Graphics.Rendering.Chart.Backend.Diagrams:calcFontMetrics from Chart-diagrams-1.5.1, A

Percentage Accurate: 88.4% → 99.6%

Time: 7.8s

Alternatives: 10

Speedup: 0.3×

Specification

Math

\[\begin{array}{l} \\ \frac{x + y}{1 - \frac{y}{z}} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(N[(x + y), $MachinePrecision] / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 88.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x + y}{1 - \frac{y}{z}} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(N[(x + y), $MachinePrecision] / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-252} \lor \neg \left(t\_0 \leq 0\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (+ x y) (- 1.0 (/ y z)))))
   (if (or (<= t_0 -5e-252) (not (<= t_0 0.0))) t_0 (* z (- -1.0 (/ x y))))))

C

double code(double x, double y, double z) {
	double t_0 = (x + y) / (1.0 - (y / z));
	double tmp;
	if ((t_0 <= -5e-252) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = z * (-1.0 - (x / y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x + y) / (1.0d0 - (y / z))
    if ((t_0 <= (-5d-252)) .or. (.not. (t_0 <= 0.0d0))) then
        tmp = t_0
    else
        tmp = z * ((-1.0d0) - (x / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (x + y) / (1.0 - (y / z));
	double tmp;
	if ((t_0 <= -5e-252) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = z * (-1.0 - (x / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x + y) / (1.0 - (y / z))
	tmp = 0
	if (t_0 <= -5e-252) or not (t_0 <= 0.0):
		tmp = t_0
	else:
		tmp = z * (-1.0 - (x / y))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x + y) / Float64(1.0 - Float64(y / z)))
	tmp = 0.0
	if ((t_0 <= -5e-252) || !(t_0 <= 0.0))
		tmp = t_0;
	else
		tmp = Float64(z * Float64(-1.0 - Float64(x / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x + y) / (1.0 - (y / z));
	tmp = 0.0;
	if ((t_0 <= -5e-252) || ~((t_0 <= 0.0)))
		tmp = t_0;
	else
		tmp = z * (-1.0 - (x / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x + y), $MachinePrecision] / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -5e-252], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], t$95$0, N[(z * N[(-1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z))) < -5.00000000000000008e-252 or -0.0 < (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z)))

   1. Initial program 99.9%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   if -5.00000000000000008e-252 < (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z))) < -0.0

   1. Initial program 6.1%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0 99.7%

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

      1. mul-1-neg 99.7%

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

      2. associate-/l* 99.9%

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

      3. distribute-rgt-neg-in 99.9%

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

      4. distribute-neg-frac2 99.9%

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

      5. +-commutative 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in y around inf 100.0%

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

   7. Taylor expanded in x around 0 100.0%

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

      1. associate-*l/ 100.0%

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

      2. associate-*r* 100.0%

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

      3. distribute-rgt-in 100.0%

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

      4. mul-1-neg 100.0%

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

      5. unsub-neg 100.0%

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

   9. Simplified 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \leq -5 \cdot 10^{-252} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \leq 0\right):\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 72.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.5 \cdot 10^{+86} \lor \neg \left(y \leq 4.5 \cdot 10^{+54}\right) \land \left(y \leq 1.15 \cdot 10^{+115} \lor \neg \left(y \leq 2.05 \cdot 10^{+142}\right)\right):\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -5.5e+86)
         (and (not (<= y 4.5e+54))
              (or (<= y 1.15e+115) (not (<= y 2.05e+142)))))
   (* z (- -1.0 (/ x y)))
   (+ x y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -5.5e+86) || (!(y <= 4.5e+54) && ((y <= 1.15e+115) || !(y <= 2.05e+142)))) {
		tmp = z * (-1.0 - (x / y));
	} else {
		tmp = x + y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((y <= (-5.5d+86)) .or. (.not. (y <= 4.5d+54)) .and. (y <= 1.15d+115) .or. (.not. (y <= 2.05d+142))) then
        tmp = z * ((-1.0d0) - (x / y))
    else
        tmp = x + y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -5.5e+86) || (!(y <= 4.5e+54) && ((y <= 1.15e+115) || !(y <= 2.05e+142)))) {
		tmp = z * (-1.0 - (x / y));
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -5.5e+86) or (not (y <= 4.5e+54) and ((y <= 1.15e+115) or not (y <= 2.05e+142))):
		tmp = z * (-1.0 - (x / y))
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -5.5e+86) || (!(y <= 4.5e+54) && ((y <= 1.15e+115) || !(y <= 2.05e+142))))
		tmp = Float64(z * Float64(-1.0 - Float64(x / y)));
	else
		tmp = Float64(x + y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -5.5e+86) || (~((y <= 4.5e+54)) && ((y <= 1.15e+115) || ~((y <= 2.05e+142)))))
		tmp = z * (-1.0 - (x / y));
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -5.5e+86], And[N[Not[LessEqual[y, 4.5e+54]], $MachinePrecision], Or[LessEqual[y, 1.15e+115], N[Not[LessEqual[y, 2.05e+142]], $MachinePrecision]]]], N[(z * N[(-1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -5.5000000000000002e86 or 4.49999999999999984e54 < y < 1.15000000000000002e115 or 2.04999999999999991e142 < y

