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

Percentage Accurate: 97.9% → 97.9%

Time: 8.3s

Alternatives: 14

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.7%

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

2. Final simplification 97.7%

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

Alternative 2: 75.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.05 \cdot 10^{+97}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;t \leq -5.2 \cdot 10^{+14}:\\ \;\;\;\;x - \frac{y}{\frac{t}{z}}\\ \mathbf{elif}\;t \leq -2300:\\ \;\;\;\;t \cdot \frac{y}{t - a}\\ \mathbf{elif}\;t \leq 5.3 \cdot 10^{-27}:\\ \;\;\;\;x + \frac{y \cdot z}{a}\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -1.05e+97)
   (+ x y)
   (if (<= t -5.2e+14)
     (- x (/ y (/ t z)))
     (if (<= t -2300.0)
       (* t (/ y (- t a)))
       (if (<= t 5.3e-27) (+ x (/ (* y z) a)) (+ x y))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -1.05e+97) {
		tmp = x + y;
	} else if (t <= -5.2e+14) {
		tmp = x - (y / (t / z));
	} else if (t <= -2300.0) {
		tmp = t * (y / (t - a));
	} else if (t <= 5.3e-27) {
		tmp = x + ((y * z) / a);
	} else {
		tmp = x + y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (t <= (-1.05d+97)) then
        tmp = x + y
    else if (t <= (-5.2d+14)) then
        tmp = x - (y / (t / z))
    else if (t <= (-2300.0d0)) then
        tmp = t * (y / (t - a))
    else if (t <= 5.3d-27) then
        tmp = x + ((y * z) / a)
    else
        tmp = x + y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -1.05e+97) {
		tmp = x + y;
	} else if (t <= -5.2e+14) {
		tmp = x - (y / (t / z));
	} else if (t <= -2300.0) {
		tmp = t * (y / (t - a));
	} else if (t <= 5.3e-27) {
		tmp = x + ((y * z) / a);
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= -1.05e+97:
		tmp = x + y
	elif t <= -5.2e+14:
		tmp = x - (y / (t / z))
	elif t <= -2300.0:
		tmp = t * (y / (t - a))
	elif t <= 5.3e-27:
		tmp = x + ((y * z) / a)
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -1.05e+97)
		tmp = Float64(x + y);
	elseif (t <= -5.2e+14)
		tmp = Float64(x - Float64(y / Float64(t / z)));
	elseif (t <= -2300.0)
		tmp = Float64(t * Float64(y / Float64(t - a)));
	elseif (t <= 5.3e-27)
		tmp = Float64(x + Float64(Float64(y * z) / a));
	else
		tmp = Float64(x + y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= -1.05e+97)
		tmp = x + y;
	elseif (t <= -5.2e+14)
		tmp = x - (y / (t / z));
	elseif (t <= -2300.0)
		tmp = t * (y / (t - a));
	elseif (t <= 5.3e-27)
		tmp = x + ((y * z) / a);
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -1.05e+97], N[(x + y), $MachinePrecision], If[LessEqual[t, -5.2e+14], N[(x - N[(y / N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -2300.0], N[(t * N[(y / N[(t - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 5.3e-27], N[(x + N[(N[(y * z), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(x + y), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -1.05000000000000006e97 or 5.30000000000000006e-27 < t

   1. Initial program 100.0%

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

   2. Taylor expanded in t around inf 78.2%

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

   if -1.05000000000000006e97 < t < -5.2e14

   1. Initial program 99.8%

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

   2. Taylor expanded in z around inf 83.9%

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

   3. Taylor expanded in a around 0 65.1%

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

      1. +-commutative 65.1%

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

      2. mul-1-neg 65.1%

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

      3. unsub-neg 65.1%

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

      4. associate-/l* 75.9%

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

   5. Simplified 75.9%

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

   if -5.2e14 < t < -2300

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

      2. *-commutative 99.7%

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

      3. associate-*l/ 99.7%

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

      4. sub-neg 99.7%

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

      5. +-commutative 99.7%

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

      6. neg-sub0 99.7%

         \[\leadsto \frac{\left(z - t\right) \cdot y}{\color{blue}{\left(0 - t\right)} + a} + x \]

      7. associate-+l- 99.7%

         \[\leadsto \frac{\left(z - t\right) \cdot y}{\color{blue}{0 - \left(t - a\right)}} + x \]

