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

Percentage Accurate: 85.6% → 98.5%

Time: 7.4s

Alternatives: 9

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

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 - z))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 85.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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 - z))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.5% accurate, 1.0× speedup

Math

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

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(Float64(Float64(y - z) / Float64(a - 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[(N[(N[(y - z), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 88.2%

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

   1. associate-*l/ 98.5%

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

3. Simplified 98.5%

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

4. Final simplification 98.5%

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

Alternative 2: 77.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.25 \cdot 10^{+128}:\\ \;\;\;\;x + t\\ \mathbf{elif}\;z \leq -1.22 \cdot 10^{-82}:\\ \;\;\;\;x - \frac{t}{\frac{z}{y}}\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{-34}:\\ \;\;\;\;x + \frac{y}{\frac{a}{t}}\\ \mathbf{else}:\\ \;\;\;\;x + t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -2.25e+128)
   (+ x t)
   (if (<= z -1.22e-82)
     (- x (/ t (/ z y)))
     (if (<= z 6.5e-34) (+ x (/ y (/ a t))) (+ x t)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.25e+128) {
		tmp = x + t;
	} else if (z <= -1.22e-82) {
		tmp = x - (t / (z / y));
	} else if (z <= 6.5e-34) {
		tmp = x + (y / (a / t));
	} else {
		tmp = x + 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 <= (-2.25d+128)) then
        tmp = x + t
    else if (z <= (-1.22d-82)) then
        tmp = x - (t / (z / y))
    else if (z <= 6.5d-34) then
        tmp = x + (y / (a / t))
    else
        tmp = x + 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 <= -2.25e+128) {
		tmp = x + t;
	} else if (z <= -1.22e-82) {
		tmp = x - (t / (z / y));
	} else if (z <= 6.5e-34) {
		tmp = x + (y / (a / t));
	} else {
		tmp = x + t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -2.25e+128:
		tmp = x + t
	elif z <= -1.22e-82:
		tmp = x - (t / (z / y))
	elif z <= 6.5e-34:
		tmp = x + (y / (a / t))
	else:
		tmp = x + t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -2.25e+128)
		tmp = Float64(x + t);
	elseif (z <= -1.22e-82)
		tmp = Float64(x - Float64(t / Float64(z / y)));
	elseif (z <= 6.5e-34)
		tmp = Float64(x + Float64(y / Float64(a / t)));
	else
		tmp = Float64(x + t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -2.25e+128)
		tmp = x + t;
	elseif (z <= -1.22e-82)
		tmp = x - (t / (z / y));
	elseif (z <= 6.5e-34)
		tmp = x + (y / (a / t));
	else
		tmp = x + t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -2.25e+128], N[(x + t), $MachinePrecision], If[LessEqual[z, -1.22e-82], N[(x - N[(t / N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 6.5e-34], N[(x + N[(y / N[(a / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + t), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.2500000000000001e128 or 6.49999999999999985e-34 < z

   1. Initial program 74.8%

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

      1. associate-*l/ 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in z around inf 80.5%

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

   if -2.2500000000000001e128 < z < -1.22000000000000001e-82

   1. Initial program 97.8%

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

      1. +-commutative 97.8%

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

      2. associate-*r/ 98.2%

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

      3. fma-def 98.2%

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

   3. Simplified 98.2%

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

   4. Taylor expanded in a around 0 73.6%

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

      1. +-commutative 73.6%

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

      2. mul-1-neg 73.6%

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

      3. unsub-neg 73.6%

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

      4. associate-/l* 73.7%

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

   6. Simplified 73.7%

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

   7. Taylor expanded in z around 0 72.1%

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

      1. *-commutative 72.1%

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

      2. associate-/l* 72.2%

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

   9. Simplified 72.2%

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

   if -1.22000000000000001e-82 < z < 6.49999999999999985e-34

   1. Initial program 99.0%

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

      1. +-commutative 99.0%

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

      2. associate-*r/ 98.9%

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

      3. fma-def 98.9%

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

   3. Simplified 98.9%

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

   4. Taylor expanded in z around 0 84.1%

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

      1. associate-/l* 85.1%

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

   6. Simplified 85.1%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.25 \cdot 10^{+128}:\\ \;\;\;\;x + t\\ \mathbf{elif}\;z \leq -1.22 \cdot 10^{-82}:\\ \;\;\;\;x - \frac{t}{\frac{z}{y}}\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{-34}:\\ \;\;\;\;x + \frac{y}{\frac{a}{t}}\\ \mathbf{else}:\\ \;\;\;\;x + t\\ \end{array} \]