   1. Initial program 66.7%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0 75.3%

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

      1. mul-1-neg 75.3%

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

      2. associate-/l* 86.4%

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

      3. distribute-rgt-neg-in 86.4%

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

      4. distribute-neg-frac2 86.4%

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

      5. +-commutative 86.4%

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

   5. Simplified 86.4%

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

   6. Taylor expanded in y around inf 86.4%

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

   7. Taylor expanded in x around 0 83.5%

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

      1. associate-*l/ 86.4%

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

      2. associate-*r* 86.4%

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

      3. distribute-rgt-in 86.4%

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

      4. mul-1-neg 86.4%

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

      5. unsub-neg 86.4%

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

   9. Simplified 86.4%

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

   if -5.5000000000000002e86 < y < 4.49999999999999984e54 or 1.15000000000000002e115 < y < 2.04999999999999991e142

   1. Initial program 99.3%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf 75.0%

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

      1. +-commutative 75.0%

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

   5. Simplified 75.0%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.5 \cdot 10^{+86} \lor \neg \left(y \leq 4.5 \cdot 10^{+54}\right) \land \left(y \leq 1.15 \cdot 10^{+115} \lor \neg \left(y \leq 2.05 \cdot 10^{+142}\right)\right):\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 71.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{if}\;z \leq -1 \cdot 10^{+50}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq -9200:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq -9.2 \cdot 10^{-140}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{elif}\;z \leq 6.2 \cdot 10^{+57}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (- -1.0 (/ x y)))))
   (if (<= z -1e+50)
     (+ x y)
     (if (<= z -9200.0)
       t_0
       (if (<= z -9.2e-140)
         (/ x (- 1.0 (/ y z)))
         (if (<= z 6.2e+57) t_0 (+ x y)))))))

