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

Percentage Accurate: 100.0% → 99.9%

Time: 6.4s

Alternatives: 8

Speedup: N/A×

• 27.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \frac{\left(x + y\right) - z}{t \cdot 2} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (/ (- (+ x y) z) (* t 2.0)))

C

double code(double x, double y, double z, double t) {
	return ((x + y) - z) / (t * 2.0);
}

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) / (t * 2.0d0)
end function

Java

public static double code(double x, double y, double z, double t) {
	return ((x + y) - z) / (t * 2.0);
}

Python

def code(x, y, z, t):
	return ((x + y) - z) / (t * 2.0)

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(x + y) - z) / Float64(t * 2.0))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = ((x + y) - z) / (t * 2.0);
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(x + y), $MachinePrecision] - z), $MachinePrecision] / N[(t * 2.0), $MachinePrecision]), $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(x + y\right) - z}{t \cdot 2} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (/ (- (+ x y) z) (* t 2.0)))

C

double code(double x, double y, double z, double t) {
	return ((x + y) - z) / (t * 2.0);
}

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) / (t * 2.0d0)
end function

Java

public static double code(double x, double y, double z, double t) {
	return ((x + y) - z) / (t * 2.0);
}

Python

def code(x, y, z, t):
	return ((x + y) - z) / (t * 2.0)

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(x + y) - z) / Float64(t * 2.0))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = ((x + y) - z) / (t * 2.0);
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(x + y), $MachinePrecision] - z), $MachinePrecision] / N[(t * 2.0), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{z - \left(x + y\right)}{t} \cdot -0.5 \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (* (/ (- z (+ x y)) t) -0.5))

C

double code(double x, double y, double z, double t) {
	return ((z - (x + y)) / t) * -0.5;
}

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 = ((z - (x + y)) / t) * (-0.5d0)
end function

Java

public static double code(double x, double y, double z, double t) {
	return ((z - (x + y)) / t) * -0.5;
}

Python

def code(x, y, z, t):
	return ((z - (x + y)) / t) * -0.5

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(z - Float64(x + y)) / t) * -0.5)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = ((z - (x + y)) / t) * -0.5;
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(z - N[(x + y), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] * -0.5), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

   1. associate-/r* 100.0%

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

   2. remove-double-neg 100.0%

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

   3. distribute-frac-neg2 100.0%

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

   4. neg-mul-1 100.0%

      \[\leadsto \frac{\color{blue}{-1 \cdot \frac{\left(x + y\right) - z}{-t}}}{2} \]

   5. *-commutative 100.0%

      \[\leadsto \frac{\color{blue}{\frac{\left(x + y\right) - z}{-t} \cdot -1}}{2} \]

   6. associate-/l* 100.0%

      \[\leadsto \color{blue}{\frac{\left(x + y\right) - z}{-t} \cdot \frac{-1}{2}} \]

   7. distribute-frac-neg2 100.0%

      \[\leadsto \color{blue}{\left(-\frac{\left(x + y\right) - z}{t}\right)} \cdot \frac{-1}{2} \]

   8. distribute-neg-frac 100.0%

      \[\leadsto \color{blue}{\frac{-\left(\left(x + y\right) - z\right)}{t}} \cdot \frac{-1}{2} \]

   9. sub-neg 100.0%

      \[\leadsto \frac{-\color{blue}{\left(\left(x + y\right) + \left(-z\right)\right)}}{t} \cdot \frac{-1}{2} \]

   10. distribute-neg-in 100.0%

       \[\leadsto \frac{\color{blue}{\left(-\left(x + y\right)\right) + \left(-\left(-z\right)\right)}}{t} \cdot \frac{-1}{2} \]

   11. remove-double-neg 100.0%

       \[\leadsto \frac{\left(-\left(x + y\right)\right) + \color{blue}{z}}{t} \cdot \frac{-1}{2} \]

   12. +-commutative 100.0%

       \[\leadsto \frac{\color{blue}{z + \left(-\left(x + y\right)\right)}}{t} \cdot \frac{-1}{2} \]

   13. sub-neg 100.0%

       \[\leadsto \frac{\color{blue}{z - \left(x + y\right)}}{t} \cdot \frac{-1}{2} \]

   14. metadata-eval 100.0%

       \[\leadsto \frac{z - \left(x + y\right)}{t} \cdot \color{blue}{-0.5} \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{z - \left(x + y\right)}{t} \cdot -0.5} \]
4. Add Preprocessing

5. Final simplification 100.0%

   \[\leadsto \frac{z - \left(x + y\right)}{t} \cdot -0.5 \]
6. Add Preprocessing

Alternative 2: 46.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{-22}:\\ \;\;\;\;\frac{x}{t} \cdot 0.5\\ \mathbf{elif}\;x \leq -4.5 \cdot 10^{-162}:\\ \;\;\;\;-0.5 \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{y}{-t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -1.05e-22)
   (* (/ x t) 0.5)
   (if (<= x -4.5e-162) (* -0.5 (/ z t)) (* -0.5 (/ y (- t))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -1.05e-22) {
		tmp = (x / t) * 0.5;
	} else if (x <= -4.5e-162) {
		tmp = -0.5 * (z / t);
	} else {
		tmp = -0.5 * (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 (x <= (-1.05d-22)) then
        tmp = (x / t) * 0.5d0
    else if (x <= (-4.5d-162)) then
        tmp = (-0.5d0) * (z / t)
    else
        tmp = (-0.5d0) * (y / -t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -1.05e-22) {
		tmp = (x / t) * 0.5;
	} else if (x <= -4.5e-162) {
		tmp = -0.5 * (z / t);
	} else {
		tmp = -0.5 * (y / -t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -1.05e-22:
		tmp = (x / t) * 0.5
	elif x <= -4.5e-162:
		tmp = -0.5 * (z / t)
	else:
		tmp = -0.5 * (y / -t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -1.05e-22)
		tmp = Float64(Float64(x / t) * 0.5);
	elseif (x <= -4.5e-162)
		tmp = Float64(-0.5 * Float64(z / t));
	else
		tmp = Float64(-0.5 * Float64(y / Float64(-t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -1.05e-22)
		tmp = (x / t) * 0.5;
	elseif (x <= -4.5e-162)
		tmp = -0.5 * (z / t);
	else
		tmp = -0.5 * (y / -t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -1.05e-22], N[(N[(x / t), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, -4.5e-162], N[(-0.5 * N[(z / t), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(y / (-t)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.05000000000000004e-22

   1. Initial program 99.9%

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

      1. associate-/r* 99.9%

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

      2. remove-double-neg 99.9%

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

      3. distribute-frac-neg2 99.9%

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

      4. neg-mul-1 99.9%

         \[\leadsto \frac{\color{blue}{-1 \cdot \frac{\left(x + y\right) - z}{-t}}}{2} \]

