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

Percentage Accurate: 98.0% → 98.0%

Time: 7.9s

Alternatives: 9

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = x + (y * ((z - t) / (z - a)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 98.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = x + (y * ((z - t) / (z - a)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.0% 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 98.2%

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

2. Final simplification 98.2%

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

Alternative 2: 82.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ t_2 := y \cdot t_1\\ \mathbf{if}\;t_1 \leq -2 \cdot 10^{+62}:\\ \;\;\;\;t \cdot \left(-\frac{y}{z - a}\right)\\ \mathbf{elif}\;t_1 \leq -2 \cdot 10^{+23}:\\ \;\;\;\;x\\ \mathbf{elif}\;t_1 \leq -3.6 \cdot 10^{-22}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 0.04:\\ \;\;\;\;x + y \cdot \frac{t}{a}\\ \mathbf{elif}\;t_1 \leq 1.00000001:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))) (t_2 (* y t_1)))
   (if (<= t_1 -2e+62)
     (* t (- (/ y (- z a))))
     (if (<= t_1 -2e+23)
       x
       (if (<= t_1 -3.6e-22)
         t_2
         (if (<= t_1 0.04)
           (+ x (* y (/ t a)))
           (if (<= t_1 1.00000001) (+ x y) t_2)))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double t_2 = y * t_1;
	double tmp;
	if (t_1 <= -2e+62) {
		tmp = t * -(y / (z - a));
	} else if (t_1 <= -2e+23) {
		tmp = x;
	} else if (t_1 <= -3.6e-22) {
		tmp = t_2;
	} else if (t_1 <= 0.04) {
		tmp = x + (y * (t / a));
	} else if (t_1 <= 1.00000001) {
		tmp = x + y;
	} else {
		tmp = t_2;
	}
	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) :: t_2
    real(8) :: tmp
    t_1 = (z - t) / (z - a)
    t_2 = y * t_1
    if (t_1 <= (-2d+62)) then
        tmp = t * -(y / (z - a))
    else if (t_1 <= (-2d+23)) then
        tmp = x
    else if (t_1 <= (-3.6d-22)) then
        tmp = t_2
    else if (t_1 <= 0.04d0) then
        tmp = x + (y * (t / a))
    else if (t_1 <= 1.00000001d0) then
        tmp = x + y
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double t_2 = y * t_1;
	double tmp;
	if (t_1 <= -2e+62) {
		tmp = t * -(y / (z - a));
	} else if (t_1 <= -2e+23) {
		tmp = x;
	} else if (t_1 <= -3.6e-22) {
		tmp = t_2;
	} else if (t_1 <= 0.04) {
		tmp = x + (y * (t / a));
	} else if (t_1 <= 1.00000001) {
		tmp = x + y;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (z - t) / (z - a)
	t_2 = y * t_1
	tmp = 0
	if t_1 <= -2e+62:
		tmp = t * -(y / (z - a))
	elif t_1 <= -2e+23:
		tmp = x
	elif t_1 <= -3.6e-22:
		tmp = t_2
	elif t_1 <= 0.04:
		tmp = x + (y * (t / a))
	elif t_1 <= 1.00000001:
		tmp = x + y
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(z - a))
	t_2 = Float64(y * t_1)
	tmp = 0.0
	if (t_1 <= -2e+62)
		tmp = Float64(t * Float64(-Float64(y / Float64(z - a))));
	elseif (t_1 <= -2e+23)
		tmp = x;
	elseif (t_1 <= -3.6e-22)
		tmp = t_2;
	elseif (t_1 <= 0.04)
		tmp = Float64(x + Float64(y * Float64(t / a)));
	elseif (t_1 <= 1.00000001)
		tmp = Float64(x + y);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (z - t) / (z - a);
	t_2 = y * t_1;
	tmp = 0.0;
	if (t_1 <= -2e+62)
		tmp = t * -(y / (z - a));
	elseif (t_1 <= -2e+23)
		tmp = x;
	elseif (t_1 <= -3.6e-22)
		tmp = t_2;
	elseif (t_1 <= 0.04)
		tmp = x + (y * (t / a));
	elseif (t_1 <= 1.00000001)
		tmp = x + y;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(y * t$95$1), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+62], N[(t * (-N[(y / N[(z - a), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], If[LessEqual[t$95$1, -2e+23], x, If[LessEqual[t$95$1, -3.6e-22], t$95$2, If[LessEqual[t$95$1, 0.04], N[(x + N[(y * N[(t / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1.00000001], N[(x + y), $MachinePrecision], t$95$2]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < -2.00000000000000007e62

   1. Initial program 93.6%

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

      1. +-commutative 93.6%

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

      2. associate-*r/ 85.2%

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

      3. associate-*l/ 99.6%

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

      4. fma-def 99.7%

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

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{z - a}, z - t, x\right)} \]
   4. Step-by-step derivation