C

double code(double x, double y, double z) {
	double t_0 = z * (-1.0 - (x / y));
	double tmp;
	if (z <= -1e+50) {
		tmp = x + y;
	} else if (z <= -9200.0) {
		tmp = t_0;
	} else if (z <= -9.2e-140) {
		tmp = x / (1.0 - (y / z));
	} else if (z <= 6.2e+57) {
		tmp = t_0;
	} else {
		tmp = x + y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = z * ((-1.0d0) - (x / y))
    if (z <= (-1d+50)) then
        tmp = x + y
    else if (z <= (-9200.0d0)) then
        tmp = t_0
    else if (z <= (-9.2d-140)) then
        tmp = x / (1.0d0 - (y / z))
    else if (z <= 6.2d+57) then
        tmp = t_0
    else
        tmp = x + y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = z * (-1.0 - (x / y));
	double tmp;
	if (z <= -1e+50) {
		tmp = x + y;
	} else if (z <= -9200.0) {
		tmp = t_0;
	} else if (z <= -9.2e-140) {
		tmp = x / (1.0 - (y / z));
	} else if (z <= 6.2e+57) {
		tmp = t_0;
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * (-1.0 - (x / y))
	tmp = 0
	if z <= -1e+50:
		tmp = x + y
	elif z <= -9200.0:
		tmp = t_0
	elif z <= -9.2e-140:
		tmp = x / (1.0 - (y / z))
	elif z <= 6.2e+57:
		tmp = t_0
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(-1.0 - Float64(x / y)))
	tmp = 0.0
	if (z <= -1e+50)
		tmp = Float64(x + y);
	elseif (z <= -9200.0)
		tmp = t_0;
	elseif (z <= -9.2e-140)
		tmp = Float64(x / Float64(1.0 - Float64(y / z)));
	elseif (z <= 6.2e+57)
		tmp = t_0;
	else
		tmp = Float64(x + y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * (-1.0 - (x / y));
	tmp = 0.0;
	if (z <= -1e+50)
		tmp = x + y;
	elseif (z <= -9200.0)
		tmp = t_0;
	elseif (z <= -9.2e-140)
		tmp = x / (1.0 - (y / z));
	elseif (z <= 6.2e+57)
		tmp = t_0;
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[(-1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1e+50], N[(x + y), $MachinePrecision], If[LessEqual[z, -9200.0], t$95$0, If[LessEqual[z, -9.2e-140], N[(x / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 6.2e+57], t$95$0, N[(x + y), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.0000000000000001e50 or 6.20000000000000026e57 < z

   1. Initial program 100.0%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf 85.9%

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

      1. +-commutative 85.9%

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

   5. Simplified 85.9%

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

   if -1.0000000000000001e50 < z < -9200 or -9.2000000000000005e-140 < z < 6.20000000000000026e57

   1. Initial program 72.7%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0 72.3%

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

      1. mul-1-neg 72.3%

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

      2. associate-/l* 76.2%

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

      3. distribute-rgt-neg-in 76.2%

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

      4. distribute-neg-frac2 76.2%

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

      5. +-commutative 76.2%

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

   5. Simplified 76.2%

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

   6. Taylor expanded in y around inf 76.2%

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

   7. Taylor expanded in x around 0 76.8%

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

      1. associate-*l/ 76.2%

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

      2. associate-*r* 76.2%

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

      3. distribute-rgt-in 76.2%

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

      4. mul-1-neg 76.2%

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

      5. unsub-neg 76.2%

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

   9. Simplified 76.2%

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

   if -9200 < z < -9.2000000000000005e-140

   1. Initial program 99.8%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 74.1%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{+50}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq -9200:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{elif}\;z \leq -9.2 \cdot 10^{-140}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{elif}\;z \leq 6.2 \cdot 10^{+57}:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 67.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.7 \cdot 10^{+86} \lor \neg \left(y \leq 1.55 \cdot 10^{+28}\right) \land \left(y \leq 1.15 \cdot 10^{+115} \lor \neg \left(y \leq 2.05 \cdot 10^{+142}\right)\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -2.7e+86)
         (and (not (<= y 1.55e+28))
              (or (<= y 1.15e+115) (not (<= y 2.05e+142)))))
   (- z)
   (+ x y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -2.7e+86) || (!(y <= 1.55e+28) && ((y <= 1.15e+115) || !(y <= 2.05e+142)))) {
		tmp = -z;
	} else {
		tmp = x + y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((y <= (-2.7d+86)) .or. (.not. (y <= 1.55d+28)) .and. (y <= 1.15d+115) .or. (.not. (y <= 2.05d+142))) then
        tmp = -z
    else
        tmp = x + y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -2.7e+86) || (!(y <= 1.55e+28) && ((y <= 1.15e+115) || !(y <= 2.05e+142)))) {
		tmp = -z;
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -2.7e+86) or (not (y <= 1.55e+28) and ((y <= 1.15e+115) or not (y <= 2.05e+142))):
		tmp = -z
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -2.7e+86) || (!(y <= 1.55e+28) && ((y <= 1.15e+115) || !(y <= 2.05e+142))))
		tmp = Float64(-z);
	else
		tmp = Float64(x + y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -2.7e+86) || (~((y <= 1.55e+28)) && ((y <= 1.15e+115) || ~((y <= 2.05e+142)))))
		tmp = -z;
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -2.7e+86], And[N[Not[LessEqual[y, 1.55e+28]], $MachinePrecision], Or[LessEqual[y, 1.15e+115], N[Not[LessEqual[y, 2.05e+142]], $MachinePrecision]]]], (-z), N[(x + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.70000000000000018e86 or 1.55e28 < y < 1.15000000000000002e115 or 2.04999999999999991e142 < y

