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

Percentage Accurate: 97.9% → 97.9%

Time: 10.4s

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 99.2%

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

2. Final simplification 99.2%

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

Alternative 2: 81.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.4 \cdot 10^{-21} \lor \neg \left(z \leq 3.2 \cdot 10^{-140}\right):\\ \;\;\;\;x + y \cdot \frac{z - t}{z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{t}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -3.4e-21) (not (<= z 3.2e-140)))
   (+ x (* y (/ (- z t) z)))
   (+ x (/ y (/ a t)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -3.4e-21) || !(z <= 3.2e-140)) {
		tmp = x + (y * ((z - t) / z));
	} else {
		tmp = x + (y / (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) :: tmp
    if ((z <= (-3.4d-21)) .or. (.not. (z <= 3.2d-140))) then
        tmp = x + (y * ((z - t) / z))
    else
        tmp = x + (y / (a / 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 <= -3.4e-21) || !(z <= 3.2e-140)) {
		tmp = x + (y * ((z - t) / z));
	} else {
		tmp = x + (y / (a / t));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (z <= -3.4e-21) or not (z <= 3.2e-140):
		tmp = x + (y * ((z - t) / z))
	else:
		tmp = x + (y / (a / t))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -3.4e-21) || !(z <= 3.2e-140))
		tmp = Float64(x + Float64(y * Float64(Float64(z - t) / z)));
	else
		tmp = Float64(x + Float64(y / Float64(a / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((z <= -3.4e-21) || ~((z <= 3.2e-140)))
		tmp = x + (y * ((z - t) / z));
	else
		tmp = x + (y / (a / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -3.4e-21], N[Not[LessEqual[z, 3.2e-140]], $MachinePrecision]], N[(x + N[(y * N[(N[(z - t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(y / N[(a / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.4e-21 or 3.2000000000000001e-140 < z

   1. Initial program 99.9%

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

   2. Taylor expanded in a around 0 87.4%

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

   if -3.4e-21 < z < 3.2000000000000001e-140

   1. Initial program 98.2%

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

   2. Taylor expanded in z around 0 81.9%

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

      1. expm1-log1p-u 58.9%

         \[\leadsto x + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{t \cdot y}{a}\right)\right)} \]

      2. expm1-udef 51.0%

         \[\leadsto x + \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{t \cdot y}{a}\right)} - 1\right)} \]

      3. div-inv 51.0%

         \[\leadsto x + \left(e^{\mathsf{log1p}\left(\color{blue}{\left(t \cdot y\right) \cdot \frac{1}{a}}\right)} - 1\right) \]

      4. associate-*l* 56.0%

         \[\leadsto x + \left(e^{\mathsf{log1p}\left(\color{blue}{t \cdot \left(y \cdot \frac{1}{a}\right)}\right)} - 1\right) \]

      5. div-inv 56.0%

         \[\leadsto x + \left(e^{\mathsf{log1p}\left(t \cdot \color{blue}{\frac{y}{a}}\right)} - 1\right) \]

   4. Applied egg-rr 56.0%

      \[\leadsto x + \color{blue}{\left(e^{\mathsf{log1p}\left(t \cdot \frac{y}{a}\right)} - 1\right)} \]
   5. Step-by-step derivation

      1. expm1-def 63.0%

         \[\leadsto x + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(t \cdot \frac{y}{a}\right)\right)} \]

      2. expm1-log1p 88.0%

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

      3. *-commutative 88.0%

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

      4. associate-*l/ 81.9%

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

      5. associate-/l* 89.6%

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

   6. Simplified 89.6%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.4 \cdot 10^{-21} \lor \neg \left(z \leq 3.2 \cdot 10^{-140}\right):\\ \;\;\;\;x + y \cdot \frac{z - t}{z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{t}}\\ \end{array} \]

