Optimisation.CirclePacking:place from circle-packing-0.1.0.4, E

Percentage Accurate: 93.0% → 97.2%

Time: 8.6s

Alternatives: 10

Speedup: 1.0×

• 25.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 93.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.5%

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

   1. associate-*l/ 97.8%

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

3. Simplified 97.8%

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

4. Final simplification 97.8%

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

Alternative 2: 45.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.02 \cdot 10^{+123}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 9 \cdot 10^{-303}:\\ \;\;\;\;\frac{y \cdot z}{a}\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{-40}:\\ \;\;\;\;y \cdot \frac{-t}{a}\\ \mathbf{elif}\;x \leq 6.6 \cdot 10^{+39}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{+108}:\\ \;\;\;\;\frac{y \cdot \left(-t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= x -1.02e+123)
   x
   (if (<= x 9e-303)
     (/ (* y z) a)
     (if (<= x 2.2e-40)
       (* y (/ (- t) a))
       (if (<= x 6.6e+39) x (if (<= x 4.2e+108) (/ (* y (- t)) a) x))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (x <= -1.02e+123) {
		tmp = x;
	} else if (x <= 9e-303) {
		tmp = (y * z) / a;
	} else if (x <= 2.2e-40) {
		tmp = y * (-t / a);
	} else if (x <= 6.6e+39) {
		tmp = x;
	} else if (x <= 4.2e+108) {
		tmp = (y * -t) / a;
	} 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 <= (-1.02d+123)) then
        tmp = x
    else if (x <= 9d-303) then
        tmp = (y * z) / a
    else if (x <= 2.2d-40) then
        tmp = y * (-t / a)
    else if (x <= 6.6d+39) then
        tmp = x
    else if (x <= 4.2d+108) then
        tmp = (y * -t) / a
    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 <= -1.02e+123) {
		tmp = x;
	} else if (x <= 9e-303) {
		tmp = (y * z) / a;
	} else if (x <= 2.2e-40) {
		tmp = y * (-t / a);
	} else if (x <= 6.6e+39) {
		tmp = x;
	} else if (x <= 4.2e+108) {
		tmp = (y * -t) / a;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if x <= -1.02e+123:
		tmp = x
	elif x <= 9e-303:
		tmp = (y * z) / a
	elif x <= 2.2e-40:
		tmp = y * (-t / a)
	elif x <= 6.6e+39:
		tmp = x
	elif x <= 4.2e+108:
		tmp = (y * -t) / a
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (x <= -1.02e+123)
		tmp = x;
	elseif (x <= 9e-303)
		tmp = Float64(Float64(y * z) / a);
	elseif (x <= 2.2e-40)
		tmp = Float64(y * Float64(Float64(-t) / a));
	elseif (x <= 6.6e+39)
		tmp = x;
	elseif (x <= 4.2e+108)
		tmp = Float64(Float64(y * Float64(-t)) / a);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (x <= -1.02e+123)
		tmp = x;
	elseif (x <= 9e-303)
		tmp = (y * z) / a;
	elseif (x <= 2.2e-40)
		tmp = y * (-t / a);
	elseif (x <= 6.6e+39)
		tmp = x;
	elseif (x <= 4.2e+108)
		tmp = (y * -t) / a;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[x, -1.02e+123], x, If[LessEqual[x, 9e-303], N[(N[(y * z), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[x, 2.2e-40], N[(y * N[((-t) / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 6.6e+39], x, If[LessEqual[x, 4.2e+108], N[(N[(y * (-t)), $MachinePrecision] / a), $MachinePrecision], x]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.02e123 or 2.20000000000000009e-40 < x < 6.60000000000000042e39 or 4.20000000000000019e108 < x

   1. Initial program 94.8%

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

      1. associate-*l/ 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around inf 68.6%

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

   if -1.02e123 < x < 9.0000000000000002e-303

   1. Initial program 90.5%

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

      1. associate-*l/ 97.0%

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

   3. Simplified 97.0%

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

      1. associate-/r/ 90.4%

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

      2. div-inv 90.3%

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

      3. associate-/r* 96.8%

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

   5. Applied egg-rr 96.8%

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

      1. +-commutative 96.8%

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

      2. associate-/r/ 97.0%

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

      3. associate-/r/ 96.8%

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

      4. div-inv 96.9%

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

      5. clear-num 96.1%

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

      6. frac-times 96.2%

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

      7. metadata-eval 96.2%

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

      8. div-inv 96.3%

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

      9. clear-num 96.4%

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

      10. flip-+ 47.9%

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

   7. Applied egg-rr 48.5%

      \[\leadsto \color{blue}{\frac{{\left(\frac{y}{a} \cdot \left(z - t\right)\right)}^{2} - x \cdot x}{\frac{y}{a} \cdot \left(z - t\right) - x}} \]

