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

Percentage Accurate: 85.3% → 91.6%

Time: 11.4s

Alternatives: 11

Speedup: 0.7×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 85.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 91.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t - z \cdot a\\ t_2 := \frac{x - y \cdot z}{t_1}\\ \mathbf{if}\;t_2 \leq -1 \cdot 10^{-155}:\\ \;\;\;\;\frac{x}{t_1} - \frac{y}{\frac{t_1}{z}}\\ \mathbf{elif}\;t_2 \leq 5 \cdot 10^{+305}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- t (* z a))) (t_2 (/ (- x (* y z)) t_1)))
   (if (<= t_2 -1e-155)
     (- (/ x t_1) (/ y (/ t_1 z)))
     (if (<= t_2 5e+305) t_2 (/ (- y (/ x z)) a)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = t - (z * a);
	double t_2 = (x - (y * z)) / t_1;
	double tmp;
	if (t_2 <= -1e-155) {
		tmp = (x / t_1) - (y / (t_1 / z));
	} else if (t_2 <= 5e+305) {
		tmp = t_2;
	} else {
		tmp = (y - (x / 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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = t - (z * a)
    t_2 = (x - (y * z)) / t_1
    if (t_2 <= (-1d-155)) then
        tmp = (x / t_1) - (y / (t_1 / z))
    else if (t_2 <= 5d+305) then
        tmp = t_2
    else
        tmp = (y - (x / z)) / 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 = t - (z * a);
	double t_2 = (x - (y * z)) / t_1;
	double tmp;
	if (t_2 <= -1e-155) {
		tmp = (x / t_1) - (y / (t_1 / z));
	} else if (t_2 <= 5e+305) {
		tmp = t_2;
	} else {
		tmp = (y - (x / z)) / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = t - (z * a)
	t_2 = (x - (y * z)) / t_1
	tmp = 0
	if t_2 <= -1e-155:
		tmp = (x / t_1) - (y / (t_1 / z))
	elif t_2 <= 5e+305:
		tmp = t_2
	else:
		tmp = (y - (x / z)) / a
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(t - Float64(z * a))
	t_2 = Float64(Float64(x - Float64(y * z)) / t_1)
	tmp = 0.0
	if (t_2 <= -1e-155)
		tmp = Float64(Float64(x / t_1) - Float64(y / Float64(t_1 / z)));
	elseif (t_2 <= 5e+305)
		tmp = t_2;
	else
		tmp = Float64(Float64(y - Float64(x / z)) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = t - (z * a);
	t_2 = (x - (y * z)) / t_1;
	tmp = 0.0;
	if (t_2 <= -1e-155)
		tmp = (x / t_1) - (y / (t_1 / z));
	elseif (t_2 <= 5e+305)
		tmp = t_2;
	else
		tmp = (y - (x / z)) / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, -1e-155], N[(N[(x / t$95$1), $MachinePrecision] - N[(y / N[(t$95$1 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e+305], t$95$2, N[(N[(y - N[(x / z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -1.00000000000000001e-155

   1. Initial program 95.1%

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

      1. *-commutative 95.1%

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

   3. Simplified 95.1%

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

   4. Taylor expanded in x around 0 95.1%

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

      1. fma-def 95.1%

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

      2. associate-/l* 99.9%

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

      3. *-commutative 99.9%

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

      4. *-commutative 99.9%

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

   6. Simplified 99.9%

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

   7. Taylor expanded in y around 0 95.1%

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

      1. +-commutative 95.1%

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

      2. sub-neg 95.1%

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

      3. mul-1-neg 95.1%

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

      4. mul-1-neg 95.1%

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

      5. sub-neg 95.1%

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

      6. mul-1-neg 95.1%

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

      7. unsub-neg 95.1%

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

      8. mul-1-neg 95.1%

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

      9. sub-neg 95.1%

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

      10. associate-/l* 99.9%

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

      11. mul-1-neg 99.9%

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

      12. sub-neg 99.9%

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

   9. Simplified 99.9%

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

   if -1.00000000000000001e-155 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 5.00000000000000009e305

   1. Initial program 91.5%

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

   if 5.00000000000000009e305 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z)))