      5. *-commutative 99.9%

         \[\leadsto \frac{\color{blue}{\frac{\left(x + y\right) - z}{-t} \cdot -1}}{2} \]

      6. associate-/l* 99.9%

         \[\leadsto \color{blue}{\frac{\left(x + y\right) - z}{-t} \cdot \frac{-1}{2}} \]

      7. distribute-frac-neg2 99.9%

         \[\leadsto \color{blue}{\left(-\frac{\left(x + y\right) - z}{t}\right)} \cdot \frac{-1}{2} \]

      8. distribute-neg-frac 99.9%

         \[\leadsto \color{blue}{\frac{-\left(\left(x + y\right) - z\right)}{t}} \cdot \frac{-1}{2} \]

      9. sub-neg 99.9%

         \[\leadsto \frac{-\color{blue}{\left(\left(x + y\right) + \left(-z\right)\right)}}{t} \cdot \frac{-1}{2} \]

      10. distribute-neg-in 99.9%

          \[\leadsto \frac{\color{blue}{\left(-\left(x + y\right)\right) + \left(-\left(-z\right)\right)}}{t} \cdot \frac{-1}{2} \]

      11. remove-double-neg 99.9%

          \[\leadsto \frac{\left(-\left(x + y\right)\right) + \color{blue}{z}}{t} \cdot \frac{-1}{2} \]

      12. +-commutative 99.9%

          \[\leadsto \frac{\color{blue}{z + \left(-\left(x + y\right)\right)}}{t} \cdot \frac{-1}{2} \]

      13. sub-neg 99.9%

          \[\leadsto \frac{\color{blue}{z - \left(x + y\right)}}{t} \cdot \frac{-1}{2} \]

      14. metadata-eval 99.9%

          \[\leadsto \frac{z - \left(x + y\right)}{t} \cdot \color{blue}{-0.5} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{z - \left(x + y\right)}{t} \cdot -0.5} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate--r+ 99.9%

         \[\leadsto \frac{\color{blue}{\left(z - x\right) - y}}{t} \cdot -0.5 \]

      2. div-sub 96.2%

         \[\leadsto \color{blue}{\left(\frac{z - x}{t} - \frac{y}{t}\right)} \cdot -0.5 \]

   6. Applied egg-rr 96.2%

      \[\leadsto \color{blue}{\left(\frac{z - x}{t} - \frac{y}{t}\right)} \cdot -0.5 \]

   7. Taylor expanded in y around 0 80.8%

      \[\leadsto \color{blue}{\left(\frac{z}{t} - \frac{x}{t}\right)} \cdot -0.5 \]

   8. Taylor expanded in z around 0 65.9%

      \[\leadsto \color{blue}{0.5 \cdot \frac{x}{t}} \]
   9. Step-by-step derivation

      1. *-commutative 65.9%

         \[\leadsto \color{blue}{\frac{x}{t} \cdot 0.5} \]

   10. Simplified 65.9%

       \[\leadsto \color{blue}{\frac{x}{t} \cdot 0.5} \]

   if -1.05000000000000004e-22 < x < -4.50000000000000023e-162

   1. Initial program 100.0%

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

      1. associate-/r* 100.0%

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

      2. remove-double-neg 100.0%

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

      3. distribute-frac-neg2 100.0%

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

      4. neg-mul-1 100.0%

         \[\leadsto \frac{\color{blue}{-1 \cdot \frac{\left(x + y\right) - z}{-t}}}{2} \]

      5. *-commutative 100.0%

         \[\leadsto \frac{\color{blue}{\frac{\left(x + y\right) - z}{-t} \cdot -1}}{2} \]

      6. associate-/l* 100.0%

         \[\leadsto \color{blue}{\frac{\left(x + y\right) - z}{-t} \cdot \frac{-1}{2}} \]

      7. distribute-frac-neg2 100.0%

         \[\leadsto \color{blue}{\left(-\frac{\left(x + y\right) - z}{t}\right)} \cdot \frac{-1}{2} \]

      8. distribute-neg-frac 100.0%

         \[\leadsto \color{blue}{\frac{-\left(\left(x + y\right) - z\right)}{t}} \cdot \frac{-1}{2} \]

      9. sub-neg 100.0%

         \[\leadsto \frac{-\color{blue}{\left(\left(x + y\right) + \left(-z\right)\right)}}{t} \cdot \frac{-1}{2} \]

      10. distribute-neg-in 100.0%

          \[\leadsto \frac{\color{blue}{\left(-\left(x + y\right)\right) + \left(-\left(-z\right)\right)}}{t} \cdot \frac{-1}{2} \]

      11. remove-double-neg 100.0%

          \[\leadsto \frac{\left(-\left(x + y\right)\right) + \color{blue}{z}}{t} \cdot \frac{-1}{2} \]

      12. +-commutative 100.0%

          \[\leadsto \frac{\color{blue}{z + \left(-\left(x + y\right)\right)}}{t} \cdot \frac{-1}{2} \]

      13. sub-neg 100.0%

          \[\leadsto \frac{\color{blue}{z - \left(x + y\right)}}{t} \cdot \frac{-1}{2} \]

      14. metadata-eval 100.0%

          \[\leadsto \frac{z - \left(x + y\right)}{t} \cdot \color{blue}{-0.5} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{z - \left(x + y\right)}{t} \cdot -0.5} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 39.1%

      \[\leadsto \color{blue}{\frac{z}{t}} \cdot -0.5 \]

   if -4.50000000000000023e-162 < x

   1. Initial program 99.3%

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

      1. associate-/r* 100.0%

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

      2. remove-double-neg 100.0%

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

      3. distribute-frac-neg2 100.0%

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

      4. neg-mul-1 100.0%

         \[\leadsto \frac{\color{blue}{-1 \cdot \frac{\left(x + y\right) - z}{-t}}}{2} \]

      5. *-commutative 100.0%

         \[\leadsto \frac{\color{blue}{\frac{\left(x + y\right) - z}{-t} \cdot -1}}{2} \]

      6. associate-/l* 100.0%

         \[\leadsto \color{blue}{\frac{\left(x + y\right) - z}{-t} \cdot \frac{-1}{2}} \]

      7. distribute-frac-neg2 100.0%

         \[\leadsto \color{blue}{\left(-\frac{\left(x + y\right) - z}{t}\right)} \cdot \frac{-1}{2} \]

      8. distribute-neg-frac 100.0%

         \[\leadsto \color{blue}{\frac{-\left(\left(x + y\right) - z\right)}{t}} \cdot \frac{-1}{2} \]

      9. sub-neg 100.0%

         \[\leadsto \frac{-\color{blue}{\left(\left(x + y\right) + \left(-z\right)\right)}}{t} \cdot \frac{-1}{2} \]

      10. distribute-neg-in 100.0%

          \[\leadsto \frac{\color{blue}{\left(-\left(x + y\right)\right) + \left(-\left(-z\right)\right)}}{t} \cdot \frac{-1}{2} \]

