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

Percentage Accurate: 88.1% → 99.9%

Time: 6.5s

Alternatives: 7

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: 99.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{-296} \lor \neg \left(t_0 \leq 0\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-z}{\frac{y}{x + y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (+ x y) (- 1.0 (/ y z)))))
   (if (or (<= t_0 -2e-296) (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 <= -2e-296) || !(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 <= (-2d-296)) .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 <= -2e-296) || !(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 <= -2e-296) 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 <= -2e-296) || !(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 <= -2e-296) || ~((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, -2e-296], 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))) < -2e-296 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 -2e-296 < (/.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 99.6%

      \[\leadsto \color{blue}{-1 \cdot \frac{z \cdot \left(x + y\right)}{y}} \]
   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* 100.0%

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

      3. distribute-neg-frac 100.0%

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

      4. +-commutative 100.0%

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

   4. Simplified 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \leq -2 \cdot 10^{-296} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \leq 0\right):\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-z}{\frac{y}{x + y}}\\ \end{array} \]

Alternative 2: 70.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y}{1 - \frac{y}{z}}\\ \mathbf{if}\;y \leq -3.7 \cdot 10^{+137}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq -4.8 \cdot 10^{+31}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{-20}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 3.3 \cdot 10^{+196}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ y (- 1.0 (/ y z)))))
   (if (<= y -3.7e+137)
     (- z)
     (if (<= y -4.8e+31)
       t_0
       (if (<= y 1.3e-20) (+ x y) (if (<= y 3.3e+196) t_0 (- z)))))))

C

double code(double x, double y, double z) {
	double t_0 = y / (1.0 - (y / z));
	double tmp;
	if (y <= -3.7e+137) {
		tmp = -z;
	} else if (y <= -4.8e+31) {
		tmp = t_0;
	} else if (y <= 1.3e-20) {
		tmp = x + y;
	} else if (y <= 3.3e+196) {
		tmp = t_0;
	} 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 = y / (1.0d0 - (y / z))
    if (y <= (-3.7d+137)) then
        tmp = -z
    else if (y <= (-4.8d+31)) then
        tmp = t_0
    else if (y <= 1.3d-20) then
        tmp = x + y
    else if (y <= 3.3d+196) then
        tmp = t_0
    else
        tmp = -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = y / (1.0 - (y / z));
	double tmp;
	if (y <= -3.7e+137) {
		tmp = -z;
	} else if (y <= -4.8e+31) {
		tmp = t_0;
	} else if (y <= 1.3e-20) {
		tmp = x + y;
	} else if (y <= 3.3e+196) {
		tmp = t_0;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y / (1.0 - (y / z))
	tmp = 0
	if y <= -3.7e+137:
		tmp = -z
	elif y <= -4.8e+31:
		tmp = t_0
	elif y <= 1.3e-20:
		tmp = x + y
	elif y <= 3.3e+196:
		tmp = t_0
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y / Float64(1.0 - Float64(y / z)))
	tmp = 0.0
	if (y <= -3.7e+137)
		tmp = Float64(-z);
	elseif (y <= -4.8e+31)
		tmp = t_0;
	elseif (y <= 1.3e-20)
		tmp = Float64(x + y);
	elseif (y <= 3.3e+196)
		tmp = t_0;
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y / (1.0 - (y / z));
	tmp = 0.0;
	if (y <= -3.7e+137)
		tmp = -z;
	elseif (y <= -4.8e+31)
		tmp = t_0;
	elseif (y <= 1.3e-20)
		tmp = x + y;
	elseif (y <= 3.3e+196)
		tmp = t_0;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -3.7e+137], (-z), If[LessEqual[y, -4.8e+31], t$95$0, If[LessEqual[y, 1.3e-20], N[(x + y), $MachinePrecision], If[LessEqual[y, 3.3e+196], t$95$0, (-z)]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.7000000000000002e137 or 3.3000000000000002e196 < y

   1. Initial program 56.0%

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

   2. Taylor expanded in y around inf 83.0%

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

      1. mul-1-neg 83.0%

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

   4. Simplified 83.0%

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

   if -3.7000000000000002e137 < y < -4.79999999999999965e31 or 1.29999999999999997e-20 < y < 3.3000000000000002e196

