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

Percentage Accurate: 88.4% → 99.7%

Time: 6.5s

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: 88.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{-266} \lor \neg \left(t_0 \leq 0\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) - \frac{z}{\frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (+ x y) (- 1.0 (/ y z)))))
   (if (or (<= t_0 -1e-266) (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 <= -1e-266) || !(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 <= (-1d-266)) .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 <= -1e-266) || !(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 <= -1e-266) 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 <= -1e-266) || !(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 <= -1e-266) || ~((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, -1e-266], 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))) < -9.9999999999999998e-267 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 -9.9999999999999998e-267 < (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z))) < -0.0

   1. Initial program 9.1%

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

   2. Taylor expanded in z around 0 97.2%

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

      1. mul-1-neg 97.2%

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

      2. +-commutative 97.2%

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

      3. *-commutative 97.2%

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

      4. +-commutative 97.2%

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

   4. Simplified 97.2%

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

   5. Taylor expanded in y around 0 99.9%

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

      1. +-commutative 99.9%

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

      2. associate-/l* 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \leq -1 \cdot 10^{-266} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \leq 0\right):\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) - \frac{z}{\frac{y}{x}}\\ \end{array} \]

Alternative 2: 68.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{y}{z}\\ t_1 := \frac{y}{t_0}\\ \mathbf{if}\;y \leq -3.6 \cdot 10^{+140}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq -2.05 \cdot 10^{+69}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq -1.4 \cdot 10^{-67}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 96000:\\ \;\;\;\;\frac{x}{t_0}\\ \mathbf{elif}\;y \leq 2.05 \cdot 10^{+111}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 3.4 \cdot 10^{+114}:\\ \;\;\;\;\frac{-z}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ y z))) (t_1 (/ y t_0)))
   (if (<= y -3.6e+140)
     (- z)
     (if (<= y -2.05e+69)
       (+ x y)
       (if (<= y -1.4e-67)
         t_1
         (if (<= y 96000.0)
           (/ x t_0)
           (if (<= y 2.05e+111)
             t_1
             (if (<= y 3.4e+114) (/ (- z) (/ y x)) (- z)))))))))

C

double code(double x, double y, double z) {
	double t_0 = 1.0 - (y / z);
	double t_1 = y / t_0;
	double tmp;
	if (y <= -3.6e+140) {
		tmp = -z;
	} else if (y <= -2.05e+69) {
		tmp = x + y;
	} else if (y <= -1.4e-67) {
		tmp = t_1;
	} else if (y <= 96000.0) {
		tmp = x / t_0;
	} else if (y <= 2.05e+111) {
		tmp = t_1;
	} else if (y <= 3.4e+114) {
		tmp = -z / (y / 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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 - (y / z)
    t_1 = y / t_0
    if (y <= (-3.6d+140)) then
        tmp = -z
    else if (y <= (-2.05d+69)) then
        tmp = x + y
    else if (y <= (-1.4d-67)) then
        tmp = t_1
    else if (y <= 96000.0d0) then
        tmp = x / t_0
    else if (y <= 2.05d+111) then
        tmp = t_1
    else if (y <= 3.4d+114) then
        tmp = -z / (y / x)
    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 t_1 = y / t_0;
	double tmp;
	if (y <= -3.6e+140) {
		tmp = -z;
	} else if (y <= -2.05e+69) {
		tmp = x + y;
	} else if (y <= -1.4e-67) {
		tmp = t_1;
	} else if (y <= 96000.0) {
		tmp = x / t_0;
	} else if (y <= 2.05e+111) {
		tmp = t_1;
	} else if (y <= 3.4e+114) {
		tmp = -z / (y / x);
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 1.0 - (y / z)
	t_1 = y / t_0
	tmp = 0
	if y <= -3.6e+140:
		tmp = -z
	elif y <= -2.05e+69:
		tmp = x + y
	elif y <= -1.4e-67:
		tmp = t_1
	elif y <= 96000.0:
		tmp = x / t_0
	elif y <= 2.05e+111:
		tmp = t_1
	elif y <= 3.4e+114:
		tmp = -z / (y / x)
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(1.0 - Float64(y / z))
	t_1 = Float64(y / t_0)
	tmp = 0.0
	if (y <= -3.6e+140)
		tmp = Float64(-z);
	elseif (y <= -2.05e+69)
		tmp = Float64(x + y);
	elseif (y <= -1.4e-67)
		tmp = t_1;
	elseif (y <= 96000.0)
		tmp = Float64(x / t_0);
	elseif (y <= 2.05e+111)
		tmp = t_1;
	elseif (y <= 3.4e+114)
		tmp = Float64(Float64(-z) / Float64(y / x));
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 1.0 - (y / z);
	t_1 = y / t_0;
	tmp = 0.0;
	if (y <= -3.6e+140)
		tmp = -z;
	elseif (y <= -2.05e+69)
		tmp = x + y;
	elseif (y <= -1.4e-67)
		tmp = t_1;
	elseif (y <= 96000.0)
		tmp = x / t_0;
	elseif (y <= 2.05e+111)
		tmp = t_1;
	elseif (y <= 3.4e+114)
		tmp = -z / (y / x);
	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]}, Block[{t$95$1 = N[(y / t$95$0), $MachinePrecision]}, If[LessEqual[y, -3.6e+140], (-z), If[LessEqual[y, -2.05e+69], N[(x + y), $MachinePrecision], If[LessEqual[y, -1.4e-67], t$95$1, If[LessEqual[y, 96000.0], N[(x / t$95$0), $MachinePrecision], If[LessEqual[y, 2.05e+111], t$95$1, If[LessEqual[y, 3.4e+114], N[((-z) / N[(y / x), $MachinePrecision]), $MachinePrecision], (-z)]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if y < -3.6e140 or 3.4000000000000001e114 < y

