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

Percentage Accurate: 92.8% → 97.6%

Time: 11.2s

Alternatives: 10

Speedup: 1.0×

• 25.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 92.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{y}{t}, z - x, x\right) \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (fma (/ y t) (- z x) x))

C

double code(double x, double y, double z, double t) {
	return fma((y / t), (z - x), x);
}

Julia

function code(x, y, z, t)
	return fma(Float64(y / t), Float64(z - x), x)
end

Wolfram

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

Derivation

1. Initial program 91.5%

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

   1. +-commutative 91.5%

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

   2. associate-*l/ 98.0%

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

   3. fma-def 98.1%

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

3. Simplified 98.1%

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

4. Final simplification 98.1%

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

Alternative 2: 85.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6.5 \cdot 10^{+49} \lor \neg \left(z \leq 2.8 \cdot 10^{-20}\right):\\ \;\;\;\;x + \frac{y}{t} \cdot z\\ \mathbf{else}:\\ \;\;\;\;x + \frac{x}{\frac{-t}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -6.5e+49) (not (<= z 2.8e-20)))
   (+ x (* (/ y t) z))
   (+ x (/ x (/ (- t) y)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -6.5e+49) || !(z <= 2.8e-20)) {
		tmp = x + ((y / t) * z);
	} else {
		tmp = x + (x / (-t / y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((z <= (-6.5d+49)) .or. (.not. (z <= 2.8d-20))) then
        tmp = x + ((y / t) * z)
    else
        tmp = x + (x / (-t / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -6.5e+49) || !(z <= 2.8e-20)) {
		tmp = x + ((y / t) * z);
	} else {
		tmp = x + (x / (-t / y));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -6.5e+49) or not (z <= 2.8e-20):
		tmp = x + ((y / t) * z)
	else:
		tmp = x + (x / (-t / y))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -6.5e+49) || !(z <= 2.8e-20))
		tmp = Float64(x + Float64(Float64(y / t) * z));
	else
		tmp = Float64(x + Float64(x / Float64(Float64(-t) / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -6.5e+49) || ~((z <= 2.8e-20)))
		tmp = x + ((y / t) * z);
	else
		tmp = x + (x / (-t / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -6.5e+49], N[Not[LessEqual[z, 2.8e-20]], $MachinePrecision]], N[(x + N[(N[(y / t), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], N[(x + N[(x / N[((-t) / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -6.5000000000000005e49 or 2.8000000000000003e-20 < z