   1. Initial program 30.8%

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

      1. *-commutative 30.8%

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

   3. Simplified 30.8%

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

   4. Taylor expanded in x around 0 30.8%

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

      1. fma-def 30.8%

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

      2. associate-/l* 51.5%

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

      3. *-commutative 51.5%

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

      4. *-commutative 51.5%

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

   6. Simplified 51.5%

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

   7. Taylor expanded in a around inf 90.0%

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

      1. neg-mul-1 90.0%

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

      2. unsub-neg 90.0%

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

   9. Simplified 90.0%

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

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y \cdot z}{t - z \cdot a} \leq -1 \cdot 10^{-155}:\\ \;\;\;\;\frac{x}{t - z \cdot a} - \frac{y}{\frac{t - z \cdot a}{z}}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq 5 \cdot 10^{+305}:\\ \;\;\;\;\frac{x - y \cdot z}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \end{array} \]

Alternative 2: 90.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.2 \cdot 10^{+80} \lor \neg \left(z \leq 6.8 \cdot 10^{+173}\right):\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y \cdot z}{t - z \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -4.2e+80) (not (<= z 6.8e+173)))
   (/ (- y (/ x z)) a)
   (/ (- x (* y z)) (- t (* z a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -4.2e+80) || !(z <= 6.8e+173)) {
		tmp = (y - (x / z)) / a;
	} else {
		tmp = (x - (y * z)) / (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+80)) .or. (.not. (z <= 6.8d+173))) then
        tmp = (y - (x / z)) / a
    else
        tmp = (x - (y * z)) / (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+80) || !(z <= 6.8e+173)) {
		tmp = (y - (x / z)) / a;
	} else {
		tmp = (x - (y * z)) / (t - (z * a));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (z <= -4.2e+80) or not (z <= 6.8e+173):
		tmp = (y - (x / z)) / a
	else:
		tmp = (x - (y * z)) / (t - (z * a))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -4.2e+80) || !(z <= 6.8e+173))
		tmp = Float64(Float64(y - Float64(x / z)) / a);
	else
		tmp = Float64(Float64(x - Float64(y * z)) / 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+80) || ~((z <= 6.8e+173)))
		tmp = (y - (x / z)) / a;
	else
		tmp = (x - (y * z)) / (t - (z * a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -4.2e+80], N[Not[LessEqual[z, 6.8e+173]], $MachinePrecision]], N[(N[(y - N[(x / z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4.20000000000000003e80 or 6.80000000000000042e173 < z

   1. Initial program 62.1%

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

      1. *-commutative 62.1%

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

   3. Simplified 62.1%

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

   4. Taylor expanded in x around 0 62.1%

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

      1. fma-def 62.1%

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

      2. associate-/l* 73.5%

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

      3. *-commutative 73.5%

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

      4. *-commutative 73.5%

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

   6. Simplified 73.5%

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

   7. Taylor expanded in a around inf 83.3%

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

      1. neg-mul-1 83.3%

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

      2. unsub-neg 83.3%

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

   9. Simplified 83.3%

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

   if -4.20000000000000003e80 < z < 6.80000000000000042e173

   1. Initial program 97.8%

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

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.2 \cdot 10^{+80} \lor \neg \left(z \leq 6.8 \cdot 10^{+173}\right):\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y \cdot z}{t - z \cdot a}\\ \end{array} \]

