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

Percentage Accurate: 88.1% → 99.7%

Time: 5.9s

Alternatives: 8

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: 88.1% 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: 99.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{-271} \lor \neg \left(t_0 \leq 0\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) - \frac{z}{\frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (+ x y) (- 1.0 (/ y z)))))
   (if (or (<= t_0 -2e-271) (not (<= t_0 0.0))) 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-271) || !(t_0 <= 0.0)) {
		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-271)) .or. (.not. (t_0 <= 0.0d0))) 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-271) || !(t_0 <= 0.0)) {
		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-271) or not (t_0 <= 0.0):
		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-271) || !(t_0 <= 0.0))
		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-271) || ~((t_0 <= 0.0)))
		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-271], N[Not[LessEqual[t$95$0, 0.0]], $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))) < -1.99999999999999993e-271 or -0.0 < (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z)))

   1. Initial program 99.8%

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

   if -1.99999999999999993e-271 < (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z))) < -0.0

   1. Initial program 5.7%

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

   2. Taylor expanded in z around 0 99.8%

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

      1. mul-1-neg 99.8%

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

      2. +-commutative 99.8%

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

      3. *-commutative 99.8%

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

      4. +-commutative 99.8%

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

   4. Simplified 99.8%

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

   5. Taylor expanded in y around 0 99.8%

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

      1. +-commutative 99.8%

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

      2. associate-/l* 99.9%

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

   7. Simplified 99.9%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \leq -2 \cdot 10^{-271} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \leq 0\right):\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) - \frac{z}{\frac{y}{x}}\\ \end{array} \]

Alternative 2: 72.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-z\right) - \frac{z}{\frac{y}{x}}\\ \mathbf{if}\;y \leq -2.2 \cdot 10^{+227}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -1.95 \cdot 10^{+191}:\\ \;\;\;\;\frac{y}{1 - \frac{y}{z}}\\ \mathbf{elif}\;y \leq -1.05 \cdot 10^{-61} \lor \neg \left(y \leq 4500000000000\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (- z) (/ z (/ y x)))))
   (if (<= y -2.2e+227)
     t_0
     (if (<= y -1.95e+191)
       (/ y (- 1.0 (/ y z)))
       (if (or (<= y -1.05e-61) (not (<= y 4500000000000.0))) t_0 (+ x y))))))

C

double code(double x, double y, double z) {
	double t_0 = -z - (z / (y / x));
	double tmp;
	if (y <= -2.2e+227) {
		tmp = t_0;
	} else if (y <= -1.95e+191) {
		tmp = y / (1.0 - (y / z));
	} else if ((y <= -1.05e-61) || !(y <= 4500000000000.0)) {
		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 - (z / (y / x))
    if (y <= (-2.2d+227)) then
        tmp = t_0
    else if (y <= (-1.95d+191)) then
        tmp = y / (1.0d0 - (y / z))
    else if ((y <= (-1.05d-61)) .or. (.not. (y <= 4500000000000.0d0))) 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 - (z / (y / x));
	double tmp;
	if (y <= -2.2e+227) {
		tmp = t_0;
	} else if (y <= -1.95e+191) {
		tmp = y / (1.0 - (y / z));
	} else if ((y <= -1.05e-61) || !(y <= 4500000000000.0)) {
		tmp = t_0;
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = -z - (z / (y / x))
	tmp = 0
	if y <= -2.2e+227:
		tmp = t_0
	elif y <= -1.95e+191:
		tmp = y / (1.0 - (y / z))
	elif (y <= -1.05e-61) or not (y <= 4500000000000.0):
		tmp = t_0
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(-z) - Float64(z / Float64(y / x)))
	tmp = 0.0
	if (y <= -2.2e+227)
		tmp = t_0;
	elseif (y <= -1.95e+191)
		tmp = Float64(y / Float64(1.0 - Float64(y / z)));
	elseif ((y <= -1.05e-61) || !(y <= 4500000000000.0))
		tmp = t_0;
	else
		tmp = Float64(x + y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = -z - (z / (y / x));
	tmp = 0.0;
	if (y <= -2.2e+227)
		tmp = t_0;
	elseif (y <= -1.95e+191)
		tmp = y / (1.0 - (y / z));
	elseif ((y <= -1.05e-61) || ~((y <= 4500000000000.0)))
		tmp = t_0;
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[((-z) - N[(z / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.2e+227], t$95$0, If[LessEqual[y, -1.95e+191], N[(y / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[y, -1.05e-61], N[Not[LessEqual[y, 4500000000000.0]], $MachinePrecision]], t$95$0, N[(x + y), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.2000000000000002e227 or -1.95e191 < y < -1.05e-61 or 4.5e12 < y

