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

Percentage Accurate: 88.5% → 99.2%

Time: 6.3s

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: 88.5% 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: 99.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{-266} \lor \neg \left(t_0 \leq 5 \cdot 10^{-253}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) - \frac{z}{\frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (+ x y) (- 1.0 (/ y z)))))
   (if (or (<= t_0 -2e-266) (not (<= t_0 5e-253)))
     t_0
     (- (- z) (/ z (/ y x))))))

C

double code(double x, double y, double z) {
	double t_0 = (x + y) / (1.0 - (y / z));
	double tmp;
	if ((t_0 <= -2e-266) || !(t_0 <= 5e-253)) {
		tmp = t_0;
	} else {
		tmp = -z - (z / (y / x));
	}
	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 <= (-2d-266)) .or. (.not. (t_0 <= 5d-253))) then
        tmp = t_0
    else
        tmp = -z - (z / (y / x))
    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 <= -2e-266) || !(t_0 <= 5e-253)) {
		tmp = t_0;
	} else {
		tmp = -z - (z / (y / x));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x + y) / (1.0 - (y / z))
	tmp = 0
	if (t_0 <= -2e-266) or not (t_0 <= 5e-253):
		tmp = t_0
	else:
		tmp = -z - (z / (y / x))
	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 <= -2e-266) || !(t_0 <= 5e-253))
		tmp = t_0;
	else
		tmp = Float64(Float64(-z) - Float64(z / Float64(y / x)));
	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 <= -2e-266) || ~((t_0 <= 5e-253)))
		tmp = t_0;
	else
		tmp = -z - (z / (y / x));
	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, -2e-266], N[Not[LessEqual[t$95$0, 5e-253]], $MachinePrecision]], t$95$0, N[((-z) - N[(z / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z))) < -2e-266 or 4.99999999999999971e-253 < (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z)))

   1. Initial program 99.8%

      \[\frac{x + y}{1 - \frac{y}{z}} \]

   if -2e-266 < (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z))) < 4.99999999999999971e-253

   1. Initial program 29.0%

      \[\frac{x + y}{1 - \frac{y}{z}} \]

   2. Taylor expanded in y around inf 100.0%

      \[\leadsto \color{blue}{\left(-1 \cdot z + -1 \cdot \frac{z \cdot x}{y}\right) - \frac{{z}^{2}}{y}} \]
   3. Step-by-step derivation

      1. sub-neg 100.0%

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

      2. mul-1-neg 100.0%

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

      3. unsub-neg 100.0%

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

      4. associate-+l- 100.0%

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

      5. mul-1-neg 100.0%

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

      6. distribute-frac-neg 100.0%

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

      7. mul-1-neg 100.0%

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

      8. div-sub 100.0%

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

      9. sub-neg 100.0%

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

      10. mul-1-neg 100.0%

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

      11. remove-double-neg 100.0%

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

      12. unpow2 100.0%

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

      13. distribute-lft-out 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in z around 0 100.0%

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

      1. associate-/l* 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \leq -2 \cdot 10^{-266} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \leq 5 \cdot 10^{-253}\right):\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) - \frac{z}{\frac{y}{x}}\\ \end{array} \]

Alternative 2: 73.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{y}{z}\\ \mathbf{if}\;y \leq -4.8 \cdot 10^{+40}:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{elif}\;y \leq 1.66 \cdot 10^{-164}:\\ \;\;\;\;\frac{x}{t_0}\\ \mathbf{elif}\;y \leq 1.62 \cdot 10^{+40}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 7 \cdot 10^{+149} \lor \neg \left(y \leq 2.45 \cdot 10^{+191}\right):\\ \;\;\;\;\left(-z\right) - z \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ y z))))
   (if (<= y -4.8e+40)
     (* z (- -1.0 (/ x y)))
     (if (<= y 1.66e-164)
       (/ x t_0)
       (if (<= y 1.62e+40)
         (+ x y)
         (if (or (<= y 7e+149) (not (<= y 2.45e+191)))
           (- (- z) (* z (/ x y)))
           (/ y t_0)))))))

C

double code(double x, double y, double z) {
	double t_0 = 1.0 - (y / z);
	double tmp;
	if (y <= -4.8e+40) {
		tmp = z * (-1.0 - (x / y));
	} else if (y <= 1.66e-164) {
		tmp = x / t_0;
	} else if (y <= 1.62e+40) {
		tmp = x + y;
	} else if ((y <= 7e+149) || !(y <= 2.45e+191)) {
		tmp = -z - (z * (x / y));
	} else {
		tmp = y / 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 = 1.0d0 - (y / z)
    if (y <= (-4.8d+40)) then
        tmp = z * ((-1.0d0) - (x / y))
    else if (y <= 1.66d-164) then
        tmp = x / t_0
    else if (y <= 1.62d+40) then
        tmp = x + y
    else if ((y <= 7d+149) .or. (.not. (y <= 2.45d+191))) then
        tmp = -z - (z * (x / y))
    else
        tmp = y / t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = 1.0 - (y / z);
	double tmp;
	if (y <= -4.8e+40) {
		tmp = z * (-1.0 - (x / y));
	} else if (y <= 1.66e-164) {
		tmp = x / t_0;
	} else if (y <= 1.62e+40) {
		tmp = x + y;
	} else if ((y <= 7e+149) || !(y <= 2.45e+191)) {
		tmp = -z - (z * (x / y));
	} else {
		tmp = y / t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 1.0 - (y / z)
	tmp = 0
	if y <= -4.8e+40:
		tmp = z * (-1.0 - (x / y))
	elif y <= 1.66e-164:
		tmp = x / t_0
	elif y <= 1.62e+40:
		tmp = x + y
	elif (y <= 7e+149) or not (y <= 2.45e+191):
		tmp = -z - (z * (x / y))
	else:
		tmp = y / t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(1.0 - Float64(y / z))
	tmp = 0.0
	if (y <= -4.8e+40)
		tmp = Float64(z * Float64(-1.0 - Float64(x / y)));
	elseif (y <= 1.66e-164)
		tmp = Float64(x / t_0);
	elseif (y <= 1.62e+40)
		tmp = Float64(x + y);
	elseif ((y <= 7e+149) || !(y <= 2.45e+191))
		tmp = Float64(Float64(-z) - Float64(z * Float64(x / y)));
	else
		tmp = Float64(y / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 1.0 - (y / z);
	tmp = 0.0;
	if (y <= -4.8e+40)
		tmp = z * (-1.0 - (x / y));
	elseif (y <= 1.66e-164)
		tmp = x / t_0;
	elseif (y <= 1.62e+40)
		tmp = x + y;
	elseif ((y <= 7e+149) || ~((y <= 2.45e+191)))
		tmp = -z - (z * (x / y));
	else
		tmp = y / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -4.8e+40], N[(z * N[(-1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.66e-164], N[(x / t$95$0), $MachinePrecision], If[LessEqual[y, 1.62e+40], N[(x + y), $MachinePrecision], If[Or[LessEqual[y, 7e+149], N[Not[LessEqual[y, 2.45e+191]], $MachinePrecision]], N[((-z) - N[(z * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y / t$95$0), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if y < -4.8e40

