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

Percentage Accurate: 85.5% → 90.2%

Time: 12.4s

Alternatives: 9

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: 90.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -7.8 \cdot 10^{+82} \lor \neg \left(z \leq 1.55 \cdot 10^{+174}\right):\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - z \cdot y}{t - z \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -7.8e+82) (not (<= z 1.55e+174)))
   (/ (- y (/ x 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 <= -7.8e+82) || !(z <= 1.55e+174)) {
		tmp = (y - (x / 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 <= (-7.8d+82)) .or. (.not. (z <= 1.55d+174))) then
        tmp = (y - (x / 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 <= -7.8e+82) || !(z <= 1.55e+174)) {
		tmp = (y - (x / z)) / a;
	} else {
		tmp = (x - (z * y)) / (t - (z * a));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (z <= -7.8e+82) or not (z <= 1.55e+174):
		tmp = (y - (x / 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 <= -7.8e+82) || !(z <= 1.55e+174))
		tmp = Float64(Float64(y - Float64(x / 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 <= -7.8e+82) || ~((z <= 1.55e+174)))
		tmp = (y - (x / 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, -7.8e+82], N[Not[LessEqual[z, 1.55e+174]], $MachinePrecision]], N[(N[(y - N[(x / z), $MachinePrecision]), $MachinePrecision] / a), $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 < -7.79999999999999951e82 or 1.55e174 < z

