Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisLine from plot-0.2.3.4, A

Percentage Accurate: 98.2% → 98.2%

Time: 8.6s

Alternatives: 12

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

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) / (z - a)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 98.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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) / (z - a)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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 * ((t - z) / (z - a)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.8%

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

2. Final simplification 98.8%

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

Alternative 2: 80.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + y \cdot \frac{z}{z - a}\\ \mathbf{if}\;z \leq -1.9 \cdot 10^{-153}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 3 \cdot 10^{-68}:\\ \;\;\;\;x + \frac{t}{\frac{a}{y}}\\ \mathbf{elif}\;z \leq 7 \cdot 10^{-13}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{+23}:\\ \;\;\;\;x + \frac{y}{\frac{a}{t}}\\ \mathbf{elif}\;z \leq 1.45 \cdot 10^{+88}:\\ \;\;\;\;x - y \cdot \frac{t}{z}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ x (* y (/ z (- z a))))))
   (if (<= z -1.9e-153)
     t_1
     (if (<= z 3e-68)
       (+ x (/ t (/ a y)))
       (if (<= z 7e-13)
         t_1
         (if (<= z 1.4e+23)
           (+ x (/ y (/ a t)))
           (if (<= z 1.45e+88) (- x (* y (/ t z))) t_1)))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x + (y * (z / (z - a)));
	double tmp;
	if (z <= -1.9e-153) {
		tmp = t_1;
	} else if (z <= 3e-68) {
		tmp = x + (t / (a / y));
	} else if (z <= 7e-13) {
		tmp = t_1;
	} else if (z <= 1.4e+23) {
		tmp = x + (y / (a / t));
	} else if (z <= 1.45e+88) {
		tmp = x - (y * (t / z));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x + (y * (z / (z - a)))
    if (z <= (-1.9d-153)) then
        tmp = t_1
    else if (z <= 3d-68) then
        tmp = x + (t / (a / y))
    else if (z <= 7d-13) then
        tmp = t_1
    else if (z <= 1.4d+23) then
        tmp = x + (y / (a / t))
    else if (z <= 1.45d+88) then
        tmp = x - (y * (t / z))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = x + (y * (z / (z - a)));
	double tmp;
	if (z <= -1.9e-153) {
		tmp = t_1;
	} else if (z <= 3e-68) {
		tmp = x + (t / (a / y));
	} else if (z <= 7e-13) {
		tmp = t_1;
	} else if (z <= 1.4e+23) {
		tmp = x + (y / (a / t));
	} else if (z <= 1.45e+88) {
		tmp = x - (y * (t / z));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = x + (y * (z / (z - a)))
	tmp = 0
	if z <= -1.9e-153:
		tmp = t_1
	elif z <= 3e-68:
		tmp = x + (t / (a / y))
	elif z <= 7e-13:
		tmp = t_1
	elif z <= 1.4e+23:
		tmp = x + (y / (a / t))
	elif z <= 1.45e+88:
		tmp = x - (y * (t / z))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x + Float64(y * Float64(z / Float64(z - a))))
	tmp = 0.0
	if (z <= -1.9e-153)
		tmp = t_1;
	elseif (z <= 3e-68)
		tmp = Float64(x + Float64(t / Float64(a / y)));
	elseif (z <= 7e-13)
		tmp = t_1;
	elseif (z <= 1.4e+23)
		tmp = Float64(x + Float64(y / Float64(a / t)));
	elseif (z <= 1.45e+88)
		tmp = Float64(x - Float64(y * Float64(t / z)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = x + (y * (z / (z - a)));
	tmp = 0.0;
	if (z <= -1.9e-153)
		tmp = t_1;
	elseif (z <= 3e-68)
		tmp = x + (t / (a / y));
	elseif (z <= 7e-13)
		tmp = t_1;
	elseif (z <= 1.4e+23)
		tmp = x + (y / (a / t));
	elseif (z <= 1.45e+88)
		tmp = x - (y * (t / z));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x + N[(y * N[(z / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.9e-153], t$95$1, If[LessEqual[z, 3e-68], N[(x + N[(t / N[(a / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 7e-13], t$95$1, If[LessEqual[z, 1.4e+23], N[(x + N[(y / N[(a / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.45e+88], N[(x - N[(y * N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -1.90000000000000011e-153 or 3e-68 < z < 7.0000000000000005e-13 or 1.45e88 < z

   1. Initial program 100.0%

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

   2. Taylor expanded in t around 0 85.0%

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

   if -1.90000000000000011e-153 < z < 3e-68

   1. Initial program 96.2%

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

      1. +-commutative 96.2%

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

      2. associate-*r/ 95.1%

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

      3. associate-*l/ 98.6%

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

      4. *-commutative 98.6%

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

      5. fma-def 98.6%

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

   3. Simplified 98.6%

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

   4. Taylor expanded in z around 0 86.4%

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

      1. associate-/l* 90.0%

         \[\leadsto \color{blue}{\frac{y}{\frac{a}{t}}} + x \]

   6. Simplified 90.0%

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

      1. clear-num 90.0%

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

      2. associate-/r/ 88.7%

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

      3. clear-num 88.7%

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

   8. Applied egg-rr 88.7%

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

      1. associate-*l/ 86.4%

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

      2. associate-/l* 90.8%

         \[\leadsto \color{blue}{\frac{t}{\frac{a}{y}}} + x \]