   1. Initial program 66.8%

      \[\frac{x + y}{1 - \frac{y}{z}} \]

   2. Taylor expanded in y around inf 82.7%

      \[\leadsto \color{blue}{\left(-1 \cdot z + -1 \cdot \frac{z \cdot x}{y}\right) - \frac{{z}^{2}}{y}} \]
   3. Step-by-step derivation

      1. sub-neg 82.7%

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

      2. mul-1-neg 82.7%

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

      3. unsub-neg 82.7%

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

      4. associate-+l- 82.7%

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

      5. mul-1-neg 82.7%

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

      6. distribute-frac-neg 82.7%

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

      7. mul-1-neg 82.7%

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

      8. div-sub 82.7%

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

      9. sub-neg 82.7%

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

      10. mul-1-neg 82.7%

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

      11. remove-double-neg 82.7%

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

      12. unpow2 82.7%

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

      13. distribute-lft-out 82.7%

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

   4. Simplified 82.7%

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

   5. Taylor expanded in z around 0 88.0%

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

      1. associate-/l* 90.4%

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

   7. Simplified 90.4%

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

   8. Taylor expanded in z around 0 88.0%

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

      1. associate-*r/ 90.3%

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

   10. Simplified 90.3%

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

   11. Taylor expanded in z around 0 90.3%

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

       1. associate-*r* 90.3%

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

       2. distribute-lft-in 90.3%

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

       3. metadata-eval 90.3%

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

       4. mul-1-neg 90.3%

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

       5. sub-neg 90.3%

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

       6. *-commutative 90.3%

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

   13. Simplified 90.3%

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

   if -4.8e40 < y < 1.6599999999999999e-164

   1. Initial program 99.8%

      \[\frac{x + y}{1 - \frac{y}{z}} \]

   2. Taylor expanded in x around inf 76.5%

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

   if 1.6599999999999999e-164 < y < 1.62e40

   1. Initial program 99.9%

      \[\frac{x + y}{1 - \frac{y}{z}} \]

   2. Taylor expanded in z around inf 74.0%

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

   if 1.62e40 < y < 7.00000000000000023e149 or 2.45e191 < y

   1. Initial program 73.5%

      \[\frac{x + y}{1 - \frac{y}{z}} \]

   2. Taylor expanded in y around inf 82.0%

      \[\leadsto \color{blue}{\left(-1 \cdot z + -1 \cdot \frac{z \cdot x}{y}\right) - \frac{{z}^{2}}{y}} \]
   3. Step-by-step derivation

      1. sub-neg 82.0%

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

      2. mul-1-neg 82.0%

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

      3. unsub-neg 82.0%

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

      4. associate-+l- 82.0%

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

      5. mul-1-neg 82.0%

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

      6. distribute-frac-neg 82.0%

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

      7. mul-1-neg 82.0%

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

      8. div-sub 82.0%

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

      9. sub-neg 82.0%

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

      10. mul-1-neg 82.0%

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

      11. remove-double-neg 82.0%

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

      12. unpow2 82.0%

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

      13. distribute-lft-out 82.1%

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

   4. Simplified 82.1%

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

   5. Taylor expanded in z around 0 84.8%

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

      1. associate-/l* 90.1%

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

   7. Simplified 90.1%

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

   8. Taylor expanded in z around 0 84.8%

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

      1. associate-*r/ 90.0%

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

   10. Simplified 90.0%

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

   if 7.00000000000000023e149 < y < 2.45e191

   1. Initial program 100.0%

      \[\frac{x + y}{1 - \frac{y}{z}} \]

   2. Taylor expanded in x around 0 90.1%

      \[\leadsto \color{blue}{\frac{y}{1 - \frac{y}{z}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{+40}:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{elif}\;y \leq 1.66 \cdot 10^{-164}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{elif}\;y \leq 1.62 \cdot 10^{+40}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 7 \cdot 10^{+149} \lor \neg \left(y \leq 2.45 \cdot 10^{+191}\right):\\ \;\;\;\;\left(-z\right) - z \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{1 - \frac{y}{z}}\\ \end{array} \]