   1. Initial program 50.2%

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

      1. *-commutative 50.2%

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

   3. Simplified 50.2%

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

   5. Taylor expanded in t around 0 44.2%

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

      1. associate-*r/ 44.2%

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

      2. neg-mul-1 44.2%

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

      3. sub-neg 44.2%

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

      4. distribute-rgt-neg-out 44.2%

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

      5. +-commutative 44.2%

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

      6. distribute-neg-in 44.2%

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

      7. distribute-rgt-neg-out 44.2%

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

      8. remove-double-neg 44.2%

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

      9. sub-neg 44.2%

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

      10. *-commutative 44.2%

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

   7. Simplified 44.2%

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

      1. associate-/r* 61.1%

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

      2. div-inv 61.1%

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

      3. *-commutative 61.1%

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

   9. Applied egg-rr 61.1%

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

   10. Taylor expanded in z around 0 83.0%

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

       1. *-commutative 83.0%

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

       2. associate-/r* 87.9%

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

       3. associate-*r/ 87.9%

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

       4. *-rgt-identity 87.9%

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

       5. *-commutative 87.9%

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

       6. metadata-eval 87.9%

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

       7. cancel-sign-sub-inv 87.9%

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

       8. associate-*r/ 87.9%

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

       9. div-sub 87.9%

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

       10. cancel-sign-sub-inv 87.9%

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

       11. metadata-eval 87.9%

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

       12. *-lft-identity 87.9%

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

       13. +-commutative 87.9%

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

       14. neg-mul-1 87.9%

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

       15. unsub-neg 87.9%

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

   12. Simplified 87.9%

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

   if -7.79999999999999951e82 < z < 1.55e174

   1. Initial program 96.1%

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

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.8 \cdot 10^{+82} \lor \neg \left(z \leq 1.55 \cdot 10^{+174}\right):\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - z \cdot y}{t - z \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 71.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y - \frac{x}{z}}{a}\\ t_2 := t - z \cdot a\\ \mathbf{if}\;z \leq -3.1 \cdot 10^{-32}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{-283}:\\ \;\;\;\;\frac{x - z \cdot y}{t}\\ \mathbf{elif}\;z \leq 3.4 \cdot 10^{+16}:\\ \;\;\;\;\frac{x}{t_2}\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{+62}:\\ \;\;\;\;z \cdot \frac{-y}{t_2}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- y (/ x z)) a)) (t_2 (- t (* z a))))
   (if (<= z -3.1e-32)
     t_1
     (if (<= z 2.3e-283)
       (/ (- x (* z y)) t)
       (if (<= z 3.4e+16)
         (/ x t_2)
         (if (<= z 1.05e+62) (* z (/ (- y) t_2)) t_1))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (y - (x / z)) / a;
	double t_2 = t - (z * a);
	double tmp;
	if (z <= -3.1e-32) {
		tmp = t_1;
	} else if (z <= 2.3e-283) {
		tmp = (x - (z * y)) / t;
	} else if (z <= 3.4e+16) {
		tmp = x / t_2;
	} else if (z <= 1.05e+62) {
		tmp = z * (-y / t_2);
	} 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) :: t_2
    real(8) :: tmp
    t_1 = (y - (x / z)) / a
    t_2 = t - (z * a)
    if (z <= (-3.1d-32)) then
        tmp = t_1
    else if (z <= 2.3d-283) then
        tmp = (x - (z * y)) / t
    else if (z <= 3.4d+16) then
        tmp = x / t_2
    else if (z <= 1.05d+62) then
        tmp = z * (-y / t_2)
    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 t_2 = t - (z * a);
	double tmp;
	if (z <= -3.1e-32) {
		tmp = t_1;
	} else if (z <= 2.3e-283) {
		tmp = (x - (z * y)) / t;
	} else if (z <= 3.4e+16) {
		tmp = x / t_2;
	} else if (z <= 1.05e+62) {
		tmp = z * (-y / t_2);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (y - (x / z)) / a
	t_2 = t - (z * a)
	tmp = 0
	if z <= -3.1e-32:
		tmp = t_1
	elif z <= 2.3e-283:
		tmp = (x - (z * y)) / t
	elif z <= 3.4e+16:
		tmp = x / t_2
	elif z <= 1.05e+62:
		tmp = z * (-y / t_2)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(y - Float64(x / z)) / a)
	t_2 = Float64(t - Float64(z * a))
	tmp = 0.0
	if (z <= -3.1e-32)
		tmp = t_1;
	elseif (z <= 2.3e-283)
		tmp = Float64(Float64(x - Float64(z * y)) / t);
	elseif (z <= 3.4e+16)
		tmp = Float64(x / t_2);
	elseif (z <= 1.05e+62)
		tmp = Float64(z * Float64(Float64(-y) / t_2));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (y - (x / z)) / a;
	t_2 = t - (z * a);
	tmp = 0.0;
	if (z <= -3.1e-32)
		tmp = t_1;
	elseif (z <= 2.3e-283)
		tmp = (x - (z * y)) / t;
	elseif (z <= 3.4e+16)
		tmp = x / t_2;
	elseif (z <= 1.05e+62)
		tmp = z * (-y / t_2);
	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]}, Block[{t$95$2 = N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -3.1e-32], t$95$1, If[LessEqual[z, 2.3e-283], N[(N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[z, 3.4e+16], N[(x / t$95$2), $MachinePrecision], If[LessEqual[z, 1.05e+62], N[(z * N[((-y) / t$95$2), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -3.10000000000000011e-32 or 1.05e62 < z