      11. remove-double-neg 100.0%

          \[\leadsto \frac{\left(-\left(x + y\right)\right) + \color{blue}{z}}{t} \cdot \frac{-1}{2} \]

      12. +-commutative 100.0%

          \[\leadsto \frac{\color{blue}{z + \left(-\left(x + y\right)\right)}}{t} \cdot \frac{-1}{2} \]

      13. sub-neg 100.0%

          \[\leadsto \frac{\color{blue}{z - \left(x + y\right)}}{t} \cdot \frac{-1}{2} \]

      14. metadata-eval 100.0%

          \[\leadsto \frac{z - \left(x + y\right)}{t} \cdot \color{blue}{-0.5} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{z - \left(x + y\right)}{t} \cdot -0.5} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf 44.4%

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{y}{t}\right)} \cdot -0.5 \]
   6. Step-by-step derivation

      1. mul-1-neg 44.4%

         \[\leadsto \color{blue}{\left(-\frac{y}{t}\right)} \cdot -0.5 \]

      2. distribute-neg-frac2 44.4%

         \[\leadsto \color{blue}{\frac{y}{-t}} \cdot -0.5 \]

   7. Simplified 44.4%

      \[\leadsto \color{blue}{\frac{y}{-t}} \cdot -0.5 \]
3. Recombined 3 regimes into one program.

4. Final simplification 50.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{-22}:\\ \;\;\;\;\frac{x}{t} \cdot 0.5\\ \mathbf{elif}\;x \leq -4.5 \cdot 10^{-162}:\\ \;\;\;\;-0.5 \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{y}{-t}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 69.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + y \leq -1 \cdot 10^{-175}:\\ \;\;\;\;-0.5 \cdot \frac{z - x}{t}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{z - y}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ x y) -1e-175) (* -0.5 (/ (- z x) t)) (* -0.5 (/ (- z y) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x + y) <= -1e-175) {
		tmp = -0.5 * ((z - x) / t);
	} else {
		tmp = -0.5 * ((z - 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 ((x + y) <= (-1d-175)) then
        tmp = (-0.5d0) * ((z - x) / t)
    else
        tmp = (-0.5d0) * ((z - y) / t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x + y) <= -1e-175) {
		tmp = -0.5 * ((z - x) / t);
	} else {
		tmp = -0.5 * ((z - y) / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x + y) <= -1e-175:
		tmp = -0.5 * ((z - x) / t)
	else:
		tmp = -0.5 * ((z - y) / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x + y) <= -1e-175)
		tmp = Float64(-0.5 * Float64(Float64(z - x) / t));
	else
		tmp = Float64(-0.5 * Float64(Float64(z - y) / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x + y) <= -1e-175)
		tmp = -0.5 * ((z - x) / t);
	else
		tmp = -0.5 * ((z - y) / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x + y), $MachinePrecision], -1e-175], N[(-0.5 * N[(N[(z - x), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(N[(z - y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x y) < -1e-175

   1. Initial program 100.0%

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

      1. associate-/r* 100.0%

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

      2. remove-double-neg 100.0%

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

      3. distribute-frac-neg2 100.0%

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

      4. neg-mul-1 100.0%

         \[\leadsto \frac{\color{blue}{-1 \cdot \frac{\left(x + y\right) - z}{-t}}}{2} \]

      5. *-commutative 100.0%

         \[\leadsto \frac{\color{blue}{\frac{\left(x + y\right) - z}{-t} \cdot -1}}{2} \]

      6. associate-/l* 100.0%

         \[\leadsto \color{blue}{\frac{\left(x + y\right) - z}{-t} \cdot \frac{-1}{2}} \]

      7. distribute-frac-neg2 100.0%

         \[\leadsto \color{blue}{\left(-\frac{\left(x + y\right) - z}{t}\right)} \cdot \frac{-1}{2} \]

      8. distribute-neg-frac 100.0%

         \[\leadsto \color{blue}{\frac{-\left(\left(x + y\right) - z\right)}{t}} \cdot \frac{-1}{2} \]

      9. sub-neg 100.0%

         \[\leadsto \frac{-\color{blue}{\left(\left(x + y\right) + \left(-z\right)\right)}}{t} \cdot \frac{-1}{2} \]

      10. distribute-neg-in 100.0%

          \[\leadsto \frac{\color{blue}{\left(-\left(x + y\right)\right) + \left(-\left(-z\right)\right)}}{t} \cdot \frac{-1}{2} \]

      11. remove-double-neg 100.0%

          \[\leadsto \frac{\left(-\left(x + y\right)\right) + \color{blue}{z}}{t} \cdot \frac{-1}{2} \]

      12. +-commutative 100.0%

          \[\leadsto \frac{\color{blue}{z + \left(-\left(x + y\right)\right)}}{t} \cdot \frac{-1}{2} \]

      13. sub-neg 100.0%

          \[\leadsto \frac{\color{blue}{z - \left(x + y\right)}}{t} \cdot \frac{-1}{2} \]

      14. metadata-eval 100.0%

          \[\leadsto \frac{z - \left(x + y\right)}{t} \cdot \color{blue}{-0.5} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{z - \left(x + y\right)}{t} \cdot -0.5} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 64.5%

      \[\leadsto \color{blue}{\frac{z - x}{t}} \cdot -0.5 \]

   if -1e-175 < (+.f64 x y)

   1. Initial program 99.2%

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

      1. associate-/r* 100.0%

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

      2. remove-double-neg 100.0%

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

      3. distribute-frac-neg2 100.0%

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

      4. neg-mul-1 100.0%

         \[\leadsto \frac{\color{blue}{-1 \cdot \frac{\left(x + y\right) - z}{-t}}}{2} \]

      5. *-commutative 100.0%

         \[\leadsto \frac{\color{blue}{\frac{\left(x + y\right) - z}{-t} \cdot -1}}{2} \]

      6. associate-/l* 100.0%

         \[\leadsto \color{blue}{\frac{\left(x + y\right) - z}{-t} \cdot \frac{-1}{2}} \]

      7. distribute-frac-neg2 100.0%

         \[\leadsto \color{blue}{\left(-\frac{\left(x + y\right) - z}{t}\right)} \cdot \frac{-1}{2} \]

      8. distribute-neg-frac 100.0%

         \[\leadsto \color{blue}{\frac{-\left(\left(x + y\right) - z\right)}{t}} \cdot \frac{-1}{2} \]

      9. sub-neg 100.0%

         \[\leadsto \frac{-\color{blue}{\left(\left(x + y\right) + \left(-z\right)\right)}}{t} \cdot \frac{-1}{2} \]

      10. distribute-neg-in 100.0%

          \[\leadsto \frac{\color{blue}{\left(-\left(x + y\right)\right) + \left(-\left(-z\right)\right)}}{t} \cdot \frac{-1}{2} \]

