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

Percentage Accurate: 98.2% → 98.2%

Time: 8.8s

Alternatives: 9

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 98.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.3%

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

2. Final simplification 97.3%

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

Alternative 2: 82.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{a - t}\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{+44}:\\ \;\;\;\;x - z \cdot \frac{y}{t}\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{-24}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z}}\\ \mathbf{elif}\;t_1 \leq 4 \cdot 10^{+81}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot z}{a - t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- a t))))
   (if (<= t_1 -1e+44)
     (- x (* z (/ y t)))
     (if (<= t_1 5e-24)
       (+ x (/ y (/ a z)))
       (if (<= t_1 4e+81) (+ x y) (/ (* y z) (- a t)))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (a - t);
	double tmp;
	if (t_1 <= -1e+44) {
		tmp = x - (z * (y / t));
	} else if (t_1 <= 5e-24) {
		tmp = x + (y / (a / z));
	} else if (t_1 <= 4e+81) {
		tmp = x + y;
	} else {
		tmp = (y * z) / (a - t);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (z - t) / (a - t)
    if (t_1 <= (-1d+44)) then
        tmp = x - (z * (y / t))
    else if (t_1 <= 5d-24) then
        tmp = x + (y / (a / z))
    else if (t_1 <= 4d+81) then
        tmp = x + y
    else
        tmp = (y * z) / (a - t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (a - t);
	double tmp;
	if (t_1 <= -1e+44) {
		tmp = x - (z * (y / t));
	} else if (t_1 <= 5e-24) {
		tmp = x + (y / (a / z));
	} else if (t_1 <= 4e+81) {
		tmp = x + y;
	} else {
		tmp = (y * z) / (a - t);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (z - t) / (a - t)
	tmp = 0
	if t_1 <= -1e+44:
		tmp = x - (z * (y / t))
	elif t_1 <= 5e-24:
		tmp = x + (y / (a / z))
	elif t_1 <= 4e+81:
		tmp = x + y
	else:
		tmp = (y * z) / (a - t)
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(a - t))
	tmp = 0.0
	if (t_1 <= -1e+44)
		tmp = Float64(x - Float64(z * Float64(y / t)));
	elseif (t_1 <= 5e-24)
		tmp = Float64(x + Float64(y / Float64(a / z)));
	elseif (t_1 <= 4e+81)
		tmp = Float64(x + y);
	else
		tmp = Float64(Float64(y * z) / Float64(a - t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (z - t) / (a - t);
	tmp = 0.0;
	if (t_1 <= -1e+44)
		tmp = x - (z * (y / t));
	elseif (t_1 <= 5e-24)
		tmp = x + (y / (a / z));
	elseif (t_1 <= 4e+81)
		tmp = x + y;
	else
		tmp = (y * z) / (a - t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+44], N[(x - N[(z * N[(y / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e-24], N[(x + N[(y / N[(a / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 4e+81], N[(x + y), $MachinePrecision], N[(N[(y * z), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 z t) (-.f64 a t)) < -1.0000000000000001e44

   1. Initial program 91.1%

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

   2. Taylor expanded in z around inf 91.1%

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

   3. Taylor expanded in a around 0 69.3%

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

      1. +-commutative 69.3%

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

      2. mul-1-neg 69.3%

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

      3. unsub-neg 69.3%

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

      4. associate-*l/ 75.3%

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

      5. *-commutative 75.3%

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

   5. Simplified 75.3%

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

   if -1.0000000000000001e44 < (/.f64 (-.f64 z t) (-.f64 a t)) < 4.9999999999999998e-24

   1. Initial program 99.0%

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

   2. Taylor expanded in t around 0 85.0%

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

      1. associate-/l* 86.9%

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

   4. Simplified 86.9%

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

   if 4.9999999999999998e-24 < (/.f64 (-.f64 z t) (-.f64 a t)) < 3.99999999999999969e81

