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

Percentage Accurate: 99.9% → 99.9%

Time: 6.3s

Alternatives: 10

Speedup: 1.0×

• 26.6% 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: 99.9% 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

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

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]

Derivation

1. Initial program 100.0%

   \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 2: 46.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z}{t} \cdot -0.5\\ t_2 := \frac{x}{t \cdot 2}\\ \mathbf{if}\;y \leq 6.4 \cdot 10^{-260}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{-235}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 9.6 \cdot 10^{-202}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq 0.43:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (/ z t) -0.5)) (t_2 (/ x (* t 2.0))))
   (if (<= y 6.4e-260)
     t_2
     (if (<= y 2.8e-235)
       t_1
       (if (<= y 9.6e-202) t_2 (if (<= y 0.43) t_1 (/ y (* t 2.0))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z / t) * -0.5;
	double t_2 = x / (t * 2.0);
	double tmp;
	if (y <= 6.4e-260) {
		tmp = t_2;
	} else if (y <= 2.8e-235) {
		tmp = t_1;
	} else if (y <= 9.6e-202) {
		tmp = t_2;
	} else if (y <= 0.43) {
		tmp = t_1;
	} else {
		tmp = y / (t * 2.0);
	}
	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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (z / t) * (-0.5d0)
    t_2 = x / (t * 2.0d0)
    if (y <= 6.4d-260) then
        tmp = t_2
    else if (y <= 2.8d-235) then
        tmp = t_1
    else if (y <= 9.6d-202) then
        tmp = t_2
    else if (y <= 0.43d0) then
        tmp = t_1
    else
        tmp = y / (t * 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (z / t) * -0.5;
	double t_2 = x / (t * 2.0);
	double tmp;
	if (y <= 6.4e-260) {
		tmp = t_2;
	} else if (y <= 2.8e-235) {
		tmp = t_1;
	} else if (y <= 9.6e-202) {
		tmp = t_2;
	} else if (y <= 0.43) {
		tmp = t_1;
	} else {
		tmp = y / (t * 2.0);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (z / t) * -0.5
	t_2 = x / (t * 2.0)
	tmp = 0
	if y <= 6.4e-260:
		tmp = t_2
	elif y <= 2.8e-235:
		tmp = t_1
	elif y <= 9.6e-202:
		tmp = t_2
	elif y <= 0.43:
		tmp = t_1
	else:
		tmp = y / (t * 2.0)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z / t) * -0.5)
	t_2 = Float64(x / Float64(t * 2.0))
	tmp = 0.0
	if (y <= 6.4e-260)
		tmp = t_2;
	elseif (y <= 2.8e-235)
		tmp = t_1;
	elseif (y <= 9.6e-202)
		tmp = t_2;
	elseif (y <= 0.43)
		tmp = t_1;
	else
		tmp = Float64(y / Float64(t * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (z / t) * -0.5;
	t_2 = x / (t * 2.0);
	tmp = 0.0;
	if (y <= 6.4e-260)
		tmp = t_2;
	elseif (y <= 2.8e-235)
		tmp = t_1;
	elseif (y <= 9.6e-202)
		tmp = t_2;
	elseif (y <= 0.43)
		tmp = t_1;
	else
		tmp = y / (t * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z / t), $MachinePrecision] * -0.5), $MachinePrecision]}, Block[{t$95$2 = N[(x / N[(t * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 6.4e-260], t$95$2, If[LessEqual[y, 2.8e-235], t$95$1, If[LessEqual[y, 9.6e-202], t$95$2, If[LessEqual[y, 0.43], t$95$1, N[(y / N[(t * 2.0), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < 6.3999999999999999e-260 or 2.79999999999999995e-235 < y < 9.6000000000000004e-202

   1. Initial program 100.0%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 39.5%

      \[\leadsto \frac{\color{blue}{x}}{t \cdot 2} \]

   if 6.3999999999999999e-260 < y < 2.79999999999999995e-235 or 9.6000000000000004e-202 < y < 0.429999999999999993