Alternative 3: 79.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.62 \cdot 10^{-34} \lor \neg \left(z \leq 3 \cdot 10^{-212}\right):\\ \;\;\;\;x + t \cdot \frac{z}{z - a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{t}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -1.62e-34) (not (<= z 3e-212)))
   (+ x (* t (/ z (- z a))))
   (+ x (/ y (/ a t)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -1.62e-34) || !(z <= 3e-212)) {
		tmp = x + (t * (z / (z - a)));
	} 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 <= (-1.62d-34)) .or. (.not. (z <= 3d-212))) then
        tmp = x + (t * (z / (z - a)))
    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 <= -1.62e-34) || !(z <= 3e-212)) {
		tmp = x + (t * (z / (z - a)));
	} else {
		tmp = x + (y / (a / t));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (z <= -1.62e-34) or not (z <= 3e-212):
		tmp = x + (t * (z / (z - a)))
	else:
		tmp = x + (y / (a / t))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -1.62e-34) || !(z <= 3e-212))
		tmp = Float64(x + Float64(t * Float64(z / Float64(z - a))));
	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 <= -1.62e-34) || ~((z <= 3e-212)))
		tmp = x + (t * (z / (z - a)));
	else
		tmp = x + (y / (a / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -1.62e-34], N[Not[LessEqual[z, 3e-212]], $MachinePrecision]], N[(x + N[(t * N[(z / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(y / N[(a / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.62000000000000006e-34 or 3.0000000000000003e-212 < z

   1. Initial program 84.0%

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

      1. associate-*l/ 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around 0 83.8%

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

      1. neg-mul-1 83.8%

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

      2. distribute-neg-frac 83.8%

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

   6. Simplified 83.8%

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

      1. frac-2neg 83.8%

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

      2. div-inv 83.7%

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

      3. remove-double-neg 83.7%

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

      4. sub-neg 83.7%

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

      5. distribute-neg-in 83.7%

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

      6. remove-double-neg 83.7%

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

   8. Applied egg-rr 83.7%

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

      1. associate-*r/ 83.8%

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

      2. *-rgt-identity 83.8%

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

      3. +-commutative 83.8%

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

      4. unsub-neg 83.8%

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

   10. Simplified 83.8%

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

   if -1.62000000000000006e-34 < z < 3.0000000000000003e-212

   1. Initial program 98.7%

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

      1. +-commutative 98.7%

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

      2. associate-*r/ 99.9%

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

      3. fma-def 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in z around 0 91.0%

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

      1. associate-/l* 92.2%

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

   6. Simplified 92.2%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.62 \cdot 10^{-34} \lor \neg \left(z \leq 3 \cdot 10^{-212}\right):\\ \;\;\;\;x + t \cdot \frac{z}{z - a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{t}}\\ \end{array} \]

Alternative 4: 84.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.45 \cdot 10^{-55} \lor \neg \left(y \leq 4.9 \cdot 10^{+67}\right):\\ \;\;\;\;x + \frac{y \cdot t}{a - z}\\ \mathbf{else}:\\ \;\;\;\;x + t \cdot \frac{z}{z - a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= y -1.45e-55) (not (<= y 4.9e+67)))
   (+ x (/ (* y t) (- a z)))
   (+ x (* t (/ z (- z a))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((y <= -1.45e-55) || !(y <= 4.9e+67)) {
		tmp = x + ((y * t) / (a - z));
	} else {
		tmp = x + (t * (z / (z - a)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((y <= (-1.45d-55)) .or. (.not. (y <= 4.9d+67))) then
        tmp = x + ((y * t) / (a - z))
    else
        tmp = x + (t * (z / (z - a)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((y <= -1.45e-55) || !(y <= 4.9e+67)) {
		tmp = x + ((y * t) / (a - z));
	} else {
		tmp = x + (t * (z / (z - a)));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (y <= -1.45e-55) or not (y <= 4.9e+67):
		tmp = x + ((y * t) / (a - z))
	else:
		tmp = x + (t * (z / (z - a)))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((y <= -1.45e-55) || !(y <= 4.9e+67))
		tmp = Float64(x + Float64(Float64(y * t) / Float64(a - z)));
	else
		tmp = Float64(x + Float64(t * Float64(z / Float64(z - a))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((y <= -1.45e-55) || ~((y <= 4.9e+67)))
		tmp = x + ((y * t) / (a - z));
	else
		tmp = x + (t * (z / (z - a)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[y, -1.45e-55], N[Not[LessEqual[y, 4.9e+67]], $MachinePrecision]], N[(x + N[(N[(y * t), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(t * N[(z / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.45e-55 or 4.8999999999999999e67 < y