      8. sub0-neg 99.7%

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

      9. neg-mul-1 99.7%

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

      10. times-frac 99.5%

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

      11. fma-def 99.5%

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

      12. sub-neg 99.5%

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

      13. +-commutative 99.5%

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

      14. neg-sub0 99.5%

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\left(0 - t\right)} + z}{-1}, \frac{y}{t - a}, x\right) \]

      15. associate-+l- 99.5%

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{0 - \left(t - z\right)}}{-1}, \frac{y}{t - a}, x\right) \]

      16. sub0-neg 99.5%

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

      17. neg-mul-1 99.5%

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

      18. *-commutative 99.5%

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

      19. associate-/l* 99.5%

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

      20. metadata-eval 99.5%

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

      21. /-rgt-identity 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in y around -inf 99.7%

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

      1. *-commutative 99.7%

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

      2. associate-/l* 99.7%

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

   6. Simplified 99.7%

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

   7. Taylor expanded in z around 0 67.9%

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

      1. *-commutative 67.9%

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

      2. associate-*r/ 67.6%

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

   9. Simplified 67.6%

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

   if -2300 < t < 5.30000000000000006e-27

   1. Initial program 94.8%

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

   2. Taylor expanded in t around 0 76.3%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.05 \cdot 10^{+97}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;t \leq -5.2 \cdot 10^{+14}:\\ \;\;\;\;x - \frac{y}{\frac{t}{z}}\\ \mathbf{elif}\;t \leq -2300:\\ \;\;\;\;t \cdot \frac{y}{t - a}\\ \mathbf{elif}\;t \leq 5.3 \cdot 10^{-27}:\\ \;\;\;\;x + \frac{y \cdot z}{a}\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 3: 76.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -9 \cdot 10^{+97}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;t \leq -5.2 \cdot 10^{+14}:\\ \;\;\;\;x - \frac{y}{\frac{t}{z}}\\ \mathbf{elif}\;t \leq -3100000:\\ \;\;\;\;\frac{y}{\frac{t}{t - z}}\\ \mathbf{elif}\;t \leq 5 \cdot 10^{-25}:\\ \;\;\;\;x + \frac{y \cdot z}{a}\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -9e+97)
   (+ x y)
   (if (<= t -5.2e+14)
     (- x (/ y (/ t z)))
     (if (<= t -3100000.0)
       (/ y (/ t (- t z)))
       (if (<= t 5e-25) (+ x (/ (* y z) a)) (+ x y))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -9e+97) {
		tmp = x + y;
	} else if (t <= -5.2e+14) {
		tmp = x - (y / (t / z));
	} else if (t <= -3100000.0) {
		tmp = y / (t / (t - z));
	} else if (t <= 5e-25) {
		tmp = x + ((y * z) / a);
	} else {
		tmp = x + y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (t <= (-9d+97)) then
        tmp = x + y
    else if (t <= (-5.2d+14)) then
        tmp = x - (y / (t / z))
    else if (t <= (-3100000.0d0)) then
        tmp = y / (t / (t - z))
    else if (t <= 5d-25) then
        tmp = x + ((y * z) / a)
    else
        tmp = x + y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -9e+97) {
		tmp = x + y;
	} else if (t <= -5.2e+14) {
		tmp = x - (y / (t / z));
	} else if (t <= -3100000.0) {
		tmp = y / (t / (t - z));
	} else if (t <= 5e-25) {
		tmp = x + ((y * z) / a);
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= -9e+97:
		tmp = x + y
	elif t <= -5.2e+14:
		tmp = x - (y / (t / z))
	elif t <= -3100000.0:
		tmp = y / (t / (t - z))
	elif t <= 5e-25:
		tmp = x + ((y * z) / a)
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -9e+97)
		tmp = Float64(x + y);
	elseif (t <= -5.2e+14)
		tmp = Float64(x - Float64(y / Float64(t / z)));
	elseif (t <= -3100000.0)
		tmp = Float64(y / Float64(t / Float64(t - z)));
	elseif (t <= 5e-25)
		tmp = Float64(x + Float64(Float64(y * z) / a));
	else
		tmp = Float64(x + y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= -9e+97)
		tmp = x + y;
	elseif (t <= -5.2e+14)
		tmp = x - (y / (t / z));
	elseif (t <= -3100000.0)
		tmp = y / (t / (t - z));
	elseif (t <= 5e-25)
		tmp = x + ((y * z) / a);
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -9e+97], N[(x + y), $MachinePrecision], If[LessEqual[t, -5.2e+14], N[(x - N[(y / N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -3100000.0], N[(y / N[(t / N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 5e-25], N[(x + N[(N[(y * z), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(x + y), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -8.99999999999999952e97 or 4.99999999999999962e-25 < t