Alternative 3: 73.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{y}{z}\\ t_1 := \left(-z\right) - \frac{z}{\frac{y}{x}}\\ \mathbf{if}\;y \leq -1.2 \cdot 10^{+41}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{-157}:\\ \;\;\;\;\frac{x}{t_0}\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{+39}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{+149}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{+190}:\\ \;\;\;\;\frac{y}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) - z \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ y z))) (t_1 (- (- z) (/ z (/ y x)))))
   (if (<= y -1.2e+41)
     t_1
     (if (<= y 1.8e-157)
       (/ x t_0)
       (if (<= y 1.9e+39)
         (+ x y)
         (if (<= y 5.8e+149)
           t_1
           (if (<= y 9.5e+190) (/ y t_0) (- (- z) (* z (/ x y))))))))))

C

double code(double x, double y, double z) {
	double t_0 = 1.0 - (y / z);
	double t_1 = -z - (z / (y / x));
	double tmp;
	if (y <= -1.2e+41) {
		tmp = t_1;
	} else if (y <= 1.8e-157) {
		tmp = x / t_0;
	} else if (y <= 1.9e+39) {
		tmp = x + y;
	} else if (y <= 5.8e+149) {
		tmp = t_1;
	} else if (y <= 9.5e+190) {
		tmp = y / t_0;
	} else {
		tmp = -z - (z * (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) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 - (y / z)
    t_1 = -z - (z / (y / x))
    if (y <= (-1.2d+41)) then
        tmp = t_1
    else if (y <= 1.8d-157) then
        tmp = x / t_0
    else if (y <= 1.9d+39) then
        tmp = x + y
    else if (y <= 5.8d+149) then
        tmp = t_1
    else if (y <= 9.5d+190) then
        tmp = y / t_0
    else
        tmp = -z - (z * (x / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = 1.0 - (y / z);
	double t_1 = -z - (z / (y / x));
	double tmp;
	if (y <= -1.2e+41) {
		tmp = t_1;
	} else if (y <= 1.8e-157) {
		tmp = x / t_0;
	} else if (y <= 1.9e+39) {
		tmp = x + y;
	} else if (y <= 5.8e+149) {
		tmp = t_1;
	} else if (y <= 9.5e+190) {
		tmp = y / t_0;
	} else {
		tmp = -z - (z * (x / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 1.0 - (y / z)
	t_1 = -z - (z / (y / x))
	tmp = 0
	if y <= -1.2e+41:
		tmp = t_1
	elif y <= 1.8e-157:
		tmp = x / t_0
	elif y <= 1.9e+39:
		tmp = x + y
	elif y <= 5.8e+149:
		tmp = t_1
	elif y <= 9.5e+190:
		tmp = y / t_0
	else:
		tmp = -z - (z * (x / y))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(1.0 - Float64(y / z))
	t_1 = Float64(Float64(-z) - Float64(z / Float64(y / x)))
	tmp = 0.0
	if (y <= -1.2e+41)
		tmp = t_1;
	elseif (y <= 1.8e-157)
		tmp = Float64(x / t_0);
	elseif (y <= 1.9e+39)
		tmp = Float64(x + y);
	elseif (y <= 5.8e+149)
		tmp = t_1;
	elseif (y <= 9.5e+190)
		tmp = Float64(y / t_0);
	else
		tmp = Float64(Float64(-z) - Float64(z * Float64(x / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 1.0 - (y / z);
	t_1 = -z - (z / (y / x));
	tmp = 0.0;
	if (y <= -1.2e+41)
		tmp = t_1;
	elseif (y <= 1.8e-157)
		tmp = x / t_0;
	elseif (y <= 1.9e+39)
		tmp = x + y;
	elseif (y <= 5.8e+149)
		tmp = t_1;
	elseif (y <= 9.5e+190)
		tmp = y / t_0;
	else
		tmp = -z - (z * (x / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[((-z) - N[(z / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.2e+41], t$95$1, If[LessEqual[y, 1.8e-157], N[(x / t$95$0), $MachinePrecision], If[LessEqual[y, 1.9e+39], N[(x + y), $MachinePrecision], If[LessEqual[y, 5.8e+149], t$95$1, If[LessEqual[y, 9.5e+190], N[(y / t$95$0), $MachinePrecision], N[((-z) - N[(z * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if y < -1.2000000000000001e41 or 1.8999999999999999e39 < y < 5.8000000000000004e149

   1. Initial program 70.9%

      \[\frac{x + y}{1 - \frac{y}{z}} \]

   2. Taylor expanded in y around inf 86.2%

      \[\leadsto \color{blue}{\left(-1 \cdot z + -1 \cdot \frac{z \cdot x}{y}\right) - \frac{{z}^{2}}{y}} \]
   3. Step-by-step derivation

      1. sub-neg 86.2%

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

      2. mul-1-neg 86.2%

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

      3. unsub-neg 86.2%

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

      4. associate-+l- 86.2%

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

      5. mul-1-neg 86.2%

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

      6. distribute-frac-neg 86.2%

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

      7. mul-1-neg 86.2%

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

      8. div-sub 86.2%

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

      9. sub-neg 86.2%

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

      10. mul-1-neg 86.2%

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

      11. remove-double-neg 86.2%

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

      12. unpow2 86.2%

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

      13. distribute-lft-out 86.3%

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

   4. Simplified 86.3%

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

   5. Taylor expanded in z around 0 89.5%

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

      1. associate-/l* 91.0%

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

   7. Simplified 91.0%

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

   if -1.2000000000000001e41 < y < 1.8e-157

   1. Initial program 99.8%

      \[\frac{x + y}{1 - \frac{y}{z}} \]

   2. Taylor expanded in x around inf 76.5%

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

   if 1.8e-157 < y < 1.8999999999999999e39

   1. Initial program 99.9%

      \[\frac{x + y}{1 - \frac{y}{z}} \]