      1. fma-udef 99.6%

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

      2. associate-*l/ 85.2%

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

   5. Applied egg-rr 85.2%

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

   6. Taylor expanded in t around inf 85.2%

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

      1. associate-*r/ 85.2%

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

      2. mul-1-neg 85.2%

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

      3. distribute-lft-neg-out 85.2%

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

      4. *-commutative 85.2%

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

   8. Simplified 85.2%

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

   9. Taylor expanded in y around 0 85.2%

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

       1. mul-1-neg 85.2%

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

       2. *-commutative 85.2%

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

       3. associate-/l* 93.8%

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

       4. sub-neg 93.8%

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

       5. associate-/r/ 99.6%

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

       6. *-commutative 99.6%

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

   11. Simplified 99.6%

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

   12. Taylor expanded in x around 0 71.5%

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

       1. mul-1-neg 71.5%

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

       2. associate-*r/ 86.0%

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

       3. distribute-rgt-neg-in 86.0%

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

       4. distribute-neg-frac 86.0%

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

   14. Simplified 86.0%

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

   if -2.00000000000000007e62 < (/.f64 (-.f64 z t) (-.f64 z a)) < -1.9999999999999998e23

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 100.0%

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

   if -1.9999999999999998e23 < (/.f64 (-.f64 z t) (-.f64 z a)) < -3.5999999999999998e-22 or 1.0000000099999999 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 96.3%

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

      1. +-commutative 96.3%

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

      2. associate-*r/ 81.6%

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

      3. associate-*l/ 93.0%

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

      4. fma-def 93.0%

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

   3. Simplified 93.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{z - a}, z - t, x\right)} \]
   4. Step-by-step derivation

      1. fma-udef 93.0%

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

      2. associate-*l/ 81.6%

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

   5. Applied egg-rr 81.6%

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

   6. Taylor expanded in y around inf 80.3%

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

      1. div-sub 80.3%

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

   8. Simplified 80.3%

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

   if -3.5999999999999998e-22 < (/.f64 (-.f64 z t) (-.f64 z a)) < 0.0400000000000000008

   1. Initial program 99.4%

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

   2. Taylor expanded in z around 0 89.5%

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

   if 0.0400000000000000008 < (/.f64 (-.f64 z t) (-.f64 z a)) < 1.0000000099999999

   1. Initial program 99.9%

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

   2. Taylor expanded in z around inf 99.5%

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

      1. +-commutative 99.5%

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

   4. Simplified 99.5%

      \[\leadsto \color{blue}{y + x} \]
3. Recombined 5 regimes into one program.

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq -2 \cdot 10^{+62}:\\ \;\;\;\;t \cdot \left(-\frac{y}{z - a}\right)\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq -2 \cdot 10^{+23}:\\ \;\;\;\;x\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq -3.6 \cdot 10^{-22}:\\ \;\;\;\;y \cdot \frac{z - t}{z - a}\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 0.04:\\ \;\;\;\;x + y \cdot \frac{t}{a}\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 1.00000001:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z - t}{z - a}\\ \end{array} \]

