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

Percentage Accurate: 88.2% → 99.7%

Time: 5.4s

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: 88.2% 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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{-276} \lor \neg \left(t_0 \leq 0\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-z}{\frac{y}{x + y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (+ x y) (- 1.0 (/ y z)))))
   (if (or (<= t_0 -1e-276) (not (<= t_0 0.0))) t_0 (/ (- z) (/ y (+ x y))))))

C

double code(double x, double y, double z) {
	double t_0 = (x + y) / (1.0 - (y / z));
	double tmp;
	if ((t_0 <= -1e-276) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = -z / (y / (x + y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x + y) / (1.0d0 - (y / z))
    if ((t_0 <= (-1d-276)) .or. (.not. (t_0 <= 0.0d0))) then
        tmp = t_0
    else
        tmp = -z / (y / (x + y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (x + y) / (1.0 - (y / z));
	double tmp;
	if ((t_0 <= -1e-276) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = -z / (y / (x + y));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x + y) / (1.0 - (y / z))
	tmp = 0
	if (t_0 <= -1e-276) or not (t_0 <= 0.0):
		tmp = t_0
	else:
		tmp = -z / (y / (x + y))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x + y) / Float64(1.0 - Float64(y / z)))
	tmp = 0.0
	if ((t_0 <= -1e-276) || !(t_0 <= 0.0))
		tmp = t_0;
	else
		tmp = Float64(Float64(-z) / Float64(y / Float64(x + y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x + y) / (1.0 - (y / z));
	tmp = 0.0;
	if ((t_0 <= -1e-276) || ~((t_0 <= 0.0)))
		tmp = t_0;
	else
		tmp = -z / (y / (x + y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x + y), $MachinePrecision] / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -1e-276], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], t$95$0, N[((-z) / N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z))) < -1e-276 or -0.0 < (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z)))

   1. Initial program 99.9%

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

   if -1e-276 < (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z))) < -0.0

   1. Initial program 5.1%

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

   2. Taylor expanded in z around 0 99.6%

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

      1. mul-1-neg 99.6%

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

      2. associate-/l* 99.9%

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

      3. distribute-neg-frac 99.9%

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

      4. +-commutative 99.9%

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

   4. Simplified 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \leq -1 \cdot 10^{-276} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \leq 0\right):\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-z}{\frac{y}{x + y}}\\ \end{array} \]

Alternative 2: 69.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{y}{z}\\ \mathbf{if}\;y \leq -5.7 \cdot 10^{+227}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq -5.7 \cdot 10^{+22}:\\ \;\;\;\;\frac{y}{t_0}\\ \mathbf{elif}\;y \leq -2 \cdot 10^{-59}:\\ \;\;\;\;\frac{x}{t_0}\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{+17}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 4 \cdot 10^{+74}:\\ \;\;\;\;\left(x + y\right) \cdot \left(-\frac{z}{y}\right)\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{+118}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ y z))))
   (if (<= y -5.7e+227)
     (- z)
     (if (<= y -5.7e+22)
       (/ y t_0)
       (if (<= y -2e-59)
         (/ x t_0)
         (if (<= y 6.2e+17)
           (+ x y)
           (if (<= y 4e+74)
             (* (+ x y) (- (/ z y)))
             (if (<= y 4.2e+118) (+ x y) (- z)))))))))