   1. Initial program 100.0%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf 61.6%

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

      1. *-commutative 61.6%

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

   5. Simplified 61.6%

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

   if 0.429999999999999993 < y

   1. Initial program 100.0%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 55.3%

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

Alternative 3: 46.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z}{t} \cdot -0.5\\ t_2 := \frac{x}{t \cdot 2}\\ \mathbf{if}\;y \leq 4.1 \cdot 10^{-259}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq 6.4 \cdot 10^{-232}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 8.6 \cdot 10^{-202}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq 0.145:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{0.5}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (/ z t) -0.5)) (t_2 (/ x (* t 2.0))))
   (if (<= y 4.1e-259)
     t_2
     (if (<= y 6.4e-232)
       t_1
       (if (<= y 8.6e-202) t_2 (if (<= y 0.145) t_1 (* y (/ 0.5 t))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z / t) * -0.5;
	double t_2 = x / (t * 2.0);
	double tmp;
	if (y <= 4.1e-259) {
		tmp = t_2;
	} else if (y <= 6.4e-232) {
		tmp = t_1;
	} else if (y <= 8.6e-202) {
		tmp = t_2;
	} else if (y <= 0.145) {
		tmp = t_1;
	} else {
		tmp = y * (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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (z / t) * (-0.5d0)
    t_2 = x / (t * 2.0d0)
    if (y <= 4.1d-259) then
        tmp = t_2
    else if (y <= 6.4d-232) then
        tmp = t_1
    else if (y <= 8.6d-202) then
        tmp = t_2
    else if (y <= 0.145d0) then
        tmp = t_1
    else
        tmp = y * (0.5d0 / t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (z / t) * -0.5;
	double t_2 = x / (t * 2.0);
	double tmp;
	if (y <= 4.1e-259) {
		tmp = t_2;
	} else if (y <= 6.4e-232) {
		tmp = t_1;
	} else if (y <= 8.6e-202) {
		tmp = t_2;
	} else if (y <= 0.145) {
		tmp = t_1;
	} else {
		tmp = y * (0.5 / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (z / t) * -0.5
	t_2 = x / (t * 2.0)
	tmp = 0
	if y <= 4.1e-259:
		tmp = t_2
	elif y <= 6.4e-232:
		tmp = t_1
	elif y <= 8.6e-202:
		tmp = t_2
	elif y <= 0.145:
		tmp = t_1
	else:
		tmp = y * (0.5 / t)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z / t) * -0.5)
	t_2 = Float64(x / Float64(t * 2.0))
	tmp = 0.0
	if (y <= 4.1e-259)
		tmp = t_2;
	elseif (y <= 6.4e-232)
		tmp = t_1;
	elseif (y <= 8.6e-202)
		tmp = t_2;
	elseif (y <= 0.145)
		tmp = t_1;
	else
		tmp = Float64(y * Float64(0.5 / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (z / t) * -0.5;
	t_2 = x / (t * 2.0);
	tmp = 0.0;
	if (y <= 4.1e-259)
		tmp = t_2;
	elseif (y <= 6.4e-232)
		tmp = t_1;
	elseif (y <= 8.6e-202)
		tmp = t_2;
	elseif (y <= 0.145)
		tmp = t_1;
	else
		tmp = y * (0.5 / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z / t), $MachinePrecision] * -0.5), $MachinePrecision]}, Block[{t$95$2 = N[(x / N[(t * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 4.1e-259], t$95$2, If[LessEqual[y, 6.4e-232], t$95$1, If[LessEqual[y, 8.6e-202], t$95$2, If[LessEqual[y, 0.145], t$95$1, N[(y * N[(0.5 / t), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < 4.0999999999999998e-259 or 6.39999999999999973e-232 < y < 8.59999999999999986e-202

   1. Initial program 100.0%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 39.5%

      \[\leadsto \frac{\color{blue}{x}}{t \cdot 2} \]

   if 4.0999999999999998e-259 < y < 6.39999999999999973e-232 or 8.59999999999999986e-202 < y < 0.14499999999999999

   1. Initial program 100.0%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf 61.6%

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

      1. *-commutative 61.6%

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

   5. Simplified 61.6%

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

   if 0.14499999999999999 < y

   1. Initial program 100.0%

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

      1. div-sub 92.3%

         \[\leadsto \color{blue}{\frac{x + y}{t \cdot 2} - \frac{z}{t \cdot 2}} \]