Alternative 3: 72.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6 \cdot 10^{+36}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{elif}\;z \leq -4.85 \cdot 10^{-82}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{elif}\;z \leq 1.65 \cdot 10^{-7}:\\ \;\;\;\;\frac{x - y \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{-y}{\frac{t}{z} - a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -6e+36)
   (/ (- y (/ x z)) a)
   (if (<= z -4.85e-82)
     (/ x (- t (* z a)))
     (if (<= z 1.65e-7) (/ (- x (* y z)) t) (/ (- y) (- (/ t z) a))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -6e+36) {
		tmp = (y - (x / z)) / a;
	} else if (z <= -4.85e-82) {
		tmp = x / (t - (z * a));
	} else if (z <= 1.65e-7) {
		tmp = (x - (y * z)) / t;
	} else {
		tmp = -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 <= (-6d+36)) then
        tmp = (y - (x / z)) / a
    else if (z <= (-4.85d-82)) then
        tmp = x / (t - (z * a))
    else if (z <= 1.65d-7) then
        tmp = (x - (y * z)) / t
    else
        tmp = -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 <= -6e+36) {
		tmp = (y - (x / z)) / a;
	} else if (z <= -4.85e-82) {
		tmp = x / (t - (z * a));
	} else if (z <= 1.65e-7) {
		tmp = (x - (y * z)) / t;
	} else {
		tmp = -y / ((t / z) - a);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -6e+36:
		tmp = (y - (x / z)) / a
	elif z <= -4.85e-82:
		tmp = x / (t - (z * a))
	elif z <= 1.65e-7:
		tmp = (x - (y * z)) / t
	else:
		tmp = -y / ((t / z) - a)
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -6e+36)
		tmp = Float64(Float64(y - Float64(x / z)) / a);
	elseif (z <= -4.85e-82)
		tmp = Float64(x / Float64(t - Float64(z * a)));
	elseif (z <= 1.65e-7)
		tmp = Float64(Float64(x - Float64(y * z)) / t);
	else
		tmp = Float64(Float64(-y) / Float64(Float64(t / z) - a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -6e+36)
		tmp = (y - (x / z)) / a;
	elseif (z <= -4.85e-82)
		tmp = x / (t - (z * a));
	elseif (z <= 1.65e-7)
		tmp = (x - (y * z)) / t;
	else
		tmp = -y / ((t / z) - a);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -6e+36], N[(N[(y - N[(x / z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[z, -4.85e-82], N[(x / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.65e-7], N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], N[((-y) / N[(N[(t / z), $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if z < -6e36

   1. Initial program 72.0%

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

      1. *-commutative 72.0%

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

   3. Simplified 72.0%

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

   4. Taylor expanded in x around 0 72.0%

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

      1. fma-def 72.0%

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

      2. associate-/l* 83.3%

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

      3. *-commutative 83.3%

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

      4. *-commutative 83.3%

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

   6. Simplified 83.3%

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

   7. Taylor expanded in a around inf 78.0%

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

      1. neg-mul-1 78.0%

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

      2. unsub-neg 78.0%

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

   9. Simplified 78.0%

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

   if -6e36 < z < -4.84999999999999993e-82

   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. Taylor expanded in x around inf 72.7%

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

      1. *-commutative 72.7%

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

   6. Simplified 72.7%

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

   if -4.84999999999999993e-82 < z < 1.6500000000000001e-7

   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. Taylor expanded in t around inf 78.5%

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

   if 1.6500000000000001e-7 < z

   1. Initial program 72.7%

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

      1. *-commutative 72.7%

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

   3. Simplified 72.7%

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

   4. Taylor expanded in x around 0 72.7%

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

      1. fma-def 72.7%

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

      2. associate-/l* 78.0%

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

      3. *-commutative 78.0%

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

      4. *-commutative 78.0%

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

   6. Simplified 78.0%

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

   7. Taylor expanded in y around 0 72.7%

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

      1. +-commutative 72.7%

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

      2. sub-neg 72.7%

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

      3. mul-1-neg 72.7%

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

      4. mul-1-neg 72.7%

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

      5. sub-neg 72.7%

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

      6. mul-1-neg 72.7%

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

      7. unsub-neg 72.7%

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

      8. mul-1-neg 72.7%

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

      9. sub-neg 72.7%

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

      10. associate-/l* 78.0%

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

      11. mul-1-neg 78.0%

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

      12. sub-neg 78.0%

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

   9. Simplified 78.0%

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

   10. Taylor expanded in x around 0 57.5%

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

       1. mul-1-neg 57.5%

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

       2. associate-/l* 62.8%

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

       3. distribute-neg-frac 62.8%

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

       4. div-sub 62.8%

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

       5. associate-/l* 76.4%

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

       6. *-inverses 76.4%

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

   12. Simplified 76.4%

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

   13. Taylor expanded in y around 0 76.4%

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

       1. mul-1-neg 76.4%

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

       2. distribute-neg-frac 76.4%

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

   15. Simplified 76.4%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6 \cdot 10^{+36}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{elif}\;z \leq -4.85 \cdot 10^{-82}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{elif}\;z \leq 1.65 \cdot 10^{-7}:\\ \;\;\;\;\frac{x - y \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{-y}{\frac{t}{z} - a}\\ \end{array} \]