      11. remove-double-neg 100.0%

          \[\leadsto \frac{\left(-\left(x + y\right)\right) + \color{blue}{z}}{t} \cdot \frac{-1}{2} \]

      12. +-commutative 100.0%

          \[\leadsto \frac{\color{blue}{z + \left(-\left(x + y\right)\right)}}{t} \cdot \frac{-1}{2} \]

      13. sub-neg 100.0%

          \[\leadsto \frac{\color{blue}{z - \left(x + y\right)}}{t} \cdot \frac{-1}{2} \]

      14. metadata-eval 100.0%

          \[\leadsto \frac{z - \left(x + y\right)}{t} \cdot \color{blue}{-0.5} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{z - \left(x + y\right)}{t} \cdot -0.5} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 66.1%

      \[\leadsto \color{blue}{\frac{z - y}{t}} \cdot -0.5 \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + y \leq -1 \cdot 10^{-175}:\\ \;\;\;\;-0.5 \cdot \frac{z - x}{t}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{z - y}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 8 \cdot 10^{+90}:\\ \;\;\;\;-0.5 \cdot \frac{z - x}{t}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{y}{-t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 8e+90) (* -0.5 (/ (- z x) t)) (* -0.5 (/ y (- t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 8e+90) {
		tmp = -0.5 * ((z - x) / t);
	} else {
		tmp = -0.5 * (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 (y <= 8d+90) then
        tmp = (-0.5d0) * ((z - x) / t)
    else
        tmp = (-0.5d0) * (y / -t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 8e+90) {
		tmp = -0.5 * ((z - x) / t);
	} else {
		tmp = -0.5 * (y / -t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= 8e+90:
		tmp = -0.5 * ((z - x) / t)
	else:
		tmp = -0.5 * (y / -t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 8e+90)
		tmp = Float64(-0.5 * Float64(Float64(z - x) / t));
	else
		tmp = Float64(-0.5 * Float64(y / Float64(-t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 8e+90)
		tmp = -0.5 * ((z - x) / t);
	else
		tmp = -0.5 * (y / -t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 8e+90], N[(-0.5 * N[(N[(z - x), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(y / (-t)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 7.99999999999999973e90

   1. Initial program 100.0%

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

      1. associate-/r* 100.0%

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

      2. remove-double-neg 100.0%

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

      3. distribute-frac-neg2 100.0%

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

      4. neg-mul-1 100.0%

         \[\leadsto \frac{\color{blue}{-1 \cdot \frac{\left(x + y\right) - z}{-t}}}{2} \]

      5. *-commutative 100.0%

         \[\leadsto \frac{\color{blue}{\frac{\left(x + y\right) - z}{-t} \cdot -1}}{2} \]

      6. associate-/l* 100.0%

         \[\leadsto \color{blue}{\frac{\left(x + y\right) - z}{-t} \cdot \frac{-1}{2}} \]

      7. distribute-frac-neg2 100.0%

         \[\leadsto \color{blue}{\left(-\frac{\left(x + y\right) - z}{t}\right)} \cdot \frac{-1}{2} \]

      8. distribute-neg-frac 100.0%

         \[\leadsto \color{blue}{\frac{-\left(\left(x + y\right) - z\right)}{t}} \cdot \frac{-1}{2} \]

      9. sub-neg 100.0%

         \[\leadsto \frac{-\color{blue}{\left(\left(x + y\right) + \left(-z\right)\right)}}{t} \cdot \frac{-1}{2} \]

      10. distribute-neg-in 100.0%

          \[\leadsto \frac{\color{blue}{\left(-\left(x + y\right)\right) + \left(-\left(-z\right)\right)}}{t} \cdot \frac{-1}{2} \]

      11. remove-double-neg 100.0%

          \[\leadsto \frac{\left(-\left(x + y\right)\right) + \color{blue}{z}}{t} \cdot \frac{-1}{2} \]

      12. +-commutative 100.0%

          \[\leadsto \frac{\color{blue}{z + \left(-\left(x + y\right)\right)}}{t} \cdot \frac{-1}{2} \]

      13. sub-neg 100.0%

          \[\leadsto \frac{\color{blue}{z - \left(x + y\right)}}{t} \cdot \frac{-1}{2} \]

      14. metadata-eval 100.0%

          \[\leadsto \frac{z - \left(x + y\right)}{t} \cdot \color{blue}{-0.5} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{z - \left(x + y\right)}{t} \cdot -0.5} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 72.3%

      \[\leadsto \color{blue}{\frac{z - x}{t}} \cdot -0.5 \]

   if 7.99999999999999973e90 < y

   1. Initial program 97.2%

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

      1. associate-/r* 100.0%

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

      2. remove-double-neg 100.0%

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

      3. distribute-frac-neg2 100.0%

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

      4. neg-mul-1 100.0%

         \[\leadsto \frac{\color{blue}{-1 \cdot \frac{\left(x + y\right) - z}{-t}}}{2} \]

      5. *-commutative 100.0%

         \[\leadsto \frac{\color{blue}{\frac{\left(x + y\right) - z}{-t} \cdot -1}}{2} \]

      6. associate-/l* 100.0%

         \[\leadsto \color{blue}{\frac{\left(x + y\right) - z}{-t} \cdot \frac{-1}{2}} \]

      7. distribute-frac-neg2 100.0%

         \[\leadsto \color{blue}{\left(-\frac{\left(x + y\right) - z}{t}\right)} \cdot \frac{-1}{2} \]

      8. distribute-neg-frac 100.0%

         \[\leadsto \color{blue}{\frac{-\left(\left(x + y\right) - z\right)}{t}} \cdot \frac{-1}{2} \]

      9. sub-neg 100.0%

         \[\leadsto \frac{-\color{blue}{\left(\left(x + y\right) + \left(-z\right)\right)}}{t} \cdot \frac{-1}{2} \]

      10. distribute-neg-in 100.0%

          \[\leadsto \frac{\color{blue}{\left(-\left(x + y\right)\right) + \left(-\left(-z\right)\right)}}{t} \cdot \frac{-1}{2} \]

      11. remove-double-neg 100.0%

          \[\leadsto \frac{\left(-\left(x + y\right)\right) + \color{blue}{z}}{t} \cdot \frac{-1}{2} \]

      12. +-commutative 100.0%

          \[\leadsto \frac{\color{blue}{z + \left(-\left(x + y\right)\right)}}{t} \cdot \frac{-1}{2} \]

      13. sub-neg 100.0%

          \[\leadsto \frac{\color{blue}{z - \left(x + y\right)}}{t} \cdot \frac{-1}{2} \]

      14. metadata-eval 100.0%

          \[\leadsto \frac{z - \left(x + y\right)}{t} \cdot \color{blue}{-0.5} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{z - \left(x + y\right)}{t} \cdot -0.5} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf 80.6%

