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

Percentage Accurate: 85.5% → 91.2%

Time: 12.0s

Alternatives: 10

Speedup: 0.5×

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: 85.5% 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: 91.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t - z \cdot a\\ \mathbf{if}\;z \leq -5.2 \cdot 10^{+80} \lor \neg \left(z \leq 1.05 \cdot 10^{+98}\right):\\ \;\;\;\;\frac{-y}{\frac{t}{z} - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t\_1} - \frac{z \cdot y}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- t (* z a))))
   (if (or (<= z -5.2e+80) (not (<= z 1.05e+98)))
     (/ (- y) (- (/ t z) a))
     (- (/ x t_1) (/ (* z y) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = t - (z * a);
	double tmp;
	if ((z <= -5.2e+80) || !(z <= 1.05e+98)) {
		tmp = -y / ((t / z) - a);
	} else {
		tmp = (x / t_1) - ((z * y) / 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 = t - (z * a)
    if ((z <= (-5.2d+80)) .or. (.not. (z <= 1.05d+98))) then
        tmp = -y / ((t / z) - a)
    else
        tmp = (x / t_1) - ((z * y) / 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 = t - (z * a);
	double tmp;
	if ((z <= -5.2e+80) || !(z <= 1.05e+98)) {
		tmp = -y / ((t / z) - a);
	} else {
		tmp = (x / t_1) - ((z * y) / t_1);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = t - (z * a)
	tmp = 0
	if (z <= -5.2e+80) or not (z <= 1.05e+98):
		tmp = -y / ((t / z) - a)
	else:
		tmp = (x / t_1) - ((z * y) / t_1)
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(t - Float64(z * a))
	tmp = 0.0
	if ((z <= -5.2e+80) || !(z <= 1.05e+98))
		tmp = Float64(Float64(-y) / Float64(Float64(t / z) - a));
	else
		tmp = Float64(Float64(x / t_1) - Float64(Float64(z * y) / t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = t - (z * a);
	tmp = 0.0;
	if ((z <= -5.2e+80) || ~((z <= 1.05e+98)))
		tmp = -y / ((t / z) - a);
	else
		tmp = (x / t_1) - ((z * y) / t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[z, -5.2e+80], N[Not[LessEqual[z, 1.05e+98]], $MachinePrecision]], N[((-y) / N[(N[(t / z), $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision], N[(N[(x / t$95$1), $MachinePrecision] - N[(N[(z * y), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -5.19999999999999963e80 or 1.05000000000000002e98 < z