   1. Initial program 67.3%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 66.8%

      \[\leadsto \color{blue}{-1 \cdot z} \]
   4. Step-by-step derivation

      1. mul-1-neg 66.8%

         \[\leadsto \color{blue}{-z} \]

   5. Simplified 66.8%

      \[\leadsto \color{blue}{-z} \]

   if -2.70000000000000018e86 < y < 1.55e28 or 1.15000000000000002e115 < y < 2.04999999999999991e142

   1. Initial program 99.3%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf 75.3%

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

      1. +-commutative 75.3%

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

   5. Simplified 75.3%

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

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.7 \cdot 10^{+86} \lor \neg \left(y \leq 1.55 \cdot 10^{+28}\right) \land \left(y \leq 1.15 \cdot 10^{+115} \lor \neg \left(y \leq 2.05 \cdot 10^{+142}\right)\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 66.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.46 \cdot 10^{+86}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 4.6 \cdot 10^{+57}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{+115}:\\ \;\;\;\;x \cdot \frac{z}{-y}\\ \mathbf{elif}\;y \leq 2.05 \cdot 10^{+142}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.46e+86)
   (- z)
   (if (<= y 4.6e+57)
     (+ x y)
     (if (<= y 1.1e+115)
       (* x (/ z (- y)))
       (if (<= y 2.05e+142) (+ x y) (- z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.46e+86) {
		tmp = -z;
	} else if (y <= 4.6e+57) {
		tmp = x + y;
	} else if (y <= 1.1e+115) {
		tmp = x * (z / -y);
	} else if (y <= 2.05e+142) {
		tmp = x + y;
	} else {
		tmp = -z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-1.46d+86)) then
        tmp = -z
    else if (y <= 4.6d+57) then
        tmp = x + y
    else if (y <= 1.1d+115) then
        tmp = x * (z / -y)
    else if (y <= 2.05d+142) then
        tmp = x + y
    else
        tmp = -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.46e+86) {
		tmp = -z;
	} else if (y <= 4.6e+57) {
		tmp = x + y;
	} else if (y <= 1.1e+115) {
		tmp = x * (z / -y);
	} else if (y <= 2.05e+142) {
		tmp = x + y;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1.46e+86:
		tmp = -z
	elif y <= 4.6e+57:
		tmp = x + y
	elif y <= 1.1e+115:
		tmp = x * (z / -y)
	elif y <= 2.05e+142:
		tmp = x + y
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.46e+86)
		tmp = Float64(-z);
	elseif (y <= 4.6e+57)
		tmp = Float64(x + y);
	elseif (y <= 1.1e+115)
		tmp = Float64(x * Float64(z / Float64(-y)));
	elseif (y <= 2.05e+142)
		tmp = Float64(x + y);
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -1.46e+86)
		tmp = -z;
	elseif (y <= 4.6e+57)
		tmp = x + y;
	elseif (y <= 1.1e+115)
		tmp = x * (z / -y);
	elseif (y <= 2.05e+142)
		tmp = x + y;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.46e+86], (-z), If[LessEqual[y, 4.6e+57], N[(x + y), $MachinePrecision], If[LessEqual[y, 1.1e+115], N[(x * N[(z / (-y)), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.05e+142], N[(x + y), $MachinePrecision], (-z)]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.46e86 or 2.04999999999999991e142 < y

   1. Initial program 63.2%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 71.4%

      \[\leadsto \color{blue}{-1 \cdot z} \]
   4. Step-by-step derivation

      1. mul-1-neg 71.4%

         \[\leadsto \color{blue}{-z} \]