   1. Initial program 89.9%

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

      1. +-commutative 89.9%

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

      2. associate-*l/ 99.1%

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

      3. fma-def 99.1%

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

   3. Simplified 99.1%

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

      1. fma-udef 99.1%

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

      2. associate-*l/ 89.9%

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

      3. *-commutative 89.9%

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

      4. associate-/l* 98.9%

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

   5. Applied egg-rr 98.9%

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

   6. Taylor expanded in z around inf 84.7%

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

      1. *-commutative 84.7%

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

      2. associate-*r/ 92.5%

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

   8. Simplified 92.5%

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

   if -6.5000000000000005e49 < z < 2.8000000000000003e-20

   1. Initial program 93.0%

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

      1. associate-*l/ 97.1%

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

   3. Simplified 97.1%

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

   4. Taylor expanded in z around 0 84.3%

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

      1. associate-*r/ 84.3%

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

      2. mul-1-neg 84.3%

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

      3. distribute-lft-neg-out 84.3%

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

      4. *-lft-identity 84.3%

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

      5. associate-*l/ 84.3%

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

      6. associate-*r* 85.2%

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

      7. *-commutative 85.2%

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

      8. associate-*l/ 85.2%

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

      9. *-commutative 85.2%

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

      10. *-rgt-identity 85.2%

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

   6. Simplified 85.2%

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

      1. associate-*r/ 84.3%

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

      2. *-commutative 84.3%

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

      3. associate-/l* 88.9%

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

      4. add-sqr-sqrt 47.6%

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

      5. sqrt-unprod 53.2%

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

      6. sqr-neg 53.2%

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

      7. sqrt-unprod 22.4%

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

      8. add-sqr-sqrt 44.3%

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

      9. frac-2neg 44.3%

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

      10. add-sqr-sqrt 21.8%

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

      11. sqrt-unprod 44.7%

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

      12. sqr-neg 44.7%

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

      13. sqrt-unprod 41.2%

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

      14. add-sqr-sqrt 88.9%

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

      15. distribute-neg-frac 88.9%

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

   8. Applied egg-rr 88.9%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.5 \cdot 10^{+49} \lor \neg \left(z \leq 2.8 \cdot 10^{-20}\right):\\ \;\;\;\;x + \frac{y}{t} \cdot z\\ \mathbf{else}:\\ \;\;\;\;x + \frac{x}{\frac{-t}{y}}\\ \end{array} \]

Alternative 3: 83.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6.5 \cdot 10^{+49} \lor \neg \left(z \leq 4.8 \cdot 10^{-20}\right):\\ \;\;\;\;x + y \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - \frac{y}{t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -6.5e+49) (not (<= z 4.8e-20)))
   (+ x (* y (/ z t)))
   (* x (- 1.0 (/ y t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -6.5e+49) || !(z <= 4.8e-20)) {
		tmp = x + (y * (z / t));
	} else {
		tmp = x * (1.0 - (y / t));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((z <= (-6.5d+49)) .or. (.not. (z <= 4.8d-20))) then
        tmp = x + (y * (z / t))
    else
        tmp = x * (1.0d0 - (y / t))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -6.5e+49) || !(z <= 4.8e-20)) {
		tmp = x + (y * (z / t));
	} else {
		tmp = x * (1.0 - (y / t));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -6.5e+49) or not (z <= 4.8e-20):
		tmp = x + (y * (z / t))
	else:
		tmp = x * (1.0 - (y / t))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -6.5e+49) || !(z <= 4.8e-20))
		tmp = Float64(x + Float64(y * Float64(z / t)));
	else
		tmp = Float64(x * Float64(1.0 - Float64(y / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -6.5e+49) || ~((z <= 4.8e-20)))
		tmp = x + (y * (z / t));
	else
		tmp = x * (1.0 - (y / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -6.5e+49], N[Not[LessEqual[z, 4.8e-20]], $MachinePrecision]], N[(x + N[(y * N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(1.0 - N[(y / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -6.5000000000000005e49 or 4.79999999999999986e-20 < z

   1. Initial program 89.9%

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

      1. associate-*l/ 99.1%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in z around inf 84.7%

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

      1. *-commutative 84.7%

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

      2. associate-/l* 92.4%

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

      3. associate-/r/ 85.7%

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

   6. Simplified 85.7%

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

   if -6.5000000000000005e49 < z < 4.79999999999999986e-20

   1. Initial program 93.0%

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

      1. +-commutative 93.0%

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

      2. associate-*l/ 97.1%

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

      3. fma-def 97.1%

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

   3. Simplified 97.1%

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

      1. fma-udef 97.1%

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

      2. associate-*l/ 93.0%

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

      3. *-commutative 93.0%

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

      4. associate-/l* 97.2%

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

   5. Applied egg-rr 97.2%

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

   6. Taylor expanded in z around 0 84.3%

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

      1. *-lft-identity 84.3%

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

      2. associate-*r/ 88.8%

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

      3. *-commutative 88.8%

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

      4. associate-*r* 88.8%

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

      5. distribute-rgt-in 88.8%

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

      6. mul-1-neg 88.8%

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

      7. unsub-neg 88.8%

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

   8. Simplified 88.8%

      \[\leadsto \color{blue}{x \cdot \left(1 - \frac{y}{t}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.5 \cdot 10^{+49} \lor \neg \left(z \leq 4.8 \cdot 10^{-20}\right):\\ \;\;\;\;x + y \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - \frac{y}{t}\right)\\ \end{array} \]