   1. Initial program 61.1%

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

      1. *-commutative 61.1%

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

   3. Simplified 61.1%

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

   5. Taylor expanded in x around 0 52.7%

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

      1. mul-1-neg 52.7%

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

      2. associate-/l* 66.7%

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

      3. associate-/r/ 63.1%

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

      4. cancel-sign-sub-inv 63.1%

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

      5. +-commutative 63.1%

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

      6. distribute-lft-neg-in 63.1%

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

      7. distribute-rgt-neg-in 63.1%

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

      8. fma-udef 63.2%

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

      9. associate-/r/ 66.7%

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

      10. fma-udef 66.7%

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

      11. distribute-rgt-neg-in 66.7%

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

      12. distribute-lft-neg-in 66.7%

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

      13. +-commutative 66.7%

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

      14. cancel-sign-sub-inv 66.7%

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

      15. *-commutative 66.7%

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

   7. Simplified 66.7%

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

   8. Taylor expanded in t around 0 86.0%

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

      1. neg-mul-1 86.0%

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

      2. +-commutative 86.0%

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

      3. unsub-neg 86.0%

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

   10. Simplified 86.0%

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

   if -5.19999999999999963e80 < z < 1.05000000000000002e98

   1. Initial program 97.0%

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

      1. *-commutative 97.0%

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

   3. Simplified 97.0%

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

   5. Taylor expanded in x around 0 97.0%

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

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.2 \cdot 10^{+80} \lor \neg \left(z \leq 1.05 \cdot 10^{+98}\right):\\ \;\;\;\;\frac{-y}{\frac{t}{z} - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - z \cdot a} - \frac{z \cdot y}{t - z \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 62.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{t - z \cdot a}\\ \mathbf{if}\;z \leq -1.7 \cdot 10^{+26}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -4.6 \cdot 10^{-24}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -2 \cdot 10^{-152}:\\ \;\;\;\;\frac{z \cdot \left(-y\right)}{t}\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{+45}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ x (- t (* z a)))))
   (if (<= z -1.7e+26)
     (/ y a)
     (if (<= z -4.6e-24)
       t_1
       (if (<= z -2e-152)
         (/ (* z (- y)) t)
         (if (<= z 1.25e+45) t_1 (/ y a)))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x / (t - (z * a));
	double tmp;
	if (z <= -1.7e+26) {
		tmp = y / a;
	} else if (z <= -4.6e-24) {
		tmp = t_1;
	} else if (z <= -2e-152) {
		tmp = (z * -y) / t;
	} else if (z <= 1.25e+45) {
		tmp = t_1;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x / (t - (z * a))
    if (z <= (-1.7d+26)) then
        tmp = y / a
    else if (z <= (-4.6d-24)) then
        tmp = t_1
    else if (z <= (-2d-152)) then
        tmp = (z * -y) / t
    else if (z <= 1.25d+45) then
        tmp = t_1
    else
        tmp = y / a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = x / (t - (z * a));
	double tmp;
	if (z <= -1.7e+26) {
		tmp = y / a;
	} else if (z <= -4.6e-24) {
		tmp = t_1;
	} else if (z <= -2e-152) {
		tmp = (z * -y) / t;
	} else if (z <= 1.25e+45) {
		tmp = t_1;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = x / (t - (z * a))
	tmp = 0
	if z <= -1.7e+26:
		tmp = y / a
	elif z <= -4.6e-24:
		tmp = t_1
	elif z <= -2e-152:
		tmp = (z * -y) / t
	elif z <= 1.25e+45:
		tmp = t_1
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x / Float64(t - Float64(z * a)))
	tmp = 0.0
	if (z <= -1.7e+26)
		tmp = Float64(y / a);
	elseif (z <= -4.6e-24)
		tmp = t_1;
	elseif (z <= -2e-152)
		tmp = Float64(Float64(z * Float64(-y)) / t);
	elseif (z <= 1.25e+45)
		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 / (t - (z * a));
	tmp = 0.0;
	if (z <= -1.7e+26)
		tmp = y / a;
	elseif (z <= -4.6e-24)
		tmp = t_1;
	elseif (z <= -2e-152)
		tmp = (z * -y) / t;
	elseif (z <= 1.25e+45)
		tmp = t_1;
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.7e+26], N[(y / a), $MachinePrecision], If[LessEqual[z, -4.6e-24], t$95$1, If[LessEqual[z, -2e-152], N[(N[(z * (-y)), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[z, 1.25e+45], t$95$1, N[(y / a), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.7000000000000001e26 or 1.25e45 < z

   1. Initial program 68.1%

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

      1. *-commutative 68.1%

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

   3. Simplified 68.1%

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

   5. Taylor expanded in z around inf 61.3%

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

   if -1.7000000000000001e26 < z < -4.6000000000000002e-24 or -2.00000000000000013e-152 < z < 1.25e45