Alternative 3: 81.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.3 \cdot 10^{-26} \lor \neg \left(z \leq 3.2 \cdot 10^{-140}\right):\\ \;\;\;\;x + \frac{y}{\frac{z}{z - t}}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{t}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -4.3e-26) (not (<= z 3.2e-140)))
   (+ x (/ y (/ z (- z t))))
   (+ x (/ y (/ a t)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -4.3e-26) || !(z <= 3.2e-140)) {
		tmp = x + (y / (z / (z - t)));
	} else {
		tmp = x + (y / (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) :: tmp
    if ((z <= (-4.3d-26)) .or. (.not. (z <= 3.2d-140))) then
        tmp = x + (y / (z / (z - t)))
    else
        tmp = x + (y / (a / 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 <= -4.3e-26) || !(z <= 3.2e-140)) {
		tmp = x + (y / (z / (z - t)));
	} else {
		tmp = x + (y / (a / t));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (z <= -4.3e-26) or not (z <= 3.2e-140):
		tmp = x + (y / (z / (z - t)))
	else:
		tmp = x + (y / (a / t))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -4.3e-26) || !(z <= 3.2e-140))
		tmp = Float64(x + Float64(y / Float64(z / Float64(z - t))));
	else
		tmp = Float64(x + Float64(y / Float64(a / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((z <= -4.3e-26) || ~((z <= 3.2e-140)))
		tmp = x + (y / (z / (z - t)));
	else
		tmp = x + (y / (a / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -4.3e-26], N[Not[LessEqual[z, 3.2e-140]], $MachinePrecision]], N[(x + N[(y / N[(z / N[(z - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(y / N[(a / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4.29999999999999988e-26 or 3.2000000000000001e-140 < z

   1. Initial program 99.9%

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

      1. add-cube-cbrt 99.6%

         \[\leadsto x + y \cdot \color{blue}{\left(\left(\sqrt[3]{\frac{z - t}{z - a}} \cdot \sqrt[3]{\frac{z - t}{z - a}}\right) \cdot \sqrt[3]{\frac{z - t}{z - a}}\right)} \]

      2. pow3 99.6%

         \[\leadsto x + y \cdot \color{blue}{{\left(\sqrt[3]{\frac{z - t}{z - a}}\right)}^{3}} \]

   3. Applied egg-rr 99.6%

      \[\leadsto x + y \cdot \color{blue}{{\left(\sqrt[3]{\frac{z - t}{z - a}}\right)}^{3}} \]

   4. Taylor expanded in a around 0 70.4%

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

      1. associate-/l* 87.4%

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

   6. Simplified 87.4%

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

   if -4.29999999999999988e-26 < z < 3.2000000000000001e-140

   1. Initial program 98.2%

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

   2. Taylor expanded in z around 0 81.9%

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

      1. expm1-log1p-u 58.9%

         \[\leadsto x + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{t \cdot y}{a}\right)\right)} \]

      2. expm1-udef 51.0%

         \[\leadsto x + \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{t \cdot y}{a}\right)} - 1\right)} \]

      3. div-inv 51.0%

         \[\leadsto x + \left(e^{\mathsf{log1p}\left(\color{blue}{\left(t \cdot y\right) \cdot \frac{1}{a}}\right)} - 1\right) \]

      4. associate-*l* 56.0%

         \[\leadsto x + \left(e^{\mathsf{log1p}\left(\color{blue}{t \cdot \left(y \cdot \frac{1}{a}\right)}\right)} - 1\right) \]

      5. div-inv 56.0%

         \[\leadsto x + \left(e^{\mathsf{log1p}\left(t \cdot \color{blue}{\frac{y}{a}}\right)} - 1\right) \]

   4. Applied egg-rr 56.0%

      \[\leadsto x + \color{blue}{\left(e^{\mathsf{log1p}\left(t \cdot \frac{y}{a}\right)} - 1\right)} \]
   5. Step-by-step derivation

      1. expm1-def 63.0%

         \[\leadsto x + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(t \cdot \frac{y}{a}\right)\right)} \]

      2. expm1-log1p 88.0%

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

      3. *-commutative 88.0%

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

      4. associate-*l/ 81.9%

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

      5. associate-/l* 89.6%

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

   6. Simplified 89.6%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.3 \cdot 10^{-26} \lor \neg \left(z \leq 3.2 \cdot 10^{-140}\right):\\ \;\;\;\;x + \frac{y}{\frac{z}{z - t}}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{t}}\\ \end{array} \]