   2. Taylor expanded in z around inf 74.0%

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

   if 5.8000000000000004e149 < y < 9.4999999999999995e190

   1. Initial program 100.0%

      \[\frac{x + y}{1 - \frac{y}{z}} \]

   2. Taylor expanded in x around 0 90.1%

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

   if 9.4999999999999995e190 < y

   1. Initial program 66.6%

      \[\frac{x + y}{1 - \frac{y}{z}} \]

   2. Taylor expanded in y around inf 64.3%

      \[\leadsto \color{blue}{\left(-1 \cdot z + -1 \cdot \frac{z \cdot x}{y}\right) - \frac{{z}^{2}}{y}} \]
   3. Step-by-step derivation

      1. sub-neg 64.3%

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

      2. mul-1-neg 64.3%

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

      3. unsub-neg 64.3%

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

      4. associate-+l- 64.3%

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

      5. mul-1-neg 64.3%

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

      6. distribute-frac-neg 64.3%

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

      7. mul-1-neg 64.3%

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

      8. div-sub 64.3%

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

      9. sub-neg 64.3%

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

      10. mul-1-neg 64.3%

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

      11. remove-double-neg 64.3%

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

      12. unpow2 64.3%

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

      13. distribute-lft-out 64.3%

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

   4. Simplified 64.3%

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

   5. Taylor expanded in z around 0 72.3%

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

      1. associate-/l* 86.7%

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

   7. Simplified 86.7%

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

   8. Taylor expanded in z around 0 72.3%

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

      1. associate-*r/ 86.7%

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

   10. Simplified 86.7%

       \[\leadsto \left(-z\right) - \color{blue}{z \cdot \frac{x}{y}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.2 \cdot 10^{+41}:\\ \;\;\;\;\left(-z\right) - \frac{z}{\frac{y}{x}}\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{-157}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{+39}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{+149}:\\ \;\;\;\;\left(-z\right) - \frac{z}{\frac{y}{x}}\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{+190}:\\ \;\;\;\;\frac{y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) - z \cdot \frac{x}{y}\\ \end{array} \]

Alternative 4: 72.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{if}\;y \leq -4.8 \cdot 10^{+40}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.02 \cdot 10^{-159}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{elif}\;y \leq 9.2 \cdot 10^{+38}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{+167} \lor \neg \left(y \leq 9.5 \cdot 10^{+190}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (- -1.0 (/ x y)))))
   (if (<= y -4.8e+40)
     t_0
     (if (<= y 1.02e-159)
       (/ x (- 1.0 (/ y z)))
       (if (<= y 9.2e+38)
         (+ x y)
         (if (or (<= y 1.9e+167) (not (<= y 9.5e+190))) t_0 y))))))

C

double code(double x, double y, double z) {
	double t_0 = z * (-1.0 - (x / y));
	double tmp;
	if (y <= -4.8e+40) {
		tmp = t_0;
	} else if (y <= 1.02e-159) {
		tmp = x / (1.0 - (y / z));
	} else if (y <= 9.2e+38) {
		tmp = x + y;
	} else if ((y <= 1.9e+167) || !(y <= 9.5e+190)) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = z * ((-1.0d0) - (x / y))
    if (y <= (-4.8d+40)) then
        tmp = t_0
    else if (y <= 1.02d-159) then
        tmp = x / (1.0d0 - (y / z))
    else if (y <= 9.2d+38) then
        tmp = x + y
    else if ((y <= 1.9d+167) .or. (.not. (y <= 9.5d+190))) then
        tmp = t_0
    else
        tmp = 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 (y <= -4.8e+40) {
		tmp = t_0;
	} else if (y <= 1.02e-159) {
		tmp = x / (1.0 - (y / z));
	} else if (y <= 9.2e+38) {
		tmp = x + y;
	} else if ((y <= 1.9e+167) || !(y <= 9.5e+190)) {
		tmp = t_0;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * (-1.0 - (x / y))
	tmp = 0
	if y <= -4.8e+40:
		tmp = t_0
	elif y <= 1.02e-159:
		tmp = x / (1.0 - (y / z))
	elif y <= 9.2e+38:
		tmp = x + y
	elif (y <= 1.9e+167) or not (y <= 9.5e+190):
		tmp = t_0
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(-1.0 - Float64(x / y)))
	tmp = 0.0
	if (y <= -4.8e+40)
		tmp = t_0;
	elseif (y <= 1.02e-159)
		tmp = Float64(x / Float64(1.0 - Float64(y / z)));
	elseif (y <= 9.2e+38)
		tmp = Float64(x + y);
	elseif ((y <= 1.9e+167) || !(y <= 9.5e+190))
		tmp = t_0;
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * (-1.0 - (x / y));
	tmp = 0.0;
	if (y <= -4.8e+40)
		tmp = t_0;
	elseif (y <= 1.02e-159)
		tmp = x / (1.0 - (y / z));
	elseif (y <= 9.2e+38)
		tmp = x + y;
	elseif ((y <= 1.9e+167) || ~((y <= 9.5e+190)))
		tmp = t_0;
	else
		tmp = 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[y, -4.8e+40], t$95$0, If[LessEqual[y, 1.02e-159], N[(x / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 9.2e+38], N[(x + y), $MachinePrecision], If[Or[LessEqual[y, 1.9e+167], N[Not[LessEqual[y, 9.5e+190]], $MachinePrecision]], t$95$0, y]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -4.8e40 or 9.2000000000000005e38 < y < 1.89999999999999997e167 or 9.4999999999999995e190 < y

   1. Initial program 71.2%

      \[\frac{x + y}{1 - \frac{y}{z}} \]