      2. div-inv 92.0%

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

      3. *-commutative 92.0%

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

      4. associate-/r* 92.0%

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

      5. metadata-eval 92.0%

         \[\leadsto \left(x + y\right) \cdot \frac{\color{blue}{0.5}}{t} - \frac{z}{t \cdot 2} \]

      6. div-inv 92.0%

         \[\leadsto \left(x + y\right) \cdot \frac{0.5}{t} - \color{blue}{z \cdot \frac{1}{t \cdot 2}} \]

      7. *-commutative 92.0%

         \[\leadsto \left(x + y\right) \cdot \frac{0.5}{t} - z \cdot \frac{1}{\color{blue}{2 \cdot t}} \]

      8. associate-/r* 92.0%

         \[\leadsto \left(x + y\right) \cdot \frac{0.5}{t} - z \cdot \color{blue}{\frac{\frac{1}{2}}{t}} \]

      9. metadata-eval 92.0%

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

   4. Applied egg-rr 92.0%

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

   5. Taylor expanded in y around inf 55.3%

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

      1. associate-*r/ 55.3%

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

      2. associate-*l/ 55.2%

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

      3. *-commutative 55.2%

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

   7. Simplified 55.2%

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

Alternative 4: 46.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z}{t} \cdot -0.5\\ t_2 := x \cdot \frac{0.5}{t}\\ \mathbf{if}\;y \leq 4 \cdot 10^{-300}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq 1.65 \cdot 10^{-237}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 6.8 \cdot 10^{-202}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq 0.29:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{0.5}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (/ z t) -0.5)) (t_2 (* x (/ 0.5 t))))
   (if (<= y 4e-300)
     t_2
     (if (<= y 1.65e-237)
       t_1
       (if (<= y 6.8e-202) t_2 (if (<= y 0.29) t_1 (* y (/ 0.5 t))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z / t) * -0.5;
	double t_2 = x * (0.5 / t);
	double tmp;
	if (y <= 4e-300) {
		tmp = t_2;
	} else if (y <= 1.65e-237) {
		tmp = t_1;
	} else if (y <= 6.8e-202) {
		tmp = t_2;
	} else if (y <= 0.29) {
		tmp = t_1;
	} else {
		tmp = y * (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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (z / t) * (-0.5d0)
    t_2 = x * (0.5d0 / t)
    if (y <= 4d-300) then
        tmp = t_2
    else if (y <= 1.65d-237) then
        tmp = t_1
    else if (y <= 6.8d-202) then
        tmp = t_2
    else if (y <= 0.29d0) then
        tmp = t_1
    else
        tmp = y * (0.5d0 / t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (z / t) * -0.5;
	double t_2 = x * (0.5 / t);
	double tmp;
	if (y <= 4e-300) {
		tmp = t_2;
	} else if (y <= 1.65e-237) {
		tmp = t_1;
	} else if (y <= 6.8e-202) {
		tmp = t_2;
	} else if (y <= 0.29) {
		tmp = t_1;
	} else {
		tmp = y * (0.5 / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (z / t) * -0.5
	t_2 = x * (0.5 / t)
	tmp = 0
	if y <= 4e-300:
		tmp = t_2
	elif y <= 1.65e-237:
		tmp = t_1
	elif y <= 6.8e-202:
		tmp = t_2
	elif y <= 0.29:
		tmp = t_1
	else:
		tmp = y * (0.5 / t)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z / t) * -0.5)
	t_2 = Float64(x * Float64(0.5 / t))
	tmp = 0.0
	if (y <= 4e-300)
		tmp = t_2;
	elseif (y <= 1.65e-237)
		tmp = t_1;
	elseif (y <= 6.8e-202)
		tmp = t_2;
	elseif (y <= 0.29)
		tmp = t_1;
	else
		tmp = Float64(y * Float64(0.5 / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (z / t) * -0.5;
	t_2 = x * (0.5 / t);
	tmp = 0.0;
	if (y <= 4e-300)
		tmp = t_2;
	elseif (y <= 1.65e-237)
		tmp = t_1;
	elseif (y <= 6.8e-202)
		tmp = t_2;
	elseif (y <= 0.29)
		tmp = t_1;
	else
		tmp = y * (0.5 / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z / t), $MachinePrecision] * -0.5), $MachinePrecision]}, Block[{t$95$2 = N[(x * N[(0.5 / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 4e-300], t$95$2, If[LessEqual[y, 1.65e-237], t$95$1, If[LessEqual[y, 6.8e-202], t$95$2, If[LessEqual[y, 0.29], t$95$1, N[(y * N[(0.5 / t), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < 4.0000000000000001e-300 or 1.6500000000000001e-237 < y < 6.80000000000000025e-202

