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

Percentage Accurate: 97.9% → 97.9%

Time: 6.3s

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: 97.9% 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: 97.9% 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 98.4%

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

2. Final simplification 98.4%

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

Alternative 2: 81.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.5 \cdot 10^{+37} \lor \neg \left(z \leq 2.4 \cdot 10^{-30}\right):\\ \;\;\;\;x + y \cdot \left(1 - \frac{t}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;x + t \cdot \frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -2.5e+37) (not (<= z 2.4e-30)))
   (+ x (* y (- 1.0 (/ t z))))
   (+ x (* t (/ y a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -2.5e+37) || !(z <= 2.4e-30)) {
		tmp = x + (y * (1.0 - (t / z)));
	} else {
		tmp = x + (t * (y / a));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((z <= (-2.5d+37)) .or. (.not. (z <= 2.4d-30))) then
        tmp = x + (y * (1.0d0 - (t / z)))
    else
        tmp = x + (t * (y / a))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -2.5e+37) || !(z <= 2.4e-30)) {
		tmp = x + (y * (1.0 - (t / z)));
	} else {
		tmp = x + (t * (y / a));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (z <= -2.5e+37) or not (z <= 2.4e-30):
		tmp = x + (y * (1.0 - (t / z)))
	else:
		tmp = x + (t * (y / a))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -2.5e+37) || !(z <= 2.4e-30))
		tmp = Float64(x + Float64(y * Float64(1.0 - Float64(t / z))));
	else
		tmp = Float64(x + Float64(t * Float64(y / a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((z <= -2.5e+37) || ~((z <= 2.4e-30)))
		tmp = x + (y * (1.0 - (t / z)));
	else
		tmp = x + (t * (y / a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -2.5e+37], N[Not[LessEqual[z, 2.4e-30]], $MachinePrecision]], N[(x + N[(y * N[(1.0 - N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(t * N[(y / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.49999999999999994e37 or 2.39999999999999985e-30 < z

   1. Initial program 99.9%

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

   2. Taylor expanded in a around 0 92.6%

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

      1. div-sub 92.6%

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

      2. *-inverses 92.6%

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

   4. Simplified 92.6%

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

   if -2.49999999999999994e37 < z < 2.39999999999999985e-30

   1. Initial program 97.0%

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

      1. +-commutative 97.0%

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

      2. associate-*r/ 98.2%

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

      3. associate-*l/ 96.3%

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

      4. *-commutative 96.3%

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

      5. fma-def 96.3%

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

   3. Simplified 96.3%

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

   4. Taylor expanded in z around 0 83.2%

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

      1. associate-/l* 83.4%

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

   6. Simplified 83.4%

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

      1. associate-/r/ 83.5%

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

   8. Applied egg-rr 83.5%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.5 \cdot 10^{+37} \lor \neg \left(z \leq 2.4 \cdot 10^{-30}\right):\\ \;\;\;\;x + y \cdot \left(1 - \frac{t}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;x + t \cdot \frac{y}{a}\\ \end{array} \]

Alternative 3: 81.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.5 \cdot 10^{+37}:\\ \;\;\;\;x + y \cdot \left(1 - \frac{t}{z}\right)\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{-30}:\\ \;\;\;\;x + t \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{z}{z - t}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -2.5e+37)
   (+ x (* y (- 1.0 (/ t z))))
   (if (<= z 1.9e-30) (+ x (* t (/ y a))) (+ x (/ y (/ z (- z t)))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.5e+37) {
		tmp = x + (y * (1.0 - (t / z)));
	} else if (z <= 1.9e-30) {
		tmp = x + (t * (y / a));
	} else {
		tmp = x + (y / (z / (z - t)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-2.5d+37)) then
        tmp = x + (y * (1.0d0 - (t / z)))
    else if (z <= 1.9d-30) then
        tmp = x + (t * (y / a))
    else
        tmp = x + (y / (z / (z - t)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.5e+37) {
		tmp = x + (y * (1.0 - (t / z)));
	} else if (z <= 1.9e-30) {
		tmp = x + (t * (y / a));
	} else {
		tmp = x + (y / (z / (z - t)));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -2.5e+37:
		tmp = x + (y * (1.0 - (t / z)))
	elif z <= 1.9e-30:
		tmp = x + (t * (y / a))
	else:
		tmp = x + (y / (z / (z - t)))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -2.5e+37)
		tmp = Float64(x + Float64(y * Float64(1.0 - Float64(t / z))));
	elseif (z <= 1.9e-30)
		tmp = Float64(x + Float64(t * Float64(y / a)));
	else
		tmp = Float64(x + Float64(y / Float64(z / Float64(z - t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -2.5e+37)
		tmp = x + (y * (1.0 - (t / z)));
	elseif (z <= 1.9e-30)
		tmp = x + (t * (y / a));
	else
		tmp = x + (y / (z / (z - t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -2.5e+37], N[(x + N[(y * N[(1.0 - N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.9e-30], N[(x + N[(t * N[(y / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(y / N[(z / N[(z - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.49999999999999994e37