   1. Initial program 70.9%

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

   2. Taylor expanded in y around inf 86.3%

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

      1. mul-1-neg 86.3%

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

   4. Simplified 86.3%

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

   if -3.6e140 < y < -2.05e69

   1. Initial program 99.7%

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

   2. Taylor expanded in z around inf 76.0%

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

   if -2.05e69 < y < -1.40000000000000005e-67 or 96000 < y < 2.04999999999999993e111

   1. Initial program 90.2%

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

   2. Taylor expanded in x around 0 58.3%

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

   if -1.40000000000000005e-67 < y < 96000

   1. Initial program 99.9%

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

   2. Taylor expanded in x around inf 85.6%

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

   if 2.04999999999999993e111 < y < 3.4000000000000001e114

   1. Initial program 53.0%

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

   2. Taylor expanded in z around 0 100.0%

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

      1. mul-1-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. *-commutative 100.0%

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

      4. +-commutative 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in y around 0 100.0%

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

      1. associate-/l* 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.6 \cdot 10^{+140}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq -2.05 \cdot 10^{+69}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq -1.4 \cdot 10^{-67}:\\ \;\;\;\;\frac{y}{1 - \frac{y}{z}}\\ \mathbf{elif}\;y \leq 96000:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{elif}\;y \leq 2.05 \cdot 10^{+111}:\\ \;\;\;\;\frac{y}{1 - \frac{y}{z}}\\ \mathbf{elif}\;y \leq 3.4 \cdot 10^{+114}:\\ \;\;\;\;\frac{-z}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 3: 71.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-z\right) - \frac{z}{\frac{y}{x}}\\ \mathbf{if}\;y \leq -3.6 \cdot 10^{+140}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -2.3 \cdot 10^{-150}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 3 \cdot 10^{-49}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{-14}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (- z) (/ z (/ y x)))))
   (if (<= y -3.6e+140)
     t_0
     (if (<= y -2.3e-150)
       (+ x y)
       (if (<= y 3e-49)
         (/ x (- 1.0 (/ y z)))
         (if (<= y 4.5e-14) (+ x y) t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = -z - (z / (y / x));
	double tmp;
	if (y <= -3.6e+140) {
		tmp = t_0;
	} else if (y <= -2.3e-150) {
		tmp = x + y;
	} else if (y <= 3e-49) {
		tmp = x / (1.0 - (y / z));
	} else if (y <= 4.5e-14) {
		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 <= (-3.6d+140)) then
        tmp = t_0
    else if (y <= (-2.3d-150)) then
        tmp = x + y
    else if (y <= 3d-49) then
        tmp = x / (1.0d0 - (y / z))
    else if (y <= 4.5d-14) 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 <= -3.6e+140) {
		tmp = t_0;
	} else if (y <= -2.3e-150) {
		tmp = x + y;
	} else if (y <= 3e-49) {
		tmp = x / (1.0 - (y / z));
	} else if (y <= 4.5e-14) {
		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 <= -3.6e+140:
		tmp = t_0
	elif y <= -2.3e-150:
		tmp = x + y
	elif y <= 3e-49:
		tmp = x / (1.0 - (y / z))
	elif y <= 4.5e-14:
		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 <= -3.6e+140)
		tmp = t_0;
	elseif (y <= -2.3e-150)
		tmp = Float64(x + y);
	elseif (y <= 3e-49)
		tmp = Float64(x / Float64(1.0 - Float64(y / z)));
	elseif (y <= 4.5e-14)
		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 <= -3.6e+140)
		tmp = t_0;
	elseif (y <= -2.3e-150)
		tmp = x + y;
	elseif (y <= 3e-49)
		tmp = x / (1.0 - (y / z));
	elseif (y <= 4.5e-14)
		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, -3.6e+140], t$95$0, If[LessEqual[y, -2.3e-150], N[(x + y), $MachinePrecision], If[LessEqual[y, 3e-49], N[(x / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.5e-14], N[(x + y), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.6e140 or 4.4999999999999998e-14 < y