   1. Initial program 100.0%

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

   2. Taylor expanded in t around inf 78.2%

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

   if -8.99999999999999952e97 < t < -5.2e14

   1. Initial program 99.8%

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

   2. Taylor expanded in z around inf 83.9%

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

   3. Taylor expanded in a around 0 65.1%

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

      1. +-commutative 65.1%

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

      2. mul-1-neg 65.1%

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

      3. unsub-neg 65.1%

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

      4. associate-/l* 75.9%

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

   5. Simplified 75.9%

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

   if -5.2e14 < t < -3.1e6

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

      2. *-commutative 99.7%

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

      3. associate-*l/ 99.7%

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

      4. sub-neg 99.7%

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

      5. +-commutative 99.7%

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

      6. neg-sub0 99.7%

         \[\leadsto \frac{\left(z - t\right) \cdot y}{\color{blue}{\left(0 - t\right)} + a} + x \]

      7. associate-+l- 99.7%

         \[\leadsto \frac{\left(z - t\right) \cdot y}{\color{blue}{0 - \left(t - a\right)}} + x \]

      8. sub0-neg 99.7%

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

      9. neg-mul-1 99.7%

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

      10. times-frac 99.4%

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

      11. fma-def 99.4%

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

      12. sub-neg 99.4%

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

      13. +-commutative 99.4%

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

      14. neg-sub0 99.4%

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\left(0 - t\right)} + z}{-1}, \frac{y}{t - a}, x\right) \]

      15. associate-+l- 99.4%

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{0 - \left(t - z\right)}}{-1}, \frac{y}{t - a}, x\right) \]

      16. sub0-neg 99.4%

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

      17. neg-mul-1 99.4%

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

      18. *-commutative 99.4%

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

      19. associate-/l* 99.4%

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

      20. metadata-eval 99.4%

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

      21. /-rgt-identity 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in y around -inf 99.7%

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

      1. *-commutative 99.7%

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

      2. associate-/l* 99.7%

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

   6. Simplified 99.7%

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

   7. Taylor expanded in a around 0 80.5%

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

      1. associate-/l* 80.8%

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

   9. Simplified 80.8%

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

   if -3.1e6 < t < 4.99999999999999962e-25

   1. Initial program 94.9%

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

   2. Taylor expanded in t around 0 75.7%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -9 \cdot 10^{+97}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;t \leq -5.2 \cdot 10^{+14}:\\ \;\;\;\;x - \frac{y}{\frac{t}{z}}\\ \mathbf{elif}\;t \leq -3100000:\\ \;\;\;\;\frac{y}{\frac{t}{t - z}}\\ \mathbf{elif}\;t \leq 5 \cdot 10^{-25}:\\ \;\;\;\;x + \frac{y \cdot z}{a}\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 4: 84.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -6 \cdot 10^{+96} \lor \neg \left(t \leq 1.9 \cdot 10^{+133}\right):\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z}{a - t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= t -6e+96) (not (<= t 1.9e+133)))
   (+ x y)
   (+ x (* y (/ z (- a t))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((t <= -6e+96) || !(t <= 1.9e+133)) {
		tmp = x + y;
	} else {
		tmp = x + (y * (z / (a - t)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((t <= (-6d+96)) .or. (.not. (t <= 1.9d+133))) then
        tmp = x + y
    else
        tmp = x + (y * (z / (a - t)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((t <= -6e+96) || !(t <= 1.9e+133)) {
		tmp = x + y;
	} else {
		tmp = x + (y * (z / (a - t)));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (t <= -6e+96) or not (t <= 1.9e+133):
		tmp = x + y
	else:
		tmp = x + (y * (z / (a - t)))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((t <= -6e+96) || !(t <= 1.9e+133))
		tmp = Float64(x + y);
	else
		tmp = Float64(x + Float64(y * Float64(z / Float64(a - t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((t <= -6e+96) || ~((t <= 1.9e+133)))
		tmp = x + y;
	else
		tmp = x + (y * (z / (a - t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[t, -6e+96], N[Not[LessEqual[t, 1.9e+133]], $MachinePrecision]], N[(x + y), $MachinePrecision], N[(x + N[(y * N[(z / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -6.0000000000000001e96 or 1.9000000000000001e133 < t