   1. Initial program 99.9%

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

   2. Taylor expanded in a around 0 94.6%

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

      1. div-sub 94.6%

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

      2. *-inverses 94.6%

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

   4. Simplified 94.6%

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

   if -2.49999999999999994e37 < z < 1.9000000000000002e-30

   1. Initial program 97.0%

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

      1. +-commutative 97.0%

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

      2. associate-*r/ 98.2%

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

      3. associate-*l/ 96.3%

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

      4. *-commutative 96.3%

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

      5. fma-def 96.3%

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

   3. Simplified 96.3%

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

   4. Taylor expanded in z around 0 83.2%

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

      1. associate-/l* 83.4%

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

   6. Simplified 83.4%

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

      1. associate-/r/ 83.5%

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

   8. Applied egg-rr 83.5%

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

   if 1.9000000000000002e-30 < 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/ 74.1%

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

      3. associate-*l/ 95.9%

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

      4. *-commutative 95.9%

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

      5. fma-def 95.9%

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

   3. Simplified 95.9%

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

   4. Taylor expanded in a around 0 69.3%

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

      1. associate-/l* 91.1%

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

   6. Simplified 91.1%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.5 \cdot 10^{+37}:\\ \;\;\;\;x + y \cdot \left(1 - \frac{t}{z}\right)\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{-30}:\\ \;\;\;\;x + t \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{z}{z - t}}\\ \end{array} \]

Alternative 4: 85.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.5 \cdot 10^{+41}:\\ \;\;\;\;x + y \cdot \left(1 - \frac{t}{z}\right)\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{-30}:\\ \;\;\;\;x - \frac{y \cdot t}{z - a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{z}{z - t}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -5.5e+41)
   (+ x (* y (- 1.0 (/ t z))))
   (if (<= z 2.1e-30) (- x (/ (* y t) (- z a))) (+ x (/ y (/ z (- z t)))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -5.5e+41) {
		tmp = x + (y * (1.0 - (t / z)));
	} else if (z <= 2.1e-30) {
		tmp = x - ((y * t) / (z - a));
	} else {
		tmp = x + (y / (z / (z - t)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-5.5d+41)) then
        tmp = x + (y * (1.0d0 - (t / z)))
    else if (z <= 2.1d-30) then
        tmp = x - ((y * t) / (z - a))
    else
        tmp = x + (y / (z / (z - t)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -5.5e+41) {
		tmp = x + (y * (1.0 - (t / z)));
	} else if (z <= 2.1e-30) {
		tmp = x - ((y * t) / (z - a));
	} else {
		tmp = x + (y / (z / (z - t)));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -5.5e+41:
		tmp = x + (y * (1.0 - (t / z)))
	elif z <= 2.1e-30:
		tmp = x - ((y * t) / (z - a))
	else:
		tmp = x + (y / (z / (z - t)))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -5.5e+41)
		tmp = Float64(x + Float64(y * Float64(1.0 - Float64(t / z))));
	elseif (z <= 2.1e-30)
		tmp = Float64(x - Float64(Float64(y * t) / Float64(z - a)));
	else
		tmp = Float64(x + Float64(y / Float64(z / Float64(z - t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -5.5e+41)
		tmp = x + (y * (1.0 - (t / z)));
	elseif (z <= 2.1e-30)
		tmp = x - ((y * t) / (z - a));
	else
		tmp = x + (y / (z / (z - t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -5.5e+41], N[(x + N[(y * N[(1.0 - N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.1e-30], N[(x - N[(N[(y * t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(y / N[(z / N[(z - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.5000000000000003e41

   1. Initial program 99.9%

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

   2. Taylor expanded in a around 0 94.3%

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

      1. div-sub 94.4%

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

      2. *-inverses 94.4%

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

   4. Simplified 94.4%

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

   if -5.5000000000000003e41 < z < 2.1000000000000002e-30

   1. Initial program 97.0%

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

   2. Taylor expanded in t around inf 91.9%

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

      1. associate-*r/ 91.9%

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

      2. associate-*r* 91.9%

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

      3. neg-mul-1 91.9%

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

   4. Simplified 91.9%

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

   if 2.1000000000000002e-30 < 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/ 74.1%

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

      3. associate-*l/ 95.9%

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

      4. *-commutative 95.9%

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

      5. fma-def 95.9%

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

   3. Simplified 95.9%

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

   4. Taylor expanded in a around 0 69.3%

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

      1. associate-/l* 91.1%

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

   6. Simplified 91.1%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.5 \cdot 10^{+41}:\\ \;\;\;\;x + y \cdot \left(1 - \frac{t}{z}\right)\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{-30}:\\ \;\;\;\;x - \frac{y \cdot t}{z - a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{z}{z - t}}\\ \end{array} \]