   1. Initial program 72.5%

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

   2. Taylor expanded in z around 0 62.9%

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

      1. mul-1-neg 62.9%

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

      2. +-commutative 62.9%

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

      3. *-commutative 62.9%

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

      4. +-commutative 62.9%

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

   4. Simplified 62.9%

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

   5. Taylor expanded in y around 0 81.8%

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

      1. +-commutative 81.8%

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

      2. associate-/l* 87.5%

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

   7. Simplified 87.5%

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

   if -3.6e140 < y < -2.30000000000000003e-150 or 3e-49 < y < 4.4999999999999998e-14

   1. Initial program 98.5%

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

   2. Taylor expanded in z around inf 66.6%

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

   if -2.30000000000000003e-150 < y < 3e-49

   1. Initial program 99.9%

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

   2. Taylor expanded in x around inf 85.7%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.6 \cdot 10^{+140}:\\ \;\;\;\;\left(-z\right) - \frac{z}{\frac{y}{x}}\\ \mathbf{elif}\;y \leq -2.3 \cdot 10^{-150}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 3 \cdot 10^{-49}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{-14}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) - \frac{z}{\frac{y}{x}}\\ \end{array} \]

Alternative 4: 67.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.6 \cdot 10^{+140}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq -8.5 \cdot 10^{-151}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{+85}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -3.6e+140)
   (- z)
   (if (<= y -8.5e-151)
     (+ x y)
     (if (<= y 1.25e+85) (/ x (- 1.0 (/ y z))) (- z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -3.6e+140) {
		tmp = -z;
	} else if (y <= -8.5e-151) {
		tmp = x + y;
	} else if (y <= 1.25e+85) {
		tmp = x / (1.0 - (y / z));
	} else {
		tmp = -z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-3.6d+140)) then
        tmp = -z
    else if (y <= (-8.5d-151)) then
        tmp = x + y
    else if (y <= 1.25d+85) then
        tmp = x / (1.0d0 - (y / z))
    else
        tmp = -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -3.6e+140) {
		tmp = -z;
	} else if (y <= -8.5e-151) {
		tmp = x + y;
	} else if (y <= 1.25e+85) {
		tmp = x / (1.0 - (y / z));
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -3.6e+140:
		tmp = -z
	elif y <= -8.5e-151:
		tmp = x + y
	elif y <= 1.25e+85:
		tmp = x / (1.0 - (y / z))
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -3.6e+140)
		tmp = Float64(-z);
	elseif (y <= -8.5e-151)
		tmp = Float64(x + y);
	elseif (y <= 1.25e+85)
		tmp = Float64(x / Float64(1.0 - Float64(y / z)));
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -3.6e+140)
		tmp = -z;
	elseif (y <= -8.5e-151)
		tmp = x + y;
	elseif (y <= 1.25e+85)
		tmp = x / (1.0 - (y / z));
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -3.6e+140], (-z), If[LessEqual[y, -8.5e-151], N[(x + y), $MachinePrecision], If[LessEqual[y, 1.25e+85], N[(x / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-z)]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.6e140 or 1.25e85 < y

