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

Percentage Accurate: 98.1% → 98.1%

Time: 5.8s

Alternatives: 8

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: 98.1% 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: 98.1% 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]

Derivation

1. Initial program 98.8%

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

2. Final simplification 98.8%

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

Alternative 2: 78.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.45 \cdot 10^{+25} \lor \neg \left(a \leq 3.8 \cdot 10^{-119}\right):\\ \;\;\;\;x + y \cdot \frac{t}{a}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z - t}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -3.45e+25) (not (<= a 3.8e-119)))
   (+ x (* y (/ t a)))
   (+ x (* y (/ (- z t) z)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -3.45e+25) || !(a <= 3.8e-119)) {
		tmp = x + (y * (t / a));
	} else {
		tmp = x + (y * ((z - t) / 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 ((a <= (-3.45d+25)) .or. (.not. (a <= 3.8d-119))) then
        tmp = x + (y * (t / a))
    else
        tmp = x + (y * ((z - t) / 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 ((a <= -3.45e+25) || !(a <= 3.8e-119)) {
		tmp = x + (y * (t / a));
	} else {
		tmp = x + (y * ((z - t) / z));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (a <= -3.45e+25) or not (a <= 3.8e-119):
		tmp = x + (y * (t / a))
	else:
		tmp = x + (y * ((z - t) / z))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -3.45e+25) || !(a <= 3.8e-119))
		tmp = Float64(x + Float64(y * Float64(t / a)));
	else
		tmp = Float64(x + Float64(y * Float64(Float64(z - t) / z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((a <= -3.45e+25) || ~((a <= 3.8e-119)))
		tmp = x + (y * (t / a));
	else
		tmp = x + (y * ((z - t) / z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -3.45e+25], N[Not[LessEqual[a, 3.8e-119]], $MachinePrecision]], N[(x + N[(y * N[(t / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(y * N[(N[(z - t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -3.4499999999999999e25 or 3.79999999999999975e-119 < a

   1. Initial program 99.9%

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

   2. Taylor expanded in z around 0 86.9%

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

   if -3.4499999999999999e25 < a < 3.79999999999999975e-119

   1. Initial program 97.4%

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

   2. Taylor expanded in a around 0 83.0%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.45 \cdot 10^{+25} \lor \neg \left(a \leq 3.8 \cdot 10^{-119}\right):\\ \;\;\;\;x + y \cdot \frac{t}{a}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z - t}{z}\\ \end{array} \]

Alternative 3: 78.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.8 \cdot 10^{+25} \lor \neg \left(a \leq 3.8 \cdot 10^{-119}\right):\\ \;\;\;\;x + y \cdot \frac{t}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{z - t}{\frac{z}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -3.8e+25) (not (<= a 3.8e-119)))
   (+ x (* y (/ t a)))
   (+ x (/ (- z t) (/ z y)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -3.8e+25) || !(a <= 3.8e-119)) {
		tmp = x + (y * (t / a));
	} else {
		tmp = x + ((z - t) / (z / 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 ((a <= (-3.8d+25)) .or. (.not. (a <= 3.8d-119))) then
        tmp = x + (y * (t / a))
    else
        tmp = x + ((z - t) / (z / 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 ((a <= -3.8e+25) || !(a <= 3.8e-119)) {
		tmp = x + (y * (t / a));
	} else {
		tmp = x + ((z - t) / (z / y));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (a <= -3.8e+25) or not (a <= 3.8e-119):
		tmp = x + (y * (t / a))
	else:
		tmp = x + ((z - t) / (z / y))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -3.8e+25) || !(a <= 3.8e-119))
		tmp = Float64(x + Float64(y * Float64(t / a)));
	else
		tmp = Float64(x + Float64(Float64(z - t) / Float64(z / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((a <= -3.8e+25) || ~((a <= 3.8e-119)))
		tmp = x + (y * (t / a));
	else
		tmp = x + ((z - t) / (z / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -3.8e+25], N[Not[LessEqual[a, 3.8e-119]], $MachinePrecision]], N[(x + N[(y * N[(t / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(z - t), $MachinePrecision] / N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -3.8e25 or 3.79999999999999975e-119 < a

   1. Initial program 99.9%

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

   2. Taylor expanded in z around 0 86.9%

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

   if -3.8e25 < a < 3.79999999999999975e-119

   1. Initial program 97.4%

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

      1. +-commutative 97.4%

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

      2. associate-*r/ 87.5%

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

      3. associate-*l/ 96.5%

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

      4. *-commutative 96.5%

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

      5. fma-def 96.5%

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

   3. Simplified 96.5%

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

   4. Taylor expanded in a around 0 73.1%

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

      1. *-commutative 73.1%

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

      2. associate-/l* 84.5%

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

   6. Simplified 84.5%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.8 \cdot 10^{+25} \lor \neg \left(a \leq 3.8 \cdot 10^{-119}\right):\\ \;\;\;\;x + y \cdot \frac{t}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{z - t}{\frac{z}{y}}\\ \end{array} \]