Alternative 5: 76.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.9 \cdot 10^{+37}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{-30}:\\ \;\;\;\;x + y \cdot \frac{t}{a}\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -3.9e+37) (+ x y) (if (<= z 2.6e-30) (+ x (* y (/ t a))) (+ x y))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -3.9e+37) {
		tmp = x + y;
	} else if (z <= 2.6e-30) {
		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 <= (-3.9d+37)) then
        tmp = x + y
    else if (z <= 2.6d-30) 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 <= -3.9e+37) {
		tmp = x + y;
	} else if (z <= 2.6e-30) {
		tmp = x + (y * (t / a));
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -3.9e+37:
		tmp = x + y
	elif z <= 2.6e-30:
		tmp = x + (y * (t / a))
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -3.9e+37)
		tmp = Float64(x + y);
	elseif (z <= 2.6e-30)
		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 <= -3.9e+37)
		tmp = x + y;
	elseif (z <= 2.6e-30)
		tmp = x + (y * (t / a));
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -3.9e+37], N[(x + y), $MachinePrecision], If[LessEqual[z, 2.6e-30], N[(x + N[(y * N[(t / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.8999999999999999e37 or 2.59999999999999987e-30 < 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/ 72.9%

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

      3. associate-*l/ 96.9%

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

      4. *-commutative 96.9%

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

      5. fma-def 96.9%

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

   3. Simplified 96.9%

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

   4. Taylor expanded in z around inf 80.6%

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

   if -3.8999999999999999e37 < z < 2.59999999999999987e-30

   1. Initial program 97.0%

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

   2. Taylor expanded in z around 0 82.7%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.9 \cdot 10^{+37}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{-30}:\\ \;\;\;\;x + y \cdot \frac{t}{a}\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 6: 76.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.2 \cdot 10^{+38}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq 7.6 \cdot 10^{-31}:\\ \;\;\;\;x + t \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -2.2e+38) (+ x y) (if (<= z 7.6e-31) (+ x (* t (/ y a))) (+ x y))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.2e+38) {
		tmp = x + y;
	} else if (z <= 7.6e-31) {
		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 <= (-2.2d+38)) then
        tmp = x + y
    else if (z <= 7.6d-31) 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 <= -2.2e+38) {
		tmp = x + y;
	} else if (z <= 7.6e-31) {
		tmp = x + (t * (y / a));
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -2.2e+38:
		tmp = x + y
	elif z <= 7.6e-31:
		tmp = x + (t * (y / a))
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -2.2e+38)
		tmp = Float64(x + y);
	elseif (z <= 7.6e-31)
		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 <= -2.2e+38)
		tmp = x + y;
	elseif (z <= 7.6e-31)
		tmp = x + (t * (y / a));
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -2.2e+38], N[(x + y), $MachinePrecision], If[LessEqual[z, 7.6e-31], N[(x + N[(t * N[(y / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.20000000000000006e38 or 7.5999999999999999e-31 < 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/ 72.9%

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

      3. associate-*l/ 96.9%

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

      4. *-commutative 96.9%

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

      5. fma-def 96.9%

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

   3. Simplified 96.9%

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

   4. Taylor expanded in z around inf 80.6%

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

   if -2.20000000000000006e38 < z < 7.5999999999999999e-31

   1. Initial program 97.0%

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

      1. +-commutative 97.0%

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

      2. associate-*r/ 98.2%

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

      3. associate-*l/ 96.3%

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

      4. *-commutative 96.3%

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

      5. fma-def 96.3%

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

   3. Simplified 96.3%

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

   4. Taylor expanded in z around 0 83.2%

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

      1. associate-/l* 83.4%

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

   6. Simplified 83.4%

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

      1. associate-/r/ 83.5%

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

   8. Applied egg-rr 83.5%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.2 \cdot 10^{+38}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq 7.6 \cdot 10^{-31}:\\ \;\;\;\;x + t \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 7: 59.6% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

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
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_] := N[(x + y), $MachinePrecision]

Derivation

1. Initial program 98.4%

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

   1. +-commutative 98.4%

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

   2. associate-*r/ 85.9%

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

   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 61.4%

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

5. Final simplification 61.4%

   \[\leadsto x + y \]

Alternative 8: 51.0% 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 98.4%

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

   1. +-commutative 98.4%

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

   2. associate-*r/ 85.9%

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

   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 y around 0 49.4%

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

5. Final simplification 49.4%

   \[\leadsto x \]

Developer target: 98.1% 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 2023224 
(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)))))