   1. Initial program 99.9%

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

      1. div-sub 94.5%

         \[\leadsto \color{blue}{\frac{x + y}{t \cdot 2} - \frac{z}{t \cdot 2}} \]

      2. div-inv 94.3%

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

      3. *-commutative 94.3%

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

      4. associate-/r* 94.3%

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

      5. metadata-eval 94.3%

         \[\leadsto \left(x + y\right) \cdot \frac{\color{blue}{0.5}}{t} - \frac{z}{t \cdot 2} \]

      6. div-inv 94.2%

         \[\leadsto \left(x + y\right) \cdot \frac{0.5}{t} - \color{blue}{z \cdot \frac{1}{t \cdot 2}} \]

      7. *-commutative 94.2%

         \[\leadsto \left(x + y\right) \cdot \frac{0.5}{t} - z \cdot \frac{1}{\color{blue}{2 \cdot t}} \]

      8. associate-/r* 94.2%

         \[\leadsto \left(x + y\right) \cdot \frac{0.5}{t} - z \cdot \color{blue}{\frac{\frac{1}{2}}{t}} \]

      9. metadata-eval 94.2%

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

   4. Applied egg-rr 94.2%

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

   5. Taylor expanded in x around inf 37.2%

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

      1. associate-*r/ 37.2%

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

      2. associate-*l/ 37.0%

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

      3. *-commutative 37.0%

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

   7. Simplified 37.0%

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

   if 4.0000000000000001e-300 < y < 1.6500000000000001e-237 or 6.80000000000000025e-202 < y < 0.28999999999999998

   1. Initial program 100.0%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf 56.0%

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

      1. *-commutative 56.0%

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

   5. Simplified 56.0%

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

   if 0.28999999999999998 < y

   1. Initial program 100.0%

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

      1. div-sub 92.3%

         \[\leadsto \color{blue}{\frac{x + y}{t \cdot 2} - \frac{z}{t \cdot 2}} \]

      2. div-inv 92.0%

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

      3. *-commutative 92.0%

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

      4. associate-/r* 92.0%

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

      5. metadata-eval 92.0%

         \[\leadsto \left(x + y\right) \cdot \frac{\color{blue}{0.5}}{t} - \frac{z}{t \cdot 2} \]

      6. div-inv 92.0%

         \[\leadsto \left(x + y\right) \cdot \frac{0.5}{t} - \color{blue}{z \cdot \frac{1}{t \cdot 2}} \]

      7. *-commutative 92.0%

         \[\leadsto \left(x + y\right) \cdot \frac{0.5}{t} - z \cdot \frac{1}{\color{blue}{2 \cdot t}} \]

      8. associate-/r* 92.0%

         \[\leadsto \left(x + y\right) \cdot \frac{0.5}{t} - z \cdot \color{blue}{\frac{\frac{1}{2}}{t}} \]

