z)) / (t - (a * z));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (y - (x / z)) / a
	tmp = 0
	if z <= -8.5e+90:
		tmp = t_1
	elif z <= 1.2e+122:
		tmp = (x - (y * z)) / (t - (a * z))
	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 <= -8.5e+90)
		tmp = t_1;
	elseif (z <= 1.2e+122)
		tmp = Float64(Float64(x - Float64(y * z)) / Float64(t - Float64(a * z)));
	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 <= -8.5e+90)
		tmp = t_1;
	elseif (z <= 1.2e+122)
		tmp = (x - (y * z)) / (t - (a * z));
	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, -8.5e+90], t$95$1, If[LessEqual[z, 1.2e+122], N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / N[(t - N[(a * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -8.5000000000000002e90 or 1.2000000000000001e122 < z

   1. Initial program 62.8%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. div-sub N/A

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

      7. sub-neg N/A

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

      8. distribute-lft-in N/A

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

      9. associate-/l* N/A

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

      10. *-inverses N/A

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

      11. *-rgt-identity N/A

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

      12. neg-mul-1 N/A

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

      13. remove-double-neg N/A

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

      14. +-commutative N/A

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

      15. mul-1-neg N/A

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

      16. unsub-neg N/A

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

      17. lower--.f64 N/A

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

      18. lower-/.f64 89.7

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

   5. Applied rewrites 89.7%

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

   if -8.5000000000000002e90 < z < 1.2000000000000001e122

   1. Initial program 98.7%

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

Alternative 3: 71.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y - \frac{x}{z}}{a}\\ \mathbf{if}\;z \leq -5.9 \cdot 10^{+59}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 9.8 \cdot 10^{-58}:\\ \;\;\;\;\frac{x}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- y (/ x z)) a)))
   (if (<= z -5.9e+59) t_1 (if (<= z 9.8e-58) (/ x (- t (* a z))) 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 <= -5.9e+59) {
		tmp = t_1;
	} else if (z <= 9.8e-58) {
		tmp = x / (t - (a * z));
	} 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 <= (-5.9d+59)) then
        tmp = t_1
    else if (z <= 9.8d-58) then
        tmp = x / (t - (a * z))
    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 <= -5.9e+59) {
		tmp = t_1;
	} else if (z <= 9.8e-58) {
		tmp = x / (t - (a * z));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (y - (x / z)) / a
	tmp = 0
	if z <= -5.9e+59:
		tmp = t_1
	elif z <= 9.8e-58:
		tmp = x / (t - (a * z))
	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 <= -5.9e+59)
		tmp = t_1;
	elseif (z <= 9.8e-58)
		tmp = Float64(x / Float64(t - Float64(a * z)));
	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 <= -5.9e+59)
		tmp = t_1;
	elseif (z <= 9.8e-58)
		tmp = x / (t - (a * z));
	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, -5.9e+59], t$95$1, If[LessEqual[z, 9.8e-58], N[(x / N[(t - N[(a * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -5.90000000000000038e59 or 9.80000000000000061e-58 < z

   1. Initial program 71.2%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. div-sub N/A

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

      7. sub-neg N/A

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

      8. distribute-lft-in N/A

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

      9. associate-/l* N/A

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

      10. *-inverses N/A

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

      11. *-rgt-identity N/A

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

      12. neg-mul-1 N/A

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

      13. remove-double-neg N/A

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

      14. +-commutative N/A

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

      15. mul-1-neg N/A

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

      16. unsub-neg N/A

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

      17. lower--.f64 N/A

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

      18. lower-/.f64 82.3

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

   5. Applied rewrites 82.3%

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

   if -5.90000000000000038e59 < z < 9.80000000000000061e-58

   1. Initial program 99.9%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   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. lower-*.f64 74.5

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

   5. Applied rewrites 74.5%

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

Alternative 4: 67.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{t - a \cdot z}\\ \mathbf{if}\;x \leq -1.2 \cdot 10^{-58}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-39}:\\ \;\;\;\;\frac{z}{\mathsf{fma}\left(a, z, -t\right)} \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ x (- t (* a z)))))
   (if (<= x -1.2e-58)
     t_1
     (if (<= x 9.5e-39) (* (/ z (fma a z (- t))) y) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x / (t - (a * z));
	double tmp;
	if (x <= -1.2e-58) {
		tmp = t_1;
	} else if (x <= 9.5e-39) {
		tmp = (z / fma(a, z, -t)) * y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x / Float64(t - Float64(a * z)))
	tmp = 0.0
	if (x <= -1.2e-58)
		tmp = t_1;
	elseif (x <= 9.5e-39)
		tmp = Float64(Float64(z / fma(a, z, Float64(-t))) * y);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x / N[(t - N[(a * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.2e-58], t$95$1, If[LessEqual[x, 9.5e-39], N[(N[(z / N[(a * z + (-t)), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.2e-58 or 9.4999999999999999e-39 < x

   1. Initial program 87.7%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   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. lower-*.f64 73.9

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

   5. Applied rewrites 73.9%

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

   if -1.2e-58 < x < 9.4999999999999999e-39

   1. Initial program 82.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{y \cdot z}{t - a \cdot z}} \]
   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. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. sub-neg N/A