   5. Simplified 71.4%

      \[\leadsto \color{blue}{-z} \]

   if -1.46e86 < y < 4.5999999999999998e57 or 1.1e115 < y < 2.04999999999999991e142

   1. Initial program 99.3%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf 74.5%

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

      1. +-commutative 74.5%

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

   5. Simplified 74.5%

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

   if 4.5999999999999998e57 < y < 1.1e115

   1. Initial program 85.9%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0 85.5%

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

      1. mul-1-neg 85.5%

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

      2. associate-/l* 85.7%

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

      3. distribute-rgt-neg-in 85.7%

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

      4. distribute-neg-frac2 85.7%

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

      5. +-commutative 85.7%

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

   5. Simplified 85.7%

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

   6. Taylor expanded in y around 0 51.9%

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

      1. mul-1-neg 51.9%

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

      2. associate-/l* 51.0%

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

      3. distribute-rgt-neg-in 51.0%

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

      4. distribute-neg-frac2 51.0%

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

   8. Simplified 51.0%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.46 \cdot 10^{+86}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 4.6 \cdot 10^{+57}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{+115}:\\ \;\;\;\;x \cdot \frac{z}{-y}\\ \mathbf{elif}\;y \leq 2.05 \cdot 10^{+142}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 66.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.8 \cdot 10^{+86}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 1.45 \cdot 10^{+58}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{+115}:\\ \;\;\;\;z \cdot \frac{x}{-y}\\ \mathbf{elif}\;y \leq 2.05 \cdot 10^{+142}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -5.8e+86)
   (- z)
   (if (<= y 1.45e+58)
     (+ x y)
     (if (<= y 1.1e+115)
       (* z (/ x (- y)))
       (if (<= y 2.05e+142) (+ x y) (- z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -5.8e+86) {
		tmp = -z;
	} else if (y <= 1.45e+58) {
		tmp = x + y;
	} else if (y <= 1.1e+115) {
		tmp = z * (x / -y);
	} else if (y <= 2.05e+142) {
		tmp = x + y;
	} else {
		tmp = -z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-5.8d+86)) then
        tmp = -z
    else if (y <= 1.45d+58) then
        tmp = x + y
    else if (y <= 1.1d+115) then
        tmp = z * (x / -y)
    else if (y <= 2.05d+142) then
        tmp = x + y
    else
        tmp = -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -5.8e+86) {
		tmp = -z;
	} else if (y <= 1.45e+58) {
		tmp = x + y;
	} else if (y <= 1.1e+115) {
		tmp = z * (x / -y);
	} else if (y <= 2.05e+142) {
		tmp = x + y;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -5.8e+86:
		tmp = -z
	elif y <= 1.45e+58:
		tmp = x + y
	elif y <= 1.1e+115:
		tmp = z * (x / -y)
	elif y <= 2.05e+142:
		tmp = x + y
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -5.8e+86)
		tmp = Float64(-z);
	elseif (y <= 1.45e+58)
		tmp = Float64(x + y);
	elseif (y <= 1.1e+115)
		tmp = Float64(z * Float64(x / Float64(-y)));
	elseif (y <= 2.05e+142)
		tmp = Float64(x + y);
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -5.8e+86)
		tmp = -z;
	elseif (y <= 1.45e+58)
		tmp = x + y;
	elseif (y <= 1.1e+115)
		tmp = z * (x / -y);
	elseif (y <= 2.05e+142)
		tmp = x + y;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -5.8e+86], (-z), If[LessEqual[y, 1.45e+58], N[(x + y), $MachinePrecision], If[LessEqual[y, 1.1e+115], N[(z * N[(x / (-y)), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.05e+142], N[(x + y), $MachinePrecision], (-z)]]]]

Derivation

1. Split input into 3 regimes

2. if y < -5.79999999999999981e86 or 2.04999999999999991e142 < y

   1. Initial program 63.2%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 71.4%

      \[\leadsto \color{blue}{-1 \cdot z} \]
   4. Step-by-step derivation

      1. mul-1-neg 71.4%

         \[\leadsto \color{blue}{-z} \]