Alternative 3: 82.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ t_2 := y \cdot t_1\\ \mathbf{if}\;t_1 \leq -2 \cdot 10^{+62}:\\ \;\;\;\;t \cdot \left(-\frac{y}{z - a}\right)\\ \mathbf{elif}\;t_1 \leq -2 \cdot 10^{+23}:\\ \;\;\;\;x\\ \mathbf{elif}\;t_1 \leq -3.6 \cdot 10^{-22}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 0.04:\\ \;\;\;\;x + y \cdot \frac{t}{a}\\ \mathbf{elif}\;t_1 \leq 1.00000001:\\ \;\;\;\;x + y \cdot \left(1 - \frac{t}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))) (t_2 (* y t_1)))
   (if (<= t_1 -2e+62)
     (* t (- (/ y (- z a))))
     (if (<= t_1 -2e+23)
       x
       (if (<= t_1 -3.6e-22)
         t_2
         (if (<= t_1 0.04)
           (+ x (* y (/ t a)))
           (if (<= t_1 1.00000001) (+ x (* y (- 1.0 (/ t z)))) t_2)))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double t_2 = y * t_1;
	double tmp;
	if (t_1 <= -2e+62) {
		tmp = t * -(y / (z - a));
	} else if (t_1 <= -2e+23) {
		tmp = x;
	} else if (t_1 <= -3.6e-22) {
		tmp = t_2;
	} else if (t_1 <= 0.04) {
		tmp = x + (y * (t / a));
	} else if (t_1 <= 1.00000001) {
		tmp = x + (y * (1.0 - (t / z)));
	} else {
		tmp = t_2;
	}
	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) :: t_2
    real(8) :: tmp
    t_1 = (z - t) / (z - a)
    t_2 = y * t_1
    if (t_1 <= (-2d+62)) then
        tmp = t * -(y / (z - a))
    else if (t_1 <= (-2d+23)) then
        tmp = x
    else if (t_1 <= (-3.6d-22)) then
        tmp = t_2
    else if (t_1 <= 0.04d0) then
        tmp = x + (y * (t / a))
    else if (t_1 <= 1.00000001d0) then
        tmp = x + (y * (1.0d0 - (t / z)))
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double t_2 = y * t_1;
	double tmp;
	if (t_1 <= -2e+62) {
		tmp = t * -(y / (z - a));
	} else if (t_1 <= -2e+23) {
		tmp = x;
	} else if (t_1 <= -3.6e-22) {
		tmp = t_2;
	} else if (t_1 <= 0.04) {
		tmp = x + (y * (t / a));
	} else if (t_1 <= 1.00000001) {
		tmp = x + (y * (1.0 - (t / z)));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (z - t) / (z - a)
	t_2 = y * t_1
	tmp = 0
	if t_1 <= -2e+62:
		tmp = t * -(y / (z - a))
	elif t_1 <= -2e+23:
		tmp = x
	elif t_1 <= -3.6e-22:
		tmp = t_2
	elif t_1 <= 0.04:
		tmp = x + (y * (t / a))
	elif t_1 <= 1.00000001:
		tmp = x + (y * (1.0 - (t / z)))
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(z - a))
	t_2 = Float64(y * t_1)
	tmp = 0.0
	if (t_1 <= -2e+62)
		tmp = Float64(t * Float64(-Float64(y / Float64(z - a))));
	elseif (t_1 <= -2e+23)
		tmp = x;
	elseif (t_1 <= -3.6e-22)
		tmp = t_2;
	elseif (t_1 <= 0.04)
		tmp = Float64(x + Float64(y * Float64(t / a)));
	elseif (t_1 <= 1.00000001)
		tmp = Float64(x + Float64(y * Float64(1.0 - Float64(t / z))));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (z - t) / (z - a);
	t_2 = y * t_1;
	tmp = 0.0;
	if (t_1 <= -2e+62)
		tmp = t * -(y / (z - a));
	elseif (t_1 <= -2e+23)
		tmp = x;
	elseif (t_1 <= -3.6e-22)
		tmp = t_2;
	elseif (t_1 <= 0.04)
		tmp = x + (y * (t / a));
	elseif (t_1 <= 1.00000001)
		tmp = x + (y * (1.0 - (t / z)));
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(y * t$95$1), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+62], N[(t * (-N[(y / N[(z - a), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], If[LessEqual[t$95$1, -2e+23], x, If[LessEqual[t$95$1, -3.6e-22], t$95$2, If[LessEqual[t$95$1, 0.04], N[(x + N[(y * N[(t / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1.00000001], N[(x + N[(y * N[(1.0 - N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < -2.00000000000000007e62

   1. Initial program 93.6%

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

      1. +-commutative 93.6%

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

      2. associate-*r/ 85.2%

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

      3. associate-*l/ 99.6%

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

      4. fma-def 99.7%

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

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{z - a}, z - t, x\right)} \]
   4. Step-by-step derivation

      1. fma-udef 99.6%

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

      2. associate-*l/ 85.2%

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

   5. Applied egg-rr 85.2%

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

   6. Taylor expanded in t around inf 85.2%

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

      1. associate-*r/ 85.2%

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

      2. mul-1-neg 85.2%

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

      3. distribute-lft-neg-out 85.2%

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

      4. *-commutative 85.2%

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

   8. Simplified 85.2%

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

   9. Taylor expanded in y around 0 85.2%

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

       1. mul-1-neg 85.2%

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

       2. *-commutative 85.2%

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

       3. associate-/l* 93.8%

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

       4. sub-neg 93.8%

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

       5. associate-/r/ 99.6%

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

       6. *-commutative 99.6%

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

   11. Simplified 99.6%

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

   12. Taylor expanded in x around 0 71.5%

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

       1. mul-1-neg 71.5%

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

       2. associate-*r/ 86.0%

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

       3. distribute-rgt-neg-in 86.0%

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

       4. distribute-neg-frac 86.0%

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

   14. Simplified 86.0%

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

   if -2.00000000000000007e62 < (/.f64 (-.f64 z t) (-.f64 z a)) < -1.9999999999999998e23

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 100.0%

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

   if -1.9999999999999998e23 < (/.f64 (-.f64 z t) (-.f64 z a)) < -3.5999999999999998e-22 or 1.0000000099999999 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 96.3%

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

      1. +-commutative 96.3%

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

      2. associate-*r/ 81.6%

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

      3. associate-*l/ 93.0%

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

      4. fma-def 93.0%

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

   3. Simplified 93.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{z - a}, z - t, x\right)} \]
   4. Step-by-step derivation

      1. fma-udef 93.0%

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

      2. associate-*l/ 81.6%

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

   5. Applied egg-rr 81.6%

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

   6. Taylor expanded in y around inf 80.3%

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

      1. div-sub 80.3%

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

   8. Simplified 80.3%

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

   if -3.5999999999999998e-22 < (/.f64 (-.f64 z t) (-.f64 z a)) < 0.0400000000000000008