Alternative 4: 64.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.75 \cdot 10^{+49}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -5.5 \cdot 10^{-83}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{elif}\;z \leq 1.55 \cdot 10^{+100}:\\ \;\;\;\;\frac{x - y \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.75e+49)
   (/ y a)
   (if (<= z -5.5e-83)
     (/ x (- t (* z a)))
     (if (<= z 1.55e+100) (/ (- x (* y z)) t) (/ y a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.75e+49) {
		tmp = y / a;
	} else if (z <= -5.5e-83) {
		tmp = x / (t - (z * a));
	} else if (z <= 1.55e+100) {
		tmp = (x - (y * z)) / t;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-1.75d+49)) then
        tmp = y / a
    else if (z <= (-5.5d-83)) then
        tmp = x / (t - (z * a))
    else if (z <= 1.55d+100) then
        tmp = (x - (y * z)) / t
    else
        tmp = y / a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.75e+49) {
		tmp = y / a;
	} else if (z <= -5.5e-83) {
		tmp = x / (t - (z * a));
	} else if (z <= 1.55e+100) {
		tmp = (x - (y * z)) / t;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.75e+49:
		tmp = y / a
	elif z <= -5.5e-83:
		tmp = x / (t - (z * a))
	elif z <= 1.55e+100:
		tmp = (x - (y * z)) / t
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.75e+49)
		tmp = Float64(y / a);
	elseif (z <= -5.5e-83)
		tmp = Float64(x / Float64(t - Float64(z * a)));
	elseif (z <= 1.55e+100)
		tmp = Float64(Float64(x - Float64(y * z)) / t);
	else
		tmp = Float64(y / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -1.75e+49)
		tmp = y / a;
	elseif (z <= -5.5e-83)
		tmp = x / (t - (z * a));
	elseif (z <= 1.55e+100)
		tmp = (x - (y * z)) / t;
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.75e+49], N[(y / a), $MachinePrecision], If[LessEqual[z, -5.5e-83], N[(x / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.55e+100], N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], N[(y / a), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.74999999999999987e49 or 1.55000000000000003e100 < z

   1. Initial program 67.3%

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

      1. *-commutative 67.3%

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

   3. Simplified 67.3%

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

   4. Taylor expanded in z around inf 60.2%

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

   if -1.74999999999999987e49 < z < -5.49999999999999964e-83

   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. Taylor expanded in x around inf 71.1%

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

      1. *-commutative 71.1%

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

   6. Simplified 71.1%

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

   if -5.49999999999999964e-83 < z < 1.55000000000000003e100

   1. Initial program 99.2%

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

      1. *-commutative 99.2%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in t around inf 75.0%

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

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.75 \cdot 10^{+49}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -5.5 \cdot 10^{-83}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{elif}\;z \leq 1.55 \cdot 10^{+100}:\\ \;\;\;\;\frac{x - y \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 5: 71.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y - \frac{x}{z}}{a}\\ \mathbf{if}\;z \leq -6.5 \cdot 10^{+34}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -4.7 \cdot 10^{-84}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{elif}\;z \leq 6.8 \cdot 10^{+21}:\\ \;\;\;\;\frac{x - y \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- y (/ x z)) a)))
   (if (<= z -6.5e+34)
     t_1
     (if (<= z -4.7e-84)
       (/ x (- t (* z a)))
       (if (<= z 6.8e+21) (/ (- x (* y z)) t) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (y - (x / z)) / a;
	double tmp;
	if (z <= -6.5e+34) {
		tmp = t_1;
	} else if (z <= -4.7e-84) {
		tmp = x / (t - (z * a));
	} else if (z <= 6.8e+21) {
		tmp = (x - (y * z)) / t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (y - (x / z)) / a
    if (z <= (-6.5d+34)) then
        tmp = t_1
    else if (z <= (-4.7d-84)) then
        tmp = x / (t - (z * a))
    else if (z <= 6.8d+21) then
        tmp = (x - (y * z)) / t
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (y - (x / z)) / a;
	double tmp;
	if (z <= -6.5e+34) {
		tmp = t_1;
	} else if (z <= -4.7e-84) {
		tmp = x / (t - (z * a));
	} else if (z <= 6.8e+21) {
		tmp = (x - (y * z)) / t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (y - (x / z)) / a
	tmp = 0
	if z <= -6.5e+34:
		tmp = t_1
	elif z <= -4.7e-84:
		tmp = x / (t - (z * a))
	elif z <= 6.8e+21:
		tmp = (x - (y * z)) / t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(y - Float64(x / z)) / a)
	tmp = 0.0
	if (z <= -6.5e+34)
		tmp = t_1;
	elseif (z <= -4.7e-84)
		tmp = Float64(x / Float64(t - Float64(z * a)));
	elseif (z <= 6.8e+21)
		tmp = Float64(Float64(x - Float64(y * z)) / t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (y - (x / z)) / a;
	tmp = 0.0;
	if (z <= -6.5e+34)
		tmp = t_1;
	elseif (z <= -4.7e-84)
		tmp = x / (t - (z * a));
	elseif (z <= 6.8e+21)
		tmp = (x - (y * z)) / t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y - N[(x / z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]}, If[LessEqual[z, -6.5e+34], t$95$1, If[LessEqual[z, -4.7e-84], N[(x / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 6.8e+21], N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -6.50000000000000017e34 or 6.8e21 < z