Alternative 4: 86.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.65 \cdot 10^{-19} \lor \neg \left(z \leq 7 \cdot 10^{+67}\right):\\ \;\;\;\;x + y \cdot \frac{z - t}{z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y \cdot t}{z - a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -1.65e-19) (not (<= z 7e+67)))
   (+ x (* y (/ (- z t) z)))
   (- x (/ (* y t) (- z a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -1.65e-19) || !(z <= 7e+67)) {
		tmp = x + (y * ((z - t) / z));
	} else {
		tmp = x - ((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 <= (-1.65d-19)) .or. (.not. (z <= 7d+67))) then
        tmp = x + (y * ((z - t) / z))
    else
        tmp = x - ((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 <= -1.65e-19) || !(z <= 7e+67)) {
		tmp = x + (y * ((z - t) / z));
	} else {
		tmp = x - ((y * t) / (z - a));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (z <= -1.65e-19) or not (z <= 7e+67):
		tmp = x + (y * ((z - t) / z))
	else:
		tmp = x - ((y * t) / (z - a))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -1.65e-19) || !(z <= 7e+67))
		tmp = Float64(x + Float64(y * Float64(Float64(z - t) / z)));
	else
		tmp = Float64(x - Float64(Float64(y * t) / Float64(z - a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((z <= -1.65e-19) || ~((z <= 7e+67)))
		tmp = x + (y * ((z - t) / z));
	else
		tmp = x - ((y * t) / (z - a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -1.65e-19], N[Not[LessEqual[z, 7e+67]], $MachinePrecision]], N[(x + N[(y * N[(N[(z - t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[(y * t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.6499999999999999e-19 or 7e67 < z

   1. Initial program 99.9%

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

   2. Taylor expanded in a around 0 88.0%

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

   if -1.6499999999999999e-19 < z < 7e67

   1. Initial program 98.0%

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

   2. Taylor expanded in t around inf 90.8%

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

      1. associate-*r/ 90.8%

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

      2. mul-1-neg 90.8%

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

      3. distribute-rgt-neg-out 90.8%

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

   4. Simplified 90.8%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.65 \cdot 10^{-19} \lor \neg \left(z \leq 7 \cdot 10^{+67}\right):\\ \;\;\;\;x + y \cdot \frac{z - t}{z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y \cdot t}{z - a}\\ \end{array} \]

Alternative 5: 79.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.4 \cdot 10^{+35}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \mathbf{elif}\;a \leq 6 \cdot 10^{-119}:\\ \;\;\;\;x + \frac{z - t}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{t}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -1.4e+35)
   (- x (/ y (/ a (- z t))))
   (if (<= a 6e-119) (+ x (/ (- z t) (/ z y))) (+ x (* y (/ t a))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -1.4e+35) {
		tmp = x - (y / (a / (z - t)));
	} else if (a <= 6e-119) {
		tmp = x + ((z - t) / (z / y));
	} else {
		tmp = x + (y * (t / 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 (a <= (-1.4d+35)) then
        tmp = x - (y / (a / (z - t)))
    else if (a <= 6d-119) then
        tmp = x + ((z - t) / (z / y))
    else
        tmp = x + (y * (t / 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 (a <= -1.4e+35) {
		tmp = x - (y / (a / (z - t)));
	} else if (a <= 6e-119) {
		tmp = x + ((z - t) / (z / y));
	} else {
		tmp = x + (y * (t / a));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if a <= -1.4e+35:
		tmp = x - (y / (a / (z - t)))
	elif a <= 6e-119:
		tmp = x + ((z - t) / (z / y))
	else:
		tmp = x + (y * (t / a))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -1.4e+35)
		tmp = Float64(x - Float64(y / Float64(a / Float64(z - t))));
	elseif (a <= 6e-119)
		tmp = Float64(x + Float64(Float64(z - t) / Float64(z / y)));
	else
		tmp = Float64(x + Float64(y * Float64(t / a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (a <= -1.4e+35)
		tmp = x - (y / (a / (z - t)));
	elseif (a <= 6e-119)
		tmp = x + ((z - t) / (z / y));
	else
		tmp = x + (y * (t / a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -1.4e+35], N[(x - N[(y / N[(a / N[(z - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 6e-119], N[(x + N[(N[(z - t), $MachinePrecision] / N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(y * N[(t / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.39999999999999999e35