   1. Initial program 99.0%

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

      1. *-commutative 99.0%

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

   3. Simplified 99.0%

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

   5. Taylor expanded in x around inf 77.7%

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

      1. *-commutative 77.7%

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

   7. Simplified 77.7%

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

   if -4.6000000000000002e-24 < z < -2.00000000000000013e-152

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0 75.4%

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

      1. mul-1-neg 75.4%

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

      2. associate-/l* 56.4%

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

      3. associate-/r/ 71.2%

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

      4. cancel-sign-sub-inv 71.2%

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

      5. +-commutative 71.2%

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

      6. distribute-lft-neg-in 71.2%

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

      7. distribute-rgt-neg-in 71.2%

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

      8. fma-udef 71.2%

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

      9. associate-/r/ 56.4%

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

      10. fma-udef 56.4%

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

      11. distribute-rgt-neg-in 56.4%

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

      12. distribute-lft-neg-in 56.4%

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

      13. +-commutative 56.4%

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

      14. cancel-sign-sub-inv 56.4%

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

      15. *-commutative 56.4%

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

   7. Simplified 56.4%

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

   8. Taylor expanded in t around 0 56.4%

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

      1. neg-mul-1 56.4%

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

      2. +-commutative 56.4%

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

      3. unsub-neg 56.4%

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

   10. Simplified 56.4%

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

   11. Taylor expanded in t around inf 66.9%

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

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.7 \cdot 10^{+26}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -4.6 \cdot 10^{-24}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{elif}\;z \leq -2 \cdot 10^{-152}:\\ \;\;\;\;\frac{z \cdot \left(-y\right)}{t}\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{+45}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-y}{\frac{t}{z} - a}\\ \mathbf{if}\;z \leq -6.7 \cdot 10^{+16}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -6.7 \cdot 10^{-155}:\\ \;\;\;\;\frac{x - z \cdot y}{t}\\ \mathbf{elif}\;z \leq 3 \cdot 10^{-53}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- y) (- (/ t z) a))))
   (if (<= z -6.7e+16)
     t_1
     (if (<= z -6.7e-155)
       (/ (- x (* z y)) t)
       (if (<= z 3e-53) (/ x (- t (* z a))) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = -y / ((t / z) - a);
	double tmp;
	if (z <= -6.7e+16) {
		tmp = t_1;
	} else if (z <= -6.7e-155) {
		tmp = (x - (z * y)) / t;
	} else if (z <= 3e-53) {
		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 / ((t / z) - a)
    if (z <= (-6.7d+16)) then
        tmp = t_1
    else if (z <= (-6.7d-155)) then
        tmp = (x - (z * y)) / t
    else if (z <= 3d-53) 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 / ((t / z) - a);
	double tmp;
	if (z <= -6.7e+16) {
		tmp = t_1;
	} else if (z <= -6.7e-155) {
		tmp = (x - (z * y)) / t;
	} else if (z <= 3e-53) {
		tmp = x / (t - (z * a));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = -y / ((t / z) - a)
	tmp = 0
	if z <= -6.7e+16:
		tmp = t_1
	elif z <= -6.7e-155:
		tmp = (x - (z * y)) / t
	elif z <= 3e-53:
		tmp = x / (t - (z * a))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(-y) / Float64(Float64(t / z) - a))
	tmp = 0.0
	if (z <= -6.7e+16)
		tmp = t_1;
	elseif (z <= -6.7e-155)
		tmp = Float64(Float64(x - Float64(z * y)) / t);
	elseif (z <= 3e-53)
		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 / ((t / z) - a);
	tmp = 0.0;
	if (z <= -6.7e+16)
		tmp = t_1;
	elseif (z <= -6.7e-155)
		tmp = (x - (z * y)) / t;
	elseif (z <= 3e-53)
		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[(N[(t / z), $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -6.7e+16], t$95$1, If[LessEqual[z, -6.7e-155], N[(N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[z, 3e-53], N[(x / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -6.7e16 or 3.0000000000000002e-53 < z