   10. Applied egg-rr 90.8%

       \[\leadsto \color{blue}{\frac{t}{\frac{a}{y}}} + x \]

   if 7.0000000000000005e-13 < z < 1.4e23

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

      2. associate-*r/ 99.7%

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

      3. associate-*l/ 99.7%

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

      4. *-commutative 99.7%

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

      5. fma-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in z around 0 99.7%

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

      1. associate-/l* 100.0%

         \[\leadsto \color{blue}{\frac{y}{\frac{a}{t}}} + x \]

   6. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{y}{\frac{a}{t}} + x} \]

   if 1.4e23 < z < 1.45e88

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. associate-*r/ 92.9%

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

      3. associate-*l/ 93.4%

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

      4. *-commutative 93.4%

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

      5. fma-def 93.1%

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

   3. Simplified 93.1%

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

   4. Taylor expanded in a around 0 92.3%

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

      1. *-commutative 92.3%

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

      2. associate-/l* 78.5%

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

   6. Simplified 78.5%

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

   7. Taylor expanded in z around 0 84.4%

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

      1. associate-*r/ 84.4%

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

      2. mul-1-neg 84.4%

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

      3. distribute-rgt-neg-in 84.4%

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

   9. Simplified 84.4%

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

   10. Taylor expanded in y around 0 84.4%

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

       1. +-commutative 84.4%

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

       2. mul-1-neg 84.4%

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

       3. associate-*r/ 84.4%

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

       4. distribute-lft-neg-in 84.4%

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

       5. cancel-sign-sub-inv 84.4%

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

   12. Simplified 84.4%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.9 \cdot 10^{-153}:\\ \;\;\;\;x + y \cdot \frac{z}{z - a}\\ \mathbf{elif}\;z \leq 3 \cdot 10^{-68}:\\ \;\;\;\;x + \frac{t}{\frac{a}{y}}\\ \mathbf{elif}\;z \leq 7 \cdot 10^{-13}:\\ \;\;\;\;x + y \cdot \frac{z}{z - a}\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{+23}:\\ \;\;\;\;x + \frac{y}{\frac{a}{t}}\\ \mathbf{elif}\;z \leq 1.45 \cdot 10^{+88}:\\ \;\;\;\;x - y \cdot \frac{t}{z}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z}{z - a}\\ \end{array} \]

Alternative 3: 81.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + y \cdot \frac{z}{z - a}\\ \mathbf{if}\;z \leq -2.9 \cdot 10^{-153}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.85 \cdot 10^{-66}:\\ \;\;\;\;x + \frac{t}{\frac{a}{y}}\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{-12}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 9.2 \cdot 10^{+22}:\\ \;\;\;\;x + \frac{y}{\frac{a}{t}}\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \frac{t - z}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ x (* y (/ z (- z a))))))
   (if (<= z -2.9e-153)
     t_1
     (if (<= z 1.85e-66)
       (+ x (/ t (/ a y)))
       (if (<= z 9.5e-12)
         t_1
         (if (<= z 9.2e+22) (+ x (/ y (/ a t))) (- x (* y (/ (- t z) z)))))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x + (y * (z / (z - a)));
	double tmp;
	if (z <= -2.9e-153) {
		tmp = t_1;
	} else if (z <= 1.85e-66) {
		tmp = x + (t / (a / y));
	} else if (z <= 9.5e-12) {
		tmp = t_1;
	} else if (z <= 9.2e+22) {
		tmp = x + (y / (a / t));
	} else {
		tmp = x - (y * ((t - z) / z));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x + (y * (z / (z - a)))
    if (z <= (-2.9d-153)) then
        tmp = t_1
    else if (z <= 1.85d-66) then
        tmp = x + (t / (a / y))
    else if (z <= 9.5d-12) then
        tmp = t_1
    else if (z <= 9.2d+22) then
        tmp = x + (y / (a / t))
    else
        tmp = x - (y * ((t - z) / z))
    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 + (y * (z / (z - a)));
	double tmp;
	if (z <= -2.9e-153) {
		tmp = t_1;
	} else if (z <= 1.85e-66) {
		tmp = x + (t / (a / y));
	} else if (z <= 9.5e-12) {
		tmp = t_1;
	} else if (z <= 9.2e+22) {
		tmp = x + (y / (a / t));
	} else {
		tmp = x - (y * ((t - z) / z));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = x + (y * (z / (z - a)))
	tmp = 0
	if z <= -2.9e-153:
		tmp = t_1
	elif z <= 1.85e-66:
		tmp = x + (t / (a / y))
	elif z <= 9.5e-12:
		tmp = t_1
	elif z <= 9.2e+22:
		tmp = x + (y / (a / t))
	else:
		tmp = x - (y * ((t - z) / z))
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x + Float64(y * Float64(z / Float64(z - a))))
	tmp = 0.0
	if (z <= -2.9e-153)
		tmp = t_1;
	elseif (z <= 1.85e-66)
		tmp = Float64(x + Float64(t / Float64(a / y)));
	elseif (z <= 9.5e-12)
		tmp = t_1;
	elseif (z <= 9.2e+22)
		tmp = Float64(x + Float64(y / Float64(a / t)));
	else
		tmp = Float64(x - Float64(y * Float64(Float64(t - z) / z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = x + (y * (z / (z - a)));
	tmp = 0.0;
	if (z <= -2.9e-153)
		tmp = t_1;
	elseif (z <= 1.85e-66)
		tmp = x + (t / (a / y));
	elseif (z <= 9.5e-12)
		tmp = t_1;
	elseif (z <= 9.2e+22)
		tmp = x + (y / (a / t));
	else
		tmp = x - (y * ((t - z) / z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x + N[(y * N[(z / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -2.9e-153], t$95$1, If[LessEqual[z, 1.85e-66], N[(x + N[(t / N[(a / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 9.5e-12], t$95$1, If[LessEqual[z, 9.2e+22], N[(x + N[(y / N[(a / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(y * N[(N[(t - z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -2.90000000000000002e-153 or 1.8500000000000001e-66 < z < 9.4999999999999995e-12