   1. Initial program 99.4%

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

   2. Taylor expanded in z around 0 89.5%

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

   if 0.0400000000000000008 < (/.f64 (-.f64 z t) (-.f64 z a)) < 1.0000000099999999

   1. Initial program 99.9%

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

   2. Taylor expanded in a around 0 99.6%

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

      1. div-sub 99.6%

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

      2. *-inverses 99.6%

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

   4. Simplified 99.6%

      \[\leadsto x + y \cdot \color{blue}{\left(1 - \frac{t}{z}\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq -2 \cdot 10^{+62}:\\ \;\;\;\;t \cdot \left(-\frac{y}{z - a}\right)\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq -2 \cdot 10^{+23}:\\ \;\;\;\;x\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq -3.6 \cdot 10^{-22}:\\ \;\;\;\;y \cdot \frac{z - t}{z - a}\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 0.04:\\ \;\;\;\;x + y \cdot \frac{t}{a}\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 1.00000001:\\ \;\;\;\;x + y \cdot \left(1 - \frac{t}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z - t}{z - a}\\ \end{array} \]

Alternative 4: 82.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ t_2 := y \cdot t_1\\ \mathbf{if}\;t_1 \leq -2 \cdot 10^{+62}:\\ \;\;\;\;t \cdot \left(-\frac{y}{z - a}\right)\\ \mathbf{elif}\;t_1 \leq -2 \cdot 10^{+23}:\\ \;\;\;\;x\\ \mathbf{elif}\;t_1 \leq -3.6 \cdot 10^{-22}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq -4 \cdot 10^{-318}:\\ \;\;\;\;x + \frac{t}{\frac{a}{y}}\\ \mathbf{elif}\;t_1 \leq 1.00000001:\\ \;\;\;\;x + y \cdot \frac{z}{z - a}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))) (t_2 (* y t_1)))
   (if (<= t_1 -2e+62)
     (* t (- (/ y (- z a))))
     (if (<= t_1 -2e+23)
       x
       (if (<= t_1 -3.6e-22)
         t_2
         (if (<= t_1 -4e-318)
           (+ x (/ t (/ a y)))
           (if (<= t_1 1.00000001) (+ x (* y (/ z (- z a)))) t_2)))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double t_2 = y * t_1;
	double tmp;
	if (t_1 <= -2e+62) {
		tmp = t * -(y / (z - a));
	} else if (t_1 <= -2e+23) {
		tmp = x;
	} else if (t_1 <= -3.6e-22) {
		tmp = t_2;
	} else if (t_1 <= -4e-318) {
		tmp = x + (t / (a / y));
	} else if (t_1 <= 1.00000001) {
		tmp = x + (y * (z / (z - a)));
	} else {
		tmp = t_2;
	}
	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) :: t_2
    real(8) :: tmp
    t_1 = (z - t) / (z - a)
    t_2 = y * t_1
    if (t_1 <= (-2d+62)) then
        tmp = t * -(y / (z - a))
    else if (t_1 <= (-2d+23)) then
        tmp = x
    else if (t_1 <= (-3.6d-22)) then
        tmp = t_2
    else if (t_1 <= (-4d-318)) then
        tmp = x + (t / (a / y))
    else if (t_1 <= 1.00000001d0) then
        tmp = x + (y * (z / (z - a)))
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double t_2 = y * t_1;
	double tmp;
	if (t_1 <= -2e+62) {
		tmp = t * -(y / (z - a));
	} else if (t_1 <= -2e+23) {
		tmp = x;
	} else if (t_1 <= -3.6e-22) {
		tmp = t_2;
	} else if (t_1 <= -4e-318) {
		tmp = x + (t / (a / y));
	} else if (t_1 <= 1.00000001) {
		tmp = x + (y * (z / (z - a)));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (z - t) / (z - a)
	t_2 = y * t_1
	tmp = 0
	if t_1 <= -2e+62:
		tmp = t * -(y / (z - a))
	elif t_1 <= -2e+23:
		tmp = x
	elif t_1 <= -3.6e-22:
		tmp = t_2
	elif t_1 <= -4e-318:
		tmp = x + (t / (a / y))
	elif t_1 <= 1.00000001:
		tmp = x + (y * (z / (z - a)))
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(z - a))
	t_2 = Float64(y * t_1)
	tmp = 0.0
	if (t_1 <= -2e+62)
		tmp = Float64(t * Float64(-Float64(y / Float64(z - a))));
	elseif (t_1 <= -2e+23)
		tmp = x;
	elseif (t_1 <= -3.6e-22)
		tmp = t_2;
	elseif (t_1 <= -4e-318)
		tmp = Float64(x + Float64(t / Float64(a / y)));
	elseif (t_1 <= 1.00000001)
		tmp = Float64(x + Float64(y * Float64(z / Float64(z - a))));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (z - t) / (z - a);
	t_2 = y * t_1;
	tmp = 0.0;
	if (t_1 <= -2e+62)
		tmp = t * -(y / (z - a));
	elseif (t_1 <= -2e+23)
		tmp = x;
	elseif (t_1 <= -3.6e-22)
		tmp = t_2;
	elseif (t_1 <= -4e-318)
		tmp = x + (t / (a / y));
	elseif (t_1 <= 1.00000001)
		tmp = x + (y * (z / (z - a)));
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(y * t$95$1), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+62], N[(t * (-N[(y / N[(z - a), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], If[LessEqual[t$95$1, -2e+23], x, If[LessEqual[t$95$1, -3.6e-22], t$95$2, If[LessEqual[t$95$1, -4e-318], N[(x + N[(t / N[(a / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1.00000001], N[(x + N[(y * N[(z / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < -2.00000000000000007e62