   1. Initial program 100.0%

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

   2. Taylor expanded in t around inf 91.0%

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

   if 3.99999999999999969e81 < (/.f64 (-.f64 z t) (-.f64 a t))

   1. Initial program 84.7%

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

   2. Taylor expanded in x around 0 80.5%

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

   3. Taylor expanded in z around inf 80.5%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{a - t} \leq -1 \cdot 10^{+44}:\\ \;\;\;\;x - z \cdot \frac{y}{t}\\ \mathbf{elif}\;\frac{z - t}{a - t} \leq 5 \cdot 10^{-24}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z}}\\ \mathbf{elif}\;\frac{z - t}{a - t} \leq 4 \cdot 10^{+81}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot z}{a - t}\\ \end{array} \]

Alternative 3: 91.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{a - t}\\ \mathbf{if}\;t_1 \leq 5 \cdot 10^{-24} \lor \neg \left(t_1 \leq 1\right):\\ \;\;\;\;x + y \cdot \frac{z}{a - t}\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- a t))))
   (if (or (<= t_1 5e-24) (not (<= t_1 1.0)))
     (+ x (* y (/ z (- a t))))
     (+ x y))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (a - t);
	double tmp;
	if ((t_1 <= 5e-24) || !(t_1 <= 1.0)) {
		tmp = x + (y * (z / (a - t)));
	} 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) :: t_1
    real(8) :: tmp
    t_1 = (z - t) / (a - t)
    if ((t_1 <= 5d-24) .or. (.not. (t_1 <= 1.0d0))) then
        tmp = x + (y * (z / (a - t)))
    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 t_1 = (z - t) / (a - t);
	double tmp;
	if ((t_1 <= 5e-24) || !(t_1 <= 1.0)) {
		tmp = x + (y * (z / (a - t)));
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (z - t) / (a - t)
	tmp = 0
	if (t_1 <= 5e-24) or not (t_1 <= 1.0):
		tmp = x + (y * (z / (a - t)))
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(a - t))
	tmp = 0.0
	if ((t_1 <= 5e-24) || !(t_1 <= 1.0))
		tmp = Float64(x + Float64(y * Float64(z / Float64(a - t))));
	else
		tmp = Float64(x + y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (z - t) / (a - t);
	tmp = 0.0;
	if ((t_1 <= 5e-24) || ~((t_1 <= 1.0)))
		tmp = x + (y * (z / (a - t)));
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, 5e-24], N[Not[LessEqual[t$95$1, 1.0]], $MachinePrecision]], N[(x + N[(y * N[(z / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 z t) (-.f64 a t)) < 4.9999999999999998e-24 or 1 < (/.f64 (-.f64 z t) (-.f64 a t))