   8. Taylor expanded in y around -inf 72.1%

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

   9. Taylor expanded in z around inf 48.7%

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

   if 9.0000000000000002e-303 < x < 2.20000000000000009e-40

   1. Initial program 89.4%

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

      1. associate-*l/ 94.4%

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

   3. Simplified 94.4%

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

   4. Taylor expanded in z around 0 65.3%

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

      1. mul-1-neg 65.3%

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

      2. associate-*l/ 65.3%

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

      3. distribute-rgt-neg-out 65.3%

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

   6. Simplified 65.3%

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

      1. add-sqr-sqrt 65.3%

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

      2. fma-def 65.3%

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

      3. distribute-rgt-neg-out 65.3%

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

      4. add-sqr-sqrt 26.5%

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

      5. sqrt-unprod 25.5%

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

      6. sqr-neg 25.5%

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

      7. sqrt-unprod 8.1%

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

      8. add-sqr-sqrt 14.2%

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

      9. *-commutative 14.2%

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

      10. *-commutative 14.2%

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

      11. fma-neg 14.2%

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

      12. add-sqr-sqrt 14.2%

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

      13. add-sqr-sqrt 8.1%

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

      14. sqrt-unprod 25.5%

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

      15. sqr-neg 25.5%

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

      16. sqrt-unprod 26.5%

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

      17. add-sqr-sqrt 65.3%

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

   8. Applied egg-rr 65.3%

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

   9. Taylor expanded in x around 0 53.0%

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

       1. mul-1-neg 53.0%

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

       2. associate-*r/ 56.5%

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

       3. distribute-rgt-neg-out 56.5%

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

   11. Simplified 56.5%

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

   if 6.60000000000000042e39 < x < 4.20000000000000019e108

   1. Initial program 79.3%

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

      1. associate-*l/ 99.9%

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

   3. Simplified 99.9%

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

      1. associate-/r/ 81.5%

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

      2. div-inv 81.4%

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

      3. associate-/r* 99.8%

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

   5. Applied egg-rr 99.8%

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

      1. +-commutative 99.8%

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

      2. associate-/r/ 99.9%

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

      3. associate-/r/ 99.8%

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

      4. div-inv 99.8%

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

      5. clear-num 99.7%

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

      6. frac-times 99.5%

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

      7. metadata-eval 99.5%

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

      8. div-inv 99.8%

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

      9. clear-num 99.8%

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

      10. flip-+ 16.4%

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

   7. Applied egg-rr 16.4%

      \[\leadsto \color{blue}{\frac{{\left(\frac{y}{a} \cdot \left(z - t\right)\right)}^{2} - x \cdot x}{\frac{y}{a} \cdot \left(z - t\right) - x}} \]

   8. Taylor expanded in y around -inf 79.3%

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

   9. Taylor expanded in z around 0 59.1%

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

       1. mul-1-neg 59.1%

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

       2. distribute-rgt-neg-out 59.1%

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

   11. Simplified 59.1%

       \[\leadsto \frac{\color{blue}{y \cdot \left(-t\right)}}{a} \]
3. Recombined 4 regimes into one program.

4. Final simplification 59.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.02 \cdot 10^{+123}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 9 \cdot 10^{-303}:\\ \;\;\;\;\frac{y \cdot z}{a}\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{-40}:\\ \;\;\;\;y \cdot \frac{-t}{a}\\ \mathbf{elif}\;x \leq 6.6 \cdot 10^{+39}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{+108}:\\ \;\;\;\;\frac{y \cdot \left(-t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 3: 45.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.75 \cdot 10^{+124}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{-33} \lor \neg \left(x \leq 1400000000000\right) \land x \leq 4.5 \cdot 10^{+106}:\\ \;\;\;\;\frac{y \cdot z}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= x -1.75e+124)
   x
   (if (or (<= x 1.7e-33) (and (not (<= x 1400000000000.0)) (<= x 4.5e+106)))
     (/ (* y z) a)
     x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (x <= -1.75e+124) {
		tmp = x;
	} else if ((x <= 1.7e-33) || (!(x <= 1400000000000.0) && (x <= 4.5e+106))) {
		tmp = (y * z) / a;
	} 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 <= (-1.75d+124)) then
        tmp = x
    else if ((x <= 1.7d-33) .or. (.not. (x <= 1400000000000.0d0)) .and. (x <= 4.5d+106)) then
        tmp = (y * z) / a
    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 <= -1.75e+124) {
		tmp = x;
	} else if ((x <= 1.7e-33) || (!(x <= 1400000000000.0) && (x <= 4.5e+106))) {
		tmp = (y * z) / a;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if x <= -1.75e+124:
		tmp = x
	elif (x <= 1.7e-33) or (not (x <= 1400000000000.0) and (x <= 4.5e+106)):
		tmp = (y * z) / a
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (x <= -1.75e+124)
		tmp = x;
	elseif ((x <= 1.7e-33) || (!(x <= 1400000000000.0) && (x <= 4.5e+106)))
		tmp = Float64(Float64(y * z) / a);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (x <= -1.75e+124)
		tmp = x;
	elseif ((x <= 1.7e-33) || (~((x <= 1400000000000.0)) && (x <= 4.5e+106)))
		tmp = (y * z) / a;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[x, -1.75e+124], x, If[Or[LessEqual[x, 1.7e-33], And[N[Not[LessEqual[x, 1400000000000.0]], $MachinePrecision], LessEqual[x, 4.5e+106]]], N[(N[(y * z), $MachinePrecision] / a), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if x < -1.7500000000000001e124 or 1.7e-33 < x < 1.4e12 or 4.4999999999999997e106 < x