Alternative 4: 83.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -8 \cdot 10^{-21}:\\ \;\;\;\;x + \frac{y}{\frac{z}{z - t}}\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{-20}:\\ \;\;\;\;x + \frac{y}{a} \cdot \left(t - z\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z - t}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -8e-21)
   (+ x (/ y (/ z (- z t))))
   (if (<= z 1.25e-20) (+ x (* (/ y a) (- t z))) (+ x (* y (/ (- z t) z))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -8e-21) {
		tmp = x + (y / (z / (z - t)));
	} else if (z <= 1.25e-20) {
		tmp = x + ((y / a) * (t - z));
	} 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 (z <= (-8d-21)) then
        tmp = x + (y / (z / (z - t)))
    else if (z <= 1.25d-20) then
        tmp = x + ((y / a) * (t - z))
    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 (z <= -8e-21) {
		tmp = x + (y / (z / (z - t)));
	} else if (z <= 1.25e-20) {
		tmp = x + ((y / a) * (t - z));
	} else {
		tmp = x + (y * ((z - t) / z));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -8e-21:
		tmp = x + (y / (z / (z - t)))
	elif z <= 1.25e-20:
		tmp = x + ((y / a) * (t - z))
	else:
		tmp = x + (y * ((z - t) / z))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -8e-21)
		tmp = Float64(x + Float64(y / Float64(z / Float64(z - t))));
	elseif (z <= 1.25e-20)
		tmp = Float64(x + Float64(Float64(y / a) * Float64(t - z)));
	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 (z <= -8e-21)
		tmp = x + (y / (z / (z - t)));
	elseif (z <= 1.25e-20)
		tmp = x + ((y / a) * (t - z));
	else
		tmp = x + (y * ((z - t) / z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -8e-21], N[(x + N[(y / N[(z / N[(z - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.25e-20], N[(x + N[(N[(y / a), $MachinePrecision] * N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(y * N[(N[(z - t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -7.99999999999999926e-21

   1. Initial program 99.9%

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

      1. add-cube-cbrt 99.7%

         \[\leadsto x + y \cdot \color{blue}{\left(\left(\sqrt[3]{\frac{z - t}{z - a}} \cdot \sqrt[3]{\frac{z - t}{z - a}}\right) \cdot \sqrt[3]{\frac{z - t}{z - a}}\right)} \]

      2. pow3 99.7%

         \[\leadsto x + y \cdot \color{blue}{{\left(\sqrt[3]{\frac{z - t}{z - a}}\right)}^{3}} \]

   3. Applied egg-rr 99.7%

      \[\leadsto x + y \cdot \color{blue}{{\left(\sqrt[3]{\frac{z - t}{z - a}}\right)}^{3}} \]

   4. Taylor expanded in a around 0 74.1%

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

      1. associate-/l* 92.4%

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

   6. Simplified 92.4%

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

   if -7.99999999999999926e-21 < z < 1.25e-20

   1. Initial program 98.5%

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

      1. +-commutative 98.5%

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

      2. associate-*r/ 92.1%

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

      3. associate-*l/ 97.1%

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

      4. fma-def 97.1%

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

   3. Simplified 97.1%

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

   4. Taylor expanded in a around inf 82.8%

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

      1. mul-1-neg 82.8%

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

      2. unsub-neg 82.8%

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

      3. associate-/l* 88.5%

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

      4. associate-/r/ 87.8%

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

   6. Simplified 87.8%

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

   if 1.25e-20 < z

   1. Initial program 100.0%

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

   2. Taylor expanded in a around 0 93.3%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8 \cdot 10^{-21}:\\ \;\;\;\;x + \frac{y}{\frac{z}{z - t}}\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{-20}:\\ \;\;\;\;x + \frac{y}{a} \cdot \left(t - z\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z - t}{z}\\ \end{array} \]