C

double code(double x, double y, double z) {
	double t_0 = 1.0 - (y / z);
	double tmp;
	if (y <= -5.7e+227) {
		tmp = -z;
	} else if (y <= -5.7e+22) {
		tmp = y / t_0;
	} else if (y <= -2e-59) {
		tmp = x / t_0;
	} else if (y <= 6.2e+17) {
		tmp = x + y;
	} else if (y <= 4e+74) {
		tmp = (x + y) * -(z / y);
	} else if (y <= 4.2e+118) {
		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) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - (y / z)
    if (y <= (-5.7d+227)) then
        tmp = -z
    else if (y <= (-5.7d+22)) then
        tmp = y / t_0
    else if (y <= (-2d-59)) then
        tmp = x / t_0
    else if (y <= 6.2d+17) then
        tmp = x + y
    else if (y <= 4d+74) then
        tmp = (x + y) * -(z / y)
    else if (y <= 4.2d+118) 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 t_0 = 1.0 - (y / z);
	double tmp;
	if (y <= -5.7e+227) {
		tmp = -z;
	} else if (y <= -5.7e+22) {
		tmp = y / t_0;
	} else if (y <= -2e-59) {
		tmp = x / t_0;
	} else if (y <= 6.2e+17) {
		tmp = x + y;
	} else if (y <= 4e+74) {
		tmp = (x + y) * -(z / y);
	} else if (y <= 4.2e+118) {
		tmp = x + y;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 1.0 - (y / z)
	tmp = 0
	if y <= -5.7e+227:
		tmp = -z
	elif y <= -5.7e+22:
		tmp = y / t_0
	elif y <= -2e-59:
		tmp = x / t_0
	elif y <= 6.2e+17:
		tmp = x + y
	elif y <= 4e+74:
		tmp = (x + y) * -(z / y)
	elif y <= 4.2e+118:
		tmp = x + y
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(1.0 - Float64(y / z))
	tmp = 0.0
	if (y <= -5.7e+227)
		tmp = Float64(-z);
	elseif (y <= -5.7e+22)
		tmp = Float64(y / t_0);
	elseif (y <= -2e-59)
		tmp = Float64(x / t_0);
	elseif (y <= 6.2e+17)
		tmp = Float64(x + y);
	elseif (y <= 4e+74)
		tmp = Float64(Float64(x + y) * Float64(-Float64(z / y)));
	elseif (y <= 4.2e+118)
		tmp = Float64(x + y);
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 1.0 - (y / z);
	tmp = 0.0;
	if (y <= -5.7e+227)
		tmp = -z;
	elseif (y <= -5.7e+22)
		tmp = y / t_0;
	elseif (y <= -2e-59)
		tmp = x / t_0;
	elseif (y <= 6.2e+17)
		tmp = x + y;
	elseif (y <= 4e+74)
		tmp = (x + y) * -(z / y);
	elseif (y <= 4.2e+118)
		tmp = x + y;
	else
		tmp = -z;
	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, -5.7e+227], (-z), If[LessEqual[y, -5.7e+22], N[(y / t$95$0), $MachinePrecision], If[LessEqual[y, -2e-59], N[(x / t$95$0), $MachinePrecision], If[LessEqual[y, 6.2e+17], N[(x + y), $MachinePrecision], If[LessEqual[y, 4e+74], N[(N[(x + y), $MachinePrecision] * (-N[(z / y), $MachinePrecision])), $MachinePrecision], If[LessEqual[y, 4.2e+118], N[(x + y), $MachinePrecision], (-z)]]]]]]]