   1. Initial program 81.1%

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

   2. Taylor expanded in z around 0 65.2%

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

      1. mul-1-neg 65.2%

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

      2. +-commutative 65.2%

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

      3. *-commutative 65.2%

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

      4. +-commutative 65.2%

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

   4. Simplified 65.2%

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

   5. Taylor expanded in y around 0 74.9%

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

      1. +-commutative 74.9%

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

      2. associate-/l* 79.4%

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

   7. Simplified 79.4%

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

   if -2.2000000000000002e227 < y < -1.95e191

   1. Initial program 91.4%

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

   2. Taylor expanded in x around 0 91.4%

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

   if -1.05e-61 < y < 4.5e12

   1. Initial program 99.9%

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

   2. Taylor expanded in z around inf 82.7%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{+227}:\\ \;\;\;\;\left(-z\right) - \frac{z}{\frac{y}{x}}\\ \mathbf{elif}\;y \leq -1.95 \cdot 10^{+191}:\\ \;\;\;\;\frac{y}{1 - \frac{y}{z}}\\ \mathbf{elif}\;y \leq -1.05 \cdot 10^{-61} \lor \neg \left(y \leq 4500000000000\right):\\ \;\;\;\;\left(-z\right) - \frac{z}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 3: 72.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-z\right) - \frac{z}{\frac{y}{x}}\\ \mathbf{if}\;y \leq -1.8 \cdot 10^{+227}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -1.02 \cdot 10^{+180}:\\ \;\;\;\;\frac{y}{1 - \frac{y}{z}}\\ \mathbf{elif}\;y \leq -1.05 \cdot 10^{-61}:\\ \;\;\;\;-\frac{z \cdot \left(x + y\right)}{y}\\ \mathbf{elif}\;y \leq 8200000000:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (- z) (/ z (/ y x)))))
   (if (<= y -1.8e+227)
     t_0
     (if (<= y -1.02e+180)
       (/ y (- 1.0 (/ y z)))
       (if (<= y -1.05e-61)
         (- (/ (* z (+ x y)) y))
         (if (<= y 8200000000.0) (+ x y) t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = -z - (z / (y / x));
	double tmp;
	if (y <= -1.8e+227) {
		tmp = t_0;
	} else if (y <= -1.02e+180) {
		tmp = y / (1.0 - (y / z));
	} else if (y <= -1.05e-61) {
		tmp = -((z * (x + y)) / y);
	} else if (y <= 8200000000.0) {
		tmp = x + y;
	} 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 = -z - (z / (y / x))
    if (y <= (-1.8d+227)) then
        tmp = t_0
    else if (y <= (-1.02d+180)) then
        tmp = y / (1.0d0 - (y / z))
    else if (y <= (-1.05d-61)) then
        tmp = -((z * (x + y)) / y)
    else if (y <= 8200000000.0d0) then
        tmp = x + y
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = -z - (z / (y / x));
	double tmp;
	if (y <= -1.8e+227) {
		tmp = t_0;
	} else if (y <= -1.02e+180) {
		tmp = y / (1.0 - (y / z));
	} else if (y <= -1.05e-61) {
		tmp = -((z * (x + y)) / y);
	} else if (y <= 8200000000.0) {
		tmp = x + y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = -z - (z / (y / x))
	tmp = 0
	if y <= -1.8e+227:
		tmp = t_0
	elif y <= -1.02e+180:
		tmp = y / (1.0 - (y / z))
	elif y <= -1.05e-61:
		tmp = -((z * (x + y)) / y)
	elif y <= 8200000000.0:
		tmp = x + y
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(-z) - Float64(z / Float64(y / x)))
	tmp = 0.0
	if (y <= -1.8e+227)
		tmp = t_0;
	elseif (y <= -1.02e+180)
		tmp = Float64(y / Float64(1.0 - Float64(y / z)));
	elseif (y <= -1.05e-61)
		tmp = Float64(-Float64(Float64(z * Float64(x + y)) / y));
	elseif (y <= 8200000000.0)
		tmp = Float64(x + y);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = -z - (z / (y / x));
	tmp = 0.0;
	if (y <= -1.8e+227)
		tmp = t_0;
	elseif (y <= -1.02e+180)
		tmp = y / (1.0 - (y / z));
	elseif (y <= -1.05e-61)
		tmp = -((z * (x + y)) / y);
	elseif (y <= 8200000000.0)
		tmp = x + y;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[((-z) - N[(z / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.8e+227], t$95$0, If[LessEqual[y, -1.02e+180], N[(y / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -1.05e-61], (-N[(N[(z * N[(x + y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), If[LessEqual[y, 8200000000.0], N[(x + y), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -1.79999999999999996e227 or 8.2e9 < y