   2. Taylor expanded in y around inf 80.6%

      \[\leadsto \color{blue}{\left(-1 \cdot z + -1 \cdot \frac{z \cdot x}{y}\right) - \frac{{z}^{2}}{y}} \]
   3. Step-by-step derivation

      1. sub-neg 80.6%

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

      2. mul-1-neg 80.6%

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

      3. unsub-neg 80.6%

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

      4. associate-+l- 80.6%

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

      5. mul-1-neg 80.6%

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

      6. distribute-frac-neg 80.6%

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

      7. mul-1-neg 80.6%

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

      8. div-sub 80.6%

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

      9. sub-neg 80.6%

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

      10. mul-1-neg 80.6%

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

      11. remove-double-neg 80.6%

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

      12. unpow2 80.6%

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

      13. distribute-lft-out 80.7%

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

   4. Simplified 80.7%

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

   5. Taylor expanded in z around 0 84.5%

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

      1. associate-/l* 89.3%

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

   7. Simplified 89.3%

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

   8. Taylor expanded in z around 0 84.5%

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

      1. associate-*r/ 89.2%

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

   10. Simplified 89.2%

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

   11. Taylor expanded in z around 0 89.2%

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

       1. associate-*r* 89.2%

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

       2. distribute-lft-in 89.2%

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

       3. metadata-eval 89.2%

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

       4. mul-1-neg 89.2%

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

       5. sub-neg 89.2%

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

       6. *-commutative 89.2%

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

   13. Simplified 89.2%

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

   if -4.8e40 < y < 1.02e-159

   1. Initial program 99.8%

      \[\frac{x + y}{1 - \frac{y}{z}} \]

   2. Taylor expanded in x around inf 76.5%

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

   if 1.02e-159 < y < 9.2000000000000005e38

   1. Initial program 99.9%

      \[\frac{x + y}{1 - \frac{y}{z}} \]

   2. Taylor expanded in z around inf 74.0%

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

   if 1.89999999999999997e167 < y < 9.4999999999999995e190

   1. Initial program 100.0%

      \[\frac{x + y}{1 - \frac{y}{z}} \]

   2. Taylor expanded in x around 0 100.0%

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

   3. Taylor expanded in y around 0 85.9%

      \[\leadsto \color{blue}{y} \]
3. Recombined 4 regimes into one program.

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{+40}:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{elif}\;y \leq 1.02 \cdot 10^{-159}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{elif}\;y \leq 9.2 \cdot 10^{+38}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{+167} \lor \neg \left(y \leq 9.5 \cdot 10^{+190}\right):\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 5: 73.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{y}{z}\\ t_1 := z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{if}\;y \leq -2 \cdot 10^{+40}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 6.4 \cdot 10^{-160}:\\ \;\;\;\;\frac{x}{t_0}\\ \mathbf{elif}\;y \leq 2.65 \cdot 10^{+42}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 7.1 \cdot 10^{+149} \lor \neg \left(y \leq 1.05 \cdot 10^{+191}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ y z))) (t_1 (* z (- -1.0 (/ x y)))))
   (if (<= y -2e+40)
     t_1
     (if (<= y 6.4e-160)
       (/ x t_0)
       (if (<= y 2.65e+42)
         (+ x y)
         (if (or (<= y 7.1e+149) (not (<= y 1.05e+191))) t_1 (/ y t_0)))))))

C

double code(double x, double y, double z) {
	double t_0 = 1.0 - (y / z);
	double t_1 = z * (-1.0 - (x / y));
	double tmp;
	if (y <= -2e+40) {
		tmp = t_1;
	} else if (y <= 6.4e-160) {
		tmp = x / t_0;
	} else if (y <= 2.65e+42) {
		tmp = x + y;
	} else if ((y <= 7.1e+149) || !(y <= 1.05e+191)) {
		tmp = t_1;
	} else {
		tmp = y / 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) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 - (y / z)
    t_1 = z * ((-1.0d0) - (x / y))
    if (y <= (-2d+40)) then
        tmp = t_1
    else if (y <= 6.4d-160) then
        tmp = x / t_0
    else if (y <= 2.65d+42) then
        tmp = x + y
    else if ((y <= 7.1d+149) .or. (.not. (y <= 1.05d+191))) then
        tmp = t_1
    else
        tmp = y / t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = 1.0 - (y / z);
	double t_1 = z * (-1.0 - (x / y));
	double tmp;
	if (y <= -2e+40) {
		tmp = t_1;
	} else if (y <= 6.4e-160) {
		tmp = x / t_0;
	} else if (y <= 2.65e+42) {
		tmp = x + y;
	} else if ((y <= 7.1e+149) || !(y <= 1.05e+191)) {
		tmp = t_1;
	} else {
		tmp = y / t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 1.0 - (y / z)
	t_1 = z * (-1.0 - (x / y))
	tmp = 0
	if y <= -2e+40:
		tmp = t_1
	elif y <= 6.4e-160:
		tmp = x / t_0
	elif y <= 2.65e+42:
		tmp = x + y
	elif (y <= 7.1e+149) or not (y <= 1.05e+191):
		tmp = t_1
	else:
		tmp = y / t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(1.0 - Float64(y / z))
	t_1 = Float64(z * Float64(-1.0 - Float64(x / y)))
	tmp = 0.0
	if (y <= -2e+40)
		tmp = t_1;
	elseif (y <= 6.4e-160)
		tmp = Float64(x / t_0);
	elseif (y <= 2.65e+42)
		tmp = Float64(x + y);
	elseif ((y <= 7.1e+149) || !(y <= 1.05e+191))
		tmp = t_1;
	else
		tmp = Float64(y / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 1.0 - (y / z);
	t_1 = z * (-1.0 - (x / y));
	tmp = 0.0;
	if (y <= -2e+40)
		tmp = t_1;
	elseif (y <= 6.4e-160)
		tmp = x / t_0;
	elseif (y <= 2.65e+42)
		tmp = x + y;
	elseif ((y <= 7.1e+149) || ~((y <= 1.05e+191)))
		tmp = t_1;
	else
		tmp = y / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(z * N[(-1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2e+40], t$95$1, If[LessEqual[y, 6.4e-160], N[(x / t$95$0), $MachinePrecision], If[LessEqual[y, 2.65e+42], N[(x + y), $MachinePrecision], If[Or[LessEqual[y, 7.1e+149], N[Not[LessEqual[y, 1.05e+191]], $MachinePrecision]], t$95$1, N[(y / t$95$0), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -2.00000000000000006e40 or 2.65000000000000014e42 < y < 7.1000000000000001e149 or 1.05e191 < y