   1. Initial program 89.6%

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

      1. associate-*l/ 97.5%

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

   3. Simplified 97.5%

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

   4. Taylor expanded in y around inf 85.4%

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

   if -1.45e-55 < y < 4.8999999999999999e67

   1. Initial program 86.9%

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

      1. associate-*l/ 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in y around 0 93.7%

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

      1. neg-mul-1 93.7%

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

      2. distribute-neg-frac 93.7%

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

   6. Simplified 93.7%

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

      1. frac-2neg 93.7%

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

      2. div-inv 93.6%

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

      3. remove-double-neg 93.6%

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

      4. sub-neg 93.6%

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

      5. distribute-neg-in 93.6%

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

      6. remove-double-neg 93.6%

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

   8. Applied egg-rr 93.6%

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

      1. associate-*r/ 93.7%

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

      2. *-rgt-identity 93.7%

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

      3. +-commutative 93.7%

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

      4. unsub-neg 93.7%

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

   10. Simplified 93.7%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.45 \cdot 10^{-55} \lor \neg \left(y \leq 4.9 \cdot 10^{+67}\right):\\ \;\;\;\;x + \frac{y \cdot t}{a - z}\\ \mathbf{else}:\\ \;\;\;\;x + t \cdot \frac{z}{z - a}\\ \end{array} \]

Alternative 5: 85.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6.6 \cdot 10^{+31}:\\ \;\;\;\;x + t \cdot \frac{z}{z - a}\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{-65}:\\ \;\;\;\;x + \frac{y \cdot t}{a - z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{t}{z} \cdot \left(z - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -6.6e+31)
   (+ x (* t (/ z (- z a))))
   (if (<= z 3.2e-65) (+ x (/ (* y t) (- a z))) (+ x (* (/ t z) (- z y))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -6.6e+31) {
		tmp = x + (t * (z / (z - a)));
	} else if (z <= 3.2e-65) {
		tmp = x + ((y * t) / (a - z));
	} else {
		tmp = x + ((t / z) * (z - y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-6.6d+31)) then
        tmp = x + (t * (z / (z - a)))
    else if (z <= 3.2d-65) then
        tmp = x + ((y * t) / (a - z))
    else
        tmp = x + ((t / z) * (z - y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -6.6e+31) {
		tmp = x + (t * (z / (z - a)));
	} else if (z <= 3.2e-65) {
		tmp = x + ((y * t) / (a - z));
	} else {
		tmp = x + ((t / z) * (z - y));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -6.6e+31:
		tmp = x + (t * (z / (z - a)))
	elif z <= 3.2e-65:
		tmp = x + ((y * t) / (a - z))
	else:
		tmp = x + ((t / z) * (z - y))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -6.6e+31)
		tmp = Float64(x + Float64(t * Float64(z / Float64(z - a))));
	elseif (z <= 3.2e-65)
		tmp = Float64(x + Float64(Float64(y * t) / Float64(a - z)));
	else
		tmp = Float64(x + Float64(Float64(t / z) * Float64(z - y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -6.6e+31)
		tmp = x + (t * (z / (z - a)));
	elseif (z <= 3.2e-65)
		tmp = x + ((y * t) / (a - z));
	else
		tmp = x + ((t / z) * (z - y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -6.6e+31], N[(x + N[(t * N[(z / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 3.2e-65], N[(x + N[(N[(y * t), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(t / z), $MachinePrecision] * N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -6.59999999999999985e31

   1. Initial program 78.4%

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

      1. associate-*l/ 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around 0 92.9%