   1. Initial program 94.7%

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

      1. associate-*l/ 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around inf 69.6%

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

   if -1.7500000000000001e124 < x < 1.7e-33 or 1.4e12 < x < 4.4999999999999997e106

   1. Initial program 89.2%

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

      1. associate-*l/ 96.4%

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

   3. Simplified 96.4%

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

      1. associate-/r/ 91.9%

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

      2. div-inv 91.8%

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

      3. associate-/r* 96.3%

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

   5. Applied egg-rr 96.3%

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

      1. +-commutative 96.3%

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

      2. associate-/r/ 96.4%

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

      3. associate-/r/ 96.3%

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

      4. div-inv 96.3%

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

      5. clear-num 95.9%

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

      6. frac-times 95.9%

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

      7. metadata-eval 95.9%

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

      8. div-inv 96.0%

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

      9. clear-num 96.1%

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

      10. flip-+ 40.8%

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

   7. Applied egg-rr 41.2%

      \[\leadsto \color{blue}{\frac{{\left(\frac{y}{a} \cdot \left(z - t\right)\right)}^{2} - x \cdot x}{\frac{y}{a} \cdot \left(z - t\right) - x}} \]

   8. Taylor expanded in y around -inf 74.4%

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

   9. Taylor expanded in z around inf 43.0%

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

4. Final simplification 53.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.75 \cdot 10^{+124}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{-33} \lor \neg \left(x \leq 1400000000000\right) \land x \leq 4.5 \cdot 10^{+106}:\\ \;\;\;\;\frac{y \cdot z}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4: 48.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \frac{-t}{a}\\ \mathbf{if}\;t \leq -3200000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 10^{-78}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 1.5 \cdot 10^{+30}:\\ \;\;\;\;\frac{y \cdot z}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* y (/ (- t) a))))
   (if (<= t -3200000.0)
     t_1
     (if (<= t 1e-78) x (if (<= t 1.5e+30) (/ (* y z) a) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = y * (-t / a);
	double tmp;
	if (t <= -3200000.0) {
		tmp = t_1;
	} else if (t <= 1e-78) {
		tmp = x;
	} else if (t <= 1.5e+30) {
		tmp = (y * z) / a;
	} 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 = y * (-t / a)
    if (t <= (-3200000.0d0)) then
        tmp = t_1
    else if (t <= 1d-78) then
        tmp = x
    else if (t <= 1.5d+30) then
        tmp = (y * z) / a
    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 = y * (-t / a);
	double tmp;
	if (t <= -3200000.0) {
		tmp = t_1;
	} else if (t <= 1e-78) {
		tmp = x;
	} else if (t <= 1.5e+30) {
		tmp = (y * z) / a;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = y * (-t / a)
	tmp = 0
	if t <= -3200000.0:
		tmp = t_1
	elif t <= 1e-78:
		tmp = x
	elif t <= 1.5e+30:
		tmp = (y * z) / a
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(y * Float64(Float64(-t) / a))
	tmp = 0.0
	if (t <= -3200000.0)
		tmp = t_1;
	elseif (t <= 1e-78)
		tmp = x;
	elseif (t <= 1.5e+30)
		tmp = Float64(Float64(y * z) / a);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = y * (-t / a);
	tmp = 0.0;
	if (t <= -3200000.0)
		tmp = t_1;
	elseif (t <= 1e-78)
		tmp = x;
	elseif (t <= 1.5e+30)
		tmp = (y * z) / a;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(y * N[((-t) / a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -3200000.0], t$95$1, If[LessEqual[t, 1e-78], x, If[LessEqual[t, 1.5e+30], N[(N[(y * z), $MachinePrecision] / a), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t < -3.2e6 or 1.49999999999999989e30 < t