Alternative 4: 83.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.62 \cdot 10^{+50} \lor \neg \left(z \leq 6.5 \cdot 10^{-20}\right):\\ \;\;\;\;x + \frac{y}{\frac{t}{z}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - \frac{y}{t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -1.62e+50) (not (<= z 6.5e-20)))
   (+ x (/ y (/ t z)))
   (* x (- 1.0 (/ y t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -1.62e+50) || !(z <= 6.5e-20)) {
		tmp = x + (y / (t / z));
	} else {
		tmp = x * (1.0 - (y / t));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((z <= (-1.62d+50)) .or. (.not. (z <= 6.5d-20))) then
        tmp = x + (y / (t / z))
    else
        tmp = x * (1.0d0 - (y / t))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -1.62e+50) || !(z <= 6.5e-20)) {
		tmp = x + (y / (t / z));
	} else {
		tmp = x * (1.0 - (y / t));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -1.62e+50) or not (z <= 6.5e-20):
		tmp = x + (y / (t / z))
	else:
		tmp = x * (1.0 - (y / t))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -1.62e+50) || !(z <= 6.5e-20))
		tmp = Float64(x + Float64(y / Float64(t / z)));
	else
		tmp = Float64(x * Float64(1.0 - Float64(y / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -1.62e+50) || ~((z <= 6.5e-20)))
		tmp = x + (y / (t / z));
	else
		tmp = x * (1.0 - (y / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -1.62e+50], N[Not[LessEqual[z, 6.5e-20]], $MachinePrecision]], N[(x + N[(y / N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(1.0 - N[(y / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.61999999999999996e50 or 6.50000000000000032e-20 < z

   1. Initial program 89.9%

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

      1. associate-*l/ 99.1%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in z around inf 84.7%

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

      1. associate-/l* 86.9%

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

   6. Simplified 86.9%

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

   if -1.61999999999999996e50 < z < 6.50000000000000032e-20

   1. Initial program 93.0%

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

      1. +-commutative 93.0%

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

      2. associate-*l/ 97.1%

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

      3. fma-def 97.1%

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

   3. Simplified 97.1%

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

      1. fma-udef 97.1%

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

      2. associate-*l/ 93.0%

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

      3. *-commutative 93.0%

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

      4. associate-/l* 97.2%

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

   5. Applied egg-rr 97.2%

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

   6. Taylor expanded in z around 0 84.3%

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

      1. *-lft-identity 84.3%

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

      2. associate-*r/ 88.8%

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

      3. *-commutative 88.8%

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

      4. associate-*r* 88.8%

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

      5. distribute-rgt-in 88.8%

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

      6. mul-1-neg 88.8%

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

      7. unsub-neg 88.8%

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

   8. Simplified 88.8%

      \[\leadsto \color{blue}{x \cdot \left(1 - \frac{y}{t}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.62 \cdot 10^{+50} \lor \neg \left(z \leq 6.5 \cdot 10^{-20}\right):\\ \;\;\;\;x + \frac{y}{\frac{t}{z}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - \frac{y}{t}\right)\\ \end{array} \]