   1. Initial program 95.9%

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

   2. Taylor expanded in z around inf 88.3%

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

   if 4.9999999999999998e-24 < (/.f64 (-.f64 z t) (-.f64 a t)) < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in t around inf 95.6%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{a - t} \leq 5 \cdot 10^{-24} \lor \neg \left(\frac{z - t}{a - t} \leq 1\right):\\ \;\;\;\;x + y \cdot \frac{z}{a - t}\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 4: 92.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{a - t}\\ \mathbf{if}\;t_1 \leq 10^{-153} \lor \neg \left(t_1 \leq 1\right):\\ \;\;\;\;x + y \cdot \frac{z}{a - t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{1 - \frac{a}{t}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- a t))))
   (if (or (<= t_1 1e-153) (not (<= t_1 1.0)))
     (+ x (* y (/ z (- a t))))
     (+ x (/ y (- 1.0 (/ a t)))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (a - t);
	double tmp;
	if ((t_1 <= 1e-153) || !(t_1 <= 1.0)) {
		tmp = x + (y * (z / (a - t)));
	} else {
		tmp = x + (y / (1.0 - (a / t)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (z - t) / (a - t)
    if ((t_1 <= 1d-153) .or. (.not. (t_1 <= 1.0d0))) then
        tmp = x + (y * (z / (a - t)))
    else
        tmp = x + (y / (1.0d0 - (a / t)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (a - t);
	double tmp;
	if ((t_1 <= 1e-153) || !(t_1 <= 1.0)) {
		tmp = x + (y * (z / (a - t)));
	} else {
		tmp = x + (y / (1.0 - (a / t)));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (z - t) / (a - t)
	tmp = 0
	if (t_1 <= 1e-153) or not (t_1 <= 1.0):
		tmp = x + (y * (z / (a - t)))
	else:
		tmp = x + (y / (1.0 - (a / t)))
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(a - t))
	tmp = 0.0
	if ((t_1 <= 1e-153) || !(t_1 <= 1.0))
		tmp = Float64(x + Float64(y * Float64(z / Float64(a - t))));
	else
		tmp = Float64(x + Float64(y / Float64(1.0 - Float64(a / t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (z - t) / (a - t);
	tmp = 0.0;
	if ((t_1 <= 1e-153) || ~((t_1 <= 1.0)))
		tmp = x + (y * (z / (a - t)));
	else
		tmp = x + (y / (1.0 - (a / t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, 1e-153], N[Not[LessEqual[t$95$1, 1.0]], $MachinePrecision]], N[(x + N[(y * N[(z / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(y / N[(1.0 - N[(a / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 z t) (-.f64 a t)) < 1.00000000000000004e-153 or 1 < (/.f64 (-.f64 z t) (-.f64 a t))

   1. Initial program 95.1%

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

   2. Taylor expanded in z around inf 89.7%

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

   if 1.00000000000000004e-153 < (/.f64 (-.f64 z t) (-.f64 a t)) < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in z around 0 75.1%

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

      1. mul-1-neg 75.1%

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

      2. associate-/l* 96.6%

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

      3. distribute-neg-frac 96.6%

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

      4. div-sub 96.6%

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

      5. *-inverses 96.6%

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

   4. Simplified 96.6%

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

      1. frac-2neg 96.6%

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

      2. div-inv 96.6%

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

      3. remove-double-neg 96.6%

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

      4. sub-neg 96.6%

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

      5. distribute-neg-in 96.6%

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

      6. metadata-eval 96.6%

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

      7. metadata-eval 96.6%

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

   6. Applied egg-rr 96.6%

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

      1. associate-*r/ 96.6%

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

      2. *-rgt-identity 96.6%

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

      3. +-commutative 96.6%

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

      4. unsub-neg 96.6%

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

   8. Simplified 96.6%

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

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{a - t} \leq 10^{-153} \lor \neg \left(\frac{z - t}{a - t} \leq 1\right):\\ \;\;\;\;x + y \cdot \frac{z}{a - t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{1 - \frac{a}{t}}\\ \end{array} \]

Alternative 5: 77.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3.1 \cdot 10^{+25}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;t \leq 10^{+29}:\\ \;\;\;\;x + y \cdot \frac{z}{a}\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -3.1e+25) (+ x y) (if (<= t 1e+29) (+ x (* y (/ z a))) (+ x y))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -3.1e+25) {
		tmp = x + y;
	} else if (t <= 1e+29) {
		tmp = x + (y * (z / 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 (t <= (-3.1d+25)) then
        tmp = x + y
    else if (t <= 1d+29) then
        tmp = x + (y * (z / 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 (t <= -3.1e+25) {
		tmp = x + y;
	} else if (t <= 1e+29) {
		tmp = x + (y * (z / a));
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= -3.1e+25:
		tmp = x + y
	elif t <= 1e+29:
		tmp = x + (y * (z / a))
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -3.1e+25)
		tmp = Float64(x + y);
	elseif (t <= 1e+29)
		tmp = Float64(x + Float64(y * Float64(z / a)));
	else
		tmp = Float64(x + y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= -3.1e+25)
		tmp = x + y;
	elseif (t <= 1e+29)
		tmp = x + (y * (z / a));
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -3.1e+25], N[(x + y), $MachinePrecision], If[LessEqual[t, 1e+29], N[(x + N[(y * N[(z / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < -3.0999999999999998e25 or 9.99999999999999914e28 < t