   1. Initial program 89.8%

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

      1. associate-*l/ 98.3%

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

   3. Simplified 98.3%

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

   4. Taylor expanded in z around 0 79.3%

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

      1. mul-1-neg 79.3%

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

      2. associate-*l/ 83.9%

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

      3. distribute-rgt-neg-out 83.9%

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

   6. Simplified 83.9%

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

      1. add-sqr-sqrt 46.7%

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

      2. fma-def 46.7%

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

      3. distribute-rgt-neg-out 46.7%

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

      4. add-sqr-sqrt 20.7%

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

      5. sqrt-unprod 15.1%

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

      6. sqr-neg 15.1%

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

      7. sqrt-unprod 5.4%

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

      8. add-sqr-sqrt 10.6%

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

      9. *-commutative 10.6%

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

      10. *-commutative 10.6%

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

      11. fma-neg 10.6%

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

      12. add-sqr-sqrt 25.2%

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

      13. add-sqr-sqrt 12.2%

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

      14. sqrt-unprod 34.6%

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

      15. sqr-neg 34.6%

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

      16. sqrt-unprod 43.7%

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

      17. add-sqr-sqrt 83.9%

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

   8. Applied egg-rr 83.9%

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

   9. Taylor expanded in x around 0 56.9%

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

       1. mul-1-neg 56.9%

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

       2. associate-*r/ 57.6%

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

       3. distribute-rgt-neg-out 57.6%

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

   11. Simplified 57.6%

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

   if -3.2e6 < t < 9.99999999999999999e-79

   1. Initial program 93.9%

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

      1. associate-*l/ 97.8%

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

   3. Simplified 97.8%

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

   4. Taylor expanded in x around inf 54.7%

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

   if 9.99999999999999999e-79 < t < 1.49999999999999989e30

   1. Initial program 88.5%

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

      1. associate-*l/ 95.8%

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

   3. Simplified 95.8%

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

      1. associate-/r/ 96.0%

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

      2. div-inv 96.1%

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

      3. associate-/r* 95.9%

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

   5. Applied egg-rr 95.9%

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

      1. +-commutative 95.9%

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

      2. associate-/r/ 95.8%

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

      3. associate-/r/ 95.9%

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

      4. div-inv 95.8%

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

      5. clear-num 95.7%

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

      6. frac-times 95.8%

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

      7. metadata-eval 95.8%

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

      8. div-inv 95.7%

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

      9. clear-num 95.8%

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

      10. flip-+ 31.4%

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

   7. Applied egg-rr 31.4%

      \[\leadsto \color{blue}{\frac{{\left(\frac{y}{a} \cdot \left(z - t\right)\right)}^{2} - x \cdot x}{\frac{y}{a} \cdot \left(z - t\right) - x}} \]

   8. Taylor expanded in y around -inf 60.8%

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

   9. Taylor expanded in z around inf 55.4%

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

4. Final simplification 56.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3200000:\\ \;\;\;\;y \cdot \frac{-t}{a}\\ \mathbf{elif}\;t \leq 10^{-78}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 1.5 \cdot 10^{+30}:\\ \;\;\;\;\frac{y \cdot z}{a}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{-t}{a}\\ \end{array} \]

Alternative 5: 74.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -4.1 \cdot 10^{+164} \lor \neg \left(t \leq 6.5 \cdot 10^{+155}\right):\\ \;\;\;\;\frac{y \cdot \left(-t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= t -4.1e+164) (not (<= t 6.5e+155)))
   (/ (* y (- t)) a)
   (+ x (* y (/ z a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((t <= -4.1e+164) || !(t <= 6.5e+155)) {
		tmp = (y * -t) / a;
	} else {
		tmp = x + (y * (z / a));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((t <= (-4.1d+164)) .or. (.not. (t <= 6.5d+155))) then
        tmp = (y * -t) / a
    else
        tmp = x + (y * (z / a))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((t <= -4.1e+164) || !(t <= 6.5e+155)) {
		tmp = (y * -t) / a;
	} else {
		tmp = x + (y * (z / a));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (t <= -4.1e+164) or not (t <= 6.5e+155):
		tmp = (y * -t) / a
	else:
		tmp = x + (y * (z / a))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((t <= -4.1e+164) || !(t <= 6.5e+155))
		tmp = Float64(Float64(y * Float64(-t)) / a);
	else
		tmp = Float64(x + Float64(y * Float64(z / a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((t <= -4.1e+164) || ~((t <= 6.5e+155)))
		tmp = (y * -t) / a;
	else
		tmp = x + (y * (z / a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[t, -4.1e+164], N[Not[LessEqual[t, 6.5e+155]], $MachinePrecision]], N[(N[(y * (-t)), $MachinePrecision] / a), $MachinePrecision], N[(x + N[(y * N[(z / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -4.10000000000000016e164 or 6.50000000000000046e155 < t