   1. Initial program 93.6%

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

      1. +-commutative 93.6%

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

      2. associate-*r/ 85.2%

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

      3. associate-*l/ 99.6%

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

      4. fma-def 99.7%

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

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{z - a}, z - t, x\right)} \]
   4. Step-by-step derivation

      1. fma-udef 99.6%

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

      2. associate-*l/ 85.2%

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

   5. Applied egg-rr 85.2%

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

   6. Taylor expanded in t around inf 85.2%

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

      1. associate-*r/ 85.2%

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

      2. mul-1-neg 85.2%

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

      3. distribute-lft-neg-out 85.2%

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

      4. *-commutative 85.2%

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

   8. Simplified 85.2%

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

   9. Taylor expanded in y around 0 85.2%

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

       1. mul-1-neg 85.2%

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

       2. *-commutative 85.2%

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

       3. associate-/l* 93.8%

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

       4. sub-neg 93.8%

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

       5. associate-/r/ 99.6%

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

       6. *-commutative 99.6%

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

   11. Simplified 99.6%

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

   12. Taylor expanded in x around 0 71.5%

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

       1. mul-1-neg 71.5%

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

       2. associate-*r/ 86.0%

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

       3. distribute-rgt-neg-in 86.0%

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

       4. distribute-neg-frac 86.0%

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

   14. Simplified 86.0%

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

   if -2.00000000000000007e62 < (/.f64 (-.f64 z t) (-.f64 z a)) < -1.9999999999999998e23

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 100.0%

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

   if -1.9999999999999998e23 < (/.f64 (-.f64 z t) (-.f64 z a)) < -3.5999999999999998e-22 or 1.0000000099999999 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 96.3%

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

      1. +-commutative 96.3%

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

      2. associate-*r/ 81.6%

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

      3. associate-*l/ 93.0%

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

      4. fma-def 93.0%

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

   3. Simplified 93.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{z - a}, z - t, x\right)} \]
   4. Step-by-step derivation

      1. fma-udef 93.0%

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

      2. associate-*l/ 81.6%

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

   5. Applied egg-rr 81.6%

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

   6. Taylor expanded in y around inf 80.3%

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

      1. div-sub 80.3%

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

   8. Simplified 80.3%

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

   if -3.5999999999999998e-22 < (/.f64 (-.f64 z t) (-.f64 z a)) < -3.9999999e-318

   1. Initial program 98.6%

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

   2. Taylor expanded in z around 0 83.7%

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

      1. +-commutative 83.7%

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

      2. associate-/l* 91.1%

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

   4. Simplified 91.1%

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

   if -3.9999999e-318 < (/.f64 (-.f64 z t) (-.f64 z a)) < 1.0000000099999999

   1. Initial program 99.9%

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

   2. Taylor expanded in t around 0 96.7%

      \[\leadsto x + y \cdot \color{blue}{\frac{z}{z - a}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq -2 \cdot 10^{+62}:\\ \;\;\;\;t \cdot \left(-\frac{y}{z - a}\right)\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq -2 \cdot 10^{+23}:\\ \;\;\;\;x\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq -3.6 \cdot 10^{-22}:\\ \;\;\;\;y \cdot \frac{z - t}{z - a}\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq -4 \cdot 10^{-318}:\\ \;\;\;\;x + \frac{t}{\frac{a}{y}}\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 1.00000001:\\ \;\;\;\;x + y \cdot \frac{z}{z - a}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z - t}{z - a}\\ \end{array} \]