   1. Initial program 100.0%

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

   2. Taylor expanded in t around inf 80.9%

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

   if -6.0000000000000001e96 < t < 1.9000000000000001e133

   1. Initial program 96.6%

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

   2. Taylor expanded in z around inf 84.9%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -6 \cdot 10^{+96} \lor \neg \left(t \leq 1.9 \cdot 10^{+133}\right):\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z}{a - t}\\ \end{array} \]

Alternative 5: 86.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6.5 \cdot 10^{+84} \lor \neg \left(z \leq 4.7 \cdot 10^{+27}\right):\\ \;\;\;\;x + y \cdot \frac{z}{a - t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{a}{t} + -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -6.5e+84) (not (<= z 4.7e+27)))
   (+ x (* y (/ z (- a t))))
   (- x (/ y (+ (/ a t) -1.0)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -6.5e+84) || !(z <= 4.7e+27)) {
		tmp = x + (y * (z / (a - t)));
	} else {
		tmp = x - (y / ((a / t) + -1.0));
	}
	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 <= (-6.5d+84)) .or. (.not. (z <= 4.7d+27))) then
        tmp = x + (y * (z / (a - t)))
    else
        tmp = x - (y / ((a / t) + (-1.0d0)))
    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 <= -6.5e+84) || !(z <= 4.7e+27)) {
		tmp = x + (y * (z / (a - t)));
	} else {
		tmp = x - (y / ((a / t) + -1.0));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (z <= -6.5e+84) or not (z <= 4.7e+27):
		tmp = x + (y * (z / (a - t)))
	else:
		tmp = x - (y / ((a / t) + -1.0))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -6.5e+84) || !(z <= 4.7e+27))
		tmp = Float64(x + Float64(y * Float64(z / Float64(a - t))));
	else
		tmp = Float64(x - Float64(y / Float64(Float64(a / t) + -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((z <= -6.5e+84) || ~((z <= 4.7e+27)))
		tmp = x + (y * (z / (a - t)));
	else
		tmp = x - (y / ((a / t) + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -6.5e+84], N[Not[LessEqual[z, 4.7e+27]], $MachinePrecision]], N[(x + N[(y * N[(z / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(y / N[(N[(a / t), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -6.50000000000000027e84 or 4.69999999999999976e27 < z

   1. Initial program 95.3%

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

   2. Taylor expanded in z around inf 89.4%

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

   if -6.50000000000000027e84 < z < 4.69999999999999976e27

   1. Initial program 99.3%

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

   2. Taylor expanded in z around 0 77.1%

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

      1. +-commutative 77.1%

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

      2. mul-1-neg 77.1%

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

      3. unsub-neg 77.1%

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

      4. associate-/l* 91.3%

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

      5. div-sub 91.3%

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

      6. *-inverses 91.3%

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

   4. Simplified 91.3%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.5 \cdot 10^{+84} \lor \neg \left(z \leq 4.7 \cdot 10^{+27}\right):\\ \;\;\;\;x + y \cdot \frac{z}{a - t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{a}{t} + -1}\\ \end{array} \]

Alternative 6: 76.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.2 \cdot 10^{-17}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;t \leq 2 \cdot 10^{-26}:\\ \;\;\;\;x + y \cdot \frac{z}{a}\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -1.2e-17) (+ x y) (if (<= t 2e-26) (+ x (* y (/ z a))) (+ x y))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -1.2e-17) {
		tmp = x + y;
	} else if (t <= 2e-26) {
		tmp = x + (y * (z / a));
	} else {
		tmp = x + y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (t <= (-1.2d-17)) then
        tmp = x + y
    else if (t <= 2d-26) then
        tmp = x + (y * (z / a))
    else
        tmp = x + y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -1.2e-17) {
		tmp = x + y;
	} else if (t <= 2e-26) {
		tmp = x + (y * (z / a));
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= -1.2e-17:
		tmp = x + y
	elif t <= 2e-26:
		tmp = x + (y * (z / a))
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -1.2e-17)
		tmp = Float64(x + y);
	elseif (t <= 2e-26)
		tmp = Float64(x + Float64(y * Float64(z / a)));
	else
		tmp = Float64(x + y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= -1.2e-17)
		tmp = x + y;
	elseif (t <= 2e-26)
		tmp = x + (y * (z / a));
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -1.2e-17], N[(x + y), $MachinePrecision], If[LessEqual[t, 2e-26], N[(x + N[(y * N[(z / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.19999999999999993e-17 or 2.0000000000000001e-26 < t