   1. Initial program 71.5%

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

      1. *-commutative 71.5%

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

   3. Simplified 71.5%

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

   5. Taylor expanded in x around 0 52.2%

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

      1. mul-1-neg 52.2%

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

      2. associate-/l* 62.6%

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

      3. associate-/r/ 60.3%

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

      4. cancel-sign-sub-inv 60.3%

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

      5. +-commutative 60.3%

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

      6. distribute-lft-neg-in 60.3%

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

      7. distribute-rgt-neg-in 60.3%

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

      8. fma-udef 60.3%

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

      9. associate-/r/ 62.6%

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

      10. fma-udef 62.6%

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

      11. distribute-rgt-neg-in 62.6%

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

      12. distribute-lft-neg-in 62.6%

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

      13. +-commutative 62.6%

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

      14. cancel-sign-sub-inv 62.6%

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

      15. *-commutative 62.6%

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

   7. Simplified 62.6%

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

   8. Taylor expanded in t around 0 76.4%

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

      1. neg-mul-1 76.4%

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

      2. +-commutative 76.4%

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

      3. unsub-neg 76.4%

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

   10. Simplified 76.4%

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

   if -6.7e16 < z < -6.70000000000000028e-155

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t around inf 87.6%

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

   if -6.70000000000000028e-155 < z < 3.0000000000000002e-53

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around inf 84.0%

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

      1. *-commutative 84.0%

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

   7. Simplified 84.0%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.7 \cdot 10^{+16}:\\ \;\;\;\;\frac{-y}{\frac{t}{z} - a}\\ \mathbf{elif}\;z \leq -6.7 \cdot 10^{-155}:\\ \;\;\;\;\frac{x - z \cdot y}{t}\\ \mathbf{elif}\;z \leq 3 \cdot 10^{-53}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-y}{\frac{t}{z} - a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-y}{\frac{t}{z} - a}\\ \mathbf{if}\;z \leq -7.8 \cdot 10^{+18}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -5.1 \cdot 10^{-154}:\\ \;\;\;\;\frac{x}{t} - \frac{z \cdot y}{t}\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{-53}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- y) (- (/ t z) a))))
   (if (<= z -7.8e+18)
     t_1
     (if (<= z -5.1e-154)
       (- (/ x t) (/ (* z y) t))
       (if (<= z 1.25e-53) (/ x (- t (* z a))) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = -y / ((t / z) - a);
	double tmp;
	if (z <= -7.8e+18) {
		tmp = t_1;
	} else if (z <= -5.1e-154) {
		tmp = (x / t) - ((z * y) / t);
	} else if (z <= 1.25e-53) {
		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 / ((t / z) - a)
    if (z <= (-7.8d+18)) then
        tmp = t_1
    else if (z <= (-5.1d-154)) then
        tmp = (x / t) - ((z * y) / t)
    else if (z <= 1.25d-53) 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 / ((t / z) - a);
	double tmp;
	if (z <= -7.8e+18) {
		tmp = t_1;
	} else if (z <= -5.1e-154) {
		tmp = (x / t) - ((z * y) / t);
	} else if (z <= 1.25e-53) {
		tmp = x / (t - (z * a));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = -y / ((t / z) - a)
	tmp = 0
	if z <= -7.8e+18:
		tmp = t_1
	elif z <= -5.1e-154:
		tmp = (x / t) - ((z * y) / t)
	elif z <= 1.25e-53:
		tmp = x / (t - (z * a))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(-y) / Float64(Float64(t / z) - a))
	tmp = 0.0
	if (z <= -7.8e+18)
		tmp = t_1;
	elseif (z <= -5.1e-154)
		tmp = Float64(Float64(x / t) - Float64(Float64(z * y) / t));
	elseif (z <= 1.25e-53)
		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 / ((t / z) - a);
	tmp = 0.0;
	if (z <= -7.8e+18)
		tmp = t_1;
	elseif (z <= -5.1e-154)
		tmp = (x / t) - ((z * y) / t);
	elseif (z <= 1.25e-53)
		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[(N[(t / z), $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -7.8e+18], t$95$1, If[LessEqual[z, -5.1e-154], N[(N[(x / t), $MachinePrecision] - N[(N[(z * y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.25e-53], N[(x / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -7.8e18 or 1.25e-53 < z