      9. metadata-eval 92.0%

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

   4. Applied egg-rr 92.0%

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

   5. Taylor expanded in y around inf 55.3%

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

      1. associate-*r/ 55.3%

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

      2. associate-*l/ 55.2%

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

      3. *-commutative 55.2%

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

   7. Simplified 55.2%

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

Alternative 5: 77.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.22 \cdot 10^{-20}:\\ \;\;\;\;\frac{x - z}{t \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - z}{t \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 2.22e-20) (/ (- x z) (* t 2.0)) (/ (- y z) (* t 2.0))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 2.22e-20) {
		tmp = (x - z) / (t * 2.0);
	} else {
		tmp = (y - z) / (t * 2.0);
	}
	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 <= 2.22d-20) then
        tmp = (x - z) / (t * 2.0d0)
    else
        tmp = (y - z) / (t * 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 2.22e-20) {
		tmp = (x - z) / (t * 2.0);
	} else {
		tmp = (y - z) / (t * 2.0);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= 2.22e-20:
		tmp = (x - z) / (t * 2.0)
	else:
		tmp = (y - z) / (t * 2.0)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 2.22e-20)
		tmp = Float64(Float64(x - z) / Float64(t * 2.0));
	else
		tmp = Float64(Float64(y - z) / Float64(t * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 2.22e-20)
		tmp = (x - z) / (t * 2.0);
	else
		tmp = (y - z) / (t * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 2.22e-20], N[(N[(x - z), $MachinePrecision] / N[(t * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(y - z), $MachinePrecision] / N[(t * 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.21999999999999997e-20

   1. Initial program 100.0%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 75.1%

      \[\leadsto \frac{\color{blue}{x - z}}{t \cdot 2} \]

   if 2.21999999999999997e-20 < y

   1. Initial program 100.0%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 69.5%

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

Alternative 6: 77.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5.6 \cdot 10^{-14}:\\ \;\;\;\;\frac{x - z}{t \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\frac{t}{y - z}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 5.6e-14) (/ (- x z) (* t 2.0)) (/ 0.5 (/ t (- y z)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 5.6e-14) {
		tmp = (x - z) / (t * 2.0);
	} else {
		tmp = 0.5 / (t / (y - z));
	}
	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 <= 5.6d-14) then
        tmp = (x - z) / (t * 2.0d0)
    else
        tmp = 0.5d0 / (t / (y - z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 5.6e-14) {
		tmp = (x - z) / (t * 2.0);
	} else {
		tmp = 0.5 / (t / (y - z));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= 5.6e-14:
		tmp = (x - z) / (t * 2.0)
	else:
		tmp = 0.5 / (t / (y - z))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 5.6e-14)
		tmp = Float64(Float64(x - z) / Float64(t * 2.0));
	else
		tmp = Float64(0.5 / Float64(t / Float64(y - z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 5.6e-14)
		tmp = (x - z) / (t * 2.0);
	else
		tmp = 0.5 / (t / (y - z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 5.6e-14], N[(N[(x - z), $MachinePrecision] / N[(t * 2.0), $MachinePrecision]), $MachinePrecision], N[(0.5 / N[(t / N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.6000000000000001e-14

   1. Initial program 100.0%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 75.4%

      \[\leadsto \frac{\color{blue}{x - z}}{t \cdot 2} \]

   if 5.6000000000000001e-14 < y

   1. Initial program 100.0%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 68.4%

      \[\leadsto \frac{\color{blue}{y - z}}{t \cdot 2} \]

   4. Taylor expanded in y around 0 64.7%

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

      1. metadata-eval 64.7%

         \[\leadsto \color{blue}{\left(0.5 \cdot -1\right)} \cdot \frac{z}{t} + 0.5 \cdot \frac{y}{t} \]

      2. associate-*r* 64.7%

         \[\leadsto \color{blue}{0.5 \cdot \left(-1 \cdot \frac{z}{t}\right)} + 0.5 \cdot \frac{y}{t} \]

      3. mul-1-neg 64.7%

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

      4. distribute-frac-neg 64.7%

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

      5. +-commutative 64.7%

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

      6. distribute-lft-out 64.7%

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

      7. distribute-frac-neg 64.7%

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

      8. sub-neg 64.7%

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

      9. div-sub 68.4%

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

      10. *-lft-identity 68.4%

          \[\leadsto 0.5 \cdot \frac{\color{blue}{1 \cdot \left(y - z\right)}}{t} \]

      11. associate-*l/ 68.3%

          \[\leadsto 0.5 \cdot \color{blue}{\left(\frac{1}{t} \cdot \left(y - z\right)\right)} \]

      12. associate-/r/ 68.2%

          \[\leadsto 0.5 \cdot \color{blue}{\frac{1}{\frac{t}{y - z}}} \]