   1. Initial program 99.9%

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

   2. Taylor expanded in t around inf 73.5%

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

   if -1.19999999999999993e-17 < t < 2.0000000000000001e-26

   1. Initial program 94.6%

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

   2. Taylor expanded in t around 0 76.1%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.2 \cdot 10^{-17}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;t \leq 2 \cdot 10^{-26}:\\ \;\;\;\;x + y \cdot \frac{z}{a}\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 7: 76.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.6 \cdot 10^{-21}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;t \leq 1.85 \cdot 10^{-22}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z}}\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -1.6e-21) (+ x y) (if (<= t 1.85e-22) (+ x (/ y (/ a z))) (+ x y))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -1.6e-21) {
		tmp = x + y;
	} else if (t <= 1.85e-22) {
		tmp = x + (y / (a / z));
	} else {
		tmp = x + y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (t <= (-1.6d-21)) then
        tmp = x + y
    else if (t <= 1.85d-22) then
        tmp = x + (y / (a / z))
    else
        tmp = x + y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -1.6e-21) {
		tmp = x + y;
	} else if (t <= 1.85e-22) {
		tmp = x + (y / (a / z));
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= -1.6e-21:
		tmp = x + y
	elif t <= 1.85e-22:
		tmp = x + (y / (a / z))
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -1.6e-21)
		tmp = Float64(x + y);
	elseif (t <= 1.85e-22)
		tmp = Float64(x + Float64(y / Float64(a / z)));
	else
		tmp = Float64(x + y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= -1.6e-21)
		tmp = x + y;
	elseif (t <= 1.85e-22)
		tmp = x + (y / (a / z));
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -1.6e-21], N[(x + y), $MachinePrecision], If[LessEqual[t, 1.85e-22], N[(x + N[(y / N[(a / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.6000000000000001e-21 or 1.85e-22 < t

   1. Initial program 99.9%

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

   2. Taylor expanded in t around inf 73.5%

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

   if -1.6000000000000001e-21 < t < 1.85e-22

   1. Initial program 94.6%

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

   2. Taylor expanded in t around 0 77.2%

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

      1. associate-/l* 76.7%

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

   4. Simplified 76.7%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.6 \cdot 10^{-21}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;t \leq 1.85 \cdot 10^{-22}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z}}\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 8: 76.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.3 \cdot 10^{-22}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;t \leq 1.2 \cdot 10^{-29}:\\ \;\;\;\;x + \frac{y \cdot z}{a}\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -2.3e-22) (+ x y) (if (<= t 1.2e-29) (+ x (/ (* y z) a)) (+ x y))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -2.3e-22) {
		tmp = x + y;
	} else if (t <= 1.2e-29) {
		tmp = x + ((y * z) / a);
	} else {
		tmp = x + y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (t <= (-2.3d-22)) then
        tmp = x + y
    else if (t <= 1.2d-29) then
        tmp = x + ((y * z) / a)
    else
        tmp = x + y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -2.3e-22) {
		tmp = x + y;
	} else if (t <= 1.2e-29) {
		tmp = x + ((y * z) / a);
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= -2.3e-22:
		tmp = x + y
	elif t <= 1.2e-29:
		tmp = x + ((y * z) / a)
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -2.3e-22)
		tmp = Float64(x + y);
	elseif (t <= 1.2e-29)
		tmp = Float64(x + Float64(Float64(y * z) / a));
	else
		tmp = Float64(x + y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= -2.3e-22)
		tmp = x + y;
	elseif (t <= 1.2e-29)
		tmp = x + ((y * z) / a);
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -2.3e-22], N[(x + y), $MachinePrecision], If[LessEqual[t, 1.2e-29], N[(x + N[(N[(y * z), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(x + y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < -2.2999999999999998e-22 or 1.19999999999999996e-29 < t