Alternative 5: 85.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -8.8 \cdot 10^{+49} \lor \neg \left(z \leq 6 \cdot 10^{-20}\right):\\ \;\;\;\;x + \frac{y}{t} \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - \frac{y}{t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -8.8e+49) (not (<= z 6e-20)))
   (+ x (* (/ y t) z))
   (* x (- 1.0 (/ y t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -8.8e+49) || !(z <= 6e-20)) {
		tmp = x + ((y / t) * z);
	} else {
		tmp = x * (1.0 - (y / t));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((z <= (-8.8d+49)) .or. (.not. (z <= 6d-20))) then
        tmp = x + ((y / t) * z)
    else
        tmp = x * (1.0d0 - (y / t))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -8.8e+49) || !(z <= 6e-20)) {
		tmp = x + ((y / t) * z);
	} else {
		tmp = x * (1.0 - (y / t));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -8.8e+49) or not (z <= 6e-20):
		tmp = x + ((y / t) * z)
	else:
		tmp = x * (1.0 - (y / t))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -8.8e+49) || !(z <= 6e-20))
		tmp = Float64(x + Float64(Float64(y / t) * z));
	else
		tmp = Float64(x * Float64(1.0 - Float64(y / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -8.8e+49) || ~((z <= 6e-20)))
		tmp = x + ((y / t) * z);
	else
		tmp = x * (1.0 - (y / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -8.8e+49], N[Not[LessEqual[z, 6e-20]], $MachinePrecision]], N[(x + N[(N[(y / t), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], N[(x * N[(1.0 - N[(y / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -8.8000000000000003e49 or 6.00000000000000057e-20 < z

   1. Initial program 89.9%

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

      1. +-commutative 89.9%

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

      2. associate-*l/ 99.1%

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

      3. fma-def 99.1%

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

   3. Simplified 99.1%

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

      1. fma-udef 99.1%

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

      2. associate-*l/ 89.9%

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

      3. *-commutative 89.9%

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

      4. associate-/l* 98.9%

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

   5. Applied egg-rr 98.9%

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

   6. Taylor expanded in z around inf 84.7%

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

      1. *-commutative 84.7%

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

      2. associate-*r/ 92.5%

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

   8. Simplified 92.5%

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

   if -8.8000000000000003e49 < z < 6.00000000000000057e-20

   1. Initial program 93.0%

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

      1. +-commutative 93.0%

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

      2. associate-*l/ 97.1%

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

      3. fma-def 97.1%

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

   3. Simplified 97.1%

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

      1. fma-udef 97.1%

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

      2. associate-*l/ 93.0%

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

      3. *-commutative 93.0%

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

      4. associate-/l* 97.2%

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

   5. Applied egg-rr 97.2%

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

   6. Taylor expanded in z around 0 84.3%

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

      1. *-lft-identity 84.3%

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

      2. associate-*r/ 88.8%

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

      3. *-commutative 88.8%

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

      4. associate-*r* 88.8%

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

      5. distribute-rgt-in 88.8%

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

      6. mul-1-neg 88.8%

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

      7. unsub-neg 88.8%

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

   8. Simplified 88.8%

      \[\leadsto \color{blue}{x \cdot \left(1 - \frac{y}{t}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8.8 \cdot 10^{+49} \lor \neg \left(z \leq 6 \cdot 10^{-20}\right):\\ \;\;\;\;x + \frac{y}{t} \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - \frac{y}{t}\right)\\ \end{array} \]

Alternative 6: 83.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.7 \cdot 10^{+51}:\\ \;\;\;\;x + \frac{y}{\frac{t}{z}}\\ \mathbf{elif}\;z \leq 4.6 \cdot 10^{-20}:\\ \;\;\;\;x \cdot \left(1 - \frac{y}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot z}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -2.7e+51)
   (+ x (/ y (/ t z)))
   (if (<= z 4.6e-20) (* x (- 1.0 (/ y t))) (+ x (/ (* y z) t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -2.7e+51) {
		tmp = x + (y / (t / z));
	} else if (z <= 4.6e-20) {
		tmp = x * (1.0 - (y / t));
	} else {
		tmp = x + ((y * z) / t);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= (-2.7d+51)) then
        tmp = x + (y / (t / z))
    else if (z <= 4.6d-20) then
        tmp = x * (1.0d0 - (y / t))
    else
        tmp = x + ((y * z) / t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -2.7e+51) {
		tmp = x + (y / (t / z));
	} else if (z <= 4.6e-20) {
		tmp = x * (1.0 - (y / t));
	} else {
		tmp = x + ((y * z) / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -2.7e+51:
		tmp = x + (y / (t / z))
	elif z <= 4.6e-20:
		tmp = x * (1.0 - (y / t))
	else:
		tmp = x + ((y * z) / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -2.7e+51)
		tmp = Float64(x + Float64(y / Float64(t / z)));
	elseif (z <= 4.6e-20)
		tmp = Float64(x * Float64(1.0 - Float64(y / t)));
	else
		tmp = Float64(x + Float64(Float64(y * z) / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -2.7e+51)
		tmp = x + (y / (t / z));
	elseif (z <= 4.6e-20)
		tmp = x * (1.0 - (y / t));
	else
		tmp = x + ((y * z) / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -2.7e+51], N[(x + N[(y / N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 4.6e-20], N[(x * N[(1.0 - N[(y / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.69999999999999992e51