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

      1. neg-mul-1 92.9%

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

      2. distribute-neg-frac 92.9%

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

   6. Simplified 92.9%

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

      1. frac-2neg 92.9%

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

      2. div-inv 92.8%

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

      3. remove-double-neg 92.8%

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

      4. sub-neg 92.8%

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

      5. distribute-neg-in 92.8%

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

      6. remove-double-neg 92.8%

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

   8. Applied egg-rr 92.8%

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

      1. associate-*r/ 92.9%

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

      2. *-rgt-identity 92.9%

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

      3. +-commutative 92.9%

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

      4. unsub-neg 92.9%

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

   10. Simplified 92.9%

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

   if -6.59999999999999985e31 < z < 3.1999999999999999e-65

   1. Initial program 99.1%

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

      1. associate-*l/ 96.7%

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

   3. Simplified 96.7%

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

   4. Taylor expanded in y around inf 91.9%

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

   if 3.1999999999999999e-65 < z

   1. Initial program 80.0%

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

      1. +-commutative 80.0%

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

      2. associate-*r/ 95.4%

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

      3. fma-def 95.4%

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

   3. Simplified 95.4%

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

   4. Taylor expanded in a around 0 71.1%

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

      1. +-commutative 71.1%

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

      2. mul-1-neg 71.1%

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

      3. unsub-neg 71.1%

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

      4. associate-/l* 88.7%

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

   6. Simplified 88.7%

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

      1. associate-/r/ 85.2%

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

   8. Applied egg-rr 85.2%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.6 \cdot 10^{+31}:\\ \;\;\;\;x + t \cdot \frac{z}{z - a}\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{-65}:\\ \;\;\;\;x + \frac{y \cdot t}{a - z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{t}{z} \cdot \left(z - y\right)\\ \end{array} \]

Alternative 6: 86.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.5 \cdot 10^{+31}:\\ \;\;\;\;x + t \cdot \frac{z}{z - a}\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{-62}:\\ \;\;\;\;x + \frac{y \cdot t}{a - z}\\ \mathbf{else}:\\ \;\;\;\;x + \left(t - \frac{t}{\frac{z}{y}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -5.5e+31)
   (+ x (* t (/ z (- z a))))
   (if (<= z 2.3e-62) (+ x (/ (* y t) (- a z))) (+ x (- t (/ t (/ z y)))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -5.5e+31) {
		tmp = x + (t * (z / (z - a)));
	} else if (z <= 2.3e-62) {
		tmp = x + ((y * t) / (a - z));
	} else {
		tmp = x + (t - (t / (z / y)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-5.5d+31)) then
        tmp = x + (t * (z / (z - a)))
    else if (z <= 2.3d-62) then
        tmp = x + ((y * t) / (a - z))
    else
        tmp = x + (t - (t / (z / y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -5.5e+31) {
		tmp = x + (t * (z / (z - a)));
	} else if (z <= 2.3e-62) {
		tmp = x + ((y * t) / (a - z));
	} else {
		tmp = x + (t - (t / (z / y)));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -5.5e+31:
		tmp = x + (t * (z / (z - a)))
	elif z <= 2.3e-62:
		tmp = x + ((y * t) / (a - z))
	else:
		tmp = x + (t - (t / (z / y)))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -5.5e+31)
		tmp = Float64(x + Float64(t * Float64(z / Float64(z - a))));
	elseif (z <= 2.3e-62)
		tmp = Float64(x + Float64(Float64(y * t) / Float64(a - z)));
	else
		tmp = Float64(x + Float64(t - Float64(t / Float64(z / y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -5.5e+31)
		tmp = x + (t * (z / (z - a)));
	elseif (z <= 2.3e-62)
		tmp = x + ((y * t) / (a - z));
	else
		tmp = x + (t - (t / (z / y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -5.5e+31], N[(x + N[(t * N[(z / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.3e-62], N[(x + N[(N[(y * t), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(t - N[(t / N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.50000000000000002e31

   1. Initial program 78.4%

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

      1. associate-*l/ 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around 0 92.9%