   1. Initial program 71.6%

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

      1. *-commutative 71.6%

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

   3. Simplified 71.6%

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

   4. Taylor expanded in x around 0 71.6%

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

      1. fma-def 71.6%

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

      2. associate-/l* 80.0%

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

      3. *-commutative 80.0%

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

      4. *-commutative 80.0%

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

   6. Simplified 80.0%

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

   7. Taylor expanded in a around inf 76.5%

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

      1. neg-mul-1 76.5%

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

      2. unsub-neg 76.5%

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

   9. Simplified 76.5%

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

   if -6.50000000000000017e34 < z < -4.7e-84

   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. Taylor expanded in x around inf 72.7%

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

      1. *-commutative 72.7%

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

   6. Simplified 72.7%

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

   if -4.7e-84 < z < 6.8e21

   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. Taylor expanded in t around inf 78.2%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.5 \cdot 10^{+34}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{elif}\;z \leq -4.7 \cdot 10^{-84}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{elif}\;z \leq 6.8 \cdot 10^{+21}:\\ \;\;\;\;\frac{x - y \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \end{array} \]

Alternative 6: 53.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.3 \cdot 10^{+36}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 2 \cdot 10^{-155}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{-82}:\\ \;\;\;\;\frac{\frac{-x}{a}}{z}\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{-8}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -2.3e+36)
   (/ y a)
   (if (<= z 2e-155)
     (/ x t)
     (if (<= z 1.6e-82)
       (/ (/ (- x) a) z)
       (if (<= z 3.5e-8) (/ x t) (/ y a))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.3e+36) {
		tmp = y / a;
	} else if (z <= 2e-155) {
		tmp = x / t;
	} else if (z <= 1.6e-82) {
		tmp = (-x / a) / z;
	} else if (z <= 3.5e-8) {
		tmp = x / t;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-2.3d+36)) then
        tmp = y / a
    else if (z <= 2d-155) then
        tmp = x / t
    else if (z <= 1.6d-82) then
        tmp = (-x / a) / z
    else if (z <= 3.5d-8) then
        tmp = x / t
    else
        tmp = y / a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.3e+36) {
		tmp = y / a;
	} else if (z <= 2e-155) {
		tmp = x / t;
	} else if (z <= 1.6e-82) {
		tmp = (-x / a) / z;
	} else if (z <= 3.5e-8) {
		tmp = x / t;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -2.3e+36:
		tmp = y / a
	elif z <= 2e-155:
		tmp = x / t
	elif z <= 1.6e-82:
		tmp = (-x / a) / z
	elif z <= 3.5e-8:
		tmp = x / t
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -2.3e+36)
		tmp = Float64(y / a);
	elseif (z <= 2e-155)
		tmp = Float64(x / t);
	elseif (z <= 1.6e-82)
		tmp = Float64(Float64(Float64(-x) / a) / z);
	elseif (z <= 3.5e-8)
		tmp = Float64(x / t);
	else
		tmp = Float64(y / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -2.3e+36)
		tmp = y / a;
	elseif (z <= 2e-155)
		tmp = x / t;
	elseif (z <= 1.6e-82)
		tmp = (-x / a) / z;
	elseif (z <= 3.5e-8)
		tmp = x / t;
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -2.3e+36], N[(y / a), $MachinePrecision], If[LessEqual[z, 2e-155], N[(x / t), $MachinePrecision], If[LessEqual[z, 1.6e-82], N[(N[((-x) / a), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[z, 3.5e-8], N[(x / t), $MachinePrecision], N[(y / a), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.29999999999999996e36 or 3.50000000000000024e-8 < z