   1. Initial program 99.9%

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

   2. Taylor expanded in t around inf 73.5%

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

   if -2.2999999999999998e-22 < t < 1.19999999999999996e-29

   1. Initial program 94.6%

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

   2. Taylor expanded in t around 0 77.2%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.3 \cdot 10^{-22}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;t \leq 1.2 \cdot 10^{-29}:\\ \;\;\;\;x + \frac{y \cdot z}{a}\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 9: 59.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -7.8 \cdot 10^{+180}:\\ \;\;\;\;\left(t - z\right) \cdot \frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -7.8e+180) (* (- t z) (/ y t)) (+ x y)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -7.8e+180) {
		tmp = (t - z) * (y / t);
	} else {
		tmp = x + y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-7.8d+180)) then
        tmp = (t - z) * (y / t)
    else
        tmp = x + y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -7.8e+180) {
		tmp = (t - z) * (y / t);
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -7.8e+180:
		tmp = (t - z) * (y / t)
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -7.8e+180)
		tmp = Float64(Float64(t - z) * Float64(y / t));
	else
		tmp = Float64(x + y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -7.8e+180)
		tmp = (t - z) * (y / t);
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -7.8e+180], N[(N[(t - z), $MachinePrecision] * N[(y / t), $MachinePrecision]), $MachinePrecision], N[(x + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -7.8000000000000002e180

   1. Initial program 93.9%

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

      1. +-commutative 93.9%

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

      2. *-commutative 93.9%

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

      3. associate-*l/ 80.2%

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

      4. sub-neg 80.2%

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

      5. +-commutative 80.2%

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

      6. neg-sub0 80.2%

         \[\leadsto \frac{\left(z - t\right) \cdot y}{\color{blue}{\left(0 - t\right)} + a} + x \]

      7. associate-+l- 80.2%

         \[\leadsto \frac{\left(z - t\right) \cdot y}{\color{blue}{0 - \left(t - a\right)}} + x \]

      8. sub0-neg 80.2%

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

      9. neg-mul-1 80.2%

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

      10. times-frac 92.5%

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

      11. fma-def 92.5%

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

      12. sub-neg 92.5%

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

      13. +-commutative 92.5%

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

      14. neg-sub0 92.5%

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\left(0 - t\right)} + z}{-1}, \frac{y}{t - a}, x\right) \]

      15. associate-+l- 92.5%

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{0 - \left(t - z\right)}}{-1}, \frac{y}{t - a}, x\right) \]

      16. sub0-neg 92.5%

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

      17. neg-mul-1 92.5%

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

      18. *-commutative 92.5%

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

      19. associate-/l* 92.5%

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

      20. metadata-eval 92.5%

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

      21. /-rgt-identity 92.5%

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

   3. Simplified 92.5%

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

   4. Taylor expanded in y around -inf 71.1%

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

      1. *-commutative 71.1%

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

      2. associate-/l* 73.7%

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

   6. Simplified 73.7%

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

   7. Taylor expanded in a around 0 44.2%

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

      1. associate-/l* 49.8%

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

   9. Simplified 49.8%

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

   10. Taylor expanded in y around 0 44.2%

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

       1. associate-/l* 49.8%

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

       2. associate-/r/ 51.6%

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

   12. Simplified 51.6%

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

   if -7.8000000000000002e180 < z

   1. Initial program 98.2%

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

   2. Taylor expanded in t around inf 64.5%

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

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.8 \cdot 10^{+180}:\\ \;\;\;\;\left(t - z\right) \cdot \frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 10: 59.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.6 \cdot 10^{+182}:\\ \;\;\;\;y \cdot \frac{-z}{t}\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.6e+182) (* y (/ (- z) t)) (+ x y)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.6e+182) {
		tmp = y * (-z / t);
	} else {
		tmp = x + y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-1.6d+182)) then
        tmp = y * (-z / t)
    else
        tmp = x + y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.6e+182) {
		tmp = y * (-z / t);
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.6e+182:
		tmp = y * (-z / t)
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.6e+182)
		tmp = Float64(y * Float64(Float64(-z) / t));
	else
		tmp = Float64(x + y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -1.6e+182)
		tmp = y * (-z / t);
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.6e+182], N[(y * N[((-z) / t), $MachinePrecision]), $MachinePrecision], N[(x + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.5999999999999999e182

   1. Initial program 93.9%

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

      1. +-commutative 93.9%

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

      2. *-commutative 93.9%

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

      3. associate-*l/ 80.2%

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

      4. sub-neg 80.2%

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

      5. +-commutative 80.2%

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

      6. neg-sub0 80.2%

         \[\leadsto \frac{\left(z - t\right) \cdot y}{\color{blue}{\left(0 - t\right)} + a} + x \]

      7. associate-+l- 80.2%

         \[\leadsto \frac{\left(z - t\right) \cdot y}{\color{blue}{0 - \left(t - a\right)}} + x \]

      8. sub0-neg 80.2%

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

      9. neg-mul-1 80.2%

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

      10. times-frac 92.5%

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

      11. fma-def 92.5%

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

      12. sub-neg 92.5%

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

      13. +-commutative 92.5%

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

      14. neg-sub0 92.5%

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\left(0 - t\right)} + z}{-1}, \frac{y}{t - a}, x\right) \]