   1. Initial program 71.5%

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

      1. *-commutative 71.5%

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

   3. Simplified 71.5%

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

   5. Taylor expanded in x around 0 52.2%

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

      1. mul-1-neg 52.2%

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

      2. associate-/l* 62.6%

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

      3. associate-/r/ 60.3%

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

      4. cancel-sign-sub-inv 60.3%

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

      5. +-commutative 60.3%

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

      6. distribute-lft-neg-in 60.3%

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

      7. distribute-rgt-neg-in 60.3%

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

      8. fma-udef 60.3%

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

      9. associate-/r/ 62.6%

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

      10. fma-udef 62.6%

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

      11. distribute-rgt-neg-in 62.6%

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

      12. distribute-lft-neg-in 62.6%

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

      13. +-commutative 62.6%

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

      14. cancel-sign-sub-inv 62.6%

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

      15. *-commutative 62.6%

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

   7. Simplified 62.6%

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

   8. Taylor expanded in t around 0 76.4%

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

      1. neg-mul-1 76.4%

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

      2. +-commutative 76.4%

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

      3. unsub-neg 76.4%

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

   10. Simplified 76.4%

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

   if -7.8e18 < z < -5.0999999999999998e-154

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 99.9%

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

   6. Taylor expanded in a around 0 87.7%

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

   if -5.0999999999999998e-154 < z < 1.25e-53

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around inf 84.0%

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

      1. *-commutative 84.0%

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

   7. Simplified 84.0%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.8 \cdot 10^{+18}:\\ \;\;\;\;\frac{-y}{\frac{t}{z} - a}\\ \mathbf{elif}\;z \leq -5.1 \cdot 10^{-154}:\\ \;\;\;\;\frac{x}{t} - \frac{z \cdot y}{t}\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{-53}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-y}{\frac{t}{z} - a}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 65.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.5 \cdot 10^{+25}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -3.8 \cdot 10^{-154}:\\ \;\;\;\;\frac{x - z \cdot y}{t}\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{+44}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -5.5e+25)
   (/ y a)
   (if (<= z -3.8e-154)
     (/ (- x (* z y)) t)
     (if (<= z 2.3e+44) (/ x (- t (* z a))) (/ y a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -5.5e+25) {
		tmp = y / a;
	} else if (z <= -3.8e-154) {
		tmp = (x - (z * y)) / t;
	} else if (z <= 2.3e+44) {
		tmp = x / (t - (z * a));
	} else {
		tmp = y / a;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-5.5d+25)) then
        tmp = y / a
    else if (z <= (-3.8d-154)) then
        tmp = (x - (z * y)) / t
    else if (z <= 2.3d+44) then
        tmp = x / (t - (z * a))
    else
        tmp = y / a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -5.5e+25) {
		tmp = y / a;
	} else if (z <= -3.8e-154) {
		tmp = (x - (z * y)) / t;
	} else if (z <= 2.3e+44) {
		tmp = x / (t - (z * a));
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -5.5e+25:
		tmp = y / a
	elif z <= -3.8e-154:
		tmp = (x - (z * y)) / t
	elif z <= 2.3e+44:
		tmp = x / (t - (z * a))
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -5.5e+25)
		tmp = Float64(y / a);
	elseif (z <= -3.8e-154)
		tmp = Float64(Float64(x - Float64(z * y)) / t);
	elseif (z <= 2.3e+44)
		tmp = Float64(x / Float64(t - Float64(z * a)));
	else
		tmp = Float64(y / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -5.5e+25)
		tmp = y / a;
	elseif (z <= -3.8e-154)
		tmp = (x - (z * y)) / t;
	elseif (z <= 2.3e+44)
		tmp = x / (t - (z * a));
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -5.5e+25], N[(y / a), $MachinePrecision], If[LessEqual[z, -3.8e-154], N[(N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[z, 2.3e+44], N[(x / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y / a), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.50000000000000018e25 or 2.30000000000000004e44 < z