   1. Initial program 90.0%

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

      1. associate-*l/ 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in z around inf 87.9%

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

      1. associate-/l* 93.7%

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

   6. Simplified 93.7%

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

   if -2.69999999999999992e51 < z < 4.5999999999999998e-20

   1. Initial program 93.0%

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

      1. +-commutative 93.0%

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

      2. associate-*l/ 97.1%

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

      3. fma-def 97.1%

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

   3. Simplified 97.1%

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

      1. fma-udef 97.1%

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

      2. associate-*l/ 93.0%

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

      3. *-commutative 93.0%

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

      4. associate-/l* 97.2%

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

   5. Applied egg-rr 97.2%

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

   6. Taylor expanded in z around 0 84.3%

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

      1. *-lft-identity 84.3%

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

      2. associate-*r/ 88.8%

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

      3. *-commutative 88.8%

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

      4. associate-*r* 88.8%

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

      5. distribute-rgt-in 88.8%

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

      6. mul-1-neg 88.8%

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

      7. unsub-neg 88.8%

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

   8. Simplified 88.8%

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

   if 4.5999999999999998e-20 < z

   1. Initial program 89.9%

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

      1. associate-*l/ 98.6%

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

   3. Simplified 98.6%

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

   4. Taylor expanded in z around inf 82.7%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.7 \cdot 10^{+51}:\\ \;\;\;\;x + \frac{y}{\frac{t}{z}}\\ \mathbf{elif}\;z \leq 4.6 \cdot 10^{-20}:\\ \;\;\;\;x \cdot \left(1 - \frac{y}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot z}{t}\\ \end{array} \]

Alternative 7: 50.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.0028:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 7.5:\\ \;\;\;\;y \cdot \frac{-x}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -0.0028) x (if (<= t 7.5) (* y (/ (- x) t)) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -0.0028) {
		tmp = x;
	} else if (t <= 7.5) {
		tmp = y * (-x / t);
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-0.0028d0)) then
        tmp = x
    else if (t <= 7.5d0) then
        tmp = y * (-x / t)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -0.0028) {
		tmp = x;
	} else if (t <= 7.5) {
		tmp = y * (-x / t);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -0.0028:
		tmp = x
	elif t <= 7.5:
		tmp = y * (-x / t)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -0.0028)
		tmp = x;
	elseif (t <= 7.5)
		tmp = Float64(y * Float64(Float64(-x) / t));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -0.0028)
		tmp = x;
	elseif (t <= 7.5)
		tmp = y * (-x / t);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -0.0028], x, If[LessEqual[t, 7.5], N[(y * N[((-x) / t), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if t < -0.00279999999999999997 or 7.5 < t

   1. Initial program 86.6%

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

      1. associate-*l/ 99.2%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in y around 0 55.5%