   1. Initial program 62.7%

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

      1. *-commutative 62.7%

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

   3. Simplified 62.7%

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

   5. Taylor expanded in t around 0 50.4%

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

      1. associate-*r/ 50.4%

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

      2. neg-mul-1 50.4%

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

      3. sub-neg 50.4%

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

      4. distribute-rgt-neg-out 50.4%

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

      5. +-commutative 50.4%

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

      6. distribute-neg-in 50.4%

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

      7. distribute-rgt-neg-out 50.4%

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

      8. remove-double-neg 50.4%

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

      9. sub-neg 50.4%

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

      10. *-commutative 50.4%

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

   7. Simplified 50.4%

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

      1. associate-/r* 64.9%

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

      2. div-inv 64.8%

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

      3. *-commutative 64.8%

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

   9. Applied egg-rr 64.8%

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

   10. Taylor expanded in z around 0 80.3%

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

       1. *-commutative 80.3%

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

       2. associate-/r* 83.6%

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

       3. associate-*r/ 83.6%

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

       4. *-rgt-identity 83.6%

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

       5. *-commutative 83.6%

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

       6. metadata-eval 83.6%

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

       7. cancel-sign-sub-inv 83.6%

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

       8. associate-*r/ 83.6%

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

       9. div-sub 83.6%

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

       10. cancel-sign-sub-inv 83.6%

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

       11. metadata-eval 83.6%

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

       12. *-lft-identity 83.6%

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

       13. +-commutative 83.6%

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

       14. neg-mul-1 83.6%

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

       15. unsub-neg 83.6%

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

   12. Simplified 83.6%

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

   if -3.10000000000000011e-32 < z < 2.2999999999999999e-283

   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 87.1%

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

   if 2.2999999999999999e-283 < z < 3.4e16

   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 76.2%

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

      1. *-commutative 76.2%

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

   7. Simplified 76.2%

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

   if 3.4e16 < z < 1.05e62

   1. Initial program 84.7%

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

      1. *-commutative 84.7%

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

   3. Simplified 84.7%

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

   5. Taylor expanded in x around 0 66.4%

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

      1. mul-1-neg 66.4%

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

      2. associate-/l* 74.2%

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

      3. associate-/r/ 74.0%

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

      4. sub-neg 74.0%

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

      5. +-commutative 74.0%

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

      6. distribute-rgt-neg-in 74.0%

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

      7. fma-udef 74.0%

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

      8. distribute-rgt-neg-in 74.0%

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

      9. fma-udef 74.0%

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

      10. distribute-rgt-neg-in 74.0%

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

      11. +-commutative 74.0%

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

      12. sub-neg 74.0%

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

      13. *-commutative 74.0%

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

   7. Simplified 74.0%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.1 \cdot 10^{-32}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{-283}:\\ \;\;\;\;\frac{x - z \cdot y}{t}\\ \mathbf{elif}\;z \leq 3.4 \cdot 10^{+16}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{+62}:\\ \;\;\;\;z \cdot \frac{-y}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 72.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - z \cdot y}{t}\\ t_2 := \frac{y - \frac{x}{z}}{a}\\ \mathbf{if}\;z \leq -1.12 \cdot 10^{-32}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 10^{-286}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 4.5 \cdot 10^{-127}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{elif}\;z \leq 1.55 \cdot 10^{-17}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- x (* z y)) t)) (t_2 (/ (- y (/ x z)) a)))
   (if (<= z -1.12e-32)
     t_2
     (if (<= z 1e-286)
       t_1
       (if (<= z 4.5e-127)
         (/ x (- t (* z a)))
         (if (<= z 1.55e-17) t_1 t_2))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (x - (z * y)) / t;
	double t_2 = (y - (x / z)) / a;
	double tmp;
	if (z <= -1.12e-32) {
		tmp = t_2;
	} else if (z <= 1e-286) {
		tmp = t_1;
	} else if (z <= 4.5e-127) {
		tmp = x / (t - (z * a));
	} else if (z <= 1.55e-17) {
		tmp = 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 = (x - (z * y)) / t
    t_2 = (y - (x / z)) / a
    if (z <= (-1.12d-32)) then
        tmp = t_2
    else if (z <= 1d-286) then
        tmp = t_1
    else if (z <= 4.5d-127) then
        tmp = x / (t - (z * a))
    else if (z <= 1.55d-17) then
        tmp = 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 = (x - (z * y)) / t;
	double t_2 = (y - (x / z)) / a;
	double tmp;
	if (z <= -1.12e-32) {
		tmp = t_2;
	} else if (z <= 1e-286) {
		tmp = t_1;
	} else if (z <= 4.5e-127) {
		tmp = x / (t - (z * a));
	} else if (z <= 1.55e-17) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (x - (z * y)) / t
	t_2 = (y - (x / z)) / a
	tmp = 0
	if z <= -1.12e-32:
		tmp = t_2
	elif z <= 1e-286:
		tmp = t_1
	elif z <= 4.5e-127:
		tmp = x / (t - (z * a))
	elif z <= 1.55e-17:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(x - Float64(z * y)) / t)
	t_2 = Float64(Float64(y - Float64(x / z)) / a)
	tmp = 0.0
	if (z <= -1.12e-32)
		tmp = t_2;
	elseif (z <= 1e-286)
		tmp = t_1;
	elseif (z <= 4.5e-127)
		tmp = Float64(x / Float64(t - Float64(z * a)));
	elseif (z <= 1.55e-17)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (x - (z * y)) / t;
	t_2 = (y - (x / z)) / a;
	tmp = 0.0;
	if (z <= -1.12e-32)
		tmp = t_2;
	elseif (z <= 1e-286)
		tmp = t_1;
	elseif (z <= 4.5e-127)
		tmp = x / (t - (z * a));
	elseif (z <= 1.55e-17)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y - N[(x / z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]}, If[LessEqual[z, -1.12e-32], t$95$2, If[LessEqual[z, 1e-286], t$95$1, If[LessEqual[z, 4.5e-127], N[(x / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.55e-17], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.12e-32 or 1.5499999999999999e-17 < z