   1. Initial program 100.0%

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

   2. Taylor expanded in t around 0 82.4%

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

   if -2.90000000000000002e-153 < z < 1.8500000000000001e-66

   1. Initial program 96.2%

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

      1. +-commutative 96.2%

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

      2. associate-*r/ 95.1%

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

      3. associate-*l/ 98.6%

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

      4. *-commutative 98.6%

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

      5. fma-def 98.6%

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

   3. Simplified 98.6%

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

   4. Taylor expanded in z around 0 86.4%

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

      1. associate-/l* 90.0%

         \[\leadsto \color{blue}{\frac{y}{\frac{a}{t}}} + x \]

   6. Simplified 90.0%

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

      1. clear-num 90.0%

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

      2. associate-/r/ 88.7%

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

      3. clear-num 88.7%

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

   8. Applied egg-rr 88.7%

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

      1. associate-*l/ 86.4%

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

      2. associate-/l* 90.8%

         \[\leadsto \color{blue}{\frac{t}{\frac{a}{y}}} + x \]

   10. Applied egg-rr 90.8%

       \[\leadsto \color{blue}{\frac{t}{\frac{a}{y}}} + x \]

   if 9.4999999999999995e-12 < z < 9.2000000000000008e22

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

      2. associate-*r/ 99.7%

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

      3. associate-*l/ 99.7%

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

      4. *-commutative 99.7%

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

      5. fma-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in z around 0 99.7%

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

      1. associate-/l* 100.0%

         \[\leadsto \color{blue}{\frac{y}{\frac{a}{t}}} + x \]

   6. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{y}{\frac{a}{t}} + x} \]

   if 9.2000000000000008e22 < z

   1. Initial program 99.9%

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

   2. Taylor expanded in a around 0 93.9%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.9 \cdot 10^{-153}:\\ \;\;\;\;x + y \cdot \frac{z}{z - a}\\ \mathbf{elif}\;z \leq 1.85 \cdot 10^{-66}:\\ \;\;\;\;x + \frac{t}{\frac{a}{y}}\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{-12}:\\ \;\;\;\;x + y \cdot \frac{z}{z - a}\\ \mathbf{elif}\;z \leq 9.2 \cdot 10^{+22}:\\ \;\;\;\;x + \frac{y}{\frac{a}{t}}\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \frac{t - z}{z}\\ \end{array} \]

Alternative 4: 86.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{y \cdot t}{a - z}\\ t_2 := x - y \cdot \frac{t - z}{z}\\ \mathbf{if}\;z \leq -9.4 \cdot 10^{-38}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -1 \cdot 10^{-144}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 7.5 \cdot 10^{-206}:\\ \;\;\;\;x + \frac{y}{a} \cdot \left(t - z\right)\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{+24}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ x (/ (* y t) (- a z)))) (t_2 (- x (* y (/ (- t z) z)))))
   (if (<= z -9.4e-38)
     t_2
     (if (<= z -1e-144)
       t_1
       (if (<= z 7.5e-206)
         (+ x (* (/ y a) (- t z)))
         (if (<= z 4.2e+24) t_1 t_2))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x + ((y * t) / (a - z));
	double t_2 = x - (y * ((t - z) / z));
	double tmp;
	if (z <= -9.4e-38) {
		tmp = t_2;
	} else if (z <= -1e-144) {
		tmp = t_1;
	} else if (z <= 7.5e-206) {
		tmp = x + ((y / a) * (t - z));
	} else if (z <= 4.2e+24) {
		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 + ((y * t) / (a - z))
    t_2 = x - (y * ((t - z) / z))
    if (z <= (-9.4d-38)) then
        tmp = t_2
    else if (z <= (-1d-144)) then
        tmp = t_1
    else if (z <= 7.5d-206) then
        tmp = x + ((y / a) * (t - z))
    else if (z <= 4.2d+24) 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 + ((y * t) / (a - z));
	double t_2 = x - (y * ((t - z) / z));
	double tmp;
	if (z <= -9.4e-38) {
		tmp = t_2;
	} else if (z <= -1e-144) {
		tmp = t_1;
	} else if (z <= 7.5e-206) {
		tmp = x + ((y / a) * (t - z));
	} else if (z <= 4.2e+24) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = x + ((y * t) / (a - z))
	t_2 = x - (y * ((t - z) / z))
	tmp = 0
	if z <= -9.4e-38:
		tmp = t_2
	elif z <= -1e-144:
		tmp = t_1
	elif z <= 7.5e-206:
		tmp = x + ((y / a) * (t - z))
	elif z <= 4.2e+24:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x + Float64(Float64(y * t) / Float64(a - z)))
	t_2 = Float64(x - Float64(y * Float64(Float64(t - z) / z)))
	tmp = 0.0
	if (z <= -9.4e-38)
		tmp = t_2;
	elseif (z <= -1e-144)
		tmp = t_1;
	elseif (z <= 7.5e-206)
		tmp = Float64(x + Float64(Float64(y / a) * Float64(t - z)));
	elseif (z <= 4.2e+24)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = x + ((y * t) / (a - z));
	t_2 = x - (y * ((t - z) / z));
	tmp = 0.0;
	if (z <= -9.4e-38)
		tmp = t_2;
	elseif (z <= -1e-144)
		tmp = t_1;
	elseif (z <= 7.5e-206)
		tmp = x + ((y / a) * (t - z));
	elseif (z <= 4.2e+24)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x + N[(N[(y * t), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x - N[(y * N[(N[(t - z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -9.4e-38], t$95$2, If[LessEqual[z, -1e-144], t$95$1, If[LessEqual[z, 7.5e-206], N[(x + N[(N[(y / a), $MachinePrecision] * N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 4.2e+24], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -9.39999999999999996e-38 or 4.2000000000000003e24 < z