Derivation

1. Split input into 5 regimes

2. if y < -5.70000000000000011e227 or 4.2e118 < y

   1. Initial program 62.6%

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

   2. Taylor expanded in y around inf 81.3%

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

      1. mul-1-neg 81.3%

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

   4. Simplified 81.3%

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

   if -5.70000000000000011e227 < y < -5.69999999999999979e22

   1. Initial program 79.1%

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

   2. Taylor expanded in x around 0 58.3%

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

   if -5.69999999999999979e22 < y < -2.0000000000000001e-59

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 63.8%

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

   if -2.0000000000000001e-59 < y < 6.2e17 or 3.99999999999999981e74 < y < 4.2e118

   1. Initial program 99.9%

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

   2. Taylor expanded in z around inf 81.3%

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

      1. +-commutative 81.3%

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

   4. Simplified 81.3%

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

   if 6.2e17 < y < 3.99999999999999981e74

   1. Initial program 86.0%

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

   2. Taylor expanded in z around 0 72.1%

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

      1. mul-1-neg 72.1%

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

      2. associate-/l* 72.1%

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

      3. associate-/r/ 60.2%

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

      4. distribute-rgt-neg-in 60.2%

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

      5. distribute-neg-in 60.2%

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

      6. unsub-neg 60.2%

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

   4. Simplified 60.2%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.7 \cdot 10^{+227}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq -5.7 \cdot 10^{+22}:\\ \;\;\;\;\frac{y}{1 - \frac{y}{z}}\\ \mathbf{elif}\;y \leq -2 \cdot 10^{-59}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{+17}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 4 \cdot 10^{+74}:\\ \;\;\;\;\left(x + y\right) \cdot \left(-\frac{z}{y}\right)\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{+118}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 3: 68.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{y}{z}\\ \mathbf{if}\;y \leq -5.5 \cdot 10^{+227}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq -1.65 \cdot 10^{+22}:\\ \;\;\;\;\frac{y}{t_0}\\ \mathbf{elif}\;y \leq -4.05 \cdot 10^{-48}:\\ \;\;\;\;\frac{x}{t_0}\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{+116}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ y z))))
   (if (<= y -5.5e+227)
     (- z)
     (if (<= y -1.65e+22)
       (/ y t_0)
       (if (<= y -4.05e-48) (/ x t_0) (if (<= y 2.5e+116) (+ x y) (- z)))))))

C

double code(double x, double y, double z) {
	double t_0 = 1.0 - (y / z);
	double tmp;
	if (y <= -5.5e+227) {
		tmp = -z;
	} else if (y <= -1.65e+22) {
		tmp = y / t_0;
	} else if (y <= -4.05e-48) {
		tmp = x / t_0;
	} else if (y <= 2.5e+116) {
		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) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - (y / z)
    if (y <= (-5.5d+227)) then
        tmp = -z
    else if (y <= (-1.65d+22)) then
        tmp = y / t_0
    else if (y <= (-4.05d-48)) then
        tmp = x / t_0
    else if (y <= 2.5d+116) 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 t_0 = 1.0 - (y / z);
	double tmp;
	if (y <= -5.5e+227) {
		tmp = -z;
	} else if (y <= -1.65e+22) {
		tmp = y / t_0;
	} else if (y <= -4.05e-48) {
		tmp = x / t_0;
	} else if (y <= 2.5e+116) {
		tmp = x + y;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 1.0 - (y / z)
	tmp = 0
	if y <= -5.5e+227:
		tmp = -z
	elif y <= -1.65e+22:
		tmp = y / t_0
	elif y <= -4.05e-48:
		tmp = x / t_0
	elif y <= 2.5e+116:
		tmp = x + y
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(1.0 - Float64(y / z))
	tmp = 0.0
	if (y <= -5.5e+227)
		tmp = Float64(-z);
	elseif (y <= -1.65e+22)
		tmp = Float64(y / t_0);
	elseif (y <= -4.05e-48)
		tmp = Float64(x / t_0);
	elseif (y <= 2.5e+116)
		tmp = Float64(x + y);
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 1.0 - (y / z);
	tmp = 0.0;
	if (y <= -5.5e+227)
		tmp = -z;
	elseif (y <= -1.65e+22)
		tmp = y / t_0;
	elseif (y <= -4.05e-48)
		tmp = x / t_0;
	elseif (y <= 2.5e+116)
		tmp = x + y;
	else
		tmp = -z;
	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, -5.5e+227], (-z), If[LessEqual[y, -1.65e+22], N[(y / t$95$0), $MachinePrecision], If[LessEqual[y, -4.05e-48], N[(x / t$95$0), $MachinePrecision], If[LessEqual[y, 2.5e+116], N[(x + y), $MachinePrecision], (-z)]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -5.5000000000000001e227 or 2.50000000000000013e116 < y