   1. Initial program 70.1%

      \[\frac{x + y}{1 - \frac{y}{z}} \]

   2. Taylor expanded in y around inf 82.4%

      \[\leadsto \color{blue}{\left(-1 \cdot z + -1 \cdot \frac{z \cdot x}{y}\right) - \frac{{z}^{2}}{y}} \]
   3. Step-by-step derivation

      1. sub-neg 82.4%

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

      2. mul-1-neg 82.4%

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

      3. unsub-neg 82.4%

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

      4. associate-+l- 82.4%

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

      5. mul-1-neg 82.4%

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

      6. distribute-frac-neg 82.4%

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

      7. mul-1-neg 82.4%

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

      8. div-sub 82.4%

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

      9. sub-neg 82.4%

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

      10. mul-1-neg 82.4%

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

      11. remove-double-neg 82.4%

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

      12. unpow2 82.4%

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

      13. distribute-lft-out 82.4%

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

   4. Simplified 82.4%

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

   5. Taylor expanded in z around 0 86.4%

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

      1. associate-/l* 90.3%

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

   7. Simplified 90.3%

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

   8. Taylor expanded in z around 0 86.4%

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

      1. associate-*r/ 90.2%

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

   10. Simplified 90.2%

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

   11. Taylor expanded in z around 0 90.1%

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

       1. associate-*r* 90.1%

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

       2. distribute-lft-in 90.1%

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

       3. metadata-eval 90.1%

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

       4. mul-1-neg 90.1%

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

       5. sub-neg 90.1%

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

       6. *-commutative 90.1%

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

   13. Simplified 90.1%

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

   if -2.00000000000000006e40 < y < 6.40000000000000018e-160

   1. Initial program 99.8%

      \[\frac{x + y}{1 - \frac{y}{z}} \]

   2. Taylor expanded in x around inf 76.5%

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

   if 6.40000000000000018e-160 < y < 2.65000000000000014e42

   1. Initial program 99.9%

      \[\frac{x + y}{1 - \frac{y}{z}} \]

   2. Taylor expanded in z around inf 74.0%

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

   if 7.1000000000000001e149 < y < 1.05e191

   1. Initial program 100.0%

      \[\frac{x + y}{1 - \frac{y}{z}} \]

   2. Taylor expanded in x around 0 90.1%

      \[\leadsto \color{blue}{\frac{y}{1 - \frac{y}{z}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{+40}:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{elif}\;y \leq 6.4 \cdot 10^{-160}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{elif}\;y \leq 2.65 \cdot 10^{+42}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 7.1 \cdot 10^{+149} \lor \neg \left(y \leq 1.05 \cdot 10^{+191}\right):\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{1 - \frac{y}{z}}\\ \end{array} \]

Alternative 6: 73.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{if}\;y \leq -11500:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{+39}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{+167} \lor \neg \left(y \leq 9.5 \cdot 10^{+190}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (- -1.0 (/ x y)))))
   (if (<= y -11500.0)
     t_0
     (if (<= y 3.6e+39)
       (+ x y)
       (if (or (<= y 1.9e+167) (not (<= y 9.5e+190))) t_0 y)))))

C

double code(double x, double y, double z) {
	double t_0 = z * (-1.0 - (x / y));
	double tmp;
	if (y <= -11500.0) {
		tmp = t_0;
	} else if (y <= 3.6e+39) {
		tmp = x + y;
	} else if ((y <= 1.9e+167) || !(y <= 9.5e+190)) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = z * ((-1.0d0) - (x / y))
    if (y <= (-11500.0d0)) then
        tmp = t_0
    else if (y <= 3.6d+39) then
        tmp = x + y
    else if ((y <= 1.9d+167) .or. (.not. (y <= 9.5d+190))) then
        tmp = t_0
    else
        tmp = 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 (y <= -11500.0) {
		tmp = t_0;
	} else if (y <= 3.6e+39) {
		tmp = x + y;
	} else if ((y <= 1.9e+167) || !(y <= 9.5e+190)) {
		tmp = t_0;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * (-1.0 - (x / y))
	tmp = 0
	if y <= -11500.0:
		tmp = t_0
	elif y <= 3.6e+39:
		tmp = x + y
	elif (y <= 1.9e+167) or not (y <= 9.5e+190):
		tmp = t_0
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(-1.0 - Float64(x / y)))
	tmp = 0.0
	if (y <= -11500.0)
		tmp = t_0;
	elseif (y <= 3.6e+39)
		tmp = Float64(x + y);
	elseif ((y <= 1.9e+167) || !(y <= 9.5e+190))
		tmp = t_0;
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * (-1.0 - (x / y));
	tmp = 0.0;
	if (y <= -11500.0)
		tmp = t_0;
	elseif (y <= 3.6e+39)
		tmp = x + y;
	elseif ((y <= 1.9e+167) || ~((y <= 9.5e+190)))
		tmp = t_0;
	else
		tmp = 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[y, -11500.0], t$95$0, If[LessEqual[y, 3.6e+39], N[(x + y), $MachinePrecision], If[Or[LessEqual[y, 1.9e+167], N[Not[LessEqual[y, 9.5e+190]], $MachinePrecision]], t$95$0, y]]]]