   1. Initial program 100.0%

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

   2. Taylor expanded in a around 0 90.8%

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

   if -9.39999999999999996e-38 < z < -9.9999999999999995e-145 or 7.5e-206 < z < 4.2000000000000003e24

   1. Initial program 99.9%

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

   2. Taylor expanded in t around inf 88.9%

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

      1. associate-*r/ 88.9%

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

      2. mul-1-neg 88.9%

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

      3. distribute-rgt-neg-out 88.9%

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

   4. Simplified 88.9%

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

      1. frac-2neg 88.9%

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

      2. div-inv 88.9%

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

      3. distribute-rgt-neg-out 88.9%

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

      4. remove-double-neg 88.9%

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

   6. Applied egg-rr 88.9%

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

   7. Taylor expanded in y around 0 88.9%

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

   if -9.9999999999999995e-145 < z < 7.5e-206

   1. Initial program 94.5%

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

      1. +-commutative 94.5%

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

      2. associate-*r/ 92.7%

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

      3. associate-*l/ 98.0%

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

      4. *-commutative 98.0%

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

      5. fma-def 98.0%

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

   3. Simplified 98.0%

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

   4. Taylor expanded in a around inf 91.0%

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

      1. +-commutative 91.0%

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

      2. mul-1-neg 91.0%

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

      3. unsub-neg 91.0%

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

      4. associate-/l* 96.4%

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

   6. Simplified 96.4%

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

   7. Taylor expanded in y around 0 91.0%

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

      1. associate-/l* 96.4%

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

      2. associate-/r/ 96.3%

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

   9. Simplified 96.3%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9.4 \cdot 10^{-38}:\\ \;\;\;\;x - y \cdot \frac{t - z}{z}\\ \mathbf{elif}\;z \leq -1 \cdot 10^{-144}:\\ \;\;\;\;x + \frac{y \cdot t}{a - z}\\ \mathbf{elif}\;z \leq 7.5 \cdot 10^{-206}:\\ \;\;\;\;x + \frac{y}{a} \cdot \left(t - z\right)\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{+24}:\\ \;\;\;\;x + \frac{y \cdot t}{a - z}\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \frac{t - z}{z}\\ \end{array} \]

Alternative 5: 86.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{y \cdot t}{a - z}\\ t_2 := x - y \cdot \frac{t - z}{z}\\ \mathbf{if}\;z \leq -3.5 \cdot 10^{-39}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -2.45 \cdot 10^{-147}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.35 \cdot 10^{-206}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{+22}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ x (/ (* y t) (- a z)))) (t_2 (- x (* y (/ (- t z) z)))))
   (if (<= z -3.5e-39)
     t_2
     (if (<= z -2.45e-147)
       t_1
       (if (<= z 2.35e-206)
         (- x (/ y (/ a (- z t))))
         (if (<= z 9.5e+22) t_1 t_2))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x + ((y * t) / (a - z));
	double t_2 = x - (y * ((t - z) / z));
	double tmp;
	if (z <= -3.5e-39) {
		tmp = t_2;
	} else if (z <= -2.45e-147) {
		tmp = t_1;
	} else if (z <= 2.35e-206) {
		tmp = x - (y / (a / (z - t)));
	} else if (z <= 9.5e+22) {
		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 + ((y * t) / (a - z))
    t_2 = x - (y * ((t - z) / z))
    if (z <= (-3.5d-39)) then
        tmp = t_2
    else if (z <= (-2.45d-147)) then
        tmp = t_1
    else if (z <= 2.35d-206) then
        tmp = x - (y / (a / (z - t)))
    else if (z <= 9.5d+22) 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 + ((y * t) / (a - z));
	double t_2 = x - (y * ((t - z) / z));
	double tmp;
	if (z <= -3.5e-39) {
		tmp = t_2;
	} else if (z <= -2.45e-147) {
		tmp = t_1;
	} else if (z <= 2.35e-206) {
		tmp = x - (y / (a / (z - t)));
	} else if (z <= 9.5e+22) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = x + ((y * t) / (a - z))
	t_2 = x - (y * ((t - z) / z))
	tmp = 0
	if z <= -3.5e-39:
		tmp = t_2
	elif z <= -2.45e-147:
		tmp = t_1
	elif z <= 2.35e-206:
		tmp = x - (y / (a / (z - t)))
	elif z <= 9.5e+22:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x + Float64(Float64(y * t) / Float64(a - z)))
	t_2 = Float64(x - Float64(y * Float64(Float64(t - z) / z)))
	tmp = 0.0
	if (z <= -3.5e-39)
		tmp = t_2;
	elseif (z <= -2.45e-147)
		tmp = t_1;
	elseif (z <= 2.35e-206)
		tmp = Float64(x - Float64(y / Float64(a / Float64(z - t))));
	elseif (z <= 9.5e+22)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = x + ((y * t) / (a - z));
	t_2 = x - (y * ((t - z) / z));
	tmp = 0.0;
	if (z <= -3.5e-39)
		tmp = t_2;
	elseif (z <= -2.45e-147)
		tmp = t_1;
	elseif (z <= 2.35e-206)
		tmp = x - (y / (a / (z - t)));
	elseif (z <= 9.5e+22)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x + N[(N[(y * t), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x - N[(y * N[(N[(t - z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -3.5e-39], t$95$2, If[LessEqual[z, -2.45e-147], t$95$1, If[LessEqual[z, 2.35e-206], N[(x - N[(y / N[(a / N[(z - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 9.5e+22], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.5e-39 or 9.49999999999999937e22 < z