      13. associate-*r/ 68.2%

          \[\leadsto \color{blue}{\frac{0.5 \cdot 1}{\frac{t}{y - z}}} \]

      14. metadata-eval 68.2%

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

   6. Simplified 68.2%

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

Alternative 7: 77.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.05 \cdot 10^{-38}:\\ \;\;\;\;\frac{x + y}{t \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\frac{t}{y - z}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -2.05e-38) (/ (+ x y) (* t 2.0)) (/ 0.5 (/ t (- y z)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -2.05e-38) {
		tmp = (x + y) / (t * 2.0);
	} else {
		tmp = 0.5 / (t / (y - z));
	}
	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 <= (-2.05d-38)) then
        tmp = (x + y) / (t * 2.0d0)
    else
        tmp = 0.5d0 / (t / (y - z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -2.05e-38) {
		tmp = (x + y) / (t * 2.0);
	} else {
		tmp = 0.5 / (t / (y - z));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -2.05e-38:
		tmp = (x + y) / (t * 2.0)
	else:
		tmp = 0.5 / (t / (y - z))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -2.05e-38)
		tmp = Float64(Float64(x + y) / Float64(t * 2.0));
	else
		tmp = Float64(0.5 / Float64(t / Float64(y - z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -2.05e-38)
		tmp = (x + y) / (t * 2.0);
	else
		tmp = 0.5 / (t / (y - z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -2.05e-38], N[(N[(x + y), $MachinePrecision] / N[(t * 2.0), $MachinePrecision]), $MachinePrecision], N[(0.5 / N[(t / N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.0499999999999999e-38

   1. Initial program 100.0%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0 77.0%

      \[\leadsto \frac{\color{blue}{x + y}}{t \cdot 2} \]
   4. Step-by-step derivation

      1. +-commutative 77.0%

         \[\leadsto \frac{\color{blue}{y + x}}{t \cdot 2} \]

   5. Simplified 77.0%

      \[\leadsto \frac{\color{blue}{y + x}}{t \cdot 2} \]

   if -2.0499999999999999e-38 < x

   1. Initial program 100.0%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 73.7%

      \[\leadsto \frac{\color{blue}{y - z}}{t \cdot 2} \]

   4. Taylor expanded in y around 0 69.4%

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

      1. metadata-eval 69.4%

         \[\leadsto \color{blue}{\left(0.5 \cdot -1\right)} \cdot \frac{z}{t} + 0.5 \cdot \frac{y}{t} \]

      2. associate-*r* 69.4%

         \[\leadsto \color{blue}{0.5 \cdot \left(-1 \cdot \frac{z}{t}\right)} + 0.5 \cdot \frac{y}{t} \]

      3. mul-1-neg 69.4%

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

      4. distribute-frac-neg 69.4%

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

      5. +-commutative 69.4%

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

      6. distribute-lft-out 69.4%

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

      7. distribute-frac-neg 69.4%

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

      8. sub-neg 69.4%

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

      9. div-sub 73.7%

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

      10. *-lft-identity 73.7%

          \[\leadsto 0.5 \cdot \frac{\color{blue}{1 \cdot \left(y - z\right)}}{t} \]

      11. associate-*l/ 73.5%

          \[\leadsto 0.5 \cdot \color{blue}{\left(\frac{1}{t} \cdot \left(y - z\right)\right)} \]

      12. associate-/r/ 73.5%

          \[\leadsto 0.5 \cdot \color{blue}{\frac{1}{\frac{t}{y - z}}} \]

      13. associate-*r/ 73.5%

          \[\leadsto \color{blue}{\frac{0.5 \cdot 1}{\frac{t}{y - z}}} \]