   1. Initial program 92.1%

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

      1. associate-*l/ 96.6%

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

   3. Simplified 96.6%

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

      1. associate-/r/ 87.4%

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

      2. div-inv 87.4%

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

      3. associate-/r* 96.6%

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

   5. Applied egg-rr 96.6%

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

      1. +-commutative 96.6%

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

      2. associate-/r/ 96.6%

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

      3. associate-/r/ 96.6%

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

      4. div-inv 96.5%

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

      5. clear-num 96.4%

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

      6. frac-times 96.4%

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

      7. metadata-eval 96.4%

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

      8. div-inv 96.6%

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

      9. clear-num 96.7%

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

      10. flip-+ 23.9%

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

   7. Applied egg-rr 23.9%

      \[\leadsto \color{blue}{\frac{{\left(\frac{y}{a} \cdot \left(z - t\right)\right)}^{2} - x \cdot x}{\frac{y}{a} \cdot \left(z - t\right) - x}} \]

   8. Taylor expanded in y around -inf 71.0%

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

   9. Taylor expanded in z around 0 68.6%

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

       1. mul-1-neg 68.6%

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

       2. distribute-rgt-neg-out 68.6%

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

   11. Simplified 68.6%

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

   if -4.10000000000000016e164 < t < 6.50000000000000046e155

   1. Initial program 91.3%

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

      1. associate-/l* 94.6%

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

   3. Simplified 94.6%

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

   4. Taylor expanded in z around inf 78.7%

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

      1. div-inv 78.7%

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

      2. clear-num 78.7%

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

   6. Applied egg-rr 78.7%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.1 \cdot 10^{+164} \lor \neg \left(t \leq 6.5 \cdot 10^{+155}\right):\\ \;\;\;\;\frac{y \cdot \left(-t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z}{a}\\ \end{array} \]

Alternative 6: 74.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3.95 \cdot 10^{+164} \lor \neg \left(t \leq 8 \cdot 10^{+155}\right):\\ \;\;\;\;\frac{y \cdot \left(-t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= t -3.95e+164) (not (<= t 8e+155)))
   (/ (* y (- t)) a)
   (+ x (/ y (/ a z)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((t <= -3.95e+164) || !(t <= 8e+155)) {
		tmp = (y * -t) / a;
	} else {
		tmp = x + (y / (a / z));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((t <= (-3.95d+164)) .or. (.not. (t <= 8d+155))) then
        tmp = (y * -t) / a
    else
        tmp = x + (y / (a / z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((t <= -3.95e+164) || !(t <= 8e+155)) {
		tmp = (y * -t) / a;
	} else {
		tmp = x + (y / (a / z));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (t <= -3.95e+164) or not (t <= 8e+155):
		tmp = (y * -t) / a
	else:
		tmp = x + (y / (a / z))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((t <= -3.95e+164) || !(t <= 8e+155))
		tmp = Float64(Float64(y * Float64(-t)) / a);
	else
		tmp = Float64(x + Float64(y / Float64(a / z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((t <= -3.95e+164) || ~((t <= 8e+155)))
		tmp = (y * -t) / a;
	else
		tmp = x + (y / (a / z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[t, -3.95e+164], N[Not[LessEqual[t, 8e+155]], $MachinePrecision]], N[(N[(y * (-t)), $MachinePrecision] / a), $MachinePrecision], N[(x + N[(y / N[(a / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -3.95000000000000019e164 or 8.00000000000000006e155 < t

   1. Initial program 92.1%

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

      1. associate-*l/ 96.6%

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

   3. Simplified 96.6%

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

      1. associate-/r/ 87.4%

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

      2. div-inv 87.4%

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

      3. associate-/r* 96.6%

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

   5. Applied egg-rr 96.6%

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

      1. +-commutative 96.6%

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

      2. associate-/r/ 96.6%

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

      3. associate-/r/ 96.6%

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

      4. div-inv 96.5%

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

      5. clear-num 96.4%

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

      6. frac-times 96.4%

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

      7. metadata-eval 96.4%

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

      8. div-inv 96.6%

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

      9. clear-num 96.7%

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

      10. flip-+ 23.9%

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

   7. Applied egg-rr 23.9%

      \[\leadsto \color{blue}{\frac{{\left(\frac{y}{a} \cdot \left(z - t\right)\right)}^{2} - x \cdot x}{\frac{y}{a} \cdot \left(z - t\right) - x}} \]