   1. Initial program 100.0%

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

   2. Taylor expanded in a around 0 90.8%

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

   if -3.5e-39 < z < -2.45000000000000002e-147 or 2.3499999999999999e-206 < z < 9.49999999999999937e22

   1. Initial program 99.9%

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

   2. Taylor expanded in t around inf 88.9%

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

      1. associate-*r/ 88.9%

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

      2. mul-1-neg 88.9%

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

      3. distribute-rgt-neg-out 88.9%

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

   4. Simplified 88.9%

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

      1. frac-2neg 88.9%

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

      2. div-inv 88.9%

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

      3. distribute-rgt-neg-out 88.9%

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

      4. remove-double-neg 88.9%

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

   6. Applied egg-rr 88.9%

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

   7. Taylor expanded in y around 0 88.9%

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

   if -2.45000000000000002e-147 < z < 2.3499999999999999e-206

   1. Initial program 94.5%

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

      1. +-commutative 94.5%

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

      2. associate-*r/ 92.7%

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

      3. associate-*l/ 98.0%

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

      4. *-commutative 98.0%

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

      5. fma-def 98.0%

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

   3. Simplified 98.0%

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

   4. Taylor expanded in a around inf 91.0%

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

      1. +-commutative 91.0%

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

      2. mul-1-neg 91.0%

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

      3. unsub-neg 91.0%

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

      4. associate-/l* 96.4%

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

   6. Simplified 96.4%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.5 \cdot 10^{-39}:\\ \;\;\;\;x - y \cdot \frac{t - z}{z}\\ \mathbf{elif}\;z \leq -2.45 \cdot 10^{-147}:\\ \;\;\;\;x + \frac{y \cdot t}{a - z}\\ \mathbf{elif}\;z \leq 2.35 \cdot 10^{-206}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{+22}:\\ \;\;\;\;x + \frac{y \cdot t}{a - z}\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \frac{t - z}{z}\\ \end{array} \]

Alternative 6: 86.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{y \cdot t}{a - z}\\ t_2 := x - y \cdot \frac{t - z}{z}\\ \mathbf{if}\;z \leq -5.2 \cdot 10^{-38}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -8.5 \cdot 10^{-146}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{-206}:\\ \;\;\;\;x + \frac{z - t}{\frac{a}{-y}}\\ \mathbf{elif}\;z \leq 5.8 \cdot 10^{+24}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ x (/ (* y t) (- a z)))) (t_2 (- x (* y (/ (- t z) z)))))
   (if (<= z -5.2e-38)
     t_2
     (if (<= z -8.5e-146)
       t_1
       (if (<= z 4.2e-206)
         (+ x (/ (- z t) (/ a (- y))))
         (if (<= z 5.8e+24) t_1 t_2))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x + ((y * t) / (a - z));
	double t_2 = x - (y * ((t - z) / z));
	double tmp;
	if (z <= -5.2e-38) {
		tmp = t_2;
	} else if (z <= -8.5e-146) {
		tmp = t_1;
	} else if (z <= 4.2e-206) {
		tmp = x + ((z - t) / (a / -y));
	} else if (z <= 5.8e+24) {
		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 + ((y * t) / (a - z))
    t_2 = x - (y * ((t - z) / z))
    if (z <= (-5.2d-38)) then
        tmp = t_2
    else if (z <= (-8.5d-146)) then
        tmp = t_1
    else if (z <= 4.2d-206) then
        tmp = x + ((z - t) / (a / -y))
    else if (z <= 5.8d+24) 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 + ((y * t) / (a - z));
	double t_2 = x - (y * ((t - z) / z));
	double tmp;
	if (z <= -5.2e-38) {
		tmp = t_2;
	} else if (z <= -8.5e-146) {
		tmp = t_1;
	} else if (z <= 4.2e-206) {
		tmp = x + ((z - t) / (a / -y));
	} else if (z <= 5.8e+24) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = x + ((y * t) / (a - z))
	t_2 = x - (y * ((t - z) / z))
	tmp = 0
	if z <= -5.2e-38:
		tmp = t_2
	elif z <= -8.5e-146:
		tmp = t_1
	elif z <= 4.2e-206:
		tmp = x + ((z - t) / (a / -y))
	elif z <= 5.8e+24:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x + Float64(Float64(y * t) / Float64(a - z)))
	t_2 = Float64(x - Float64(y * Float64(Float64(t - z) / z)))
	tmp = 0.0
	if (z <= -5.2e-38)
		tmp = t_2;
	elseif (z <= -8.5e-146)
		tmp = t_1;
	elseif (z <= 4.2e-206)
		tmp = Float64(x + Float64(Float64(z - t) / Float64(a / Float64(-y))));
	elseif (z <= 5.8e+24)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = x + ((y * t) / (a - z));
	t_2 = x - (y * ((t - z) / z));
	tmp = 0.0;
	if (z <= -5.2e-38)
		tmp = t_2;
	elseif (z <= -8.5e-146)
		tmp = t_1;
	elseif (z <= 4.2e-206)
		tmp = x + ((z - t) / (a / -y));
	elseif (z <= 5.8e+24)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x + N[(N[(y * t), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x - N[(y * N[(N[(t - z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -5.2e-38], t$95$2, If[LessEqual[z, -8.5e-146], t$95$1, If[LessEqual[z, 4.2e-206], N[(x + N[(N[(z - t), $MachinePrecision] / N[(a / (-y)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 5.8e+24], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.20000000000000022e-38 or 5.79999999999999958e24 < z