      14. metadata-eval 73.5%

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

   6. Simplified 73.5%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.05 \cdot 10^{-38}:\\ \;\;\;\;\frac{x + y}{t \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\frac{t}{y - z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6 \cdot 10^{+82}:\\ \;\;\;\;\frac{x}{t \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\frac{t}{y - z}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -6e+82) (/ x (* t 2.0)) (/ 0.5 (/ t (- y z)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -6e+82) {
		tmp = x / (t * 2.0);
	} else {
		tmp = 0.5 / (t / (y - z));
	}
	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 <= (-6d+82)) then
        tmp = x / (t * 2.0d0)
    else
        tmp = 0.5d0 / (t / (y - z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -6e+82) {
		tmp = x / (t * 2.0);
	} else {
		tmp = 0.5 / (t / (y - z));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -6e+82:
		tmp = x / (t * 2.0)
	else:
		tmp = 0.5 / (t / (y - z))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -6e+82)
		tmp = Float64(x / Float64(t * 2.0));
	else
		tmp = Float64(0.5 / Float64(t / Float64(y - z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -6e+82)
		tmp = x / (t * 2.0);
	else
		tmp = 0.5 / (t / (y - z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -6e+82], N[(x / N[(t * 2.0), $MachinePrecision]), $MachinePrecision], N[(0.5 / N[(t / N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.99999999999999978e82

   1. Initial program 99.9%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 71.6%

      \[\leadsto \frac{\color{blue}{x}}{t \cdot 2} \]

   if -5.99999999999999978e82 < x

   1. Initial program 100.0%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 72.8%

      \[\leadsto \frac{\color{blue}{y - z}}{t \cdot 2} \]

   4. Taylor expanded in y around 0 69.1%

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

      1. metadata-eval 69.1%

         \[\leadsto \color{blue}{\left(0.5 \cdot -1\right)} \cdot \frac{z}{t} + 0.5 \cdot \frac{y}{t} \]

      2. associate-*r* 69.1%

         \[\leadsto \color{blue}{0.5 \cdot \left(-1 \cdot \frac{z}{t}\right)} + 0.5 \cdot \frac{y}{t} \]

      3. mul-1-neg 69.1%

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

      4. distribute-frac-neg 69.1%

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

      5. +-commutative 69.1%

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

      6. distribute-lft-out 69.1%

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

      7. distribute-frac-neg 69.1%

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

      8. sub-neg 69.1%

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

      9. div-sub 72.8%

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

      10. *-lft-identity 72.8%

          \[\leadsto 0.5 \cdot \frac{\color{blue}{1 \cdot \left(y - z\right)}}{t} \]

      11. associate-*l/ 72.6%

          \[\leadsto 0.5 \cdot \color{blue}{\left(\frac{1}{t} \cdot \left(y - z\right)\right)} \]

      12. associate-/r/ 72.6%

          \[\leadsto 0.5 \cdot \color{blue}{\frac{1}{\frac{t}{y - z}}} \]

      13. associate-*r/ 72.6%

          \[\leadsto \color{blue}{\frac{0.5 \cdot 1}{\frac{t}{y - z}}} \]

      14. metadata-eval 72.6%

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

   6. Simplified 72.6%

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

Alternative 9: 45.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.05 \cdot 10^{-17}:\\ \;\;\;\;x \cdot \frac{0.5}{t}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{0.5}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 2.05e-17) (* x (/ 0.5 t)) (* y (/ 0.5 t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 2.05e-17) {
		tmp = x * (0.5 / t);
	} else {
		tmp = y * (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 (y <= 2.05d-17) then
        tmp = x * (0.5d0 / t)
    else
        tmp = y * (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 (y <= 2.05e-17) {
		tmp = x * (0.5 / t);
	} else {
		tmp = y * (0.5 / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= 2.05e-17:
		tmp = x * (0.5 / t)
	else:
		tmp = y * (0.5 / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 2.05e-17)
		tmp = Float64(x * Float64(0.5 / t));
	else
		tmp = Float64(y * Float64(0.5 / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 2.05e-17)
		tmp = x * (0.5 / t);
	else
		tmp = y * (0.5 / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 2.05e-17], N[(x * N[(0.5 / t), $MachinePrecision]), $MachinePrecision], N[(y * N[(0.5 / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.05e-17

   1. Initial program 100.0%

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

      1. div-sub 95.5%

         \[\leadsto \color{blue}{\frac{x + y}{t \cdot 2} - \frac{z}{t \cdot 2}} \]