      15. associate-+l- 92.5%

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{0 - \left(t - z\right)}}{-1}, \frac{y}{t - a}, x\right) \]

      16. sub0-neg 92.5%

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

      17. neg-mul-1 92.5%

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

      18. *-commutative 92.5%

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

      19. associate-/l* 92.5%

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

      20. metadata-eval 92.5%

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

      21. /-rgt-identity 92.5%

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

   3. Simplified 92.5%

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

   4. Taylor expanded in y around -inf 71.1%

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

      1. *-commutative 71.1%

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

      2. associate-/l* 73.7%

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

   6. Simplified 73.7%

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

   7. Taylor expanded in a around 0 44.2%

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

      1. associate-/l* 49.8%

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

   9. Simplified 49.8%

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

   10. Taylor expanded in t around 0 43.7%

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

       1. mul-1-neg 43.7%

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

       2. associate-*r/ 46.5%

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

       3. distribute-lft-neg-out 46.5%

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

   12. Simplified 46.5%

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

   if -1.5999999999999999e182 < z

   1. Initial program 98.2%

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

   2. Taylor expanded in t around inf 64.5%

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

4. Final simplification 62.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.6 \cdot 10^{+182}:\\ \;\;\;\;y \cdot \frac{-z}{t}\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 11: 59.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.5 \cdot 10^{+181}:\\ \;\;\;\;\frac{y}{\frac{-t}{z}}\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -4.5e+181) (/ y (/ (- t) z)) (+ x y)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -4.5e+181) {
		tmp = y / (-t / z);
	} else {
		tmp = x + y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-4.5d+181)) then
        tmp = y / (-t / z)
    else
        tmp = x + y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -4.5e+181) {
		tmp = y / (-t / z);
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -4.5e+181:
		tmp = y / (-t / z)
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -4.5e+181)
		tmp = Float64(y / Float64(Float64(-t) / z));
	else
		tmp = Float64(x + y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -4.5e+181)
		tmp = y / (-t / z);
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -4.5e+181], N[(y / N[((-t) / z), $MachinePrecision]), $MachinePrecision], N[(x + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4.5e181

   1. Initial program 93.9%

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

      1. +-commutative 93.9%

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

      2. *-commutative 93.9%

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

      3. associate-*l/ 80.2%

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

      4. sub-neg 80.2%

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

      5. +-commutative 80.2%

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

      6. neg-sub0 80.2%

         \[\leadsto \frac{\left(z - t\right) \cdot y}{\color{blue}{\left(0 - t\right)} + a} + x \]

      7. associate-+l- 80.2%

         \[\leadsto \frac{\left(z - t\right) \cdot y}{\color{blue}{0 - \left(t - a\right)}} + x \]

      8. sub0-neg 80.2%

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

      9. neg-mul-1 80.2%

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

      10. times-frac 92.5%

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

      11. fma-def 92.5%

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

      12. sub-neg 92.5%

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

      13. +-commutative 92.5%

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

      14. neg-sub0 92.5%

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\left(0 - t\right)} + z}{-1}, \frac{y}{t - a}, x\right) \]

      15. associate-+l- 92.5%

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{0 - \left(t - z\right)}}{-1}, \frac{y}{t - a}, x\right) \]