Alternative 5: 91.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ t_2 := x - t \cdot \frac{y}{z - a}\\ \mathbf{if}\;t_1 \leq -2 \cdot 10^{+23}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq -3.6 \cdot 10^{-22}:\\ \;\;\;\;y \cdot t_1\\ \mathbf{elif}\;t_1 \leq -4 \cdot 10^{-318}:\\ \;\;\;\;x + \frac{t}{\frac{a}{y}}\\ \mathbf{elif}\;t_1 \leq 10:\\ \;\;\;\;x + y \cdot \frac{z}{z - a}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))) (t_2 (- x (* t (/ y (- z a))))))
   (if (<= t_1 -2e+23)
     t_2
     (if (<= t_1 -3.6e-22)
       (* y t_1)
       (if (<= t_1 -4e-318)
         (+ x (/ t (/ a y)))
         (if (<= t_1 10.0) (+ x (* y (/ z (- z a)))) t_2))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double t_2 = x - (t * (y / (z - a)));
	double tmp;
	if (t_1 <= -2e+23) {
		tmp = t_2;
	} else if (t_1 <= -3.6e-22) {
		tmp = y * t_1;
	} else if (t_1 <= -4e-318) {
		tmp = x + (t / (a / y));
	} else if (t_1 <= 10.0) {
		tmp = x + (y * (z / (z - a)));
	} else {
		tmp = t_2;
	}
	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) :: t_2
    real(8) :: tmp
    t_1 = (z - t) / (z - a)
    t_2 = x - (t * (y / (z - a)))
    if (t_1 <= (-2d+23)) then
        tmp = t_2
    else if (t_1 <= (-3.6d-22)) then
        tmp = y * t_1
    else if (t_1 <= (-4d-318)) then
        tmp = x + (t / (a / y))
    else if (t_1 <= 10.0d0) then
        tmp = x + (y * (z / (z - a)))
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double t_2 = x - (t * (y / (z - a)));
	double tmp;
	if (t_1 <= -2e+23) {
		tmp = t_2;
	} else if (t_1 <= -3.6e-22) {
		tmp = y * t_1;
	} else if (t_1 <= -4e-318) {
		tmp = x + (t / (a / y));
	} else if (t_1 <= 10.0) {
		tmp = x + (y * (z / (z - a)));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (z - t) / (z - a)
	t_2 = x - (t * (y / (z - a)))
	tmp = 0
	if t_1 <= -2e+23:
		tmp = t_2
	elif t_1 <= -3.6e-22:
		tmp = y * t_1
	elif t_1 <= -4e-318:
		tmp = x + (t / (a / y))
	elif t_1 <= 10.0:
		tmp = x + (y * (z / (z - a)))
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(z - a))
	t_2 = Float64(x - Float64(t * Float64(y / Float64(z - a))))
	tmp = 0.0
	if (t_1 <= -2e+23)
		tmp = t_2;
	elseif (t_1 <= -3.6e-22)
		tmp = Float64(y * t_1);
	elseif (t_1 <= -4e-318)
		tmp = Float64(x + Float64(t / Float64(a / y)));
	elseif (t_1 <= 10.0)
		tmp = Float64(x + Float64(y * Float64(z / Float64(z - a))));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (z - t) / (z - a);
	t_2 = x - (t * (y / (z - a)));
	tmp = 0.0;
	if (t_1 <= -2e+23)
		tmp = t_2;
	elseif (t_1 <= -3.6e-22)
		tmp = y * t_1;
	elseif (t_1 <= -4e-318)
		tmp = x + (t / (a / y));
	elseif (t_1 <= 10.0)
		tmp = x + (y * (z / (z - a)));
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x - N[(t * N[(y / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+23], t$95$2, If[LessEqual[t$95$1, -3.6e-22], N[(y * t$95$1), $MachinePrecision], If[LessEqual[t$95$1, -4e-318], N[(x + N[(t / N[(a / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 10.0], N[(x + N[(y * N[(z / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < -1.9999999999999998e23 or 10 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 94.9%

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

      1. +-commutative 94.9%

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

      2. associate-*r/ 90.4%

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

      3. associate-*l/ 97.3%

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

      4. fma-def 97.3%

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

   3. Simplified 97.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{z - a}, z - t, x\right)} \]
   4. Step-by-step derivation

      1. fma-udef 97.3%

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

      2. associate-*l/ 90.4%

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

   5. Applied egg-rr 90.4%

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

   6. Taylor expanded in t around inf 90.4%

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

      1. associate-*r/ 90.4%

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

      2. mul-1-neg 90.4%

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

      3. distribute-lft-neg-out 90.4%

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

      4. *-commutative 90.4%

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

   8. Simplified 90.4%

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

   9. Taylor expanded in y around 0 90.4%

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

       1. mul-1-neg 90.4%

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

       2. *-commutative 90.4%

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

       3. associate-/l* 95.0%

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

       4. sub-neg 95.0%

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

       5. associate-/r/ 97.3%

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

       6. *-commutative 97.3%

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

   11. Simplified 97.3%

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

   if -1.9999999999999998e23 < (/.f64 (-.f64 z t) (-.f64 z a)) < -3.5999999999999998e-22

   1. Initial program 99.5%

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

      1. +-commutative 99.5%

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

      2. associate-*r/ 42.4%

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

      3. associate-*l/ 84.9%

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

      4. fma-def 85.0%

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

   3. Simplified 85.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{z - a}, z - t, x\right)} \]
   4. Step-by-step derivation