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

      1. neg-mul-1 92.9%

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

      2. distribute-neg-frac 92.9%

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

   6. Simplified 92.9%

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

      1. frac-2neg 92.9%

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

      2. div-inv 92.8%

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

      3. remove-double-neg 92.8%

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

      4. sub-neg 92.8%

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

      5. distribute-neg-in 92.8%

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

      6. remove-double-neg 92.8%

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

   8. Applied egg-rr 92.8%

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

      1. associate-*r/ 92.9%

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

      2. *-rgt-identity 92.9%

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

      3. +-commutative 92.9%

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

      4. unsub-neg 92.9%

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

   10. Simplified 92.9%

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

   if -5.50000000000000002e31 < z < 2.3e-62

   1. Initial program 99.1%

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

      1. associate-*l/ 96.7%

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

   3. Simplified 96.7%

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

   4. Taylor expanded in y around inf 91.9%

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

   if 2.3e-62 < z

   1. Initial program 80.0%

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

      1. +-commutative 80.0%

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

      2. associate-*r/ 95.4%

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

      3. fma-def 95.4%

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

   3. Simplified 95.4%

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

   4. Taylor expanded in a around 0 71.1%

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

      1. +-commutative 71.1%

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

      2. mul-1-neg 71.1%

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

      3. unsub-neg 71.1%

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

      4. associate-/l* 88.7%

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

   6. Simplified 88.7%

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

   7. Taylor expanded in z around 0 83.1%

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

      1. neg-mul-1 83.1%

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

      2. +-commutative 83.1%

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

      3. unsub-neg 83.1%

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

      4. *-commutative 83.1%

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

      5. associate-/l* 88.7%

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

   9. Simplified 88.7%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.5 \cdot 10^{+31}:\\ \;\;\;\;x + t \cdot \frac{z}{z - a}\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{-62}:\\ \;\;\;\;x + \frac{y \cdot t}{a - z}\\ \mathbf{else}:\\ \;\;\;\;x + \left(t - \frac{t}{\frac{z}{y}}\right)\\ \end{array} \]

Alternative 7: 77.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.1 \cdot 10^{+18}:\\ \;\;\;\;x + t\\ \mathbf{elif}\;z \leq 5.1 \cdot 10^{-34}:\\ \;\;\;\;x + \frac{y}{\frac{a}{t}}\\ \mathbf{else}:\\ \;\;\;\;x + t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -4.1e+18) (+ x t) (if (<= z 5.1e-34) (+ x (/ y (/ a t))) (+ x t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -4.1e+18) {
		tmp = x + t;
	} else if (z <= 5.1e-34) {
		tmp = x + (y / (a / t));
	} else {
		tmp = x + 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 <= (-4.1d+18)) then
        tmp = x + t
    else if (z <= 5.1d-34) then
        tmp = x + (y / (a / t))
    else
        tmp = x + 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 <= -4.1e+18) {
		tmp = x + t;
	} else if (z <= 5.1e-34) {
		tmp = x + (y / (a / t));
	} else {
		tmp = x + t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -4.1e+18:
		tmp = x + t
	elif z <= 5.1e-34:
		tmp = x + (y / (a / t))
	else:
		tmp = x + t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -4.1e+18)
		tmp = Float64(x + t);
	elseif (z <= 5.1e-34)
		tmp = Float64(x + Float64(y / Float64(a / t)));
	else
		tmp = Float64(x + t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -4.1e+18)
		tmp = x + t;
	elseif (z <= 5.1e-34)
		tmp = x + (y / (a / t));
	else
		tmp = x + t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -4.1e+18], N[(x + t), $MachinePrecision], If[LessEqual[z, 5.1e-34], N[(x + N[(y / N[(a / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + t), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -4.1e18 or 5.1000000000000001e-34 < z

   1. Initial program 78.9%

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

      1. associate-*l/ 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in z around inf 76.1%

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

   if -4.1e18 < z < 5.1000000000000001e-34

   1. Initial program 99.1%

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

      1. +-commutative 99.1%

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

      2. associate-*r/ 99.1%

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

      3. fma-def 99.1%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in z around 0 79.6%

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

      1. associate-/l* 80.4%

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

   6. Simplified 80.4%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.1 \cdot 10^{+18}:\\ \;\;\;\;x + t\\ \mathbf{elif}\;z \leq 5.1 \cdot 10^{-34}:\\ \;\;\;\;x + \frac{y}{\frac{a}{t}}\\ \mathbf{else}:\\ \;\;\;\;x + t\\ \end{array} \]