   1. Initial program 70.3%

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

   2. Taylor expanded in y around inf 80.9%

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

      1. mul-1-neg 80.9%

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

   4. Simplified 80.9%

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

   if -3.6e140 < y < -8.49999999999999999e-151

   1. Initial program 98.4%

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

   2. Taylor expanded in z around inf 64.9%

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

   if -8.49999999999999999e-151 < y < 1.25e85

   1. Initial program 97.3%

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

   2. Taylor expanded in x around inf 79.9%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.6 \cdot 10^{+140}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq -8.5 \cdot 10^{-151}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{+85}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 5: 56.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.6 \cdot 10^{+140}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq -3.8 \cdot 10^{+16}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq -9 \cdot 10^{-49}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 4 \cdot 10^{-11}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -3.6e+140)
   (- z)
   (if (<= y -3.8e+16) y (if (<= y -9e-49) (- z) (if (<= y 4e-11) x (- z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -3.6e+140) {
		tmp = -z;
	} else if (y <= -3.8e+16) {
		tmp = y;
	} else if (y <= -9e-49) {
		tmp = -z;
	} else if (y <= 4e-11) {
		tmp = x;
	} else {
		tmp = -z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-3.6d+140)) then
        tmp = -z
    else if (y <= (-3.8d+16)) then
        tmp = y
    else if (y <= (-9d-49)) then
        tmp = -z
    else if (y <= 4d-11) then
        tmp = x
    else
        tmp = -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -3.6e+140) {
		tmp = -z;
	} else if (y <= -3.8e+16) {
		tmp = y;
	} else if (y <= -9e-49) {
		tmp = -z;
	} else if (y <= 4e-11) {
		tmp = x;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -3.6e+140:
		tmp = -z
	elif y <= -3.8e+16:
		tmp = y
	elif y <= -9e-49:
		tmp = -z
	elif y <= 4e-11:
		tmp = x
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -3.6e+140)
		tmp = Float64(-z);
	elseif (y <= -3.8e+16)
		tmp = y;
	elseif (y <= -9e-49)
		tmp = Float64(-z);
	elseif (y <= 4e-11)
		tmp = x;
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -3.6e+140)
		tmp = -z;
	elseif (y <= -3.8e+16)
		tmp = y;
	elseif (y <= -9e-49)
		tmp = -z;
	elseif (y <= 4e-11)
		tmp = x;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -3.6e+140], (-z), If[LessEqual[y, -3.8e+16], y, If[LessEqual[y, -9e-49], (-z), If[LessEqual[y, 4e-11], x, (-z)]]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.6e140 or -3.8e16 < y < -9.0000000000000004e-49 or 3.99999999999999976e-11 < y

   1. Initial program 76.2%

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

   2. Taylor expanded in y around inf 70.4%

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

      1. mul-1-neg 70.4%

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

   4. Simplified 70.4%

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

   if -3.6e140 < y < -3.8e16

   1. Initial program 96.4%

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

   2. Taylor expanded in x around 0 59.6%

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

   3. Taylor expanded in y around 0 44.9%

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

   if -9.0000000000000004e-49 < y < 3.99999999999999976e-11

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 68.5%

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

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.6 \cdot 10^{+140}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq -3.8 \cdot 10^{+16}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq -9 \cdot 10^{-49}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 4 \cdot 10^{-11}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 6: 67.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.6 \cdot 10^{+140}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{+99}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -3.6e+140) (- z) (if (<= y 1.55e+99) (+ x y) (- z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -3.6e+140) {
		tmp = -z;
	} else if (y <= 1.55e+99) {
		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 <= (-3.6d+140)) then
        tmp = -z
    else if (y <= 1.55d+99) 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 <= -3.6e+140) {
		tmp = -z;
	} else if (y <= 1.55e+99) {
		tmp = x + y;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -3.6e+140:
		tmp = -z
	elif y <= 1.55e+99:
		tmp = x + y
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -3.6e+140)
		tmp = Float64(-z);
	elseif (y <= 1.55e+99)
		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 <= -3.6e+140)
		tmp = -z;
	elseif (y <= 1.55e+99)
		tmp = x + y;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -3.6e+140], (-z), If[LessEqual[y, 1.55e+99], N[(x + y), $MachinePrecision], (-z)]]

Derivation

1. Split input into 2 regimes

2. if y < -3.6e140 or 1.55e99 < y

   1. Initial program 71.1%

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

   2. Taylor expanded in y around inf 83.5%

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

      1. mul-1-neg 83.5%

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

   4. Simplified 83.5%

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

   if -3.6e140 < y < 1.55e99

   1. Initial program 96.7%

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

   2. Taylor expanded in z around inf 70.0%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.6 \cdot 10^{+140}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{+99}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 7: 40.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{-117}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.85 \cdot 10^{-23}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.6e-117) x (if (<= x 1.85e-23) y x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.6e-117) {
		tmp = x;
	} else if (x <= 1.85e-23) {
		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 <= (-1.6d-117)) then
        tmp = x
    else if (x <= 1.85d-23) 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 <= -1.6e-117) {
		tmp = x;
	} else if (x <= 1.85e-23) {
		tmp = y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.6e-117:
		tmp = x
	elif x <= 1.85e-23:
		tmp = y
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.6e-117)
		tmp = x;
	elseif (x <= 1.85e-23)
		tmp = y;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.6e-117)
		tmp = x;
	elseif (x <= 1.85e-23)
		tmp = y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.6e-117], x, If[LessEqual[x, 1.85e-23], y, x]]

Derivation

1. Split input into 2 regimes

2. if x < -1.59999999999999998e-117 or 1.8500000000000001e-23 < x

   1. Initial program 87.9%

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

   2. Taylor expanded in y around 0 47.8%

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

   if -1.59999999999999998e-117 < x < 1.8500000000000001e-23

   1. Initial program 89.4%

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

   2. Taylor expanded in x around 0 68.9%

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

   3. Taylor expanded in y around 0 31.2%

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

4. Final simplification 41.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{-117}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.85 \cdot 10^{-23}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 8: 34.8% 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 88.5%

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

2. Taylor expanded in y around 0 36.3%

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

3. Final simplification 36.3%

   \[\leadsto x \]

Developer target: 93.9% 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 2023196 
(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))))