      2. div-inv 95.3%

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

      3. *-commutative 95.3%

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

      4. associate-/r* 95.3%

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

      5. metadata-eval 95.3%

         \[\leadsto \left(x + y\right) \cdot \frac{\color{blue}{0.5}}{t} - \frac{z}{t \cdot 2} \]

      6. div-inv 95.1%

         \[\leadsto \left(x + y\right) \cdot \frac{0.5}{t} - \color{blue}{z \cdot \frac{1}{t \cdot 2}} \]

      7. *-commutative 95.1%

         \[\leadsto \left(x + y\right) \cdot \frac{0.5}{t} - z \cdot \frac{1}{\color{blue}{2 \cdot t}} \]

      8. associate-/r* 95.1%

         \[\leadsto \left(x + y\right) \cdot \frac{0.5}{t} - z \cdot \color{blue}{\frac{\frac{1}{2}}{t}} \]

      9. metadata-eval 95.1%

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

   4. Applied egg-rr 95.1%

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

   5. Taylor expanded in x around inf 40.2%

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

      1. associate-*r/ 40.2%

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

      2. associate-*l/ 40.0%

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

      3. *-commutative 40.0%

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

   7. Simplified 40.0%

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

   if 2.05e-17 < y

   1. Initial program 100.0%

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

      1. div-sub 92.8%

         \[\leadsto \color{blue}{\frac{x + y}{t \cdot 2} - \frac{z}{t \cdot 2}} \]

      2. div-inv 92.6%

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

      3. *-commutative 92.6%

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

      4. associate-/r* 92.6%

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

      5. metadata-eval 92.6%

         \[\leadsto \left(x + y\right) \cdot \frac{\color{blue}{0.5}}{t} - \frac{z}{t \cdot 2} \]

      6. div-inv 92.6%

         \[\leadsto \left(x + y\right) \cdot \frac{0.5}{t} - \color{blue}{z \cdot \frac{1}{t \cdot 2}} \]

      7. *-commutative 92.6%

         \[\leadsto \left(x + y\right) \cdot \frac{0.5}{t} - z \cdot \frac{1}{\color{blue}{2 \cdot t}} \]

      8. associate-/r* 92.6%

         \[\leadsto \left(x + y\right) \cdot \frac{0.5}{t} - z \cdot \color{blue}{\frac{\frac{1}{2}}{t}} \]

      9. metadata-eval 92.6%

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

   4. Applied egg-rr 92.6%

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

   5. Taylor expanded in y around inf 51.7%

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

      1. associate-*r/ 51.7%

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

      2. associate-*l/ 51.6%

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

      3. *-commutative 51.6%

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

   7. Simplified 51.6%

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

Alternative 10: 36.6% 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 100.0%

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

   1. div-sub 94.9%

      \[\leadsto \color{blue}{\frac{x + y}{t \cdot 2} - \frac{z}{t \cdot 2}} \]

   2. div-inv 94.7%

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

   3. *-commutative 94.7%

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

   4. associate-/r* 94.7%

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

   5. metadata-eval 94.7%

      \[\leadsto \left(x + y\right) \cdot \frac{\color{blue}{0.5}}{t} - \frac{z}{t \cdot 2} \]

   6. div-inv 94.6%

      \[\leadsto \left(x + y\right) \cdot \frac{0.5}{t} - \color{blue}{z \cdot \frac{1}{t \cdot 2}} \]

   7. *-commutative 94.6%

      \[\leadsto \left(x + y\right) \cdot \frac{0.5}{t} - z \cdot \frac{1}{\color{blue}{2 \cdot t}} \]

   8. associate-/r* 94.6%

      \[\leadsto \left(x + y\right) \cdot \frac{0.5}{t} - z \cdot \color{blue}{\frac{\frac{1}{2}}{t}} \]

   9. metadata-eval 94.6%

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

4. Applied egg-rr 94.6%

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

5. Taylor expanded in x around inf 39.1%

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

   1. associate-*r/ 39.1%

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

   2. associate-*l/ 39.0%

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

   3. *-commutative 39.0%

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

7. Simplified 39.0%

   \[\leadsto \color{blue}{x \cdot \frac{0.5}{t}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024100 
(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)))