   1. Initial program 62.6%

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

   2. Taylor expanded in y around inf 81.3%

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

      1. mul-1-neg 81.3%

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

   4. Simplified 81.3%

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

   if -5.5000000000000001e227 < y < -1.6499999999999999e22

   1. Initial program 79.1%

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

   2. Taylor expanded in x around 0 58.3%

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

   if -1.6499999999999999e22 < y < -4.0499999999999997e-48

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 63.8%

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

   if -4.0499999999999997e-48 < y < 2.50000000000000013e116

   1. Initial program 98.5%

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

   2. Taylor expanded in z around inf 75.7%

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

      1. +-commutative 75.7%

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

   4. Simplified 75.7%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.5 \cdot 10^{+227}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq -1.65 \cdot 10^{+22}:\\ \;\;\;\;\frac{y}{1 - \frac{y}{z}}\\ \mathbf{elif}\;y \leq -4.05 \cdot 10^{-48}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{+116}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 4: 74.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.85 \cdot 10^{-11} \lor \neg \left(y \leq 1.15 \cdot 10^{+18}\right):\\ \;\;\;\;\frac{-z}{\frac{y}{x + y}}\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -2.85e-11) (not (<= y 1.15e+18)))
   (/ (- z) (/ y (+ x y)))
   (+ x y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -2.85e-11) || !(y <= 1.15e+18)) {
		tmp = -z / (y / (x + y));
	} else {
		tmp = x + y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((y <= (-2.85d-11)) .or. (.not. (y <= 1.15d+18))) then
        tmp = -z / (y / (x + y))
    else
        tmp = x + y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -2.85e-11) || !(y <= 1.15e+18)) {
		tmp = -z / (y / (x + y));
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -2.85e-11) or not (y <= 1.15e+18):
		tmp = -z / (y / (x + y))
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -2.85e-11) || !(y <= 1.15e+18))
		tmp = Float64(Float64(-z) / Float64(y / Float64(x + y)));
	else
		tmp = Float64(x + y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -2.85e-11) || ~((y <= 1.15e+18)))
		tmp = -z / (y / (x + y));
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -2.85e-11], N[Not[LessEqual[y, 1.15e+18]], $MachinePrecision]], N[((-z) / N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.8499999999999999e-11 or 1.15e18 < y

   1. Initial program 74.2%

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

   2. Taylor expanded in z around 0 61.8%

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

      1. mul-1-neg 61.8%

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

      2. associate-/l* 76.0%

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

      3. distribute-neg-frac 76.0%

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

      4. +-commutative 76.0%

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

   4. Simplified 76.0%

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

   if -2.8499999999999999e-11 < y < 1.15e18

   1. Initial program 99.9%

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

   2. Taylor expanded in z around inf 80.3%

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

      1. +-commutative 80.3%

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

   4. Simplified 80.3%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.85 \cdot 10^{-11} \lor \neg \left(y \leq 1.15 \cdot 10^{+18}\right):\\ \;\;\;\;\frac{-z}{\frac{y}{x + y}}\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 5: 56.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.35 \cdot 10^{-12}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 1.46 \cdot 10^{-83}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{+114}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.35e-12)
   (- z)
   (if (<= y 1.46e-83) x (if (<= y 1.9e+114) y (- z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.35e-12) {
		tmp = -z;
	} else if (y <= 1.46e-83) {
		tmp = x;
	} else if (y <= 1.9e+114) {
		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 <= (-1.35d-12)) then
        tmp = -z
    else if (y <= 1.46d-83) then
        tmp = x
    else if (y <= 1.9d+114) 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 <= -1.35e-12) {
		tmp = -z;
	} else if (y <= 1.46e-83) {
		tmp = x;
	} else if (y <= 1.9e+114) {
		tmp = y;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1.35e-12:
		tmp = -z
	elif y <= 1.46e-83:
		tmp = x
	elif y <= 1.9e+114:
		tmp = y
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.35e-12)
		tmp = Float64(-z);
	elseif (y <= 1.46e-83)
		tmp = x;
	elseif (y <= 1.9e+114)
		tmp = y;
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -1.35e-12)
		tmp = -z;
	elseif (y <= 1.46e-83)
		tmp = x;
	elseif (y <= 1.9e+114)
		tmp = y;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.35e-12], (-z), If[LessEqual[y, 1.46e-83], x, If[LessEqual[y, 1.9e+114], y, (-z)]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.3499999999999999e-12 or 1.9e114 < y