   1. Initial program 66.4%

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

      1. *-commutative 66.4%

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

   3. Simplified 66.4%

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

   5. Taylor expanded in t around 0 50.3%

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

      1. associate-*r/ 50.3%

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

      2. neg-mul-1 50.3%

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

      3. sub-neg 50.3%

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

      4. distribute-rgt-neg-out 50.3%

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

      5. +-commutative 50.3%

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

      6. distribute-neg-in 50.3%

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

      7. distribute-rgt-neg-out 50.3%

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

      8. remove-double-neg 50.3%

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

      9. sub-neg 50.3%

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

      10. *-commutative 50.3%

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

   7. Simplified 50.3%

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

      1. associate-/r* 63.5%

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

      2. div-inv 63.4%

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

      3. *-commutative 63.4%

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

   9. Applied egg-rr 63.4%

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

   10. Taylor expanded in z around 0 76.7%

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

       1. *-commutative 76.7%

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

       2. associate-/r* 79.6%

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

       3. associate-*r/ 79.6%

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

       4. *-rgt-identity 79.6%

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

       5. *-commutative 79.6%

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

       6. metadata-eval 79.6%

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

       7. cancel-sign-sub-inv 79.6%

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

       8. associate-*r/ 79.6%

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

       9. div-sub 79.6%

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

       10. cancel-sign-sub-inv 79.6%

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

       11. metadata-eval 79.6%

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

       12. *-lft-identity 79.6%

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

       13. +-commutative 79.6%

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

       14. neg-mul-1 79.6%

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

       15. unsub-neg 79.6%

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

   12. Simplified 79.6%

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

   if -1.12e-32 < z < 1.00000000000000005e-286 or 4.4999999999999999e-127 < z < 1.5499999999999999e-17

   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 t around inf 80.7%

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

   if 1.00000000000000005e-286 < z < 4.4999999999999999e-127

   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 inf 91.4%

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

      1. *-commutative 91.4%

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

   7. Simplified 91.4%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.12 \cdot 10^{-32}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{elif}\;z \leq 10^{-286}:\\ \;\;\;\;\frac{x - z \cdot y}{t}\\ \mathbf{elif}\;z \leq 4.5 \cdot 10^{-127}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{elif}\;z \leq 1.55 \cdot 10^{-17}:\\ \;\;\;\;\frac{x - z \cdot y}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 54.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.15 \cdot 10^{-51}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 4.3 \cdot 10^{-122}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;z \leq 3.3 \cdot 10^{-80}:\\ \;\;\;\;y \cdot \frac{z}{-t}\\ \mathbf{elif}\;z \leq 62000000000000:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.15e-51)
   (/ y a)
   (if (<= z 4.3e-122)
     (/ x t)
     (if (<= z 3.3e-80)
       (* y (/ z (- t)))
       (if (<= z 62000000000000.0) (/ x t) (/ y a))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.15e-51) {
		tmp = y / a;
	} else if (z <= 4.3e-122) {
		tmp = x / t;
	} else if (z <= 3.3e-80) {
		tmp = y * (z / -t);
	} else if (z <= 62000000000000.0) {
		tmp = x / t;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-1.15d-51)) then
        tmp = y / a
    else if (z <= 4.3d-122) then
        tmp = x / t
    else if (z <= 3.3d-80) then
        tmp = y * (z / -t)
    else if (z <= 62000000000000.0d0) then
        tmp = x / t
    else
        tmp = y / a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.15e-51) {
		tmp = y / a;
	} else if (z <= 4.3e-122) {
		tmp = x / t;
	} else if (z <= 3.3e-80) {
		tmp = y * (z / -t);
	} else if (z <= 62000000000000.0) {
		tmp = x / t;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.15e-51:
		tmp = y / a
	elif z <= 4.3e-122:
		tmp = x / t
	elif z <= 3.3e-80:
		tmp = y * (z / -t)
	elif z <= 62000000000000.0:
		tmp = x / t
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.15e-51)
		tmp = Float64(y / a);
	elseif (z <= 4.3e-122)
		tmp = Float64(x / t);
	elseif (z <= 3.3e-80)
		tmp = Float64(y * Float64(z / Float64(-t)));
	elseif (z <= 62000000000000.0)
		tmp = Float64(x / t);
	else
		tmp = Float64(y / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -1.15e-51)
		tmp = y / a;
	elseif (z <= 4.3e-122)
		tmp = x / t;
	elseif (z <= 3.3e-80)
		tmp = y * (z / -t);
	elseif (z <= 62000000000000.0)
		tmp = x / t;
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.15e-51], N[(y / a), $MachinePrecision], If[LessEqual[z, 4.3e-122], N[(x / t), $MachinePrecision], If[LessEqual[z, 3.3e-80], N[(y * N[(z / (-t)), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 62000000000000.0], N[(x / t), $MachinePrecision], N[(y / a), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.15000000000000001e-51 or 6.2e13 < z