   1. Initial program 91.1%

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

   2. Taylor expanded in x around 0 73.6%

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

   if -4.79999999999999965e31 < y < 1.29999999999999997e-20

   1. Initial program 99.9%

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

   2. Taylor expanded in z around inf 73.8%

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

      1. +-commutative 73.8%

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

   4. Simplified 73.8%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.7 \cdot 10^{+137}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq -4.8 \cdot 10^{+31}:\\ \;\;\;\;\frac{y}{1 - \frac{y}{z}}\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{-20}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 3.3 \cdot 10^{+196}:\\ \;\;\;\;\frac{y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 3: 74.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-z}{\frac{y}{x + y}}\\ \mathbf{if}\;y \leq -3 \cdot 10^{+128}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -2.35 \cdot 10^{+32}:\\ \;\;\;\;\frac{y}{1 - \frac{y}{z}}\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{-20}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (- z) (/ y (+ x y)))))
   (if (<= y -3e+128)
     t_0
     (if (<= y -2.35e+32)
       (/ y (- 1.0 (/ y z)))
       (if (<= y 4.8e-20) (+ x y) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = -z / (y / (x + y));
	double tmp;
	if (y <= -3e+128) {
		tmp = t_0;
	} else if (y <= -2.35e+32) {
		tmp = y / (1.0 - (y / z));
	} else if (y <= 4.8e-20) {
		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 / (y / (x + y))
    if (y <= (-3d+128)) then
        tmp = t_0
    else if (y <= (-2.35d+32)) then
        tmp = y / (1.0d0 - (y / z))
    else if (y <= 4.8d-20) 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 / (y / (x + y));
	double tmp;
	if (y <= -3e+128) {
		tmp = t_0;
	} else if (y <= -2.35e+32) {
		tmp = y / (1.0 - (y / z));
	} else if (y <= 4.8e-20) {
		tmp = x + y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = -z / (y / (x + y))
	tmp = 0
	if y <= -3e+128:
		tmp = t_0
	elif y <= -2.35e+32:
		tmp = y / (1.0 - (y / z))
	elif y <= 4.8e-20:
		tmp = x + y
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(-z) / Float64(y / Float64(x + y)))
	tmp = 0.0
	if (y <= -3e+128)
		tmp = t_0;
	elseif (y <= -2.35e+32)
		tmp = Float64(y / Float64(1.0 - Float64(y / z)));
	elseif (y <= 4.8e-20)
		tmp = Float64(x + y);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = -z / (y / (x + y));
	tmp = 0.0;
	if (y <= -3e+128)
		tmp = t_0;
	elseif (y <= -2.35e+32)
		tmp = y / (1.0 - (y / z));
	elseif (y <= 4.8e-20)
		tmp = x + y;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[((-z) / N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -3e+128], t$95$0, If[LessEqual[y, -2.35e+32], N[(y / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.8e-20], N[(x + y), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.9999999999999998e128 or 4.79999999999999986e-20 < y

   1. Initial program 69.1%

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

   2. Taylor expanded in z around 0 65.4%

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

      1. mul-1-neg 65.4%

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

      2. associate-/l* 81.0%

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

      3. distribute-neg-frac 81.0%

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

      4. +-commutative 81.0%

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

   4. Simplified 81.0%

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

   if -2.9999999999999998e128 < y < -2.35000000000000012e32

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 88.1%

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

   if -2.35000000000000012e32 < y < 4.79999999999999986e-20

   1. Initial program 99.9%

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

   2. Taylor expanded in z around inf 73.8%

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

      1. +-commutative 73.8%

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

   4. Simplified 73.8%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3 \cdot 10^{+128}:\\ \;\;\;\;\frac{-z}{\frac{y}{x + y}}\\ \mathbf{elif}\;y \leq -2.35 \cdot 10^{+32}:\\ \;\;\;\;\frac{y}{1 - \frac{y}{z}}\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{-20}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;\frac{-z}{\frac{y}{x + y}}\\ \end{array} \]