   1. Initial program 99.9%

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

   2. Taylor expanded in t around inf 79.5%

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

   if -3.0999999999999998e25 < t < 9.99999999999999914e28

   1. Initial program 95.2%

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

   2. Taylor expanded in t around 0 81.4%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.1 \cdot 10^{+25}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;t \leq 10^{+29}:\\ \;\;\;\;x + y \cdot \frac{z}{a}\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 6: 77.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.65 \cdot 10^{+26}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{+29}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z}}\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -1.65e+26)
   (+ x y)
   (if (<= t 1.05e+29) (+ x (/ y (/ a z))) (+ x y))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -1.65e+26) {
		tmp = x + y;
	} else if (t <= 1.05e+29) {
		tmp = x + (y / (a / z));
	} 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 (t <= (-1.65d+26)) then
        tmp = x + y
    else if (t <= 1.05d+29) then
        tmp = x + (y / (a / z))
    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 (t <= -1.65e+26) {
		tmp = x + y;
	} else if (t <= 1.05e+29) {
		tmp = x + (y / (a / z));
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= -1.65e+26:
		tmp = x + y
	elif t <= 1.05e+29:
		tmp = x + (y / (a / z))
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -1.65e+26)
		tmp = Float64(x + y);
	elseif (t <= 1.05e+29)
		tmp = Float64(x + Float64(y / Float64(a / z)));
	else
		tmp = Float64(x + y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= -1.65e+26)
		tmp = x + y;
	elseif (t <= 1.05e+29)
		tmp = x + (y / (a / z));
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -1.65e+26], N[(x + y), $MachinePrecision], If[LessEqual[t, 1.05e+29], N[(x + N[(y / N[(a / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.64999999999999997e26 or 1.0500000000000001e29 < t

   1. Initial program 99.9%

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

   2. Taylor expanded in t around inf 79.5%

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

   if -1.64999999999999997e26 < t < 1.0500000000000001e29

   1. Initial program 95.2%

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

   2. Taylor expanded in t around 0 80.3%

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

      1. associate-/l* 81.9%

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

   4. Simplified 81.9%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.65 \cdot 10^{+26}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{+29}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z}}\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 7: 62.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -9.8 \cdot 10^{-18}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 10^{+141}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -9.8e-18) x (if (<= a 1e+141) (+ x y) x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -9.8e-18) {
		tmp = x;
	} else if (a <= 1e+141) {
		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 <= (-9.8d-18)) then
        tmp = x
    else if (a <= 1d+141) 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 <= -9.8e-18) {
		tmp = x;
	} else if (a <= 1e+141) {
		tmp = x + y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if a <= -9.8e-18:
		tmp = x
	elif a <= 1e+141:
		tmp = x + y
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -9.8e-18)
		tmp = x;
	elseif (a <= 1e+141)
		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 <= -9.8e-18)
		tmp = x;
	elseif (a <= 1e+141)
		tmp = x + y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -9.8e-18], x, If[LessEqual[a, 1e+141], N[(x + y), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if a < -9.8000000000000002e-18 or 1.00000000000000002e141 < a

   1. Initial program 99.0%

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

   2. Taylor expanded in x around inf 69.6%

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

   if -9.8000000000000002e-18 < a < 1.00000000000000002e141

   1. Initial program 96.1%

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

   2. Taylor expanded in t around inf 66.6%

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

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -9.8 \cdot 10^{-18}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 10^{+141}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 8: 52.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{-252}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{-287}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= x -9.5e-252) x (if (<= x 1.2e-287) y x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (x <= -9.5e-252) {
		tmp = x;
	} else if (x <= 1.2e-287) {
		tmp = 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 (x <= (-9.5d-252)) then
        tmp = x
    else if (x <= 1.2d-287) then
        tmp = 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 (x <= -9.5e-252) {
		tmp = x;
	} else if (x <= 1.2e-287) {
		tmp = y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if x <= -9.5e-252:
		tmp = x
	elif x <= 1.2e-287:
		tmp = y
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (x <= -9.5e-252)
		tmp = x;
	elseif (x <= 1.2e-287)
		tmp = y;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (x <= -9.5e-252)
		tmp = x;
	elseif (x <= 1.2e-287)
		tmp = y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[x, -9.5e-252], x, If[LessEqual[x, 1.2e-287], y, x]]