   1. Initial program 100.0%

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

   2. Taylor expanded in a around 0 90.8%

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

   if -5.20000000000000022e-38 < z < -8.4999999999999997e-146 or 4.2000000000000002e-206 < z < 5.79999999999999958e24

   1. Initial program 99.9%

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

   2. Taylor expanded in t around inf 88.9%

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

      1. associate-*r/ 88.9%

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

      2. mul-1-neg 88.9%

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

      3. distribute-rgt-neg-out 88.9%

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

   4. Simplified 88.9%

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

      1. frac-2neg 88.9%

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

      2. div-inv 88.9%

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

      3. distribute-rgt-neg-out 88.9%

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

      4. remove-double-neg 88.9%

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

   6. Applied egg-rr 88.9%

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

   7. Taylor expanded in y around 0 88.9%

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

   if -8.4999999999999997e-146 < z < 4.2000000000000002e-206

   1. Initial program 94.5%

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

      1. associate-*r/ 92.7%

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

   3. Simplified 92.7%

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

      1. associate-/l* 96.4%

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

      2. associate-/r/ 98.0%

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

   5. Applied egg-rr 98.0%

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

   6. Taylor expanded in a around inf 91.0%

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

      1. associate-*r/ 91.0%

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

      2. neg-mul-1 91.0%

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

      3. distribute-lft-neg-in 91.0%

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

      4. *-commutative 91.0%

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

      5. associate-/l* 97.6%

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

   8. Simplified 97.6%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.2 \cdot 10^{-38}:\\ \;\;\;\;x - y \cdot \frac{t - z}{z}\\ \mathbf{elif}\;z \leq -8.5 \cdot 10^{-146}:\\ \;\;\;\;x + \frac{y \cdot t}{a - z}\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{-206}:\\ \;\;\;\;x + \frac{z - t}{\frac{a}{-y}}\\ \mathbf{elif}\;z \leq 5.8 \cdot 10^{+24}:\\ \;\;\;\;x + \frac{y \cdot t}{a - z}\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \frac{t - z}{z}\\ \end{array} \]

Alternative 7: 86.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -9.2 \cdot 10^{-38} \lor \neg \left(z \leq 4.2 \cdot 10^{+24}\right):\\ \;\;\;\;x - y \cdot \frac{t - z}{z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot t}{a - z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -9.2e-38) (not (<= z 4.2e+24)))
   (- x (* y (/ (- t z) z)))
   (+ x (/ (* y t) (- a z)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -9.2e-38) || !(z <= 4.2e+24)) {
		tmp = x - (y * ((t - z) / z));
	} else {
		tmp = x + ((y * t) / (a - z));
	}
	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 <= (-9.2d-38)) .or. (.not. (z <= 4.2d+24))) then
        tmp = x - (y * ((t - z) / z))
    else
        tmp = x + ((y * t) / (a - z))
    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 <= -9.2e-38) || !(z <= 4.2e+24)) {
		tmp = x - (y * ((t - z) / z));
	} else {
		tmp = x + ((y * t) / (a - z));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (z <= -9.2e-38) or not (z <= 4.2e+24):
		tmp = x - (y * ((t - z) / z))
	else:
		tmp = x + ((y * t) / (a - z))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -9.2e-38) || !(z <= 4.2e+24))
		tmp = Float64(x - Float64(y * Float64(Float64(t - z) / z)));
	else
		tmp = Float64(x + Float64(Float64(y * t) / Float64(a - z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((z <= -9.2e-38) || ~((z <= 4.2e+24)))
		tmp = x - (y * ((t - z) / z));
	else
		tmp = x + ((y * t) / (a - z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -9.2e-38], N[Not[LessEqual[z, 4.2e+24]], $MachinePrecision]], N[(x - N[(y * N[(N[(t - z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(y * t), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -9.20000000000000007e-38 or 4.2000000000000003e24 < z