Alternative 5: 88.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.6 \cdot 10^{+20}:\\ \;\;\;\;x + \frac{y}{\frac{z}{z - t}}\\ \mathbf{elif}\;z \leq 4100000000:\\ \;\;\;\;x - t \cdot \frac{y}{z - a}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z - t}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -2.6e+20)
   (+ x (/ y (/ z (- z t))))
   (if (<= z 4100000000.0)
     (- x (* t (/ y (- z a))))
     (+ x (* y (/ (- z t) z))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.6e+20) {
		tmp = x + (y / (z / (z - t)));
	} else if (z <= 4100000000.0) {
		tmp = x - (t * (y / (z - 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 (z <= (-2.6d+20)) then
        tmp = x + (y / (z / (z - t)))
    else if (z <= 4100000000.0d0) then
        tmp = x - (t * (y / (z - 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 (z <= -2.6e+20) {
		tmp = x + (y / (z / (z - t)));
	} else if (z <= 4100000000.0) {
		tmp = x - (t * (y / (z - a)));
	} else {
		tmp = x + (y * ((z - t) / z));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -2.6e+20:
		tmp = x + (y / (z / (z - t)))
	elif z <= 4100000000.0:
		tmp = x - (t * (y / (z - a)))
	else:
		tmp = x + (y * ((z - t) / z))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -2.6e+20)
		tmp = Float64(x + Float64(y / Float64(z / Float64(z - t))));
	elseif (z <= 4100000000.0)
		tmp = Float64(x - Float64(t * Float64(y / Float64(z - 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 (z <= -2.6e+20)
		tmp = x + (y / (z / (z - t)));
	elseif (z <= 4100000000.0)
		tmp = x - (t * (y / (z - a)));
	else
		tmp = x + (y * ((z - t) / z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -2.6e+20], N[(x + N[(y / N[(z / N[(z - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 4100000000.0], N[(x - N[(t * N[(y / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(y * N[(N[(z - t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.6e20

   1. Initial program 100.0%

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

      1. add-cube-cbrt 99.9%

         \[\leadsto x + y \cdot \color{blue}{\left(\left(\sqrt[3]{\frac{z - t}{z - a}} \cdot \sqrt[3]{\frac{z - t}{z - a}}\right) \cdot \sqrt[3]{\frac{z - t}{z - a}}\right)} \]

      2. pow3 99.9%

         \[\leadsto x + y \cdot \color{blue}{{\left(\sqrt[3]{\frac{z - t}{z - a}}\right)}^{3}} \]

   3. Applied egg-rr 99.9%

      \[\leadsto x + y \cdot \color{blue}{{\left(\sqrt[3]{\frac{z - t}{z - a}}\right)}^{3}} \]

   4. Taylor expanded in a around 0 72.7%

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

      1. associate-/l* 94.4%

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

   6. Simplified 94.4%

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

   if -2.6e20 < z < 4.1e9

   1. Initial program 98.7%

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

   2. Taylor expanded in t around inf 85.5%

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

      1. mul-1-neg 85.5%

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

      2. *-commutative 85.5%

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

      3. associate-*l/ 90.3%

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

      4. distribute-rgt-neg-out 90.3%

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

   4. Simplified 90.3%

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

   if 4.1e9 < z

   1. Initial program 100.0%

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

   2. Taylor expanded in a around 0 94.5%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.6 \cdot 10^{+20}:\\ \;\;\;\;x + \frac{y}{\frac{z}{z - t}}\\ \mathbf{elif}\;z \leq 4100000000:\\ \;\;\;\;x - t \cdot \frac{y}{z - a}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z - t}{z}\\ \end{array} \]

Alternative 6: 76.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.45 \cdot 10^{+20}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq 1.65 \cdot 10^{-16}:\\ \;\;\;\;x + \frac{y}{\frac{a}{t}}\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -2.45e+20)
   (+ x y)
   (if (<= z 1.65e-16) (+ x (/ y (/ a t))) (+ x y))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.45e+20) {
		tmp = x + y;
	} else if (z <= 1.65e-16) {
		tmp = x + (y / (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) :: tmp
    if (z <= (-2.45d+20)) then
        tmp = x + y
    else if (z <= 1.65d-16) then
        tmp = x + (y / (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 tmp;
	if (z <= -2.45e+20) {
		tmp = x + y;
	} else if (z <= 1.65e-16) {
		tmp = x + (y / (a / t));
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -2.45e+20:
		tmp = x + y
	elif z <= 1.65e-16:
		tmp = x + (y / (a / t))
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -2.45e+20)
		tmp = Float64(x + y);
	elseif (z <= 1.65e-16)
		tmp = Float64(x + Float64(y / Float64(a / t)));
	else
		tmp = Float64(x + y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -2.45e+20)
		tmp = x + y;
	elseif (z <= 1.65e-16)
		tmp = x + (y / (a / t));
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -2.45e+20], N[(x + y), $MachinePrecision], If[LessEqual[z, 1.65e-16], N[(x + N[(y / N[(a / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.45e20 or 1.64999999999999994e-16 < z