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{y}{t}\right)} \cdot -0.5 \]
   6. Step-by-step derivation

      1. mul-1-neg 80.6%

         \[\leadsto \color{blue}{\left(-\frac{y}{t}\right)} \cdot -0.5 \]

      2. distribute-neg-frac2 80.6%

         \[\leadsto \color{blue}{\frac{y}{-t}} \cdot -0.5 \]

   7. Simplified 80.6%

      \[\leadsto \color{blue}{\frac{y}{-t}} \cdot -0.5 \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8 \cdot 10^{+90}:\\ \;\;\;\;-0.5 \cdot \frac{z - x}{t}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{y}{-t}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 46.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{-22}:\\ \;\;\;\;x \cdot \frac{0.5}{t}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{-0.5}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -1.1e-22) (* x (/ 0.5 t)) (* z (/ -0.5 t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -1.1e-22) {
		tmp = x * (0.5 / t);
	} else {
		tmp = z * (-0.5 / 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 (x <= (-1.1d-22)) then
        tmp = x * (0.5d0 / t)
    else
        tmp = z * ((-0.5d0) / t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -1.1e-22) {
		tmp = x * (0.5 / t);
	} else {
		tmp = z * (-0.5 / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -1.1e-22:
		tmp = x * (0.5 / t)
	else:
		tmp = z * (-0.5 / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -1.1e-22)
		tmp = Float64(x * Float64(0.5 / t));
	else
		tmp = Float64(z * Float64(-0.5 / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -1.1e-22)
		tmp = x * (0.5 / t);
	else
		tmp = z * (-0.5 / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -1.1e-22], N[(x * N[(0.5 / t), $MachinePrecision]), $MachinePrecision], N[(z * N[(-0.5 / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.1e-22

   1. Initial program 99.9%

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

      1. associate-/r* 99.9%

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

      2. remove-double-neg 99.9%

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

      3. distribute-frac-neg2 99.9%

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

      4. neg-mul-1 99.9%

         \[\leadsto \frac{\color{blue}{-1 \cdot \frac{\left(x + y\right) - z}{-t}}}{2} \]

      5. *-commutative 99.9%

         \[\leadsto \frac{\color{blue}{\frac{\left(x + y\right) - z}{-t} \cdot -1}}{2} \]

      6. associate-/l* 99.9%

         \[\leadsto \color{blue}{\frac{\left(x + y\right) - z}{-t} \cdot \frac{-1}{2}} \]

      7. distribute-frac-neg2 99.9%

         \[\leadsto \color{blue}{\left(-\frac{\left(x + y\right) - z}{t}\right)} \cdot \frac{-1}{2} \]

      8. distribute-neg-frac 99.9%

         \[\leadsto \color{blue}{\frac{-\left(\left(x + y\right) - z\right)}{t}} \cdot \frac{-1}{2} \]

      9. sub-neg 99.9%

         \[\leadsto \frac{-\color{blue}{\left(\left(x + y\right) + \left(-z\right)\right)}}{t} \cdot \frac{-1}{2} \]

      10. distribute-neg-in 99.9%

          \[\leadsto \frac{\color{blue}{\left(-\left(x + y\right)\right) + \left(-\left(-z\right)\right)}}{t} \cdot \frac{-1}{2} \]

      11. remove-double-neg 99.9%

          \[\leadsto \frac{\left(-\left(x + y\right)\right) + \color{blue}{z}}{t} \cdot \frac{-1}{2} \]

      12. +-commutative 99.9%

          \[\leadsto \frac{\color{blue}{z + \left(-\left(x + y\right)\right)}}{t} \cdot \frac{-1}{2} \]

      13. sub-neg 99.9%

          \[\leadsto \frac{\color{blue}{z - \left(x + y\right)}}{t} \cdot \frac{-1}{2} \]

      14. metadata-eval 99.9%

          \[\leadsto \frac{z - \left(x + y\right)}{t} \cdot \color{blue}{-0.5} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{z - \left(x + y\right)}{t} \cdot -0.5} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate--r+ 99.9%

         \[\leadsto \frac{\color{blue}{\left(z - x\right) - y}}{t} \cdot -0.5 \]

      2. div-sub 96.2%

         \[\leadsto \color{blue}{\left(\frac{z - x}{t} - \frac{y}{t}\right)} \cdot -0.5 \]

   6. Applied egg-rr 96.2%

      \[\leadsto \color{blue}{\left(\frac{z - x}{t} - \frac{y}{t}\right)} \cdot -0.5 \]

   7. Taylor expanded in y around 0 80.8%

      \[\leadsto \color{blue}{\left(\frac{z}{t} - \frac{x}{t}\right)} \cdot -0.5 \]

   8. Taylor expanded in z around 0 65.9%

      \[\leadsto \color{blue}{0.5 \cdot \frac{x}{t}} \]
   9. Step-by-step derivation

      1. *-commutative 65.9%

         \[\leadsto \color{blue}{\frac{x}{t} \cdot 0.5} \]

      2. associate-*l/ 65.9%

         \[\leadsto \color{blue}{\frac{x \cdot 0.5}{t}} \]

      3. associate-*r/ 65.8%

         \[\leadsto \color{blue}{x \cdot \frac{0.5}{t}} \]

   10. Simplified 65.8%

       \[\leadsto \color{blue}{x \cdot \frac{0.5}{t}} \]

   if -1.1e-22 < x

   1. Initial program 99.5%

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

      1. associate-/r* 100.0%

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

      2. remove-double-neg 100.0%

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

      3. distribute-frac-neg2 100.0%

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

      4. neg-mul-1 100.0%

         \[\leadsto \frac{\color{blue}{-1 \cdot \frac{\left(x + y\right) - z}{-t}}}{2} \]

      5. *-commutative 100.0%

         \[\leadsto \frac{\color{blue}{\frac{\left(x + y\right) - z}{-t} \cdot -1}}{2} \]

      6. associate-/l* 100.0%

         \[\leadsto \color{blue}{\frac{\left(x + y\right) - z}{-t} \cdot \frac{-1}{2}} \]

      7. distribute-frac-neg2 100.0%

         \[\leadsto \color{blue}{\left(-\frac{\left(x + y\right) - z}{t}\right)} \cdot \frac{-1}{2} \]

      8. distribute-neg-frac 100.0%

         \[\leadsto \color{blue}{\frac{-\left(\left(x + y\right) - z\right)}{t}} \cdot \frac{-1}{2} \]

      9. sub-neg 100.0%

         \[\leadsto \frac{-\color{blue}{\left(\left(x + y\right) + \left(-z\right)\right)}}{t} \cdot \frac{-1}{2} \]

      10. distribute-neg-in 100.0%

          \[\leadsto \frac{\color{blue}{\left(-\left(x + y\right)\right) + \left(-\left(-z\right)\right)}}{t} \cdot \frac{-1}{2} \]