      1. fma-udef 84.9%

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

      2. associate-*l/ 42.4%

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

   5. Applied egg-rr 42.4%

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

   6. Taylor expanded in y around inf 98.3%

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

      1. div-sub 98.1%

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

   8. Simplified 98.1%

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

   if -3.5999999999999998e-22 < (/.f64 (-.f64 z t) (-.f64 z a)) < -3.9999999e-318

   1. Initial program 98.6%

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

   2. Taylor expanded in z around 0 83.7%

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

      1. +-commutative 83.7%

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

      2. associate-/l* 91.1%

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

   4. Simplified 91.1%

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

   if -3.9999999e-318 < (/.f64 (-.f64 z t) (-.f64 z a)) < 10

   1. Initial program 99.9%

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

   2. Taylor expanded in t around 0 96.6%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq -2 \cdot 10^{+23}:\\ \;\;\;\;x - t \cdot \frac{y}{z - a}\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq -3.6 \cdot 10^{-22}:\\ \;\;\;\;y \cdot \frac{z - t}{z - a}\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq -4 \cdot 10^{-318}:\\ \;\;\;\;x + \frac{t}{\frac{a}{y}}\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 10:\\ \;\;\;\;x + y \cdot \frac{z}{z - a}\\ \mathbf{else}:\\ \;\;\;\;x - t \cdot \frac{y}{z - a}\\ \end{array} \]

Alternative 6: 97.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ t_2 := x - t \cdot \frac{y}{z - a}\\ \mathbf{if}\;t_1 \leq -200:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 0.04:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \mathbf{elif}\;t_1 \leq 10:\\ \;\;\;\;x + \frac{y}{1 - \frac{a}{z}}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))) (t_2 (- x (* t (/ y (- z a))))))
   (if (<= t_1 -200.0)
     t_2
     (if (<= t_1 0.04)
       (- x (/ y (/ a (- z t))))
       (if (<= t_1 10.0) (+ x (/ y (- 1.0 (/ a z)))) t_2)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double t_2 = x - (t * (y / (z - a)));
	double tmp;
	if (t_1 <= -200.0) {
		tmp = t_2;
	} else if (t_1 <= 0.04) {
		tmp = x - (y / (a / (z - t)));
	} else if (t_1 <= 10.0) {
		tmp = x + (y / (1.0 - (a / z)));
	} else {
		tmp = t_2;
	}
	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) :: t_2
    real(8) :: tmp
    t_1 = (z - t) / (z - a)
    t_2 = x - (t * (y / (z - a)))
    if (t_1 <= (-200.0d0)) then
        tmp = t_2
    else if (t_1 <= 0.04d0) then
        tmp = x - (y / (a / (z - t)))
    else if (t_1 <= 10.0d0) then
        tmp = x + (y / (1.0d0 - (a / z)))
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double t_2 = x - (t * (y / (z - a)));
	double tmp;
	if (t_1 <= -200.0) {
		tmp = t_2;
	} else if (t_1 <= 0.04) {
		tmp = x - (y / (a / (z - t)));
	} else if (t_1 <= 10.0) {
		tmp = x + (y / (1.0 - (a / z)));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (z - t) / (z - a)
	t_2 = x - (t * (y / (z - a)))
	tmp = 0
	if t_1 <= -200.0:
		tmp = t_2
	elif t_1 <= 0.04:
		tmp = x - (y / (a / (z - t)))
	elif t_1 <= 10.0:
		tmp = x + (y / (1.0 - (a / z)))
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(z - a))
	t_2 = Float64(x - Float64(t * Float64(y / Float64(z - a))))
	tmp = 0.0
	if (t_1 <= -200.0)
		tmp = t_2;
	elseif (t_1 <= 0.04)
		tmp = Float64(x - Float64(y / Float64(a / Float64(z - t))));
	elseif (t_1 <= 10.0)
		tmp = Float64(x + Float64(y / Float64(1.0 - Float64(a / z))));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (z - t) / (z - a);
	t_2 = x - (t * (y / (z - a)));
	tmp = 0.0;
	if (t_1 <= -200.0)
		tmp = t_2;
	elseif (t_1 <= 0.04)
		tmp = x - (y / (a / (z - t)));
	elseif (t_1 <= 10.0)
		tmp = x + (y / (1.0 - (a / z)));
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x - N[(t * N[(y / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -200.0], t$95$2, If[LessEqual[t$95$1, 0.04], N[(x - N[(y / N[(a / N[(z - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 10.0], N[(x + N[(y / N[(1.0 - N[(a / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < -200 or 10 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 95.1%

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

      1. +-commutative 95.1%

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

      2. associate-*r/ 87.6%

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

      3. associate-*l/ 96.3%

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

      4. fma-def 96.3%

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

   3. Simplified 96.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{z - a}, z - t, x\right)} \]
   4. Step-by-step derivation