   5. Simplified 71.4%

      \[\leadsto \color{blue}{-z} \]

   if -5.79999999999999981e86 < y < 1.45000000000000001e58 or 1.1e115 < y < 2.04999999999999991e142

   1. Initial program 99.3%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf 74.5%

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

      1. +-commutative 74.5%

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

   5. Simplified 74.5%

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

   if 1.45000000000000001e58 < y < 1.1e115

   1. Initial program 85.9%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0 85.5%

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

      1. mul-1-neg 85.5%

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

      2. associate-/l* 85.7%

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

      3. distribute-rgt-neg-in 85.7%

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

      4. distribute-neg-frac2 85.7%

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

      5. +-commutative 85.7%

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

   5. Simplified 85.7%

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

   6. Taylor expanded in y around 0 51.8%

      \[\leadsto z \cdot \color{blue}{\left(-1 \cdot \frac{x}{y}\right)} \]
   7. Step-by-step derivation

      1. associate-*r/ 51.8%

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

      2. mul-1-neg 51.8%

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

   8. Simplified 51.8%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.8 \cdot 10^{+86}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 1.45 \cdot 10^{+58}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{+115}:\\ \;\;\;\;z \cdot \frac{x}{-y}\\ \mathbf{elif}\;y \leq 2.05 \cdot 10^{+142}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.45 \cdot 10^{+87}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 1.95 \cdot 10^{+62}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{+115}:\\ \;\;\;\;\frac{x \cdot \left(-z\right)}{y}\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{+143}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.45e+87)
   (- z)
   (if (<= y 1.95e+62)
     (+ x y)
     (if (<= y 1.1e+115)
       (/ (* x (- z)) y)
       (if (<= y 2.6e+143) (+ x y) (- z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.45e+87) {
		tmp = -z;
	} else if (y <= 1.95e+62) {
		tmp = x + y;
	} else if (y <= 1.1e+115) {
		tmp = (x * -z) / y;
	} else if (y <= 2.6e+143) {
		tmp = x + y;
	} else {
		tmp = -z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-1.45d+87)) then
        tmp = -z
    else if (y <= 1.95d+62) then
        tmp = x + y
    else if (y <= 1.1d+115) then
        tmp = (x * -z) / y
    else if (y <= 2.6d+143) then
        tmp = x + y
    else
        tmp = -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.45e+87) {
		tmp = -z;
	} else if (y <= 1.95e+62) {
		tmp = x + y;
	} else if (y <= 1.1e+115) {
		tmp = (x * -z) / y;
	} else if (y <= 2.6e+143) {
		tmp = x + y;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1.45e+87:
		tmp = -z
	elif y <= 1.95e+62:
		tmp = x + y
	elif y <= 1.1e+115:
		tmp = (x * -z) / y
	elif y <= 2.6e+143:
		tmp = x + y
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.45e+87)
		tmp = Float64(-z);
	elseif (y <= 1.95e+62)
		tmp = Float64(x + y);
	elseif (y <= 1.1e+115)
		tmp = Float64(Float64(x * Float64(-z)) / y);
	elseif (y <= 2.6e+143)
		tmp = Float64(x + y);
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -1.45e+87)
		tmp = -z;
	elseif (y <= 1.95e+62)
		tmp = x + y;
	elseif (y <= 1.1e+115)
		tmp = (x * -z) / y;
	elseif (y <= 2.6e+143)
		tmp = x + y;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.45e+87], (-z), If[LessEqual[y, 1.95e+62], N[(x + y), $MachinePrecision], If[LessEqual[y, 1.1e+115], N[(N[(x * (-z)), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[y, 2.6e+143], N[(x + y), $MachinePrecision], (-z)]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.4499999999999999e87 or 2.5999999999999999e143 < y

   1. Initial program 63.2%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 71.4%

      \[\leadsto \color{blue}{-1 \cdot z} \]
   4. Step-by-step derivation

      1. mul-1-neg 71.4%

         \[\leadsto \color{blue}{-z} \]