   1. Initial program 68.1%

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

      1. *-commutative 68.1%

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

   3. Simplified 68.1%

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

   5. Taylor expanded in z around inf 61.3%

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

   if -5.50000000000000018e25 < z < -3.8000000000000001e-154

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t around inf 87.6%

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

   if -3.8000000000000001e-154 < z < 2.30000000000000004e44

   1. Initial program 98.9%

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

      1. *-commutative 98.9%

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

   3. Simplified 98.9%

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

   5. Taylor expanded in x around inf 77.7%

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

      1. *-commutative 77.7%

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

   7. Simplified 77.7%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.5 \cdot 10^{+25}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -3.8 \cdot 10^{-154}:\\ \;\;\;\;\frac{x - z \cdot y}{t}\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{+44}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 72.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y - \frac{x}{z}}{a}\\ \mathbf{if}\;z \leq -1.06 \cdot 10^{+26}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -1.65 \cdot 10^{-155}:\\ \;\;\;\;\frac{x - z \cdot y}{t}\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{+38}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- y (/ x z)) a)))
   (if (<= z -1.06e+26)
     t_1
     (if (<= z -1.65e-155)
       (/ (- x (* z y)) t)
       (if (<= z 2.3e+38) (/ x (- t (* z a))) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (y - (x / z)) / a;
	double tmp;
	if (z <= -1.06e+26) {
		tmp = t_1;
	} else if (z <= -1.65e-155) {
		tmp = (x - (z * y)) / t;
	} else if (z <= 2.3e+38) {
		tmp = x / (t - (z * a));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (y - (x / z)) / a
    if (z <= (-1.06d+26)) then
        tmp = t_1
    else if (z <= (-1.65d-155)) then
        tmp = (x - (z * y)) / t
    else if (z <= 2.3d+38) then
        tmp = x / (t - (z * a))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (y - (x / z)) / a;
	double tmp;
	if (z <= -1.06e+26) {
		tmp = t_1;
	} else if (z <= -1.65e-155) {
		tmp = (x - (z * y)) / t;
	} else if (z <= 2.3e+38) {
		tmp = x / (t - (z * a));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (y - (x / z)) / a
	tmp = 0
	if z <= -1.06e+26:
		tmp = t_1
	elif z <= -1.65e-155:
		tmp = (x - (z * y)) / t
	elif z <= 2.3e+38:
		tmp = x / (t - (z * a))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(y - Float64(x / z)) / a)
	tmp = 0.0
	if (z <= -1.06e+26)
		tmp = t_1;
	elseif (z <= -1.65e-155)
		tmp = Float64(Float64(x - Float64(z * y)) / t);
	elseif (z <= 2.3e+38)
		tmp = Float64(x / Float64(t - Float64(z * a)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (y - (x / z)) / a;
	tmp = 0.0;
	if (z <= -1.06e+26)
		tmp = t_1;
	elseif (z <= -1.65e-155)
		tmp = (x - (z * y)) / t;
	elseif (z <= 2.3e+38)
		tmp = x / (t - (z * a));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y - N[(x / z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]}, If[LessEqual[z, -1.06e+26], t$95$1, If[LessEqual[z, -1.65e-155], N[(N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[z, 2.3e+38], N[(x / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.05999999999999997e26 or 2.3000000000000001e38 < z