   1. Initial program 65.4%

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

      1. *-commutative 65.4%

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

   3. Simplified 65.4%

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

   5. Taylor expanded in z around inf 59.6%

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

   if -1.15000000000000001e-51 < z < 4.30000000000000019e-122 or 3.3e-80 < z < 6.2e13

   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 z around 0 64.8%

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

   if 4.30000000000000019e-122 < z < 3.3e-80

   1. Initial program 99.5%

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

      1. *-commutative 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in t around inf 61.9%

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

   6. Taylor expanded in x around 0 55.1%

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

      1. associate-*r/ 55.1%

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

      2. mul-1-neg 55.1%

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

      3. *-commutative 55.1%

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

      4. distribute-rgt-neg-in 55.1%

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

   8. Simplified 55.1%

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

      1. frac-2neg 55.1%

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

      2. div-inv 55.2%

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

      3. distribute-rgt-neg-out 55.2%

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

      4. remove-double-neg 55.2%

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

      5. *-commutative 55.2%

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

   10. Applied egg-rr 55.2%

       \[\leadsto \color{blue}{\left(y \cdot z\right) \cdot \frac{1}{-t}} \]
   11. Step-by-step derivation

       1. associate-*l* 49.8%

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

       2. associate-*r/ 49.9%

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

       3. *-rgt-identity 49.9%

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

   12. Simplified 49.9%

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

4. Final simplification 61.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.15 \cdot 10^{-51}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 4.3 \cdot 10^{-122}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;z \leq 3.3 \cdot 10^{-80}:\\ \;\;\;\;y \cdot \frac{z}{-t}\\ \mathbf{elif}\;z \leq 62000000000000:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 54.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.55 \cdot 10^{-51}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 9 \cdot 10^{-122}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;z \leq 3.3 \cdot 10^{-80}:\\ \;\;\;\;\frac{z \cdot \left(-y\right)}{t}\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{+16}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -2.55e-51)
   (/ y a)
   (if (<= z 9e-122)
     (/ x t)
     (if (<= z 3.3e-80)
       (/ (* z (- y)) t)
       (if (<= z 1.9e+16) (/ x t) (/ y a))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.55e-51) {
		tmp = y / a;
	} else if (z <= 9e-122) {
		tmp = x / t;
	} else if (z <= 3.3e-80) {
		tmp = (z * -y) / t;
	} else if (z <= 1.9e+16) {
		tmp = x / t;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-2.55d-51)) then
        tmp = y / a
    else if (z <= 9d-122) then
        tmp = x / t
    else if (z <= 3.3d-80) then
        tmp = (z * -y) / t
    else if (z <= 1.9d+16) then
        tmp = x / t
    else
        tmp = y / a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.55e-51) {
		tmp = y / a;
	} else if (z <= 9e-122) {
		tmp = x / t;
	} else if (z <= 3.3e-80) {
		tmp = (z * -y) / t;
	} else if (z <= 1.9e+16) {
		tmp = x / t;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -2.55e-51:
		tmp = y / a
	elif z <= 9e-122:
		tmp = x / t
	elif z <= 3.3e-80:
		tmp = (z * -y) / t
	elif z <= 1.9e+16:
		tmp = x / t
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -2.55e-51)
		tmp = Float64(y / a);
	elseif (z <= 9e-122)
		tmp = Float64(x / t);
	elseif (z <= 3.3e-80)
		tmp = Float64(Float64(z * Float64(-y)) / t);
	elseif (z <= 1.9e+16)
		tmp = Float64(x / t);
	else
		tmp = Float64(y / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -2.55e-51)
		tmp = y / a;
	elseif (z <= 9e-122)
		tmp = x / t;
	elseif (z <= 3.3e-80)
		tmp = (z * -y) / t;
	elseif (z <= 1.9e+16)
		tmp = x / t;
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -2.55e-51], N[(y / a), $MachinePrecision], If[LessEqual[z, 9e-122], N[(x / t), $MachinePrecision], If[LessEqual[z, 3.3e-80], N[(N[(z * (-y)), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[z, 1.9e+16], N[(x / t), $MachinePrecision], N[(y / a), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.5499999999999999e-51 or 1.9e16 < z