      1. fma-udef 96.3%

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

      2. associate-*l/ 87.6%

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

   5. Applied egg-rr 87.6%

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

   6. Taylor expanded in t around inf 87.0%

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

      1. associate-*r/ 87.0%

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

      2. mul-1-neg 87.0%

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

      3. distribute-lft-neg-out 87.0%

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

      4. *-commutative 87.0%

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

   8. Simplified 87.0%

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

   9. Taylor expanded in y around 0 87.0%

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

       1. mul-1-neg 87.0%

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

       2. *-commutative 87.0%

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

       3. associate-/l* 93.1%

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

       4. sub-neg 93.1%

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

       5. associate-/r/ 94.1%

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

       6. *-commutative 94.1%

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

   11. Simplified 94.1%

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

   if -200 < (/.f64 (-.f64 z t) (-.f64 z a)) < 0.0400000000000000008

   1. Initial program 99.4%

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

   2. Taylor expanded in a around inf 84.3%

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

      1. mul-1-neg 84.3%

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

      2. unsub-neg 84.3%

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

      3. associate-/l* 98.3%

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

   4. Simplified 98.3%

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

   if 0.0400000000000000008 < (/.f64 (-.f64 z t) (-.f64 z a)) < 10

   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/ 70.4%

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

      3. associate-*l/ 96.7%

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

      4. fma-def 96.7%

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

   3. Simplified 96.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{z - a}, z - t, x\right)} \]
   4. Step-by-step derivation

      1. fma-udef 96.7%

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

      2. associate-*l/ 70.4%

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

   5. Applied egg-rr 70.4%

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

   6. Taylor expanded in t around 0 70.1%

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

      1. associate-/l* 99.6%

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

      2. div-sub 99.7%

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

      3. *-inverses 99.7%

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

   8. Simplified 99.7%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq -200:\\ \;\;\;\;x - t \cdot \frac{y}{z - a}\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 0.04:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 10:\\ \;\;\;\;x + \frac{y}{1 - \frac{a}{z}}\\ \mathbf{else}:\\ \;\;\;\;x - t \cdot \frac{y}{z - a}\\ \end{array} \]

Alternative 7: 76.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.8 \cdot 10^{+58} \lor \neg \left(z \leq 1.25 \cdot 10^{-8}\right):\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{t}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -4.8e+58) (not (<= z 1.25e-8))) (+ x y) (+ x (* y (/ t a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -4.8e+58) || !(z <= 1.25e-8)) {
		tmp = x + y;
	} else {
		tmp = x + (y * (t / a));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((z <= (-4.8d+58)) .or. (.not. (z <= 1.25d-8))) then
        tmp = x + y
    else
        tmp = x + (y * (t / a))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -4.8e+58) || !(z <= 1.25e-8)) {
		tmp = x + y;
	} else {
		tmp = x + (y * (t / a));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (z <= -4.8e+58) or not (z <= 1.25e-8):
		tmp = x + y
	else:
		tmp = x + (y * (t / a))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -4.8e+58) || !(z <= 1.25e-8))
		tmp = Float64(x + y);
	else
		tmp = Float64(x + Float64(y * Float64(t / a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((z <= -4.8e+58) || ~((z <= 1.25e-8)))
		tmp = x + y;
	else
		tmp = x + (y * (t / a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -4.8e+58], N[Not[LessEqual[z, 1.25e-8]], $MachinePrecision]], N[(x + y), $MachinePrecision], N[(x + N[(y * N[(t / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4.8e58 or 1.2499999999999999e-8 < z

   1. Initial program 99.9%

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

   2. Taylor expanded in z around inf 79.0%

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

      1. +-commutative 79.0%

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

   4. Simplified 79.0%

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

   if -4.8e58 < z < 1.2499999999999999e-8

   1. Initial program 97.1%

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

   2. Taylor expanded in z around 0 80.5%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.8 \cdot 10^{+58} \lor \neg \left(z \leq 1.25 \cdot 10^{-8}\right):\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{t}{a}\\ \end{array} \]

Alternative 8: 63.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.2 \cdot 10^{-59}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq 6 \cdot 10^{-21}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -2.2e-59) (+ x y) (if (<= z 6e-21) x (+ x y))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.2e-59) {
		tmp = x + y;
	} else if (z <= 6e-21) {
		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 <= (-2.2d-59)) then
        tmp = x + y
    else if (z <= 6d-21) 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 <= -2.2e-59) {
		tmp = x + y;
	} else if (z <= 6e-21) {
		tmp = x;
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -2.2e-59:
		tmp = x + y
	elif z <= 6e-21:
		tmp = x
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -2.2e-59)
		tmp = Float64(x + y);
	elseif (z <= 6e-21)
		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 <= -2.2e-59)
		tmp = x + y;
	elseif (z <= 6e-21)
		tmp = x;
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -2.2e-59], N[(x + y), $MachinePrecision], If[LessEqual[z, 6e-21], x, N[(x + y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.1999999999999999e-59 or 5.99999999999999982e-21 < z

   1. Initial program 99.9%

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

   2. Taylor expanded in z around inf 71.7%

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

      1. +-commutative 71.7%

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

   4. Simplified 71.7%

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

   if -2.1999999999999999e-59 < z < 5.99999999999999982e-21

   1. Initial program 96.6%

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

   2. Taylor expanded in x around inf 54.6%

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

4. Final simplification 62.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.2 \cdot 10^{-59}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq 6 \cdot 10^{-21}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 9: 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 98.2%

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

2. Taylor expanded in x around inf 48.2%

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

3. Final simplification 48.2%

   \[\leadsto x \]

Developer target: 98.2% 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 2023290 
(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)))))