   1. Initial program 72.4%

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

      1. *-commutative 72.4%

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

   3. Simplified 72.4%

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

   4. Taylor expanded in z around inf 56.9%

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

   if -2.29999999999999996e36 < z < 2.00000000000000003e-155 or 1.6000000000000001e-82 < z < 3.50000000000000024e-8

   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. Taylor expanded in z around 0 61.6%

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

   if 2.00000000000000003e-155 < z < 1.6000000000000001e-82

   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. Taylor expanded in x around 0 99.9%

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

      1. fma-def 99.9%

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

      2. associate-/l* 99.7%

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

      3. *-commutative 99.7%

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

      4. *-commutative 99.7%

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

   6. Simplified 99.7%

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

   7. Taylor expanded in a around inf 43.8%

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

      1. neg-mul-1 43.8%

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

      2. unsub-neg 43.8%

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

   9. Simplified 43.8%

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

   10. Taylor expanded in y around 0 51.5%

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

       1. mul-1-neg 51.5%

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

       2. associate-/r* 51.7%

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

   12. Simplified 51.7%

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

4. Final simplification 59.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.3 \cdot 10^{+36}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 2 \cdot 10^{-155}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{-82}:\\ \;\;\;\;\frac{\frac{-x}{a}}{z}\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{-8}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 7: 52.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -8 \cdot 10^{+36}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 4.9 \cdot 10^{-170}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;z \leq 4.5 \cdot 10^{-74}:\\ \;\;\;\;\frac{y}{t} \cdot \left(-z\right)\\ \mathbf{elif}\;z \leq 2.15 \cdot 10^{-8}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -8e+36)
   (/ y a)
   (if (<= z 4.9e-170)
     (/ x t)
     (if (<= z 4.5e-74)
       (* (/ y t) (- z))
       (if (<= z 2.15e-8) (/ x t) (/ y a))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -8e+36) {
		tmp = y / a;
	} else if (z <= 4.9e-170) {
		tmp = x / t;
	} else if (z <= 4.5e-74) {
		tmp = (y / t) * -z;
	} else if (z <= 2.15e-8) {
		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 <= (-8d+36)) then
        tmp = y / a
    else if (z <= 4.9d-170) then
        tmp = x / t
    else if (z <= 4.5d-74) then
        tmp = (y / t) * -z
    else if (z <= 2.15d-8) 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 <= -8e+36) {
		tmp = y / a;
	} else if (z <= 4.9e-170) {
		tmp = x / t;
	} else if (z <= 4.5e-74) {
		tmp = (y / t) * -z;
	} else if (z <= 2.15e-8) {
		tmp = x / t;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -8e+36:
		tmp = y / a
	elif z <= 4.9e-170:
		tmp = x / t
	elif z <= 4.5e-74:
		tmp = (y / t) * -z
	elif z <= 2.15e-8:
		tmp = x / t
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -8e+36)
		tmp = Float64(y / a);
	elseif (z <= 4.9e-170)
		tmp = Float64(x / t);
	elseif (z <= 4.5e-74)
		tmp = Float64(Float64(y / t) * Float64(-z));
	elseif (z <= 2.15e-8)
		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 <= -8e+36)
		tmp = y / a;
	elseif (z <= 4.9e-170)
		tmp = x / t;
	elseif (z <= 4.5e-74)
		tmp = (y / t) * -z;
	elseif (z <= 2.15e-8)
		tmp = x / t;
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -8e+36], N[(y / a), $MachinePrecision], If[LessEqual[z, 4.9e-170], N[(x / t), $MachinePrecision], If[LessEqual[z, 4.5e-74], N[(N[(y / t), $MachinePrecision] * (-z)), $MachinePrecision], If[LessEqual[z, 2.15e-8], N[(x / t), $MachinePrecision], N[(y / a), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -8.00000000000000034e36 or 2.1500000000000001e-8 < z

   1. Initial program 72.4%

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

      1. *-commutative 72.4%

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

   3. Simplified 72.4%

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

   4. Taylor expanded in z around inf 56.9%

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

   if -8.00000000000000034e36 < z < 4.8999999999999996e-170 or 4.4999999999999999e-74 < z < 2.1500000000000001e-8