   1. Initial program 67.6%

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

      1. *-commutative 67.6%

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

   3. Simplified 67.6%

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

   5. Taylor expanded in x around 0 67.6%

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

   6. Taylor expanded in a around inf 75.9%

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

      1. mul-1-neg 75.9%

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

      2. unsub-neg 75.9%

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

   8. Simplified 75.9%

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

   if -1.05999999999999997e26 < z < -1.64999999999999993e-155

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t around inf 87.6%

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

   if -1.64999999999999993e-155 < z < 2.3000000000000001e38

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around inf 78.3%

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

      1. *-commutative 78.3%

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

   7. Simplified 78.3%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.06 \cdot 10^{+26}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{elif}\;z \leq -1.65 \cdot 10^{-155}:\\ \;\;\;\;\frac{x - z \cdot y}{t}\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{+38}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 91.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.2 \cdot 10^{+78} \lor \neg \left(z \leq 6.8 \cdot 10^{+103}\right):\\ \;\;\;\;\frac{-y}{\frac{t}{z} - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - z \cdot y}{t - z \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -4.2e+78) (not (<= z 6.8e+103)))
   (/ (- y) (- (/ t z) a))
   (/ (- x (* z y)) (- t (* z a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -4.2e+78) || !(z <= 6.8e+103)) {
		tmp = -y / ((t / z) - a);
	} else {
		tmp = (x - (z * y)) / (t - (z * a));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((z <= (-4.2d+78)) .or. (.not. (z <= 6.8d+103))) then
        tmp = -y / ((t / z) - a)
    else
        tmp = (x - (z * y)) / (t - (z * a))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -4.2e+78) || !(z <= 6.8e+103)) {
		tmp = -y / ((t / z) - a);
	} else {
		tmp = (x - (z * y)) / (t - (z * a));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (z <= -4.2e+78) or not (z <= 6.8e+103):
		tmp = -y / ((t / z) - a)
	else:
		tmp = (x - (z * y)) / (t - (z * a))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -4.2e+78) || !(z <= 6.8e+103))
		tmp = Float64(Float64(-y) / Float64(Float64(t / z) - a));
	else
		tmp = Float64(Float64(x - Float64(z * y)) / Float64(t - Float64(z * a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((z <= -4.2e+78) || ~((z <= 6.8e+103)))
		tmp = -y / ((t / z) - a);
	else
		tmp = (x - (z * y)) / (t - (z * a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -4.2e+78], N[Not[LessEqual[z, 6.8e+103]], $MachinePrecision]], N[((-y) / N[(N[(t / z), $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision], N[(N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision] / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4.2000000000000002e78 or 6.7999999999999997e103 < z

   1. Initial program 61.1%

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

      1. *-commutative 61.1%

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

   3. Simplified 61.1%

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

   5. Taylor expanded in x around 0 52.7%

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

      1. mul-1-neg 52.7%

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

      2. associate-/l* 66.7%

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

      3. associate-/r/ 63.1%

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

      4. cancel-sign-sub-inv 63.1%

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

      5. +-commutative 63.1%

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

      6. distribute-lft-neg-in 63.1%

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

      7. distribute-rgt-neg-in 63.1%

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

      8. fma-udef 63.2%

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

      9. associate-/r/ 66.7%

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

      10. fma-udef 66.7%

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

      11. distribute-rgt-neg-in 66.7%

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

      12. distribute-lft-neg-in 66.7%

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

      13. +-commutative 66.7%

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

      14. cancel-sign-sub-inv 66.7%

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

      15. *-commutative 66.7%

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

   7. Simplified 66.7%

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

   8. Taylor expanded in t around 0 86.0%

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

      1. neg-mul-1 86.0%

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

      2. +-commutative 86.0%

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

      3. unsub-neg 86.0%

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

   10. Simplified 86.0%

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

   if -4.2000000000000002e78 < z < 6.7999999999999997e103

   1. Initial program 97.0%

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

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.2 \cdot 10^{+78} \lor \neg \left(z \leq 6.8 \cdot 10^{+103}\right):\\ \;\;\;\;\frac{-y}{\frac{t}{z} - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - z \cdot y}{t - z \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 54.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.5 \cdot 10^{+23}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -2 \cdot 10^{-152}:\\ \;\;\;\;\frac{z \cdot \left(-y\right)}{t}\\ \mathbf{elif}\;z \leq 0.032:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -5.5e+23)
   (/ y a)
   (if (<= z -2e-152) (/ (* z (- y)) t) (if (<= z 0.032) (/ x t) (/ y a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -5.5e+23) {
		tmp = y / a;
	} else if (z <= -2e-152) {
		tmp = (z * -y) / t;
	} else if (z <= 0.032) {
		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 <= (-5.5d+23)) then
        tmp = y / a
    else if (z <= (-2d-152)) then
        tmp = (z * -y) / t
    else if (z <= 0.032d0) 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 <= -5.5e+23) {
		tmp = y / a;
	} else if (z <= -2e-152) {
		tmp = (z * -y) / t;
	} else if (z <= 0.032) {
		tmp = x / t;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -5.5e+23:
		tmp = y / a
	elif z <= -2e-152:
		tmp = (z * -y) / t
	elif z <= 0.032:
		tmp = x / t
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -5.5e+23)
		tmp = Float64(y / a);
	elseif (z <= -2e-152)
		tmp = Float64(Float64(z * Float64(-y)) / t);
	elseif (z <= 0.032)
		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 <= -5.5e+23)
		tmp = y / a;
	elseif (z <= -2e-152)
		tmp = (z * -y) / t;
	elseif (z <= 0.032)
		tmp = x / t;
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -5.5e+23], N[(y / a), $MachinePrecision], If[LessEqual[z, -2e-152], N[(N[(z * (-y)), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[z, 0.032], N[(x / t), $MachinePrecision], N[(y / a), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.50000000000000004e23 or 0.032000000000000001 < z