   8. Taylor expanded in y around -inf 71.0%

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

   9. Taylor expanded in z around 0 68.6%

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

       1. mul-1-neg 68.6%

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

       2. distribute-rgt-neg-out 68.6%

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

   11. Simplified 68.6%

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

   if -3.95000000000000019e164 < t < 8.00000000000000006e155

   1. Initial program 91.3%

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

      1. associate-/l* 94.6%

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

   3. Simplified 94.6%

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

   4. Taylor expanded in z around inf 78.7%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.95 \cdot 10^{+164} \lor \neg \left(t \leq 8 \cdot 10^{+155}\right):\\ \;\;\;\;\frac{y \cdot \left(-t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z}}\\ \end{array} \]

Alternative 7: 76.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.35 \cdot 10^{+171} \lor \neg \left(t \leq 8 \cdot 10^{+155}\right):\\ \;\;\;\;\frac{y \cdot \left(-t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{a} \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= t -1.35e+171) (not (<= t 8e+155)))
   (/ (* y (- t)) a)
   (+ x (* (/ y a) z))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((t <= -1.35e+171) || !(t <= 8e+155)) {
		tmp = (y * -t) / a;
	} else {
		tmp = x + ((y / a) * z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((t <= (-1.35d+171)) .or. (.not. (t <= 8d+155))) then
        tmp = (y * -t) / a
    else
        tmp = x + ((y / a) * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((t <= -1.35e+171) || !(t <= 8e+155)) {
		tmp = (y * -t) / a;
	} else {
		tmp = x + ((y / a) * z);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (t <= -1.35e+171) or not (t <= 8e+155):
		tmp = (y * -t) / a
	else:
		tmp = x + ((y / a) * z)
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((t <= -1.35e+171) || !(t <= 8e+155))
		tmp = Float64(Float64(y * Float64(-t)) / a);
	else
		tmp = Float64(x + Float64(Float64(y / a) * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((t <= -1.35e+171) || ~((t <= 8e+155)))
		tmp = (y * -t) / a;
	else
		tmp = x + ((y / a) * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[t, -1.35e+171], N[Not[LessEqual[t, 8e+155]], $MachinePrecision]], N[(N[(y * (-t)), $MachinePrecision] / a), $MachinePrecision], N[(x + N[(N[(y / a), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.3499999999999999e171 or 8.00000000000000006e155 < t

   1. Initial program 91.9%

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

      1. associate-*l/ 96.5%

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

   3. Simplified 96.5%

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

      1. associate-/r/ 88.7%

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

      2. div-inv 88.7%

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

      3. associate-/r* 96.5%

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

   5. Applied egg-rr 96.5%

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

      1. +-commutative 96.5%

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

      2. associate-/r/ 96.5%

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

      3. associate-/r/ 96.5%

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

      4. div-inv 96.4%

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

      5. clear-num 96.3%

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

      6. frac-times 96.3%

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

      7. metadata-eval 96.3%

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

      8. div-inv 96.5%

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

      9. clear-num 96.6%

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

      10. flip-+ 24.8%

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

   7. Applied egg-rr 24.7%

      \[\leadsto \color{blue}{\frac{{\left(\frac{y}{a} \cdot \left(z - t\right)\right)}^{2} - x \cdot x}{\frac{y}{a} \cdot \left(z - t\right) - x}} \]

   8. Taylor expanded in y around -inf 71.8%

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

   9. Taylor expanded in z around 0 69.3%

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

       1. mul-1-neg 69.3%

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

       2. distribute-rgt-neg-out 69.3%

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

   11. Simplified 69.3%

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

   if -1.3499999999999999e171 < t < 8.00000000000000006e155

   1. Initial program 91.4%

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

      1. associate-*l/ 98.2%

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

   3. Simplified 98.2%

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

   4. Taylor expanded in t around 0 77.0%

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

      1. associate-*l/ 82.9%

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

      2. *-commutative 82.9%

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

   6. Simplified 82.9%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.35 \cdot 10^{+171} \lor \neg \left(t \leq 8 \cdot 10^{+155}\right):\\ \;\;\;\;\frac{y \cdot \left(-t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{a} \cdot z\\ \end{array} \]