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

      7. mul-1-neg N/A

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

      8. +-commutative N/A

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

      9. distribute-neg-in N/A

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

      10. associate-*r* N/A

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

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

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

      12. mul-1-neg N/A

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

      13. remove-double-neg N/A

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

      14. mul-1-neg N/A

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

      15. lower-fma.f64 N/A

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

      16. mul-1-neg N/A

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

      17. lower-neg.f64 73.0

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

   5. Applied rewrites 73.0%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 74.9%

      \[\leadsto y \cdot \color{blue}{\frac{z}{\mathsf{fma}\left(a, z, -t\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{-58}:\\ \;\;\;\;\frac{x}{t - a \cdot z}\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-39}:\\ \;\;\;\;\frac{z}{\mathsf{fma}\left(a, z, -t\right)} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{t - a \cdot z}\\ \mathbf{if}\;x \leq -1.2 \cdot 10^{-58}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-39}:\\ \;\;\;\;\frac{z}{a \cdot z - t} \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ x (- t (* a z)))))
   (if (<= x -1.2e-58) t_1 (if (<= x 9.5e-39) (* (/ z (- (* a z) t)) y) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x / (t - (a * z));
	double tmp;
	if (x <= -1.2e-58) {
		tmp = t_1;
	} else if (x <= 9.5e-39) {
		tmp = (z / ((a * z) - t)) * y;
	} 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 = x / (t - (a * z))
    if (x <= (-1.2d-58)) then
        tmp = t_1
    else if (x <= 9.5d-39) then
        tmp = (z / ((a * z) - t)) * y
    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 = x / (t - (a * z));
	double tmp;
	if (x <= -1.2e-58) {
		tmp = t_1;
	} else if (x <= 9.5e-39) {
		tmp = (z / ((a * z) - t)) * y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = x / (t - (a * z))
	tmp = 0
	if x <= -1.2e-58:
		tmp = t_1
	elif x <= 9.5e-39:
		tmp = (z / ((a * z) - t)) * y
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x / Float64(t - Float64(a * z)))
	tmp = 0.0
	if (x <= -1.2e-58)
		tmp = t_1;
	elseif (x <= 9.5e-39)
		tmp = Float64(Float64(z / Float64(Float64(a * z) - t)) * y);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = x / (t - (a * z));
	tmp = 0.0;
	if (x <= -1.2e-58)
		tmp = t_1;
	elseif (x <= 9.5e-39)
		tmp = (z / ((a * z) - t)) * y;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x / N[(t - N[(a * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.2e-58], t$95$1, If[LessEqual[x, 9.5e-39], N[(N[(z / N[(N[(a * z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.2e-58 or 9.4999999999999999e-39 < x

   1. Initial program 87.7%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   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. lower-*.f64 73.9

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

   5. Applied rewrites 73.9%

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

   if -1.2e-58 < x < 9.4999999999999999e-39

   1. Initial program 82.9%

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

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 84.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 74.9%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{-58}:\\ \;\;\;\;\frac{x}{t - a \cdot z}\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-39}:\\ \;\;\;\;\frac{z}{a \cdot z - t} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 62.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.4 \cdot 10^{+93}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{-57}:\\ \;\;\;\;\frac{x}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -5.4e+93) (/ y a) (if (<= z 1.05e-57) (/ x (- t (* a z))) (/ y a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -5.4e+93) {
		tmp = y / a;
	} else if (z <= 1.05e-57) {
		tmp = x / (t - (a * z));
	} 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 <= (-5.4d+93)) then
        tmp = y / a
    else if (z <= 1.05d-57) then
        tmp = x / (t - (a * z))
    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 <= -5.4e+93) {
		tmp = y / a;
	} else if (z <= 1.05e-57) {
		tmp = x / (t - (a * z));
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -5.4e+93:
		tmp = y / a
	elif z <= 1.05e-57:
		tmp = x / (t - (a * z))
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -5.4e+93)
		tmp = Float64(y / a);
	elseif (z <= 1.05e-57)
		tmp = Float64(x / Float64(t - Float64(a * z)));
	else
		tmp = Float64(y / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -5.4e+93)
		tmp = y / a;
	elseif (z <= 1.05e-57)
		tmp = x / (t - (a * z));
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -5.4e+93], N[(y / a), $MachinePrecision], If[LessEqual[z, 1.05e-57], N[(x / N[(t - N[(a * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y / a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -5.3999999999999999e93 or 1.05e-57 < z

   1. Initial program 69.6%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 65.6

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

   5. Applied rewrites 65.6%

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

   if -5.3999999999999999e93 < z < 1.05e-57

   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. lower-*.f64 73.2

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

   5. Applied rewrites 73.2%

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

Alternative 7: 54.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.9 \cdot 10^{+59}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 9.8 \cdot 10^{-58}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -5.9e+59) (/ y a) (if (<= z 9.8e-58) (/ x t) (/ y a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -5.9e+59) {
		tmp = y / a;
	} else if (z <= 9.8e-58) {
		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 <= (-5.9d+59)) then
        tmp = y / a
    else if (z <= 9.8d-58) 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 <= -5.9e+59) {
		tmp = y / a;
	} else if (z <= 9.8e-58) {
		tmp = x / t;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -5.9e+59:
		tmp = y / a
	elif z <= 9.8e-58:
		tmp = x / t
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -5.9e+59)
		tmp = Float64(y / a);
	elseif (z <= 9.8e-58)
		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 <= -5.9e+59)
		tmp = y / a;
	elseif (z <= 9.8e-58)
		tmp = x / t;
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -5.9e+59], N[(y / a), $MachinePrecision], If[LessEqual[z, 9.8e-58], N[(x / t), $MachinePrecision], N[(y / a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -5.90000000000000038e59 or 9.80000000000000061e-58 < z

   1. Initial program 71.2%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 64.2

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

   5. Applied rewrites 64.2%

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

   if -5.90000000000000038e59 < z < 9.80000000000000061e-58

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 54.5

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

   5. Applied rewrites 54.5%

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

Alternative 8: 35.6% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.8%

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

3. Taylor expanded in z around 0

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

   1. lower-/.f64 33.9

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

5. Applied rewrites 33.9%

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

Developer Target 1: 97.5% 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 2024308 
(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))))