   1. Initial program 75.8%

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

   2. Taylor expanded in z around 0 62.1%

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

      1. mul-1-neg 62.1%

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

      2. +-commutative 62.1%

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

      3. *-commutative 62.1%

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

      4. +-commutative 62.1%

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

   4. Simplified 62.1%

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

   5. Taylor expanded in y around 0 77.9%

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

      1. +-commutative 77.9%

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

      2. associate-/l* 86.2%

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

   7. Simplified 86.2%

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

   if -1.79999999999999996e227 < y < -1.02e180

   1. Initial program 92.1%

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

   2. Taylor expanded in x around 0 92.1%

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

   if -1.02e180 < y < -1.05e-61

   1. Initial program 87.7%

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

   2. Taylor expanded in z around 0 70.4%

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

      1. mul-1-neg 70.4%

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

      2. +-commutative 70.4%

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

      3. *-commutative 70.4%

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

      4. +-commutative 70.4%

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

   4. Simplified 70.4%

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

   if -1.05e-61 < y < 8.2e9

   1. Initial program 99.9%

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

   2. Taylor expanded in z around inf 82.7%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.8 \cdot 10^{+227}:\\ \;\;\;\;\left(-z\right) - \frac{z}{\frac{y}{x}}\\ \mathbf{elif}\;y \leq -1.02 \cdot 10^{+180}:\\ \;\;\;\;\frac{y}{1 - \frac{y}{z}}\\ \mathbf{elif}\;y \leq -1.05 \cdot 10^{-61}:\\ \;\;\;\;-\frac{z \cdot \left(x + y\right)}{y}\\ \mathbf{elif}\;y \leq 8200000000:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) - \frac{z}{\frac{y}{x}}\\ \end{array} \]

Alternative 4: 67.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.15 \cdot 10^{+80}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq -3.4 \cdot 10^{-107}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{elif}\;y \leq 7 \cdot 10^{+19}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.15e+80)
   (- z)
   (if (<= y -3.4e-107)
     (/ x (- 1.0 (/ y z)))
     (if (<= y 7e+19) (+ x y) (- z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.15e+80) {
		tmp = -z;
	} else if (y <= -3.4e-107) {
		tmp = x / (1.0 - (y / z));
	} else if (y <= 7e+19) {
		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 <= (-1.15d+80)) then
        tmp = -z
    else if (y <= (-3.4d-107)) then
        tmp = x / (1.0d0 - (y / z))
    else if (y <= 7d+19) 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 <= -1.15e+80) {
		tmp = -z;
	} else if (y <= -3.4e-107) {
		tmp = x / (1.0 - (y / z));
	} else if (y <= 7e+19) {
		tmp = x + y;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1.15e+80:
		tmp = -z
	elif y <= -3.4e-107:
		tmp = x / (1.0 - (y / z))
	elif y <= 7e+19:
		tmp = x + y
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.15e+80)
		tmp = Float64(-z);
	elseif (y <= -3.4e-107)
		tmp = Float64(x / Float64(1.0 - Float64(y / z)));
	elseif (y <= 7e+19)
		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 <= -1.15e+80)
		tmp = -z;
	elseif (y <= -3.4e-107)
		tmp = x / (1.0 - (y / z));
	elseif (y <= 7e+19)
		tmp = x + y;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.15e+80], (-z), If[LessEqual[y, -3.4e-107], N[(x / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 7e+19], N[(x + y), $MachinePrecision], (-z)]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.15000000000000002e80 or 7e19 < y