   1. Initial program 100.0%

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

   2. Taylor expanded in a around 0 90.8%

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

   if -9.20000000000000007e-38 < z < 4.2000000000000003e24

   1. Initial program 97.5%

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

   2. Taylor expanded in t around inf 88.2%

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

      1. associate-*r/ 88.2%

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

      2. mul-1-neg 88.2%

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

      3. distribute-rgt-neg-out 88.2%

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

   4. Simplified 88.2%

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

      1. frac-2neg 88.2%

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

      2. div-inv 88.1%

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

      3. distribute-rgt-neg-out 88.1%

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

      4. remove-double-neg 88.1%

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

   6. Applied egg-rr 88.1%

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

   7. Taylor expanded in y around 0 88.2%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9.2 \cdot 10^{-38} \lor \neg \left(z \leq 4.2 \cdot 10^{+24}\right):\\ \;\;\;\;x - y \cdot \frac{t - z}{z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot t}{a - z}\\ \end{array} \]

Alternative 8: 77.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.5 \cdot 10^{-38}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{+23}:\\ \;\;\;\;x + y \cdot \frac{t}{a}\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -5.5e-38) (+ x y) (if (<= z 2.9e+23) (+ x (* y (/ t a))) (+ x y))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -5.5e-38) {
		tmp = x + y;
	} else if (z <= 2.9e+23) {
		tmp = x + (y * (t / a));
	} else {
		tmp = x + y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-5.5d-38)) then
        tmp = x + y
    else if (z <= 2.9d+23) then
        tmp = x + (y * (t / a))
    else
        tmp = x + y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -5.5e-38) {
		tmp = x + y;
	} else if (z <= 2.9e+23) {
		tmp = x + (y * (t / a));
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -5.5e-38:
		tmp = x + y
	elif z <= 2.9e+23:
		tmp = x + (y * (t / a))
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -5.5e-38)
		tmp = Float64(x + y);
	elseif (z <= 2.9e+23)
		tmp = Float64(x + Float64(y * Float64(t / a)));
	else
		tmp = Float64(x + y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -5.5e-38)
		tmp = x + y;
	elseif (z <= 2.9e+23)
		tmp = x + (y * (t / a));
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -5.5e-38], N[(x + y), $MachinePrecision], If[LessEqual[z, 2.9e+23], N[(x + N[(y * N[(t / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -5.50000000000000005e-38 or 2.90000000000000013e23 < z

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. associate-*r/ 77.9%

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

      3. associate-*l/ 97.0%

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

      4. *-commutative 97.0%

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

      5. fma-def 97.0%

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

   3. Simplified 97.0%

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

   4. Taylor expanded in z around inf 81.4%

      \[\leadsto \color{blue}{y + x} \]

   if -5.50000000000000005e-38 < z < 2.90000000000000013e23

   1. Initial program 97.5%

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

   2. Taylor expanded in z around 0 79.1%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.5 \cdot 10^{-38}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{+23}:\\ \;\;\;\;x + y \cdot \frac{t}{a}\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 9: 77.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -9.4 \cdot 10^{-38}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{+23}:\\ \;\;\;\;x + t \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -9.4e-38) (+ x y) (if (<= z 1.25e+23) (+ x (* t (/ y a))) (+ x y))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -9.4e-38) {
		tmp = x + y;
	} else if (z <= 1.25e+23) {
		tmp = x + (t * (y / a));
	} else {
		tmp = x + y;
	}
	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 <= (-9.4d-38)) then
        tmp = x + y
    else if (z <= 1.25d+23) then
        tmp = x + (t * (y / a))
    else
        tmp = x + y
    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 <= -9.4e-38) {
		tmp = x + y;
	} else if (z <= 1.25e+23) {
		tmp = x + (t * (y / a));
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -9.4e-38:
		tmp = x + y
	elif z <= 1.25e+23:
		tmp = x + (t * (y / a))
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -9.4e-38)
		tmp = Float64(x + y);
	elseif (z <= 1.25e+23)
		tmp = Float64(x + Float64(t * Float64(y / a)));
	else
		tmp = Float64(x + y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -9.4e-38)
		tmp = x + y;
	elseif (z <= 1.25e+23)
		tmp = x + (t * (y / a));
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -9.4e-38], N[(x + y), $MachinePrecision], If[LessEqual[z, 1.25e+23], N[(x + N[(t * N[(y / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -9.39999999999999996e-38 or 1.25e23 < z

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. associate-*r/ 77.9%

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

      3. associate-*l/ 97.0%

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

      4. *-commutative 97.0%

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

      5. fma-def 97.0%

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

   3. Simplified 97.0%

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

   4. Taylor expanded in z around inf 81.4%

      \[\leadsto \color{blue}{y + x} \]

   if -9.39999999999999996e-38 < z < 1.25e23

   1. Initial program 97.5%

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

      1. +-commutative 97.5%

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

      2. associate-*r/ 96.0%

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

      3. associate-*l/ 98.8%

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

      4. *-commutative 98.8%

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

      5. fma-def 98.8%

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

   3. Simplified 98.8%

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

   4. Taylor expanded in z around 0 76.8%

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

      1. associate-/l* 80.0%

         \[\leadsto \color{blue}{\frac{y}{\frac{a}{t}}} + x \]