   5. Simplified 71.4%

      \[\leadsto \color{blue}{-z} \]

   if -1.4499999999999999e87 < y < 1.95e62 or 1.1e115 < y < 2.5999999999999999e143

   1. Initial program 99.3%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf 74.5%

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

      1. +-commutative 74.5%

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

   5. Simplified 74.5%

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

   if 1.95e62 < y < 1.1e115

   1. Initial program 85.9%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0 85.5%

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

      1. mul-1-neg 85.5%

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

      2. associate-/l* 85.7%

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

      3. distribute-rgt-neg-in 85.7%

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

      4. distribute-neg-frac2 85.7%

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

      5. +-commutative 85.7%

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

   5. Simplified 85.7%

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

   6. Taylor expanded in y around 0 51.9%

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

      1. associate-*r/ 51.9%

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

      2. associate-*r* 51.9%

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

      3. mul-1-neg 51.9%

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

   8. Simplified 51.9%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.45 \cdot 10^{+87}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 1.95 \cdot 10^{+62}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{+115}:\\ \;\;\;\;\frac{x \cdot \left(-z\right)}{y}\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{+143}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 57.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.1 \cdot 10^{+85}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq -9.6 \cdot 10^{+48}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -1.6 \cdot 10^{-87}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 2.9:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -3.1e+85)
   (- z)
   (if (<= y -9.6e+48) x (if (<= y -1.6e-87) y (if (<= y 2.9) x (- z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -3.1e+85) {
		tmp = -z;
	} else if (y <= -9.6e+48) {
		tmp = x;
	} else if (y <= -1.6e-87) {
		tmp = y;
	} else if (y <= 2.9) {
		tmp = x;
	} else {
		tmp = -z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-3.1d+85)) then
        tmp = -z
    else if (y <= (-9.6d+48)) then
        tmp = x
    else if (y <= (-1.6d-87)) then
        tmp = y
    else if (y <= 2.9d0) then
        tmp = x
    else
        tmp = -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -3.1e+85) {
		tmp = -z;
	} else if (y <= -9.6e+48) {
		tmp = x;
	} else if (y <= -1.6e-87) {
		tmp = y;
	} else if (y <= 2.9) {
		tmp = x;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -3.1e+85:
		tmp = -z
	elif y <= -9.6e+48:
		tmp = x
	elif y <= -1.6e-87:
		tmp = y
	elif y <= 2.9:
		tmp = x
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -3.1e+85)
		tmp = Float64(-z);
	elseif (y <= -9.6e+48)
		tmp = x;
	elseif (y <= -1.6e-87)
		tmp = y;
	elseif (y <= 2.9)
		tmp = x;
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -3.1e+85)
		tmp = -z;
	elseif (y <= -9.6e+48)
		tmp = x;
	elseif (y <= -1.6e-87)
		tmp = y;
	elseif (y <= 2.9)
		tmp = x;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -3.1e+85], (-z), If[LessEqual[y, -9.6e+48], x, If[LessEqual[y, -1.6e-87], y, If[LessEqual[y, 2.9], x, (-z)]]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.10000000000000011e85 or 2.89999999999999991 < y

   1. Initial program 70.7%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 61.9%

      \[\leadsto \color{blue}{-1 \cdot z} \]
   4. Step-by-step derivation

      1. mul-1-neg 61.9%

         \[\leadsto \color{blue}{-z} \]

   5. Simplified 61.9%

      \[\leadsto \color{blue}{-z} \]

   if -3.10000000000000011e85 < y < -9.6000000000000004e48 or -1.59999999999999989e-87 < y < 2.89999999999999991

   1. Initial program 99.9%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 66.9%

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

   if -9.6000000000000004e48 < y < -1.59999999999999989e-87

   1. Initial program 95.7%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 63.6%

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

   4. Taylor expanded in y around 0 39.1%

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

4. Final simplification 62.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.1 \cdot 10^{+85}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq -9.6 \cdot 10^{+48}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -1.6 \cdot 10^{-87}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 2.9:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 36.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8 \cdot 10^{-111}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{-110}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -8e-111) y (if (<= y 8.5e-110) x y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -8e-111) {
		tmp = y;
	} else if (y <= 8.5e-110) {
		tmp = x;
	} else {
		tmp = y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-8d-111)) then
        tmp = y
    else if (y <= 8.5d-110) then
        tmp = x
    else
        tmp = y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -8e-111) {
		tmp = y;
	} else if (y <= 8.5e-110) {
		tmp = x;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -8e-111:
		tmp = y
	elif y <= 8.5e-110:
		tmp = x
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -8e-111)
		tmp = y;
	elseif (y <= 8.5e-110)
		tmp = x;
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -8e-111)
		tmp = y;
	elseif (y <= 8.5e-110)
		tmp = x;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -8e-111], y, If[LessEqual[y, 8.5e-110], x, y]]