Derivation

1. Split input into 3 regimes

2. if y < -11500 or 3.59999999999999984e39 < y < 1.89999999999999997e167 or 9.4999999999999995e190 < y

   1. Initial program 73.5%

      \[\frac{x + y}{1 - \frac{y}{z}} \]

   2. Taylor expanded in y around inf 78.8%

      \[\leadsto \color{blue}{\left(-1 \cdot z + -1 \cdot \frac{z \cdot x}{y}\right) - \frac{{z}^{2}}{y}} \]
   3. Step-by-step derivation

      1. sub-neg 78.8%

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

      2. mul-1-neg 78.8%

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

      3. unsub-neg 78.8%

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

      4. associate-+l- 78.8%

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

      5. mul-1-neg 78.8%

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

      6. distribute-frac-neg 78.8%

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

      7. mul-1-neg 78.8%

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

      8. div-sub 78.8%

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

      9. sub-neg 78.8%

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

      10. mul-1-neg 78.8%

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

      11. remove-double-neg 78.8%

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

      12. unpow2 78.8%

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

      13. distribute-lft-out 79.0%

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

   4. Simplified 79.0%

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

   5. Taylor expanded in z around 0 82.4%

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

      1. associate-/l* 87.0%

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

   7. Simplified 87.0%

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

   8. Taylor expanded in z around 0 82.4%

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

      1. associate-*r/ 86.8%

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

   10. Simplified 86.8%

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

   11. Taylor expanded in z around 0 86.8%

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

       1. associate-*r* 86.8%

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

       2. distribute-lft-in 86.8%

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

       3. metadata-eval 86.8%

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

       4. mul-1-neg 86.8%

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

       5. sub-neg 86.8%

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

       6. *-commutative 86.8%

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

   13. Simplified 86.8%

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

   if -11500 < y < 3.59999999999999984e39

   1. Initial program 99.9%

      \[\frac{x + y}{1 - \frac{y}{z}} \]

   2. Taylor expanded in z around inf 73.5%

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

   if 1.89999999999999997e167 < y < 9.4999999999999995e190

   1. Initial program 100.0%

      \[\frac{x + y}{1 - \frac{y}{z}} \]

   2. Taylor expanded in x around 0 100.0%

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

   3. Taylor expanded in y around 0 85.9%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -11500:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{+39}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{+167} \lor \neg \left(y \leq 9.5 \cdot 10^{+190}\right):\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 7: 56.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.18 \cdot 10^{+41}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 10^{-53}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.02 \cdot 10^{+40}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 7.1 \cdot 10^{+149}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{+190}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.18e+41)
   (- z)
   (if (<= y 1e-53)
     x
     (if (<= y 1.02e+40)
       y
       (if (<= y 7.1e+149) (- z) (if (<= y 9.5e+190) y (- z)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.18e+41) {
		tmp = -z;
	} else if (y <= 1e-53) {
		tmp = x;
	} else if (y <= 1.02e+40) {
		tmp = y;
	} else if (y <= 7.1e+149) {
		tmp = -z;
	} else if (y <= 9.5e+190) {
		tmp = 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 <= (-1.18d+41)) then
        tmp = -z
    else if (y <= 1d-53) then
        tmp = x
    else if (y <= 1.02d+40) then
        tmp = y
    else if (y <= 7.1d+149) then
        tmp = -z
    else if (y <= 9.5d+190) then
        tmp = 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 <= -1.18e+41) {
		tmp = -z;
	} else if (y <= 1e-53) {
		tmp = x;
	} else if (y <= 1.02e+40) {
		tmp = y;
	} else if (y <= 7.1e+149) {
		tmp = -z;
	} else if (y <= 9.5e+190) {
		tmp = y;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1.18e+41:
		tmp = -z
	elif y <= 1e-53:
		tmp = x
	elif y <= 1.02e+40:
		tmp = y
	elif y <= 7.1e+149:
		tmp = -z
	elif y <= 9.5e+190:
		tmp = y
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.18e+41)
		tmp = Float64(-z);
	elseif (y <= 1e-53)
		tmp = x;
	elseif (y <= 1.02e+40)
		tmp = y;
	elseif (y <= 7.1e+149)
		tmp = Float64(-z);
	elseif (y <= 9.5e+190)
		tmp = y;
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -1.18e+41)
		tmp = -z;
	elseif (y <= 1e-53)
		tmp = x;
	elseif (y <= 1.02e+40)
		tmp = y;
	elseif (y <= 7.1e+149)
		tmp = -z;
	elseif (y <= 9.5e+190)
		tmp = y;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.18e+41], (-z), If[LessEqual[y, 1e-53], x, If[LessEqual[y, 1.02e+40], y, If[LessEqual[y, 7.1e+149], (-z), If[LessEqual[y, 9.5e+190], y, (-z)]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.17999999999999998e41 or 1.02e40 < y < 7.1000000000000001e149 or 9.4999999999999995e190 < y

   1. Initial program 70.1%

      \[\frac{x + y}{1 - \frac{y}{z}} \]

   2. Taylor expanded in y around inf 69.8%

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

      1. mul-1-neg 69.8%

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

   4. Simplified 69.8%

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

   if -1.17999999999999998e41 < y < 1.00000000000000003e-53

   1. Initial program 99.9%

      \[\frac{x + y}{1 - \frac{y}{z}} \]