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

   if -0.00279999999999999997 < t < 7.5

   1. Initial program 96.9%

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

      1. associate-*l/ 96.8%

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

   3. Simplified 96.8%

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

   4. Taylor expanded in z around 0 55.4%

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

      1. associate-*r/ 55.4%

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

      2. mul-1-neg 55.4%

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

      3. distribute-lft-neg-out 55.4%

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

      4. *-lft-identity 55.4%

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

      5. associate-*l/ 55.3%

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

      6. associate-*r* 50.8%

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

      7. *-commutative 50.8%

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

      8. associate-*l/ 50.8%

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

      9. *-commutative 50.8%

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

      10. *-rgt-identity 50.8%

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

   6. Simplified 50.8%

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

      1. distribute-frac-neg 50.8%

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

      2. distribute-rgt-neg-out 50.8%

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

      3. add-sqr-sqrt 20.7%

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

      4. sqrt-unprod 25.7%

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

      5. sqr-neg 25.7%

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

      6. sqrt-unprod 7.7%

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

      7. add-sqr-sqrt 14.6%

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

      8. sub-neg 14.6%

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

      9. add-sqr-sqrt 7.7%

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

      10. sqrt-unprod 25.7%

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

      11. sqr-neg 25.7%

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

      12. sqrt-unprod 20.7%

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

      13. add-sqr-sqrt 50.8%

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

   8. Applied egg-rr 50.8%

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

   9. Taylor expanded in y around 0 55.4%

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

       1. *-commutative 55.4%

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

       2. associate-/l* 50.8%

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

   11. Simplified 50.8%

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

   12. Taylor expanded in y around inf 46.5%

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

       1. mul-1-neg 46.5%

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

       2. *-commutative 46.5%

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

       3. distribute-frac-neg 46.5%

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

       4. distribute-rgt-neg-out 46.5%

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

       5. associate-*r/ 42.9%

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

   14. Simplified 42.9%

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

4. Final simplification 49.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.0028:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 7.5:\\ \;\;\;\;y \cdot \frac{-x}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 8: 97.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.5%

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

   1. associate-*l/ 98.0%

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

3. Simplified 98.0%

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

4. Final simplification 98.0%

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

Alternative 9: 65.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(1 - \frac{y}{t}\right) \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (* x (- 1.0 (/ y t))))

C

double code(double x, double y, double z, double t) {
	return x * (1.0 - (y / t));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x * (1.0d0 - (y / t))
end function

Java

public static double code(double x, double y, double z, double t) {
	return x * (1.0 - (y / t));
}

Python

def code(x, y, z, t):
	return x * (1.0 - (y / t))

Julia

function code(x, y, z, t)
	return Float64(x * Float64(1.0 - Float64(y / t)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = x * (1.0 - (y / t));
end

Wolfram

code[x_, y_, z_, t_] := N[(x * N[(1.0 - N[(y / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 91.5%

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

   1. +-commutative 91.5%

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

   2. associate-*l/ 98.0%

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

   3. fma-def 98.1%

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

3. Simplified 98.1%

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

   1. fma-udef 98.0%

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

   2. associate-*l/ 91.5%

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

   3. *-commutative 91.5%

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

   4. associate-/l* 98.0%

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

5. Applied egg-rr 98.0%

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

6. Taylor expanded in z around 0 59.0%

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

   1. *-lft-identity 59.0%

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

   2. associate-*r/ 62.4%

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

   3. *-commutative 62.4%

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

   4. associate-*r* 62.4%

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

   5. distribute-rgt-in 62.4%

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

   6. mul-1-neg 62.4%

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

   7. unsub-neg 62.4%

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

8. Simplified 62.4%

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

9. Final simplification 62.4%

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

Alternative 10: 38.9% accurate, 9.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.5%

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

   1. associate-*l/ 98.0%

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

3. Simplified 98.0%

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

4. Taylor expanded in y around 0 34.4%

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

5. Final simplification 34.4%

   \[\leadsto x \]

Developer target: 90.4% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023301 
(FPCore (x y z t)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, D"
  :precision binary64

  :herbie-target
  (- x (+ (* x (/ y t)) (* (- z) (/ y t))))

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