Derivation

1. Split input into 2 regimes

2. if y < -8.00000000000000071e-111 or 8.50000000000000029e-110 < y

   1. Initial program 79.4%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 50.5%

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

   4. Taylor expanded in y around 0 23.6%

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

   if -8.00000000000000071e-111 < y < 8.50000000000000029e-110

   1. Initial program 100.0%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 78.2%

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

4. Final simplification 41.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8 \cdot 10^{-111}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{-110}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 35.6% accurate, 9.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := x

Derivation

1. Initial program 86.3%

   \[\frac{x + y}{1 - \frac{y}{z}} \]
2. Add Preprocessing

3. Taylor expanded in y around 0 34.9%

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

4. Final simplification 34.9%

   \[\leadsto x \]
5. Add Preprocessing

Developer target: 93.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y + x}{-y} \cdot z\\ \mathbf{if}\;y < -3.7429310762689856 \cdot 10^{+171}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y < 3.5534662456086734 \cdot 10^{+168}:\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (/ (+ y x) (- y)) z)))
   (if (< y -3.7429310762689856e+171)
     t_0
     (if (< y 3.5534662456086734e+168) (/ (+ x y) (- 1.0 (/ y z))) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = ((y + x) / -y) * z;
	double tmp;
	if (y < -3.7429310762689856e+171) {
		tmp = t_0;
	} else if (y < 3.5534662456086734e+168) {
		tmp = (x + y) / (1.0 - (y / z));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((y + x) / -y) * z
    if (y < (-3.7429310762689856d+171)) then
        tmp = t_0
    else if (y < 3.5534662456086734d+168) then
        tmp = (x + y) / (1.0d0 - (y / z))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = ((y + x) / -y) * z;
	double tmp;
	if (y < -3.7429310762689856e+171) {
		tmp = t_0;
	} else if (y < 3.5534662456086734e+168) {
		tmp = (x + y) / (1.0 - (y / z));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = ((y + x) / -y) * z
	tmp = 0
	if y < -3.7429310762689856e+171:
		tmp = t_0
	elif y < 3.5534662456086734e+168:
		tmp = (x + y) / (1.0 - (y / z))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(y + x) / Float64(-y)) * z)
	tmp = 0.0
	if (y < -3.7429310762689856e+171)
		tmp = t_0;
	elseif (y < 3.5534662456086734e+168)
		tmp = Float64(Float64(x + y) / Float64(1.0 - Float64(y / z)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = ((y + x) / -y) * z;
	tmp = 0.0;
	if (y < -3.7429310762689856e+171)
		tmp = t_0;
	elseif (y < 3.5534662456086734e+168)
		tmp = (x + y) / (1.0 - (y / z));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(y + x), $MachinePrecision] / (-y)), $MachinePrecision] * z), $MachinePrecision]}, If[Less[y, -3.7429310762689856e+171], t$95$0, If[Less[y, 3.5534662456086734e+168], N[(N[(x + y), $MachinePrecision] / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Reproduce

herbie shell --seed 2024082 
(FPCore (x y z)
  :name "Graphics.Rendering.Chart.Backend.Diagrams:calcFontMetrics from Chart-diagrams-1.5.1, A"
  :precision binary64

  :alt
  (if (< y -3.7429310762689856e+171) (* (/ (+ y x) (- y)) z) (if (< y 3.5534662456086734e+168) (/ (+ x y) (- 1.0 (/ y z))) (* (/ (+ y x) (- y)) z)))

  (/ (+ x y) (- 1.0 (/ y z))))