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. associate-*r/ 84.2%

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

      3. associate-*l/ 97.5%

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

      4. *-commutative 97.5%

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

      5. fma-def 97.5%

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

   3. Simplified 97.5%

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

   4. Taylor expanded in a around inf 78.6%

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

      1. +-commutative 78.6%

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

      2. mul-1-neg 78.6%

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

      3. unsub-neg 78.6%

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

      4. associate-/l* 88.1%

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

   6. Simplified 88.1%

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

   if -1.39999999999999999e35 < a < 6.0000000000000004e-119

   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/ 87.8%

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

      3. associate-*l/ 96.6%

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

      4. *-commutative 96.6%

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

      5. fma-def 96.6%

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

   3. Simplified 96.6%

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

   4. Taylor expanded in a around 0 73.0%

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

      1. *-commutative 73.0%

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

      2. associate-/l* 84.1%

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

   6. Simplified 84.1%

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

   if 6.0000000000000004e-119 < a

   1. Initial program 100.0%

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

   2. Taylor expanded in z around 0 88.6%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.4 \cdot 10^{+35}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \mathbf{elif}\;a \leq 6 \cdot 10^{-119}:\\ \;\;\;\;x + \frac{z - t}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{t}{a}\\ \end{array} \]

Alternative 6: 76.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.2 \cdot 10^{+87}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{+67}:\\ \;\;\;\;x + y \cdot \frac{t}{a}\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.2e+87) (+ x y) (if (<= z 9.5e+67) (+ x (* y (/ t a))) (+ x y))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.2e+87) {
		tmp = x + y;
	} else if (z <= 9.5e+67) {
		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 <= (-1.2d+87)) then
        tmp = x + y
    else if (z <= 9.5d+67) 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 <= -1.2e+87) {
		tmp = x + y;
	} else if (z <= 9.5e+67) {
		tmp = x + (y * (t / a));
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.2e+87:
		tmp = x + y
	elif z <= 9.5e+67:
		tmp = x + (y * (t / a))
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.2e+87)
		tmp = Float64(x + y);
	elseif (z <= 9.5e+67)
		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 <= -1.2e+87)
		tmp = x + y;
	elseif (z <= 9.5e+67)
		tmp = x + (y * (t / a));
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.2e+87], N[(x + y), $MachinePrecision], If[LessEqual[z, 9.5e+67], N[(x + N[(y * N[(t / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.19999999999999991e87 or 9.5000000000000002e67 < z

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. associate-*r/ 73.4%

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

      3. associate-*l/ 96.6%

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

      4. *-commutative 96.6%

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

      5. fma-def 96.6%

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

   3. Simplified 96.6%

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

   4. Taylor expanded in z around inf 83.5%

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

   if -1.19999999999999991e87 < z < 9.5000000000000002e67

   1. Initial program 98.2%

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

   2. Taylor expanded in z around 0 79.0%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.2 \cdot 10^{+87}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{+67}:\\ \;\;\;\;x + y \cdot \frac{t}{a}\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 7: 62.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.4 \cdot 10^{+100}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 8 \cdot 10^{+130}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -2.4e+100) x (if (<= a 8e+130) (+ x y) x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -2.4e+100) {
		tmp = x;
	} else if (a <= 8e+130) {
		tmp = x + y;
	} else {
		tmp = x;
	}
	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 (a <= (-2.4d+100)) then
        tmp = x
    else if (a <= 8d+130) then
        tmp = x + y
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -2.4e+100) {
		tmp = x;
	} else if (a <= 8e+130) {
		tmp = x + y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if a <= -2.4e+100:
		tmp = x
	elif a <= 8e+130:
		tmp = x + y
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -2.4e+100)
		tmp = x;
	elseif (a <= 8e+130)
		tmp = Float64(x + y);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (a <= -2.4e+100)
		tmp = x;
	elseif (a <= 8e+130)
		tmp = x + y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -2.4e+100], x, If[LessEqual[a, 8e+130], N[(x + y), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if a < -2.40000000000000012e100 or 8.0000000000000005e130 < a

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. associate-*r/ 87.0%

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

      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 y around 0 74.5%

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

   if -2.40000000000000012e100 < a < 8.0000000000000005e130

   1. Initial program 98.2%

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

      1. +-commutative 98.2%

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

      2. associate-*r/ 87.1%

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

      3. associate-*l/ 96.5%

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

      4. *-commutative 96.5%

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

      5. fma-def 96.5%

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

   3. Simplified 96.5%

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

   4. Taylor expanded in z around inf 67.0%

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

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.4 \cdot 10^{+100}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 8 \cdot 10^{+130}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 8: 50.1% 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/ 87.0%

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

   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 y around 0 58.7%

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

5. Final simplification 58.7%

   \[\leadsto x \]

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 2023199 
(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)))))