   1. Initial program 76.0%

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

   2. Taylor expanded in y around inf 70.1%

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

      1. mul-1-neg 70.1%

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

   4. Simplified 70.1%

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

   if -1.15000000000000002e80 < y < -3.39999999999999994e-107

   1. Initial program 99.8%

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

   2. Taylor expanded in x around inf 67.4%

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

   if -3.39999999999999994e-107 < y < 7e19

   1. Initial program 99.9%

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

   2. Taylor expanded in z around inf 85.7%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.15 \cdot 10^{+80}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq -3.4 \cdot 10^{-107}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{elif}\;y \leq 7 \cdot 10^{+19}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 5: 67.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{+77}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 8 \cdot 10^{+19}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -4e+77) (- z) (if (<= y 8e+19) (+ x y) (- z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -4e+77) {
		tmp = -z;
	} else if (y <= 8e+19) {
		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 <= (-4d+77)) then
        tmp = -z
    else if (y <= 8d+19) 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 <= -4e+77) {
		tmp = -z;
	} else if (y <= 8e+19) {
		tmp = x + y;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -4e+77:
		tmp = -z
	elif y <= 8e+19:
		tmp = x + y
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -4e+77)
		tmp = Float64(-z);
	elseif (y <= 8e+19)
		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 <= -4e+77)
		tmp = -z;
	elseif (y <= 8e+19)
		tmp = x + y;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -4e+77], (-z), If[LessEqual[y, 8e+19], N[(x + y), $MachinePrecision], (-z)]]

Derivation

1. Split input into 2 regimes

2. if y < -3.99999999999999993e77 or 8e19 < y

   1. Initial program 76.2%

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

   2. Taylor expanded in y around inf 69.4%

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

      1. mul-1-neg 69.4%

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

   4. Simplified 69.4%

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

   if -3.99999999999999993e77 < y < 8e19

   1. Initial program 99.9%

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

   2. Taylor expanded in z around inf 72.9%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{+77}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 8 \cdot 10^{+19}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 6: 58.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.5 \cdot 10^{-34}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 58000000000000:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -4.5e-34) (- z) (if (<= y 58000000000000.0) x (- z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -4.5e-34) {
		tmp = -z;
	} else if (y <= 58000000000000.0) {
		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 <= (-4.5d-34)) then
        tmp = -z
    else if (y <= 58000000000000.0d0) 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 <= -4.5e-34) {
		tmp = -z;
	} else if (y <= 58000000000000.0) {
		tmp = x;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -4.5e-34:
		tmp = -z
	elif y <= 58000000000000.0:
		tmp = x
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -4.5e-34)
		tmp = Float64(-z);
	elseif (y <= 58000000000000.0)
		tmp = x;
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -4.5e-34)
		tmp = -z;
	elseif (y <= 58000000000000.0)
		tmp = x;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -4.5e-34], (-z), If[LessEqual[y, 58000000000000.0], x, (-z)]]

Derivation

1. Split input into 2 regimes

2. if y < -4.50000000000000042e-34 or 5.8e13 < y

   1. Initial program 80.6%

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

   2. Taylor expanded in y around inf 61.5%

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

      1. mul-1-neg 61.5%

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

   4. Simplified 61.5%

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

   if -4.50000000000000042e-34 < y < 5.8e13

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 62.0%

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

4. Final simplification 61.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.5 \cdot 10^{-34}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 58000000000000:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 7: 40.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3 \cdot 10^{-179}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{-124}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -3e-179) x (if (<= x 3.2e-124) y x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -3e-179) {
		tmp = x;
	} else if (x <= 3.2e-124) {
		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 <= (-3d-179)) then
        tmp = x
    else if (x <= 3.2d-124) 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 <= -3e-179) {
		tmp = x;
	} else if (x <= 3.2e-124) {
		tmp = y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -3e-179:
		tmp = x
	elif x <= 3.2e-124:
		tmp = y
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -3e-179)
		tmp = x;
	elseif (x <= 3.2e-124)
		tmp = y;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -3e-179)
		tmp = x;
	elseif (x <= 3.2e-124)
		tmp = y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -3e-179], x, If[LessEqual[x, 3.2e-124], y, x]]

Derivation

1. Split input into 2 regimes

2. if x < -3.00000000000000006e-179 or 3.20000000000000004e-124 < x

   1. Initial program 91.1%

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

   2. Taylor expanded in y around 0 43.6%

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

   if -3.00000000000000006e-179 < x < 3.20000000000000004e-124

   1. Initial program 88.2%

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

   2. Taylor expanded in x around 0 72.6%

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

   3. Taylor expanded in y around 0 37.2%

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

4. Final simplification 41.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3 \cdot 10^{-179}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{-124}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 8: 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 90.3%

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

2. Taylor expanded in y around 0 35.7%

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

3. Final simplification 35.7%

   \[\leadsto x \]

Developer target: 93.5% 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 2023222 
(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))))