      11. remove-double-neg 100.0%

          \[\leadsto \frac{\left(-\left(x + y\right)\right) + \color{blue}{z}}{t} \cdot \frac{-1}{2} \]

      12. +-commutative 100.0%

          \[\leadsto \frac{\color{blue}{z + \left(-\left(x + y\right)\right)}}{t} \cdot \frac{-1}{2} \]

      13. sub-neg 100.0%

          \[\leadsto \frac{\color{blue}{z - \left(x + y\right)}}{t} \cdot \frac{-1}{2} \]

      14. metadata-eval 100.0%

          \[\leadsto \frac{z - \left(x + y\right)}{t} \cdot \color{blue}{-0.5} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{z - \left(x + y\right)}{t} \cdot -0.5} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 33.9%

      \[\leadsto \color{blue}{\frac{z}{t}} \cdot -0.5 \]

   6. Taylor expanded in z around 0 33.9%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{z}{t}} \]
   7. Step-by-step derivation

      1. associate-*r/ 33.9%

         \[\leadsto \color{blue}{\frac{-0.5 \cdot z}{t}} \]

      2. *-commutative 33.9%

         \[\leadsto \frac{\color{blue}{z \cdot -0.5}}{t} \]

      3. associate-*r/ 33.8%

         \[\leadsto \color{blue}{z \cdot \frac{-0.5}{t}} \]

   8. Simplified 33.8%

      \[\leadsto \color{blue}{z \cdot \frac{-0.5}{t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 43.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{-22}:\\ \;\;\;\;x \cdot \frac{0.5}{t}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{-0.5}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 46.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{-23}:\\ \;\;\;\;\frac{x}{t} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{-0.5}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -1.6e-23) (* (/ x t) 0.5) (* z (/ -0.5 t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -1.6e-23) {
		tmp = (x / t) * 0.5;
	} else {
		tmp = z * (-0.5 / 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 (x <= (-1.6d-23)) then
        tmp = (x / t) * 0.5d0
    else
        tmp = z * ((-0.5d0) / t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -1.6e-23) {
		tmp = (x / t) * 0.5;
	} else {
		tmp = z * (-0.5 / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -1.6e-23:
		tmp = (x / t) * 0.5
	else:
		tmp = z * (-0.5 / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -1.6e-23)
		tmp = Float64(Float64(x / t) * 0.5);
	else
		tmp = Float64(z * Float64(-0.5 / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -1.6e-23)
		tmp = (x / t) * 0.5;
	else
		tmp = z * (-0.5 / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -1.6e-23], N[(N[(x / t), $MachinePrecision] * 0.5), $MachinePrecision], N[(z * N[(-0.5 / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.59999999999999988e-23

   1. Initial program 99.9%

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

      1. associate-/r* 99.9%

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

      2. remove-double-neg 99.9%

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

      3. distribute-frac-neg2 99.9%

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

      4. neg-mul-1 99.9%

         \[\leadsto \frac{\color{blue}{-1 \cdot \frac{\left(x + y\right) - z}{-t}}}{2} \]

      5. *-commutative 99.9%

         \[\leadsto \frac{\color{blue}{\frac{\left(x + y\right) - z}{-t} \cdot -1}}{2} \]

      6. associate-/l* 99.9%

         \[\leadsto \color{blue}{\frac{\left(x + y\right) - z}{-t} \cdot \frac{-1}{2}} \]

      7. distribute-frac-neg2 99.9%

         \[\leadsto \color{blue}{\left(-\frac{\left(x + y\right) - z}{t}\right)} \cdot \frac{-1}{2} \]

      8. distribute-neg-frac 99.9%

         \[\leadsto \color{blue}{\frac{-\left(\left(x + y\right) - z\right)}{t}} \cdot \frac{-1}{2} \]

      9. sub-neg 99.9%

         \[\leadsto \frac{-\color{blue}{\left(\left(x + y\right) + \left(-z\right)\right)}}{t} \cdot \frac{-1}{2} \]

      10. distribute-neg-in 99.9%

          \[\leadsto \frac{\color{blue}{\left(-\left(x + y\right)\right) + \left(-\left(-z\right)\right)}}{t} \cdot \frac{-1}{2} \]

      11. remove-double-neg 99.9%

          \[\leadsto \frac{\left(-\left(x + y\right)\right) + \color{blue}{z}}{t} \cdot \frac{-1}{2} \]

      12. +-commutative 99.9%

          \[\leadsto \frac{\color{blue}{z + \left(-\left(x + y\right)\right)}}{t} \cdot \frac{-1}{2} \]

      13. sub-neg 99.9%

          \[\leadsto \frac{\color{blue}{z - \left(x + y\right)}}{t} \cdot \frac{-1}{2} \]

      14. metadata-eval 99.9%

          \[\leadsto \frac{z - \left(x + y\right)}{t} \cdot \color{blue}{-0.5} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{z - \left(x + y\right)}{t} \cdot -0.5} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate--r+ 99.9%

         \[\leadsto \frac{\color{blue}{\left(z - x\right) - y}}{t} \cdot -0.5 \]

      2. div-sub 96.2%

         \[\leadsto \color{blue}{\left(\frac{z - x}{t} - \frac{y}{t}\right)} \cdot -0.5 \]

   6. Applied egg-rr 96.2%

      \[\leadsto \color{blue}{\left(\frac{z - x}{t} - \frac{y}{t}\right)} \cdot -0.5 \]

   7. Taylor expanded in y around 0 80.8%

      \[\leadsto \color{blue}{\left(\frac{z}{t} - \frac{x}{t}\right)} \cdot -0.5 \]

   8. Taylor expanded in z around 0 65.9%

      \[\leadsto \color{blue}{0.5 \cdot \frac{x}{t}} \]
   9. Step-by-step derivation

      1. *-commutative 65.9%

         \[\leadsto \color{blue}{\frac{x}{t} \cdot 0.5} \]

   10. Simplified 65.9%

       \[\leadsto \color{blue}{\frac{x}{t} \cdot 0.5} \]

   if -1.59999999999999988e-23 < x

   1. Initial program 99.5%

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

      1. associate-/r* 100.0%

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

      2. remove-double-neg 100.0%

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

      3. distribute-frac-neg2 100.0%

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

      4. neg-mul-1 100.0%

         \[\leadsto \frac{\color{blue}{-1 \cdot \frac{\left(x + y\right) - z}{-t}}}{2} \]

      5. *-commutative 100.0%

         \[\leadsto \frac{\color{blue}{\frac{\left(x + y\right) - z}{-t} \cdot -1}}{2} \]

      6. associate-/l* 100.0%

         \[\leadsto \color{blue}{\frac{\left(x + y\right) - z}{-t} \cdot \frac{-1}{2}} \]