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. associate-*r/ 73.4%

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

      3. associate-*l/ 94.3%

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

      4. fma-def 94.3%

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

   3. Simplified 94.3%

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

   4. Taylor expanded in z around inf 85.3%

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

      1. +-commutative 85.3%

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

   6. Simplified 85.3%

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

   if -2.45e20 < z < 1.64999999999999994e-16

   1. Initial program 98.6%

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

   2. Taylor expanded in z around 0 76.5%

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

      1. expm1-log1p-u 56.6%

         \[\leadsto x + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{t \cdot y}{a}\right)\right)} \]

      2. expm1-udef 50.8%

         \[\leadsto x + \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{t \cdot y}{a}\right)} - 1\right)} \]

      3. div-inv 50.8%

         \[\leadsto x + \left(e^{\mathsf{log1p}\left(\color{blue}{\left(t \cdot y\right) \cdot \frac{1}{a}}\right)} - 1\right) \]

      4. associate-*l* 54.5%

         \[\leadsto x + \left(e^{\mathsf{log1p}\left(\color{blue}{t \cdot \left(y \cdot \frac{1}{a}\right)}\right)} - 1\right) \]

      5. div-inv 54.5%

         \[\leadsto x + \left(e^{\mathsf{log1p}\left(t \cdot \color{blue}{\frac{y}{a}}\right)} - 1\right) \]

   4. Applied egg-rr 54.5%

      \[\leadsto x + \color{blue}{\left(e^{\mathsf{log1p}\left(t \cdot \frac{y}{a}\right)} - 1\right)} \]
   5. Step-by-step derivation

      1. expm1-def 59.7%

         \[\leadsto x + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(t \cdot \frac{y}{a}\right)\right)} \]

      2. expm1-log1p 81.0%

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

      3. *-commutative 81.0%

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

      4. associate-*l/ 76.5%

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

      5. associate-/l* 81.6%

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

   6. Simplified 81.6%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.45 \cdot 10^{+20}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq 1.65 \cdot 10^{-16}:\\ \;\;\;\;x + \frac{y}{\frac{a}{t}}\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 7: 64.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.2 \cdot 10^{-39}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq 4.9 \cdot 10^{-51}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -4.2e-39) (+ x y) (if (<= z 4.9e-51) x (+ x y))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -4.2e-39) {
		tmp = x + y;
	} else if (z <= 4.9e-51) {
		tmp = x;
	} else {
		tmp = x + y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-4.2d-39)) then
        tmp = x + y
    else if (z <= 4.9d-51) then
        tmp = x
    else
        tmp = x + y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -4.2e-39) {
		tmp = x + y;
	} else if (z <= 4.9e-51) {
		tmp = x;
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -4.2e-39:
		tmp = x + y
	elif z <= 4.9e-51:
		tmp = x
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -4.2e-39)
		tmp = Float64(x + y);
	elseif (z <= 4.9e-51)
		tmp = x;
	else
		tmp = Float64(x + y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -4.2e-39)
		tmp = x + y;
	elseif (z <= 4.9e-51)
		tmp = x;
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -4.2e-39], N[(x + y), $MachinePrecision], If[LessEqual[z, 4.9e-51], x, N[(x + y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -4.19999999999999987e-39 or 4.89999999999999974e-51 < z

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. associate-*r/ 77.5%

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

      3. associate-*l/ 95.3%

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

      4. fma-def 95.3%

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

   3. Simplified 95.3%

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

   4. Taylor expanded in z around inf 79.1%

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

      1. +-commutative 79.1%

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

   6. Simplified 79.1%

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

   if -4.19999999999999987e-39 < z < 4.89999999999999974e-51

   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/ 92.2%

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

      3. associate-*l/ 96.9%

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

      4. fma-def 96.9%

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

   3. Simplified 96.9%

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

   4. Taylor expanded in y around 0 51.6%

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

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.2 \cdot 10^{-39}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq 4.9 \cdot 10^{-51}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 8: 50.3% 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 99.2%

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

   1. +-commutative 99.2%

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

   2. associate-*r/ 84.7%

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

   3. associate-*l/ 96.1%

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

   4. fma-def 96.1%

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

3. Simplified 96.1%

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

4. Taylor expanded in y around 0 53.1%

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

5. Final simplification 53.1%

   \[\leadsto x \]

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 2023279 
(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)))))