Alternative 8: 62.3% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -9 \cdot 10^{+141}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x + t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 (if (<= a -9e+141) x (+ x t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -9e+141) {
		tmp = x;
	} else {
		tmp = x + 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 (a <= (-9d+141)) then
        tmp = x
    else
        tmp = x + 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 (a <= -9e+141) {
		tmp = x;
	} else {
		tmp = x + t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if a <= -9e+141:
		tmp = x
	else:
		tmp = x + t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -9e+141)
		tmp = x;
	else
		tmp = Float64(x + t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (a <= -9e+141)
		tmp = x;
	else
		tmp = x + t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -9e+141], x, N[(x + t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -9.0000000000000003e141

   1. Initial program 76.0%

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

      1. +-commutative 76.0%

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

      2. associate-*r/ 97.4%

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

      3. fma-def 97.4%

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

   3. Simplified 97.4%

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

   4. Taylor expanded in t around 0 73.0%

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

   if -9.0000000000000003e141 < a

   1. Initial program 90.4%

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

      1. associate-*l/ 98.6%

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

   3. Simplified 98.6%

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

   4. Taylor expanded in z around inf 65.1%

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

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -9 \cdot 10^{+141}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x + t\\ \end{array} \]

Alternative 9: 51.4% 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 88.2%

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

   1. +-commutative 88.2%

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

   2. associate-*r/ 96.9%

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

   3. fma-def 96.9%

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

3. Simplified 96.9%

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

4. Taylor expanded in t around 0 52.3%

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

5. Final simplification 52.3%

   \[\leadsto x \]

Developer target: 99.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{y - z}{a - z} \cdot t\\ \mathbf{if}\;t < -1.0682974490174067 \cdot 10^{-39}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t < 3.9110949887586375 \cdot 10^{-141}:\\ \;\;\;\;x + \frac{\left(y - z\right) \cdot t}{a - z}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ x (* (/ (- y z) (- a z)) t))))
   (if (< t -1.0682974490174067e-39)
     t_1
     (if (< t 3.9110949887586375e-141) (+ x (/ (* (- y z) t) (- a z))) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x + (((y - z) / (a - z)) * t);
	double tmp;
	if (t < -1.0682974490174067e-39) {
		tmp = t_1;
	} else if (t < 3.9110949887586375e-141) {
		tmp = x + (((y - z) * t) / (a - z));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x + (((y - z) / (a - z)) * t)
    if (t < (-1.0682974490174067d-39)) then
        tmp = t_1
    else if (t < 3.9110949887586375d-141) then
        tmp = x + (((y - z) * t) / (a - z))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = x + (((y - z) / (a - z)) * t);
	double tmp;
	if (t < -1.0682974490174067e-39) {
		tmp = t_1;
	} else if (t < 3.9110949887586375e-141) {
		tmp = x + (((y - z) * t) / (a - z));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = x + (((y - z) / (a - z)) * t)
	tmp = 0
	if t < -1.0682974490174067e-39:
		tmp = t_1
	elif t < 3.9110949887586375e-141:
		tmp = x + (((y - z) * t) / (a - z))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x + Float64(Float64(Float64(y - z) / Float64(a - z)) * t))
	tmp = 0.0
	if (t < -1.0682974490174067e-39)
		tmp = t_1;
	elseif (t < 3.9110949887586375e-141)
		tmp = Float64(x + Float64(Float64(Float64(y - z) * t) / Float64(a - z)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = x + (((y - z) / (a - z)) * t);
	tmp = 0.0;
	if (t < -1.0682974490174067e-39)
		tmp = t_1;
	elseif (t < 3.9110949887586375e-141)
		tmp = x + (((y - z) * t) / (a - z));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x + N[(N[(N[(y - z), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]}, If[Less[t, -1.0682974490174067e-39], t$95$1, If[Less[t, 3.9110949887586375e-141], N[(x + N[(N[(N[(y - z), $MachinePrecision] * t), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Reproduce

herbie shell --seed 2023230 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTick from plot-0.2.3.4, A"
  :precision binary64

  :herbie-target
  (if (< t -1.0682974490174067e-39) (+ x (* (/ (- y z) (- a z)) t)) (if (< t 3.9110949887586375e-141) (+ x (/ (* (- y z) t) (- a z))) (+ x (* (/ (- y z) (- a z)) t))))

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