      7. distribute-frac-neg2 100.0%

         \[\leadsto \color{blue}{\left(-\frac{\left(x + y\right) - z}{t}\right)} \cdot \frac{-1}{2} \]

      8. distribute-neg-frac 100.0%

         \[\leadsto \color{blue}{\frac{-\left(\left(x + y\right) - z\right)}{t}} \cdot \frac{-1}{2} \]

      9. sub-neg 100.0%

         \[\leadsto \frac{-\color{blue}{\left(\left(x + y\right) + \left(-z\right)\right)}}{t} \cdot \frac{-1}{2} \]

      10. distribute-neg-in 100.0%

          \[\leadsto \frac{\color{blue}{\left(-\left(x + y\right)\right) + \left(-\left(-z\right)\right)}}{t} \cdot \frac{-1}{2} \]

      11. remove-double-neg 100.0%

          \[\leadsto \frac{\left(-\left(x + y\right)\right) + \color{blue}{z}}{t} \cdot \frac{-1}{2} \]

      12. +-commutative 100.0%

          \[\leadsto \frac{\color{blue}{z + \left(-\left(x + y\right)\right)}}{t} \cdot \frac{-1}{2} \]

      13. sub-neg 100.0%

          \[\leadsto \frac{\color{blue}{z - \left(x + y\right)}}{t} \cdot \frac{-1}{2} \]

      14. metadata-eval 100.0%

          \[\leadsto \frac{z - \left(x + y\right)}{t} \cdot \color{blue}{-0.5} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{z - \left(x + y\right)}{t} \cdot -0.5} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 33.9%

      \[\leadsto \color{blue}{\frac{z}{t}} \cdot -0.5 \]

   6. Taylor expanded in z around 0 33.9%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{z}{t}} \]
   7. Step-by-step derivation

      1. associate-*r/ 33.9%

         \[\leadsto \color{blue}{\frac{-0.5 \cdot z}{t}} \]

      2. *-commutative 33.9%

         \[\leadsto \frac{\color{blue}{z \cdot -0.5}}{t} \]

      3. associate-*r/ 33.8%

         \[\leadsto \color{blue}{z \cdot \frac{-0.5}{t}} \]

   8. Simplified 33.8%

      \[\leadsto \color{blue}{z \cdot \frac{-0.5}{t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 43.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{-23}:\\ \;\;\;\;\frac{x}{t} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{-0.5}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 46.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.2 \cdot 10^{-23}:\\ \;\;\;\;\frac{x}{t} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{z}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -4.2e-23) (* (/ x t) 0.5) (* -0.5 (/ z t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -4.2e-23) {
		tmp = (x / t) * 0.5;
	} else {
		tmp = -0.5 * (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 (x <= (-4.2d-23)) then
        tmp = (x / t) * 0.5d0
    else
        tmp = (-0.5d0) * (z / t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -4.2e-23) {
		tmp = (x / t) * 0.5;
	} else {
		tmp = -0.5 * (z / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -4.2e-23:
		tmp = (x / t) * 0.5
	else:
		tmp = -0.5 * (z / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -4.2e-23)
		tmp = Float64(Float64(x / t) * 0.5);
	else
		tmp = Float64(-0.5 * Float64(z / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -4.2e-23)
		tmp = (x / t) * 0.5;
	else
		tmp = -0.5 * (z / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -4.2e-23], N[(N[(x / t), $MachinePrecision] * 0.5), $MachinePrecision], N[(-0.5 * N[(z / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.2000000000000002e-23

   1. Initial program 99.9%

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

      1. associate-/r* 99.9%

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

      2. remove-double-neg 99.9%

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

      3. distribute-frac-neg2 99.9%

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

      4. neg-mul-1 99.9%

         \[\leadsto \frac{\color{blue}{-1 \cdot \frac{\left(x + y\right) - z}{-t}}}{2} \]

      5. *-commutative 99.9%

         \[\leadsto \frac{\color{blue}{\frac{\left(x + y\right) - z}{-t} \cdot -1}}{2} \]

      6. associate-/l* 99.9%

         \[\leadsto \color{blue}{\frac{\left(x + y\right) - z}{-t} \cdot \frac{-1}{2}} \]

      7. distribute-frac-neg2 99.9%

         \[\leadsto \color{blue}{\left(-\frac{\left(x + y\right) - z}{t}\right)} \cdot \frac{-1}{2} \]

      8. distribute-neg-frac 99.9%

         \[\leadsto \color{blue}{\frac{-\left(\left(x + y\right) - z\right)}{t}} \cdot \frac{-1}{2} \]

      9. sub-neg 99.9%

         \[\leadsto \frac{-\color{blue}{\left(\left(x + y\right) + \left(-z\right)\right)}}{t} \cdot \frac{-1}{2} \]

      10. distribute-neg-in 99.9%

          \[\leadsto \frac{\color{blue}{\left(-\left(x + y\right)\right) + \left(-\left(-z\right)\right)}}{t} \cdot \frac{-1}{2} \]

      11. remove-double-neg 99.9%

          \[\leadsto \frac{\left(-\left(x + y\right)\right) + \color{blue}{z}}{t} \cdot \frac{-1}{2} \]

      12. +-commutative 99.9%

          \[\leadsto \frac{\color{blue}{z + \left(-\left(x + y\right)\right)}}{t} \cdot \frac{-1}{2} \]

      13. sub-neg 99.9%

          \[\leadsto \frac{\color{blue}{z - \left(x + y\right)}}{t} \cdot \frac{-1}{2} \]

      14. metadata-eval 99.9%

          \[\leadsto \frac{z - \left(x + y\right)}{t} \cdot \color{blue}{-0.5} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{z - \left(x + y\right)}{t} \cdot -0.5} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate--r+ 99.9%

         \[\leadsto \frac{\color{blue}{\left(z - x\right) - y}}{t} \cdot -0.5 \]

      2. div-sub 96.2%

         \[\leadsto \color{blue}{\left(\frac{z - x}{t} - \frac{y}{t}\right)} \cdot -0.5 \]

   6. Applied egg-rr 96.2%

      \[\leadsto \color{blue}{\left(\frac{z - x}{t} - \frac{y}{t}\right)} \cdot -0.5 \]

   7. Taylor expanded in y around 0 80.8%

      \[\leadsto \color{blue}{\left(\frac{z}{t} - \frac{x}{t}\right)} \cdot -0.5 \]

   8. Taylor expanded in z around 0 65.9%

      \[\leadsto \color{blue}{0.5 \cdot \frac{x}{t}} \]
   9. Step-by-step derivation

      1. *-commutative 65.9%

         \[\leadsto \color{blue}{\frac{x}{t} \cdot 0.5} \]