   1. Initial program 65.4%

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

      1. *-commutative 65.4%

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

   3. Simplified 65.4%

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

   5. Taylor expanded in z around inf 59.6%

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

   if -2.5499999999999999e-51 < z < 8.99999999999999959e-122 or 3.3e-80 < z < 1.9e16

   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 z around 0 64.8%

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

   if 8.99999999999999959e-122 < z < 3.3e-80

   1. Initial program 99.5%

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

      1. *-commutative 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in t around inf 61.9%

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

   6. Taylor expanded in x around 0 55.1%

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

      1. associate-*r/ 55.1%

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

      2. mul-1-neg 55.1%

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

      3. *-commutative 55.1%

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

      4. distribute-rgt-neg-in 55.1%

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

   8. Simplified 55.1%

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

4. Final simplification 61.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.55 \cdot 10^{-51}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 9 \cdot 10^{-122}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;z \leq 3.3 \cdot 10^{-80}:\\ \;\;\;\;\frac{z \cdot \left(-y\right)}{t}\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{+16}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 64.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{-7}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 2.8 \cdot 10^{-282}:\\ \;\;\;\;\frac{x - z \cdot y}{t}\\ \mathbf{elif}\;z \leq 9200000000:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -5e-7)
   (/ y a)
   (if (<= z 2.8e-282)
     (/ (- x (* z y)) t)
     (if (<= z 9200000000.0) (/ x (- t (* z a))) (/ y a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -5e-7) {
		tmp = y / a;
	} else if (z <= 2.8e-282) {
		tmp = (x - (z * y)) / t;
	} else if (z <= 9200000000.0) {
		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 <= (-5d-7)) then
        tmp = y / a
    else if (z <= 2.8d-282) then
        tmp = (x - (z * y)) / t
    else if (z <= 9200000000.0d0) 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 <= -5e-7) {
		tmp = y / a;
	} else if (z <= 2.8e-282) {
		tmp = (x - (z * y)) / t;
	} else if (z <= 9200000000.0) {
		tmp = x / (t - (z * a));
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -5e-7:
		tmp = y / a
	elif z <= 2.8e-282:
		tmp = (x - (z * y)) / t
	elif z <= 9200000000.0:
		tmp = x / (t - (z * a))
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -5e-7)
		tmp = Float64(y / a);
	elseif (z <= 2.8e-282)
		tmp = Float64(Float64(x - Float64(z * y)) / t);
	elseif (z <= 9200000000.0)
		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 <= -5e-7)
		tmp = y / a;
	elseif (z <= 2.8e-282)
		tmp = (x - (z * y)) / t;
	elseif (z <= 9200000000.0)
		tmp = x / (t - (z * a));
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -5e-7], N[(y / a), $MachinePrecision], If[LessEqual[z, 2.8e-282], N[(N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[z, 9200000000.0], N[(x / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y / a), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.99999999999999977e-7 or 9.2e9 < z