   1. Initial program 69.7%

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

      1. *-commutative 69.7%

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

   3. Simplified 69.7%

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

   5. Taylor expanded in z around inf 59.4%

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

   if -5.50000000000000004e23 < z < -2.00000000000000013e-152

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 61.5%

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

      1. mul-1-neg 61.5%

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

      2. associate-/l* 47.7%

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

      3. associate-/r/ 58.5%

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

      4. cancel-sign-sub-inv 58.5%

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

      5. +-commutative 58.5%

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

      6. distribute-lft-neg-in 58.5%

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

      7. distribute-rgt-neg-in 58.5%

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

      8. fma-udef 58.5%

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

      9. associate-/r/ 47.7%

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

      10. fma-udef 47.7%

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

      11. distribute-rgt-neg-in 47.7%

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

      12. distribute-lft-neg-in 47.7%

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

      13. +-commutative 47.7%

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

      14. cancel-sign-sub-inv 47.7%

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

      15. *-commutative 47.7%

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

   7. Simplified 47.7%

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

   8. Taylor expanded in t around 0 47.7%

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

      1. neg-mul-1 47.7%

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

      2. +-commutative 47.7%

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

      3. unsub-neg 47.7%

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

   10. Simplified 47.7%

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

   11. Taylor expanded in t around inf 55.2%

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

   if -2.00000000000000013e-152 < z < 0.032000000000000001

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in z around 0 57.6%

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

4. Final simplification 58.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.5 \cdot 10^{+23}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -2 \cdot 10^{-152}:\\ \;\;\;\;\frac{z \cdot \left(-y\right)}{t}\\ \mathbf{elif}\;z \leq 0.032:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 55.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{+23} \lor \neg \left(z \leq 310\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -1.3e+23) (not (<= z 310.0))) (/ y a) (/ x t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -1.3e+23) || !(z <= 310.0)) {
		tmp = y / a;
	} else {
		tmp = x / t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((z <= (-1.3d+23)) .or. (.not. (z <= 310.0d0))) then
        tmp = y / a
    else
        tmp = x / t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -1.3e+23) || !(z <= 310.0)) {
		tmp = y / a;
	} else {
		tmp = x / t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (z <= -1.3e+23) or not (z <= 310.0):
		tmp = y / a
	else:
		tmp = x / t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -1.3e+23) || !(z <= 310.0))
		tmp = Float64(y / a);
	else
		tmp = Float64(x / t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((z <= -1.3e+23) || ~((z <= 310.0)))
		tmp = y / a;
	else
		tmp = x / t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -1.3e+23], N[Not[LessEqual[z, 310.0]], $MachinePrecision]], N[(y / a), $MachinePrecision], N[(x / t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.29999999999999996e23 or 310 < z

   1. Initial program 69.7%

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

      1. *-commutative 69.7%

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

   3. Simplified 69.7%

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

   5. Taylor expanded in z around inf 59.4%

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

   if -1.29999999999999996e23 < z < 310

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in z around 0 52.5%

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

4. Final simplification 55.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{+23} \lor \neg \left(z \leq 310\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 35.3% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.2%

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

   1. *-commutative 85.2%

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

3. Simplified 85.2%

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

5. Taylor expanded in z around 0 35.7%

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

6. Final simplification 35.7%

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

Developer target: 97.4% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = t - (a * z)
    t_2 = (x / t_1) - (y / ((t / z) - a))
    if (z < (-32113435955957344.0d0)) then
        tmp = t_2
    else if (z < 3.5139522372978296d-86) then
        tmp = (x - (y * z)) * (1.0d0 / t_1)
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024026 
(FPCore (x y z t a)
  :name "Diagrams.Solve.Tridiagonal:solveTriDiagonal from diagrams-solve-0.1, A"
  :precision binary64

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

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