   10. Simplified 65.9%

       \[\leadsto \color{blue}{\frac{x}{t} \cdot 0.5} \]

   if -4.2000000000000002e-23 < x

   1. Initial program 99.5%

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

      1. associate-/r* 100.0%

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

      2. remove-double-neg 100.0%

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

      3. distribute-frac-neg2 100.0%

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

      4. neg-mul-1 100.0%

         \[\leadsto \frac{\color{blue}{-1 \cdot \frac{\left(x + y\right) - z}{-t}}}{2} \]

      5. *-commutative 100.0%

         \[\leadsto \frac{\color{blue}{\frac{\left(x + y\right) - z}{-t} \cdot -1}}{2} \]

      6. associate-/l* 100.0%

         \[\leadsto \color{blue}{\frac{\left(x + y\right) - z}{-t} \cdot \frac{-1}{2}} \]

      7. distribute-frac-neg2 100.0%

         \[\leadsto \color{blue}{\left(-\frac{\left(x + y\right) - z}{t}\right)} \cdot \frac{-1}{2} \]

      8. distribute-neg-frac 100.0%

         \[\leadsto \color{blue}{\frac{-\left(\left(x + y\right) - z\right)}{t}} \cdot \frac{-1}{2} \]

      9. sub-neg 100.0%

         \[\leadsto \frac{-\color{blue}{\left(\left(x + y\right) + \left(-z\right)\right)}}{t} \cdot \frac{-1}{2} \]

      10. distribute-neg-in 100.0%

          \[\leadsto \frac{\color{blue}{\left(-\left(x + y\right)\right) + \left(-\left(-z\right)\right)}}{t} \cdot \frac{-1}{2} \]

      11. remove-double-neg 100.0%

          \[\leadsto \frac{\left(-\left(x + y\right)\right) + \color{blue}{z}}{t} \cdot \frac{-1}{2} \]

      12. +-commutative 100.0%

          \[\leadsto \frac{\color{blue}{z + \left(-\left(x + y\right)\right)}}{t} \cdot \frac{-1}{2} \]

      13. sub-neg 100.0%

          \[\leadsto \frac{\color{blue}{z - \left(x + y\right)}}{t} \cdot \frac{-1}{2} \]

      14. metadata-eval 100.0%

          \[\leadsto \frac{z - \left(x + y\right)}{t} \cdot \color{blue}{-0.5} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{z - \left(x + y\right)}{t} \cdot -0.5} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 33.9%

      \[\leadsto \color{blue}{\frac{z}{t}} \cdot -0.5 \]
3. Recombined 2 regimes into one program.

4. Final simplification 43.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.2 \cdot 10^{-23}:\\ \;\;\;\;\frac{x}{t} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{z}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 37.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ x \cdot \frac{0.5}{t} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (* x (/ 0.5 t)))

C

double code(double x, double y, double z, double t) {
	return x * (0.5 / 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 * (0.5d0 / t)
end function

Java

public static double code(double x, double y, double z, double t) {
	return x * (0.5 / t);
}

Python

def code(x, y, z, t):
	return x * (0.5 / t)

Julia

function code(x, y, z, t)
	return Float64(x * Float64(0.5 / t))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = x * (0.5 / t);
end

Wolfram

code[x_, y_, z_, t_] := N[(x * N[(0.5 / t), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

   1. associate-/r* 100.0%

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

   2. remove-double-neg 100.0%

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

   3. distribute-frac-neg2 100.0%

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

   4. neg-mul-1 100.0%

      \[\leadsto \frac{\color{blue}{-1 \cdot \frac{\left(x + y\right) - z}{-t}}}{2} \]

   5. *-commutative 100.0%

      \[\leadsto \frac{\color{blue}{\frac{\left(x + y\right) - z}{-t} \cdot -1}}{2} \]

   6. associate-/l* 100.0%

      \[\leadsto \color{blue}{\frac{\left(x + y\right) - z}{-t} \cdot \frac{-1}{2}} \]

   7. distribute-frac-neg2 100.0%

      \[\leadsto \color{blue}{\left(-\frac{\left(x + y\right) - z}{t}\right)} \cdot \frac{-1}{2} \]

   8. distribute-neg-frac 100.0%

      \[\leadsto \color{blue}{\frac{-\left(\left(x + y\right) - z\right)}{t}} \cdot \frac{-1}{2} \]

   9. sub-neg 100.0%

      \[\leadsto \frac{-\color{blue}{\left(\left(x + y\right) + \left(-z\right)\right)}}{t} \cdot \frac{-1}{2} \]

   10. distribute-neg-in 100.0%

       \[\leadsto \frac{\color{blue}{\left(-\left(x + y\right)\right) + \left(-\left(-z\right)\right)}}{t} \cdot \frac{-1}{2} \]

   11. remove-double-neg 100.0%

       \[\leadsto \frac{\left(-\left(x + y\right)\right) + \color{blue}{z}}{t} \cdot \frac{-1}{2} \]

   12. +-commutative 100.0%

       \[\leadsto \frac{\color{blue}{z + \left(-\left(x + y\right)\right)}}{t} \cdot \frac{-1}{2} \]

   13. sub-neg 100.0%

       \[\leadsto \frac{\color{blue}{z - \left(x + y\right)}}{t} \cdot \frac{-1}{2} \]

   14. metadata-eval 100.0%

       \[\leadsto \frac{z - \left(x + y\right)}{t} \cdot \color{blue}{-0.5} \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{z - \left(x + y\right)}{t} \cdot -0.5} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate--r+ 100.0%

      \[\leadsto \frac{\color{blue}{\left(z - x\right) - y}}{t} \cdot -0.5 \]

   2. div-sub 97.3%

      \[\leadsto \color{blue}{\left(\frac{z - x}{t} - \frac{y}{t}\right)} \cdot -0.5 \]

6. Applied egg-rr 97.3%

   \[\leadsto \color{blue}{\left(\frac{z - x}{t} - \frac{y}{t}\right)} \cdot -0.5 \]

7. Taylor expanded in y around 0 64.7%

   \[\leadsto \color{blue}{\left(\frac{z}{t} - \frac{x}{t}\right)} \cdot -0.5 \]

8. Taylor expanded in z around 0 40.6%

   \[\leadsto \color{blue}{0.5 \cdot \frac{x}{t}} \]
9. Step-by-step derivation

   1. *-commutative 40.6%

      \[\leadsto \color{blue}{\frac{x}{t} \cdot 0.5} \]

   2. associate-*l/ 40.6%

      \[\leadsto \color{blue}{\frac{x \cdot 0.5}{t}} \]

   3. associate-*r/ 40.5%

      \[\leadsto \color{blue}{x \cdot \frac{0.5}{t}} \]

10. Simplified 40.5%

    \[\leadsto \color{blue}{x \cdot \frac{0.5}{t}} \]

11. Final simplification 40.5%

    \[\leadsto x \cdot \frac{0.5}{t} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024077 
(FPCore (x y z t)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, B"
  :precision binary64
  (/ (- (+ x y) z) (* t 2.0)))