Alternative 4: 58.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.3 \cdot 10^{+67}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq -6.2 \cdot 10^{-69}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{-22}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2.3e+67) (- z) (if (<= y -6.2e-69) y (if (<= y 4.2e-22) x (- z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.3e+67) {
		tmp = -z;
	} else if (y <= -6.2e-69) {
		tmp = y;
	} else if (y <= 4.2e-22) {
		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 <= (-2.3d+67)) then
        tmp = -z
    else if (y <= (-6.2d-69)) then
        tmp = y
    else if (y <= 4.2d-22) 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 <= -2.3e+67) {
		tmp = -z;
	} else if (y <= -6.2e-69) {
		tmp = y;
	} else if (y <= 4.2e-22) {
		tmp = x;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -2.3e+67:
		tmp = -z
	elif y <= -6.2e-69:
		tmp = y
	elif y <= 4.2e-22:
		tmp = x
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -2.3e+67)
		tmp = Float64(-z);
	elseif (y <= -6.2e-69)
		tmp = y;
	elseif (y <= 4.2e-22)
		tmp = x;
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -2.3e+67)
		tmp = -z;
	elseif (y <= -6.2e-69)
		tmp = y;
	elseif (y <= 4.2e-22)
		tmp = x;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -2.3e+67], (-z), If[LessEqual[y, -6.2e-69], y, If[LessEqual[y, 4.2e-22], x, (-z)]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.2999999999999999e67 or 4.20000000000000016e-22 < y

   1. Initial program 72.7%

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

   2. Taylor expanded in y around inf 66.8%

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

      1. mul-1-neg 66.8%

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

   4. Simplified 66.8%

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

   if -2.2999999999999999e67 < y < -6.1999999999999999e-69

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 61.0%

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

   3. Taylor expanded in y around 0 37.1%

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

   if -6.1999999999999999e-69 < y < 4.20000000000000016e-22

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 63.7%

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

4. Final simplification 60.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.3 \cdot 10^{+67}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq -6.2 \cdot 10^{-69}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{-22}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 5: 68.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.85 \cdot 10^{+87}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{+54}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2.85e+87) (- z) (if (<= y 4.2e+54) (+ x y) (- z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.85e+87) {
		tmp = -z;
	} else if (y <= 4.2e+54) {
		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 <= (-2.85d+87)) then
        tmp = -z
    else if (y <= 4.2d+54) 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 <= -2.85e+87) {
		tmp = -z;
	} else if (y <= 4.2e+54) {
		tmp = x + y;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -2.85e+87:
		tmp = -z
	elif y <= 4.2e+54:
		tmp = x + y
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -2.85e+87)
		tmp = Float64(-z);
	elseif (y <= 4.2e+54)
		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 <= -2.85e+87)
		tmp = -z;
	elseif (y <= 4.2e+54)
		tmp = x + y;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -2.85e+87], (-z), If[LessEqual[y, 4.2e+54], N[(x + y), $MachinePrecision], (-z)]]

Derivation

1. Split input into 2 regimes

2. if y < -2.85000000000000019e87 or 4.19999999999999972e54 < y

   1. Initial program 65.6%

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

   2. Taylor expanded in y around inf 75.0%

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

      1. mul-1-neg 75.0%

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

   4. Simplified 75.0%

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

   if -2.85000000000000019e87 < y < 4.19999999999999972e54

   1. Initial program 99.9%

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

   2. Taylor expanded in z around inf 69.6%

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

      1. +-commutative 69.6%

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

   4. Simplified 69.6%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.85 \cdot 10^{+87}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{+54}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 6: 41.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.32 \cdot 10^{-86}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{-71}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.32e-86) x (if (<= x 1.55e-71) y x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.32e-86) {
		tmp = x;
	} else if (x <= 1.55e-71) {
		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 <= (-1.32d-86)) then
        tmp = x
    else if (x <= 1.55d-71) 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 <= -1.32e-86) {
		tmp = x;
	} else if (x <= 1.55e-71) {
		tmp = y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.32e-86:
		tmp = x
	elif x <= 1.55e-71:
		tmp = y
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.32e-86)
		tmp = x;
	elseif (x <= 1.55e-71)
		tmp = y;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.32e-86)
		tmp = x;
	elseif (x <= 1.55e-71)
		tmp = y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.32e-86], x, If[LessEqual[x, 1.55e-71], y, x]]

Derivation

1. Split input into 2 regimes

2. if x < -1.32e-86 or 1.55000000000000001e-71 < x

   1. Initial program 88.6%

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

   2. Taylor expanded in y around 0 44.3%

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

   if -1.32e-86 < x < 1.55000000000000001e-71

   1. Initial program 87.4%

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

   2. Taylor expanded in x around 0 74.5%

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

   3. Taylor expanded in y around 0 43.7%

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

4. Final simplification 44.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.32 \cdot 10^{-86}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{-71}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 7: 35.6% accurate, 9.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := x

Derivation

1. Initial program 88.1%

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

2. Taylor expanded in y around 0 31.5%

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

3. Final simplification 31.5%

   \[\leadsto x \]

Developer target: 93.7% 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 2023279 
(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))))