      16. sub0-neg 92.5%

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

      17. neg-mul-1 92.5%

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

      18. *-commutative 92.5%

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

      19. associate-/l* 92.5%

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

      20. metadata-eval 92.5%

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

      21. /-rgt-identity 92.5%

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

   3. Simplified 92.5%

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

   4. Taylor expanded in y around -inf 71.1%

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

      1. *-commutative 71.1%

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

      2. associate-/l* 73.7%

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

   6. Simplified 73.7%

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

   7. Taylor expanded in a around 0 44.2%

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

      1. associate-/l* 49.8%

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

   9. Simplified 49.8%

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

   10. Taylor expanded in t around 0 46.6%

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

       1. associate-*r/ 46.6%

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

       2. neg-mul-1 46.6%

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

   12. Simplified 46.6%

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

   if -4.5e181 < z

   1. Initial program 98.2%

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

   2. Taylor expanded in t around inf 64.5%

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

4. Final simplification 62.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.5 \cdot 10^{+181}:\\ \;\;\;\;\frac{y}{\frac{-t}{z}}\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 12: 52.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.32 \cdot 10^{-141}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{-201}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= x -1.32e-141) x (if (<= x 3.2e-201) y x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (x <= -1.32e-141) {
		tmp = x;
	} else if (x <= 3.2e-201) {
		tmp = y;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (x <= (-1.32d-141)) then
        tmp = x
    else if (x <= 3.2d-201) then
        tmp = y
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (x <= -1.32e-141) {
		tmp = x;
	} else if (x <= 3.2e-201) {
		tmp = y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if x <= -1.32e-141:
		tmp = x
	elif x <= 3.2e-201:
		tmp = y
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (x <= -1.32e-141)
		tmp = x;
	elseif (x <= 3.2e-201)
		tmp = y;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (x <= -1.32e-141)
		tmp = x;
	elseif (x <= 3.2e-201)
		tmp = y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[x, -1.32e-141], x, If[LessEqual[x, 3.2e-201], y, x]]

Derivation

1. Split input into 2 regimes

2. if x < -1.3199999999999999e-141 or 3.2000000000000001e-201 < x

   1. Initial program 98.5%

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

   2. Taylor expanded in x around inf 57.6%

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

   if -1.3199999999999999e-141 < x < 3.2000000000000001e-201

   1. Initial program 94.8%

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

      1. +-commutative 94.8%

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

      2. *-commutative 94.8%

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

      3. associate-*l/ 86.6%

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

      4. sub-neg 86.6%

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

      5. +-commutative 86.6%

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

      6. neg-sub0 86.6%

         \[\leadsto \frac{\left(z - t\right) \cdot y}{\color{blue}{\left(0 - t\right)} + a} + x \]

      7. associate-+l- 86.6%

         \[\leadsto \frac{\left(z - t\right) \cdot y}{\color{blue}{0 - \left(t - a\right)}} + x \]

      8. sub0-neg 86.6%

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

      9. neg-mul-1 86.6%

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

      10. times-frac 90.4%

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

      11. fma-def 90.4%

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

      12. sub-neg 90.4%

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

      13. +-commutative 90.4%

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

      14. neg-sub0 90.4%

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\left(0 - t\right)} + z}{-1}, \frac{y}{t - a}, x\right) \]

      15. associate-+l- 90.4%

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{0 - \left(t - z\right)}}{-1}, \frac{y}{t - a}, x\right) \]

      16. sub0-neg 90.4%

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

      17. neg-mul-1 90.4%

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

      18. *-commutative 90.4%

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

      19. associate-/l* 90.4%

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

      20. metadata-eval 90.4%

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

      21. /-rgt-identity 90.4%

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

   3. Simplified 90.4%

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

   4. Taylor expanded in y around -inf 73.8%

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

      1. *-commutative 73.8%

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

      2. associate-/l* 76.6%

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

   6. Simplified 76.6%

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

   7. Taylor expanded in t around inf 39.2%

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

4. Final simplification 53.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.32 \cdot 10^{-141}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{-201}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 13: 61.9% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -9 \cdot 10^{+128}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 (if (<= a -9e+128) x (+ x y)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -9e+128) {
		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 (a <= (-9d+128)) 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 (a <= -9e+128) {
		tmp = x;
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if a <= -9e+128:
		tmp = x
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -9e+128)
		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 (a <= -9e+128)
		tmp = x;
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -9e+128], x, N[(x + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -9.0000000000000003e128

   1. Initial program 97.6%

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

   2. Taylor expanded in x around inf 73.7%

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

   if -9.0000000000000003e128 < a

   1. Initial program 97.7%

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

   2. Taylor expanded in t around inf 60.9%

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

4. Final simplification 62.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -9 \cdot 10^{+128}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 14: 50.1% accurate, 11.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_] := x

Derivation

1. Initial program 97.7%

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

2. Taylor expanded in x around inf 48.2%

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

3. Final simplification 48.2%

   \[\leadsto x \]

Developer target: 99.4% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = x + (y * ((z - t) / (a - t)));
	tmp = 0.0;
	if (y < -8.508084860551241e-17)
		tmp = t_1;
	elseif (y < 2.894426862792089e-49)
		tmp = x + ((y * (z - t)) * (1.0 / (a - t)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x + N[(y * N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Less[y, -8.508084860551241e-17], t$95$1, If[Less[y, 2.894426862792089e-49], N[(x + N[(N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Reproduce

herbie shell --seed 2023173 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisLine from plot-0.2.3.4, B"
  :precision binary64

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

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