   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. Taylor expanded in z around 0 62.3%

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

   if 4.8999999999999996e-170 < z < 4.4999999999999999e-74

   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. Taylor expanded in x around 0 55.3%

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

      1. mul-1-neg 55.3%

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

      2. sub-neg 55.3%

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

      3. +-commutative 55.3%

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

      4. distribute-rgt-neg-in 55.3%

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

      5. fma-udef 55.3%

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

      6. associate-*l/ 55.3%

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

      7. distribute-rgt-neg-in 55.3%

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

      8. fma-udef 55.3%

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

      9. distribute-rgt-neg-in 55.3%

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

      10. +-commutative 55.3%

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

      11. sub-neg 55.3%

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

      12. *-commutative 55.3%

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

   6. Simplified 55.3%

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

   7. Taylor expanded in t around inf 54.9%

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

4. Final simplification 59.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8 \cdot 10^{+36}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 4.9 \cdot 10^{-170}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;z \leq 4.5 \cdot 10^{-74}:\\ \;\;\;\;\frac{y}{t} \cdot \left(-z\right)\\ \mathbf{elif}\;z \leq 2.15 \cdot 10^{-8}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 8: 52.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.2 \cdot 10^{+36}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 4.9 \cdot 10^{-170}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;z \leq 7.5 \cdot 10^{-74}:\\ \;\;\;\;\frac{y \cdot \left(-z\right)}{t}\\ \mathbf{elif}\;z \leq 3.9 \cdot 10^{-8}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -3.2e+36)
   (/ y a)
   (if (<= z 4.9e-170)
     (/ x t)
     (if (<= z 7.5e-74)
       (/ (* y (- z)) t)
       (if (<= z 3.9e-8) (/ x t) (/ y a))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -3.2e+36) {
		tmp = y / a;
	} else if (z <= 4.9e-170) {
		tmp = x / t;
	} else if (z <= 7.5e-74) {
		tmp = (y * -z) / t;
	} else if (z <= 3.9e-8) {
		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 <= (-3.2d+36)) then
        tmp = y / a
    else if (z <= 4.9d-170) then
        tmp = x / t
    else if (z <= 7.5d-74) then
        tmp = (y * -z) / t
    else if (z <= 3.9d-8) 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 <= -3.2e+36) {
		tmp = y / a;
	} else if (z <= 4.9e-170) {
		tmp = x / t;
	} else if (z <= 7.5e-74) {
		tmp = (y * -z) / t;
	} else if (z <= 3.9e-8) {
		tmp = x / t;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -3.2e+36:
		tmp = y / a
	elif z <= 4.9e-170:
		tmp = x / t
	elif z <= 7.5e-74:
		tmp = (y * -z) / t
	elif z <= 3.9e-8:
		tmp = x / t
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -3.2e+36)
		tmp = Float64(y / a);
	elseif (z <= 4.9e-170)
		tmp = Float64(x / t);
	elseif (z <= 7.5e-74)
		tmp = Float64(Float64(y * Float64(-z)) / t);
	elseif (z <= 3.9e-8)
		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 <= -3.2e+36)
		tmp = y / a;
	elseif (z <= 4.9e-170)
		tmp = x / t;
	elseif (z <= 7.5e-74)
		tmp = (y * -z) / t;
	elseif (z <= 3.9e-8)
		tmp = x / t;
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -3.2e+36], N[(y / a), $MachinePrecision], If[LessEqual[z, 4.9e-170], N[(x / t), $MachinePrecision], If[LessEqual[z, 7.5e-74], N[(N[(y * (-z)), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[z, 3.9e-8], N[(x / t), $MachinePrecision], N[(y / a), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.1999999999999999e36 or 3.89999999999999985e-8 < z

   1. Initial program 72.4%

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

      1. *-commutative 72.4%

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

   3. Simplified 72.4%

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

   4. Taylor expanded in z around inf 56.9%

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

   if -3.1999999999999999e36 < z < 4.8999999999999996e-170 or 7.5e-74 < z < 3.89999999999999985e-8

   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. Taylor expanded in z around 0 62.3%

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

   if 4.8999999999999996e-170 < z < 7.5e-74

   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. Taylor expanded in x around 0 99.8%