Alternative 8: 83.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.4 \cdot 10^{-36} \lor \neg \left(z \leq 1.9 \cdot 10^{-40}\right):\\ \;\;\;\;x + \frac{y}{a} \cdot z\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \frac{t}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -5.4e-36) (not (<= z 1.9e-40)))
   (+ x (* (/ y a) z))
   (- x (* y (/ t a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -5.4e-36) || !(z <= 1.9e-40)) {
		tmp = x + ((y / a) * z);
	} 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 <= (-5.4d-36)) .or. (.not. (z <= 1.9d-40))) then
        tmp = x + ((y / a) * z)
    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 <= -5.4e-36) || !(z <= 1.9e-40)) {
		tmp = x + ((y / a) * z);
	} else {
		tmp = x - (y * (t / a));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (z <= -5.4e-36) or not (z <= 1.9e-40):
		tmp = x + ((y / a) * z)
	else:
		tmp = x - (y * (t / a))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -5.4e-36) || !(z <= 1.9e-40))
		tmp = Float64(x + Float64(Float64(y / a) * z));
	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 <= -5.4e-36) || ~((z <= 1.9e-40)))
		tmp = x + ((y / a) * z);
	else
		tmp = x - (y * (t / a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -5.4e-36], N[Not[LessEqual[z, 1.9e-40]], $MachinePrecision]], N[(x + N[(N[(y / a), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], N[(x - N[(y * N[(t / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -5.40000000000000015e-36 or 1.8999999999999999e-40 < z

   1. Initial program 88.3%

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

      1. associate-*l/ 97.6%

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

   3. Simplified 97.6%

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

   4. Taylor expanded in t around 0 79.0%

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

      1. associate-*l/ 85.7%

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

      2. *-commutative 85.7%

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

   6. Simplified 85.7%

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

   if -5.40000000000000015e-36 < z < 1.8999999999999999e-40

   1. Initial program 95.6%

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

      1. associate-*l/ 98.1%

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

   3. Simplified 98.1%

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

      1. associate-/r/ 98.2%

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

      2. div-inv 98.2%

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

      3. associate-/r* 98.1%

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

   5. Applied egg-rr 98.1%

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

   6. Taylor expanded in z around 0 91.3%

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

      1. +-commutative 91.3%

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

      2. mul-1-neg 91.3%

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

      3. associate-*r/ 93.9%

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

      4. distribute-lft-neg-out 93.9%

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

      5. cancel-sign-sub-inv 93.9%

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

   8. Simplified 93.9%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.4 \cdot 10^{-36} \lor \neg \left(z \leq 1.9 \cdot 10^{-40}\right):\\ \;\;\;\;x + \frac{y}{a} \cdot z\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \frac{t}{a}\\ \end{array} \]

Alternative 9: 85.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.8 \cdot 10^{+26} \lor \neg \left(z \leq 1.9 \cdot 10^{-40}\right):\\ \;\;\;\;x + \frac{y}{a} \cdot z\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{a} \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -1.8e+26) (not (<= z 1.9e-40)))
   (+ x (* (/ y a) z))
   (- x (* (/ y a) t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -1.8e+26) || !(z <= 1.9e-40)) {
		tmp = x + ((y / a) * z);
	} else {
		tmp = x - ((y / a) * t);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((z <= (-1.8d+26)) .or. (.not. (z <= 1.9d-40))) then
        tmp = x + ((y / a) * z)
    else
        tmp = x - ((y / a) * t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -1.8e+26) || !(z <= 1.9e-40)) {
		tmp = x + ((y / a) * z);
	} else {
		tmp = x - ((y / a) * t);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (z <= -1.8e+26) or not (z <= 1.9e-40):
		tmp = x + ((y / a) * z)
	else:
		tmp = x - ((y / a) * t)
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -1.8e+26) || !(z <= 1.9e-40))
		tmp = Float64(x + Float64(Float64(y / a) * z));
	else
		tmp = Float64(x - Float64(Float64(y / a) * t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((z <= -1.8e+26) || ~((z <= 1.9e-40)))
		tmp = x + ((y / a) * z);
	else
		tmp = x - ((y / a) * t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -1.8e+26], N[Not[LessEqual[z, 1.9e-40]], $MachinePrecision]], N[(x + N[(N[(y / a), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[(y / a), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.80000000000000012e26 or 1.8999999999999999e-40 < z