Derivation

1. Split input into 2 regimes

2. if x < -9.4999999999999993e-252 or 1.2e-287 < x

   1. Initial program 97.0%

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

   2. Taylor expanded in x around inf 62.8%

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

   if -9.4999999999999993e-252 < x < 1.2e-287

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 62.8%

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

   3. Taylor expanded in t around inf 54.1%

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

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{-252}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{-287}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 9: 50.9% 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 97.3%

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

2. Taylor expanded in x around inf 57.8%

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

3. Final simplification 57.8%

   \[\leadsto x \]

Developer target: 99.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + y \cdot \frac{z - t}{a - t}\\ \mathbf{if}\;y < -8.508084860551241 \cdot 10^{-17}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y < 2.894426862792089 \cdot 10^{-49}:\\ \;\;\;\;x + \left(y \cdot \left(z - t\right)\right) \cdot \frac{1}{a - t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ x (* y (/ (- z t) (- a t))))))
   (if (< y -8.508084860551241e-17)
     t_1
     (if (< y 2.894426862792089e-49)
       (+ x (* (* y (- z t)) (/ 1.0 (- a t))))
       t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x + (y * ((z - t) / (a - t)));
	double tmp;
	if (y < -8.508084860551241e-17) {
		tmp = t_1;
	} else if (y < 2.894426862792089e-49) {
		tmp = x + ((y * (z - t)) * (1.0 / (a - t)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x + (y * ((z - t) / (a - t)))
    if (y < (-8.508084860551241d-17)) then
        tmp = t_1
    else if (y < 2.894426862792089d-49) then
        tmp = x + ((y * (z - t)) * (1.0d0 / (a - t)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = x + (y * ((z - t) / (a - t)));
	double tmp;
	if (y < -8.508084860551241e-17) {
		tmp = t_1;
	} else if (y < 2.894426862792089e-49) {
		tmp = x + ((y * (z - t)) * (1.0 / (a - t)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = x + (y * ((z - t) / (a - t)))
	tmp = 0
	if y < -8.508084860551241e-17:
		tmp = t_1
	elif y < 2.894426862792089e-49:
		tmp = x + ((y * (z - t)) * (1.0 / (a - t)))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x + Float64(y * Float64(Float64(z - t) / Float64(a - t))))
	tmp = 0.0
	if (y < -8.508084860551241e-17)
		tmp = t_1;
	elseif (y < 2.894426862792089e-49)
		tmp = Float64(x + Float64(Float64(y * Float64(z - t)) * Float64(1.0 / Float64(a - t))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = x + (y * ((z - t) / (a - t)));
	tmp = 0.0;
	if (y < -8.508084860551241e-17)
		tmp = t_1;
	elseif (y < 2.894426862792089e-49)
		tmp = x + ((y * (z - t)) * (1.0 / (a - t)));
	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[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Less[y, -8.508084860551241e-17], t$95$1, If[Less[y, 2.894426862792089e-49], N[(x + N[(N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Reproduce

herbie shell --seed 2023279 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisLine from plot-0.2.3.4, B"
  :precision binary64

  :herbie-target
  (if (< y -8.508084860551241e-17) (+ x (* y (/ (- z t) (- a t)))) (if (< y 2.894426862792089e-49) (+ x (* (* y (- z t)) (/ 1.0 (- a t)))) (+ x (* y (/ (- z t) (- a t))))))

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