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

      1. fma-def 99.8%

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

      2. associate-/l* 88.6%

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

      3. *-commutative 88.6%

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

      4. *-commutative 88.6%

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

   6. Simplified 88.6%

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

   7. Taylor expanded in t around inf 65.6%

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

      1. mul-1-neg 65.6%

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

      2. sub-neg 65.6%

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

      3. *-commutative 65.6%

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

   9. Simplified 65.6%

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

   10. Taylor expanded in x around 0 55.1%

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

       1. associate-*r/ 55.1%

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

       2. associate-*r* 55.1%

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

       3. neg-mul-1 55.1%

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

   12. Simplified 55.1%

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

4. Final simplification 59.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.2 \cdot 10^{+36}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 4.9 \cdot 10^{-170}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;z \leq 7.5 \cdot 10^{-74}:\\ \;\;\;\;\frac{y \cdot \left(-z\right)}{t}\\ \mathbf{elif}\;z \leq 3.9 \cdot 10^{-8}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 9: 64.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.75 \cdot 10^{+49} \lor \neg \left(z \leq 4.5 \cdot 10^{+104}\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -1.75e+49) (not (<= z 4.5e+104))) (/ y a) (/ x (- t (* z a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -1.75e+49) || !(z <= 4.5e+104)) {
		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 <= (-1.75d+49)) .or. (.not. (z <= 4.5d+104))) 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 <= -1.75e+49) || !(z <= 4.5e+104)) {
		tmp = y / a;
	} else {
		tmp = x / (t - (z * a));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (z <= -1.75e+49) or not (z <= 4.5e+104):
		tmp = y / a
	else:
		tmp = x / (t - (z * a))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -1.75e+49) || !(z <= 4.5e+104))
		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 <= -1.75e+49) || ~((z <= 4.5e+104)))
		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, -1.75e+49], N[Not[LessEqual[z, 4.5e+104]], $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 < -1.74999999999999987e49 or 4.4999999999999998e104 < z

   1. Initial program 67.3%

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

      1. *-commutative 67.3%

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

   3. Simplified 67.3%

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

   4. Taylor expanded in z around inf 60.2%

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

   if -1.74999999999999987e49 < z < 4.4999999999999998e104

   1. Initial program 99.3%

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

      1. *-commutative 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in x around inf 70.5%

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

      1. *-commutative 70.5%

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

   6. Simplified 70.5%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.75 \cdot 10^{+49} \lor \neg \left(z \leq 4.5 \cdot 10^{+104}\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \end{array} \]

Alternative 10: 54.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.1 \cdot 10^{+34} \lor \neg \left(z \leq 1.55 \cdot 10^{-7}\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -2.1e+34) (not (<= z 1.55e-7))) (/ y a) (/ x t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -2.1e+34) || !(z <= 1.55e-7)) {
		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 <= (-2.1d+34)) .or. (.not. (z <= 1.55d-7))) 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 <= -2.1e+34) || !(z <= 1.55e-7)) {
		tmp = y / a;
	} else {
		tmp = x / t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (z <= -2.1e+34) or not (z <= 1.55e-7):
		tmp = y / a
	else:
		tmp = x / t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -2.1e+34) || !(z <= 1.55e-7))
		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 <= -2.1e+34) || ~((z <= 1.55e-7)))
		tmp = y / a;
	else
		tmp = x / t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -2.1e+34], N[Not[LessEqual[z, 1.55e-7]], $MachinePrecision]], N[(y / a), $MachinePrecision], N[(x / t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.10000000000000017e34 or 1.55e-7 < z

   1. Initial program 72.4%

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

      1. *-commutative 72.4%

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

   3. Simplified 72.4%

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

   4. Taylor expanded in z around inf 56.9%

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

   if -2.10000000000000017e34 < z < 1.55e-7

   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. Taylor expanded in z around 0 57.5%

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

4. Final simplification 57.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.1 \cdot 10^{+34} \lor \neg \left(z \leq 1.55 \cdot 10^{-7}\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t}\\ \end{array} \]

Alternative 11: 34.7% 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 88.2%

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

   1. *-commutative 88.2%

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

3. Simplified 88.2%

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

4. Taylor expanded in z around 0 38.6%

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

5. Final simplification 38.6%

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

Developer target: 97.4% accurate, 0.6× 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 2023310 
(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))))