   1. Initial program 88.4%

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

      1. associate-*l/ 97.5%

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

   3. Simplified 97.5%

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

   4. Taylor expanded in t around 0 79.4%

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

      1. associate-*l/ 86.5%

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

      2. *-commutative 86.5%

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

   6. Simplified 86.5%

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

   if -1.80000000000000012e26 < z < 1.8999999999999999e-40

   1. Initial program 95.1%

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

      1. associate-*l/ 98.2%

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

   3. Simplified 98.2%

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

   4. Taylor expanded in z around 0 90.2%

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

      1. mul-1-neg 90.2%

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

      2. associate-*l/ 93.3%

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

      3. distribute-rgt-neg-out 93.3%

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

   6. Simplified 93.3%

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

      1. add-sqr-sqrt 59.4%

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

      2. fma-def 59.4%

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

      3. distribute-rgt-neg-out 59.4%

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

      4. add-sqr-sqrt 24.9%

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

      5. sqrt-unprod 34.2%

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

      6. sqr-neg 34.2%

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

      7. sqrt-unprod 15.9%

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

      8. add-sqr-sqrt 28.7%

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

      9. *-commutative 28.7%

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

      10. *-commutative 28.7%

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

      11. fma-neg 28.7%

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

      12. add-sqr-sqrt 43.9%

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

      13. add-sqr-sqrt 24.0%

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

      14. sqrt-unprod 58.0%

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

      15. sqr-neg 58.0%

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

      16. sqrt-unprod 43.9%

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

      17. add-sqr-sqrt 93.3%

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

   8. Applied egg-rr 93.3%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.8 \cdot 10^{+26} \lor \neg \left(z \leq 1.9 \cdot 10^{-40}\right):\\ \;\;\;\;x + \frac{y}{a} \cdot z\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{a} \cdot t\\ \end{array} \]

Alternative 10: 39.1% accurate, 9.0× 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 91.5%

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

   1. associate-*l/ 97.8%

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

3. Simplified 97.8%

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

4. Taylor expanded in x around inf 38.6%

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

5. Final simplification 38.6%

   \[\leadsto x \]

Developer target: 99.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a}{z - t}\\ \mathbf{if}\;y < -1.0761266216389975 \cdot 10^{-10}:\\ \;\;\;\;x + \frac{1}{\frac{t_1}{y}}\\ \mathbf{elif}\;y < 2.894426862792089 \cdot 10^{-49}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{t_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ a (- z t))))
   (if (< y -1.0761266216389975e-10)
     (+ x (/ 1.0 (/ t_1 y)))
     (if (< y 2.894426862792089e-49)
       (+ x (/ (* y (- z t)) a))
       (+ x (/ y t_1))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = a / (z - t);
	double tmp;
	if (y < -1.0761266216389975e-10) {
		tmp = x + (1.0 / (t_1 / y));
	} else if (y < 2.894426862792089e-49) {
		tmp = x + ((y * (z - t)) / a);
	} else {
		tmp = x + (y / 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 = a / (z - t)
    if (y < (-1.0761266216389975d-10)) then
        tmp = x + (1.0d0 / (t_1 / y))
    else if (y < 2.894426862792089d-49) then
        tmp = x + ((y * (z - t)) / a)
    else
        tmp = x + (y / 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 = a / (z - t);
	double tmp;
	if (y < -1.0761266216389975e-10) {
		tmp = x + (1.0 / (t_1 / y));
	} else if (y < 2.894426862792089e-49) {
		tmp = x + ((y * (z - t)) / a);
	} else {
		tmp = x + (y / t_1);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = a / (z - t)
	tmp = 0
	if y < -1.0761266216389975e-10:
		tmp = x + (1.0 / (t_1 / y))
	elif y < 2.894426862792089e-49:
		tmp = x + ((y * (z - t)) / a)
	else:
		tmp = x + (y / t_1)
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(a / Float64(z - t))
	tmp = 0.0
	if (y < -1.0761266216389975e-10)
		tmp = Float64(x + Float64(1.0 / Float64(t_1 / y)));
	elseif (y < 2.894426862792089e-49)
		tmp = Float64(x + Float64(Float64(y * Float64(z - t)) / a));
	else
		tmp = Float64(x + Float64(y / t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = a / (z - t);
	tmp = 0.0;
	if (y < -1.0761266216389975e-10)
		tmp = x + (1.0 / (t_1 / y));
	elseif (y < 2.894426862792089e-49)
		tmp = x + ((y * (z - t)) / a);
	else
		tmp = x + (y / t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(a / N[(z - t), $MachinePrecision]), $MachinePrecision]}, If[Less[y, -1.0761266216389975e-10], N[(x + N[(1.0 / N[(t$95$1 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Less[y, 2.894426862792089e-49], N[(x + N[(N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(x + N[(y / t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Reproduce

herbie shell --seed 2023262 
(FPCore (x y z t a)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, E"
  :precision binary64

  :herbie-target
  (if (< y -1.0761266216389975e-10) (+ x (/ 1.0 (/ (/ a (- z t)) y))) (if (< y 2.894426862792089e-49) (+ x (/ (* y (- z t)) a)) (+ x (/ y (/ a (- z t))))))

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