   2. Taylor expanded in y around 0 56.8%

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

   if 1.00000000000000003e-53 < y < 1.02e40 or 7.1000000000000001e149 < y < 9.4999999999999995e190

   1. Initial program 99.9%

      \[\frac{x + y}{1 - \frac{y}{z}} \]

   2. Taylor expanded in x around 0 74.8%

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

   3. Taylor expanded in y around 0 54.9%

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

4. Final simplification 60.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.18 \cdot 10^{+41}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 10^{-53}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.02 \cdot 10^{+40}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 7.1 \cdot 10^{+149}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{+190}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 8: 65.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -45000000000:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 1.45 \cdot 10^{+41}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 7.1 \cdot 10^{+149}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{+190}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -45000000000.0)
   (- z)
   (if (<= y 1.45e+41)
     (+ x y)
     (if (<= y 7.1e+149) (- z) (if (<= y 9.5e+190) y (- z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -45000000000.0) {
		tmp = -z;
	} else if (y <= 1.45e+41) {
		tmp = x + y;
	} else if (y <= 7.1e+149) {
		tmp = -z;
	} else if (y <= 9.5e+190) {
		tmp = 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 <= (-45000000000.0d0)) then
        tmp = -z
    else if (y <= 1.45d+41) then
        tmp = x + y
    else if (y <= 7.1d+149) then
        tmp = -z
    else if (y <= 9.5d+190) then
        tmp = 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 <= -45000000000.0) {
		tmp = -z;
	} else if (y <= 1.45e+41) {
		tmp = x + y;
	} else if (y <= 7.1e+149) {
		tmp = -z;
	} else if (y <= 9.5e+190) {
		tmp = y;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -45000000000.0:
		tmp = -z
	elif y <= 1.45e+41:
		tmp = x + y
	elif y <= 7.1e+149:
		tmp = -z
	elif y <= 9.5e+190:
		tmp = y
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -45000000000.0)
		tmp = Float64(-z);
	elseif (y <= 1.45e+41)
		tmp = Float64(x + y);
	elseif (y <= 7.1e+149)
		tmp = Float64(-z);
	elseif (y <= 9.5e+190)
		tmp = y;
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -45000000000.0)
		tmp = -z;
	elseif (y <= 1.45e+41)
		tmp = x + y;
	elseif (y <= 7.1e+149)
		tmp = -z;
	elseif (y <= 9.5e+190)
		tmp = y;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -45000000000.0], (-z), If[LessEqual[y, 1.45e+41], N[(x + y), $MachinePrecision], If[LessEqual[y, 7.1e+149], (-z), If[LessEqual[y, 9.5e+190], y, (-z)]]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.5e10 or 1.44999999999999994e41 < y < 7.1000000000000001e149 or 9.4999999999999995e190 < y

   1. Initial program 71.9%

      \[\frac{x + y}{1 - \frac{y}{z}} \]

   2. Taylor expanded in y around inf 68.2%

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

      1. mul-1-neg 68.2%

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

   4. Simplified 68.2%

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

   if -4.5e10 < y < 1.44999999999999994e41

   1. Initial program 99.9%

      \[\frac{x + y}{1 - \frac{y}{z}} \]

   2. Taylor expanded in z around inf 73.2%

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

   if 7.1000000000000001e149 < y < 9.4999999999999995e190

   1. Initial program 100.0%

      \[\frac{x + y}{1 - \frac{y}{z}} \]

   2. Taylor expanded in x around 0 90.1%

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

   3. Taylor expanded in y around 0 71.5%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -45000000000:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 1.45 \cdot 10^{+41}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 7.1 \cdot 10^{+149}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{+190}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 9: 39.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{-117}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 7 \cdot 10^{-244}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.25e-117) x (if (<= x 7e-244) y x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.25e-117) {
		tmp = x;
	} else if (x <= 7e-244) {
		tmp = y;
	} else {
		tmp = x;
	}
	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 (x <= (-1.25d-117)) then
        tmp = x
    else if (x <= 7d-244) then
        tmp = y
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.25e-117) {
		tmp = x;
	} else if (x <= 7e-244) {
		tmp = y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.25e-117:
		tmp = x
	elif x <= 7e-244:
		tmp = y
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.25e-117)
		tmp = x;
	elseif (x <= 7e-244)
		tmp = y;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.25e-117)
		tmp = x;
	elseif (x <= 7e-244)
		tmp = y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.25e-117], x, If[LessEqual[x, 7e-244], y, x]]

Derivation

1. Split input into 2 regimes

2. if x < -1.25e-117 or 6.99999999999999984e-244 < x

   1. Initial program 89.3%

      \[\frac{x + y}{1 - \frac{y}{z}} \]

   2. Taylor expanded in y around 0 41.3%

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

   if -1.25e-117 < x < 6.99999999999999984e-244

   1. Initial program 96.3%

      \[\frac{x + y}{1 - \frac{y}{z}} \]

   2. Taylor expanded in x around 0 86.6%

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

   3. Taylor expanded in y around 0 52.4%

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

4. Final simplification 43.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{-117}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 7 \cdot 10^{-244}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 10: 34.3% 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 90.7%

   \[\frac{x + y}{1 - \frac{y}{z}} \]

2. Taylor expanded in y around 0 35.5%

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

3. Final simplification 35.5%

   \[\leadsto x \]

Developer target: 93.8% accurate, 0.7× 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 2023208 
(FPCore (x y z)
  :name "Graphics.Rendering.Chart.Backend.Diagrams:calcFontMetrics from Chart-diagrams-1.5.1, A"
  :precision binary64

  :herbie-target
  (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))))