   1. Initial program 70.4%

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

   2. Taylor expanded in y around inf 64.7%

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

      1. mul-1-neg 64.7%

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

   4. Simplified 64.7%

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

   if -1.3499999999999999e-12 < y < 1.4600000000000001e-83

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 68.0%

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

   if 1.4600000000000001e-83 < y < 1.9e114

   1. Initial program 95.1%

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

   2. Taylor expanded in x around 0 53.0%

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

   3. Taylor expanded in y around 0 34.5%

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

4. Final simplification 61.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.35 \cdot 10^{-12}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 1.46 \cdot 10^{-83}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{+114}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 6: 65.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.5 \cdot 10^{+200} \lor \neg \left(y \leq 8 \cdot 10^{+118}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -7.5e+200) (not (<= y 8e+118))) (- z) (+ x y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -7.5e+200) || !(y <= 8e+118)) {
		tmp = -z;
	} else {
		tmp = x + y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((y <= (-7.5d+200)) .or. (.not. (y <= 8d+118))) then
        tmp = -z
    else
        tmp = x + y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -7.5e+200) || !(y <= 8e+118)) {
		tmp = -z;
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -7.5e+200) or not (y <= 8e+118):
		tmp = -z
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -7.5e+200) || !(y <= 8e+118))
		tmp = Float64(-z);
	else
		tmp = Float64(x + y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -7.5e+200) || ~((y <= 8e+118)))
		tmp = -z;
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -7.5e+200], N[Not[LessEqual[y, 8e+118]], $MachinePrecision]], (-z), N[(x + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -7.50000000000000062e200 or 7.99999999999999973e118 < y

   1. Initial program 64.8%

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

   2. Taylor expanded in y around inf 82.3%

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

      1. mul-1-neg 82.3%

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

   4. Simplified 82.3%

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

   if -7.50000000000000062e200 < y < 7.99999999999999973e118

   1. Initial program 95.3%

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

   2. Taylor expanded in z around inf 68.6%

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

      1. +-commutative 68.6%

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

   4. Simplified 68.6%

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

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.5 \cdot 10^{+200} \lor \neg \left(y \leq 8 \cdot 10^{+118}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 7: 39.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{-193}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-164}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -2.9e-193) x (if (<= x 4e-164) y x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -2.9e-193) {
		tmp = x;
	} else if (x <= 4e-164) {
		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 <= (-2.9d-193)) then
        tmp = x
    else if (x <= 4d-164) 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 <= -2.9e-193) {
		tmp = x;
	} else if (x <= 4e-164) {
		tmp = y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -2.9e-193:
		tmp = x
	elif x <= 4e-164:
		tmp = y
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -2.9e-193)
		tmp = x;
	elseif (x <= 4e-164)
		tmp = y;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -2.9e-193)
		tmp = x;
	elseif (x <= 4e-164)
		tmp = y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -2.9e-193], x, If[LessEqual[x, 4e-164], y, x]]

Derivation

1. Split input into 2 regimes

2. if x < -2.90000000000000007e-193 or 3.99999999999999985e-164 < x

   1. Initial program 87.4%

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

   2. Taylor expanded in y around 0 43.0%

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

   if -2.90000000000000007e-193 < x < 3.99999999999999985e-164

   1. Initial program 85.3%

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

   2. Taylor expanded in x around 0 75.6%

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

   3. Taylor expanded in y around 0 49.4%

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

4. Final simplification 44.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{-193}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-164}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

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 86.9%

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

2. Taylor expanded in y around 0 35.5%

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

3. Final simplification 35.5%

   \[\leadsto x \]

Developer target: 93.6% 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 2023298 
(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))))