   6. Simplified 80.0%

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

      1. associate-/r/ 79.6%

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

   8. Applied egg-rr 79.6%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9.4 \cdot 10^{-38}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{+23}:\\ \;\;\;\;x + t \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 10: 77.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.2 \cdot 10^{-38}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq 9.2 \cdot 10^{+22}:\\ \;\;\;\;x + \frac{t}{\frac{a}{y}}\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.2e-38) (+ x y) (if (<= z 9.2e+22) (+ x (/ t (/ a y))) (+ x y))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.2e-38) {
		tmp = x + y;
	} else if (z <= 9.2e+22) {
		tmp = x + (t / (a / y));
	} else {
		tmp = x + y;
	}
	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 <= (-1.2d-38)) then
        tmp = x + y
    else if (z <= 9.2d+22) then
        tmp = x + (t / (a / y))
    else
        tmp = x + y
    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 <= -1.2e-38) {
		tmp = x + y;
	} else if (z <= 9.2e+22) {
		tmp = x + (t / (a / y));
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.2e-38:
		tmp = x + y
	elif z <= 9.2e+22:
		tmp = x + (t / (a / y))
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.2e-38)
		tmp = Float64(x + y);
	elseif (z <= 9.2e+22)
		tmp = Float64(x + Float64(t / Float64(a / y)));
	else
		tmp = Float64(x + y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -1.2e-38)
		tmp = x + y;
	elseif (z <= 9.2e+22)
		tmp = x + (t / (a / y));
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.2e-38], N[(x + y), $MachinePrecision], If[LessEqual[z, 9.2e+22], N[(x + N[(t / N[(a / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.20000000000000011e-38 or 9.2000000000000008e22 < z

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. associate-*r/ 77.9%

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

      3. associate-*l/ 97.0%

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

      4. *-commutative 97.0%

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

      5. fma-def 97.0%

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

   3. Simplified 97.0%

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

   4. Taylor expanded in z around inf 81.4%

      \[\leadsto \color{blue}{y + x} \]

   if -1.20000000000000011e-38 < z < 9.2000000000000008e22

   1. Initial program 97.5%

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

      1. +-commutative 97.5%

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

      2. associate-*r/ 96.0%

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

      3. associate-*l/ 98.8%

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

      4. *-commutative 98.8%

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

      5. fma-def 98.8%

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

   3. Simplified 98.8%

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

   4. Taylor expanded in z around 0 76.8%

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

      1. associate-/l* 80.0%

         \[\leadsto \color{blue}{\frac{y}{\frac{a}{t}}} + x \]

   6. Simplified 80.0%

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

      1. clear-num 80.0%

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

      2. associate-/r/ 79.1%

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

      3. clear-num 79.1%

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

   8. Applied egg-rr 79.1%

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

      1. associate-*l/ 76.8%

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

      2. associate-/l* 80.2%

         \[\leadsto \color{blue}{\frac{t}{\frac{a}{y}}} + x \]

   10. Applied egg-rr 80.2%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.2 \cdot 10^{-38}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq 9.2 \cdot 10^{+22}:\\ \;\;\;\;x + \frac{t}{\frac{a}{y}}\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 11: 64.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -7.8 \cdot 10^{-38}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq 1.15 \cdot 10^{+30}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -7.8e-38) (+ x y) (if (<= z 1.15e+30) x (+ x y))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -7.8e-38) {
		tmp = x + y;
	} else if (z <= 1.15e+30) {
		tmp = x;
	} else {
		tmp = x + y;
	}
	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-38)) then
        tmp = x + y
    else if (z <= 1.15d+30) then
        tmp = x
    else
        tmp = x + y
    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-38) {
		tmp = x + y;
	} else if (z <= 1.15e+30) {
		tmp = x;
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -7.8e-38:
		tmp = x + y
	elif z <= 1.15e+30:
		tmp = x
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -7.8e-38)
		tmp = Float64(x + y);
	elseif (z <= 1.15e+30)
		tmp = x;
	else
		tmp = Float64(x + y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -7.8e-38)
		tmp = x + y;
	elseif (z <= 1.15e+30)
		tmp = x;
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -7.8e-38], N[(x + y), $MachinePrecision], If[LessEqual[z, 1.15e+30], x, N[(x + y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -7.7999999999999998e-38 or 1.15e30 < z

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. associate-*r/ 77.7%

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

      3. associate-*l/ 97.0%

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

      4. *-commutative 97.0%

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

      5. fma-def 97.0%

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

   3. Simplified 97.0%

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

   4. Taylor expanded in z around inf 82.0%

      \[\leadsto \color{blue}{y + x} \]

   if -7.7999999999999998e-38 < z < 1.15e30

   1. Initial program 97.5%

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

      1. +-commutative 97.5%

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

      2. associate-*r/ 96.0%

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

      3. associate-*l/ 98.8%

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

      4. *-commutative 98.8%

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

      5. fma-def 98.8%

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

   3. Simplified 98.8%

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

   4. Taylor expanded in y around 0 44.0%

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

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.8 \cdot 10^{-38}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq 1.15 \cdot 10^{+30}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 12: 50.7% accurate, 11.0× speedup

Math

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

FPCore

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

C

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

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
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_] := x

Derivation

1. Initial program 98.8%

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

   1. +-commutative 98.8%

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

   2. associate-*r/ 86.4%

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

   3. associate-*l/ 97.8%

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

   4. *-commutative 97.8%

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

   5. fma-def 97.8%

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

3. Simplified 97.8%

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

4. Taylor expanded in y around 0 50.6%

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

5. Final simplification 50.6%

   \[\leadsto x \]

Developer target: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t, double a) {
	return x + (y / ((z - a) / (z - 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 + (y / ((z - a) / (z - t)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023185 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisLine from plot-0.2.3.4, A"
  :precision binary64

  :herbie-target
  (+ x (/ y (/ (- z a) (- z t))))

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