   1. Initial program 62.9%

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

      1. *-commutative 62.9%

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

   3. Simplified 62.9%

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

   5. Taylor expanded in z around inf 62.1%

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

   if -4.99999999999999977e-7 < z < 2.7999999999999999e-282

   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 82.7%

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

   if 2.7999999999999999e-282 < z < 9.2e9

   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 76.2%

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

      1. *-commutative 76.2%

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

   7. Simplified 76.2%

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

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{-7}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 2.8 \cdot 10^{-282}:\\ \;\;\;\;\frac{x - z \cdot y}{t}\\ \mathbf{elif}\;z \leq 9200000000:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 64.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.1 \cdot 10^{-5} \lor \neg \left(z \leq 5.8 \cdot 10^{+17}\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -5.1e-5) (not (<= z 5.8e+17))) (/ y a) (/ x (- t (* z a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -5.1e-5) || !(z <= 5.8e+17)) {
		tmp = y / a;
	} else {
		tmp = x / (t - (z * a));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((z <= (-5.1d-5)) .or. (.not. (z <= 5.8d+17))) then
        tmp = y / a
    else
        tmp = x / (t - (z * a))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -5.1e-5) || !(z <= 5.8e+17)) {
		tmp = y / a;
	} else {
		tmp = x / (t - (z * a));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (z <= -5.1e-5) or not (z <= 5.8e+17):
		tmp = y / a
	else:
		tmp = x / (t - (z * a))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -5.1e-5) || !(z <= 5.8e+17))
		tmp = Float64(y / a);
	else
		tmp = Float64(x / Float64(t - Float64(z * a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((z <= -5.1e-5) || ~((z <= 5.8e+17)))
		tmp = y / a;
	else
		tmp = x / (t - (z * a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -5.1e-5], N[Not[LessEqual[z, 5.8e+17]], $MachinePrecision]], N[(y / a), $MachinePrecision], N[(x / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -5.09999999999999996e-5 or 5.8e17 < z

   1. Initial program 62.6%

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

      1. *-commutative 62.6%

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

   3. Simplified 62.6%

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

   5. Taylor expanded in z around inf 62.6%

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

   if -5.09999999999999996e-5 < z < 5.8e17

   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 73.5%

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

      1. *-commutative 73.5%

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

   7. Simplified 73.5%

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

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.1 \cdot 10^{-5} \lor \neg \left(z \leq 5.8 \cdot 10^{+17}\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 55.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.5 \cdot 10^{-51} \lor \neg \left(z \leq 7800000000000\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -2.5e-51) (not (<= z 7800000000000.0))) (/ y a) (/ x t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -2.5e-51) || !(z <= 7800000000000.0)) {
		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 <= (-2.5d-51)) .or. (.not. (z <= 7800000000000.0d0))) 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 <= -2.5e-51) || !(z <= 7800000000000.0)) {
		tmp = y / a;
	} else {
		tmp = x / t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (z <= -2.5e-51) or not (z <= 7800000000000.0):
		tmp = y / a
	else:
		tmp = x / t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -2.5e-51) || !(z <= 7800000000000.0))
		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 <= -2.5e-51) || ~((z <= 7800000000000.0)))
		tmp = y / a;
	else
		tmp = x / t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -2.5e-51], N[Not[LessEqual[z, 7800000000000.0]], $MachinePrecision]], N[(y / a), $MachinePrecision], N[(x / t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.50000000000000002e-51 or 7.8e12 < z

   1. Initial program 65.4%

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

      1. *-commutative 65.4%

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

   3. Simplified 65.4%

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

   5. Taylor expanded in z around inf 59.6%

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

   if -2.50000000000000002e-51 < z < 7.8e12

   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 59.0%

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

4. Final simplification 59.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.5 \cdot 10^{-51} \lor \neg \left(z \leq 7800000000000\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 35.8% 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 82.1%

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

   1. *-commutative 82.1%

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

3. Simplified 82.1%

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

5. Taylor expanded in z around 0 33.4%

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

6. Final simplification 33.4%

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

Developer target: 97.5% 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 2024011 
(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))))