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

Percentage Accurate: 88.0% → 99.8%

Time: 5.6s

Alternatives: 12

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: 88.0% 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: 99.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{y}{z}\\ t_1 := \frac{x + y}{t_0}\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{-280}:\\ \;\;\;\;\frac{1}{t_0} \cdot \left(x + y\right)\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;\left(\left(-z\right) - \frac{z}{\frac{y}{x}}\right) - \frac{z}{\frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ y z))) (t_1 (/ (+ x y) t_0)))
   (if (<= t_1 -5e-280)
     (* (/ 1.0 t_0) (+ x y))
     (if (<= t_1 0.0) (- (- (- z) (/ z (/ y x))) (/ z (/ y z))) t_1))))

C

double code(double x, double y, double z) {
	double t_0 = 1.0 - (y / z);
	double t_1 = (x + y) / t_0;
	double tmp;
	if (t_1 <= -5e-280) {
		tmp = (1.0 / t_0) * (x + y);
	} else if (t_1 <= 0.0) {
		tmp = (-z - (z / (y / x))) - (z / (y / z));
	} else {
		tmp = t_1;
	}
	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 = (x + y) / t_0
    if (t_1 <= (-5d-280)) then
        tmp = (1.0d0 / t_0) * (x + y)
    else if (t_1 <= 0.0d0) then
        tmp = (-z - (z / (y / x))) - (z / (y / z))
    else
        tmp = t_1
    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 = (x + y) / t_0;
	double tmp;
	if (t_1 <= -5e-280) {
		tmp = (1.0 / t_0) * (x + y);
	} else if (t_1 <= 0.0) {
		tmp = (-z - (z / (y / x))) - (z / (y / z));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 1.0 - (y / z)
	t_1 = (x + y) / t_0
	tmp = 0
	if t_1 <= -5e-280:
		tmp = (1.0 / t_0) * (x + y)
	elif t_1 <= 0.0:
		tmp = (-z - (z / (y / x))) - (z / (y / z))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(1.0 - Float64(y / z))
	t_1 = Float64(Float64(x + y) / t_0)
	tmp = 0.0
	if (t_1 <= -5e-280)
		tmp = Float64(Float64(1.0 / t_0) * Float64(x + y));
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(Float64(-z) - Float64(z / Float64(y / x))) - Float64(z / Float64(y / z)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 1.0 - (y / z);
	t_1 = (x + y) / t_0;
	tmp = 0.0;
	if (t_1 <= -5e-280)
		tmp = (1.0 / t_0) * (x + y);
	elseif (t_1 <= 0.0)
		tmp = (-z - (z / (y / x))) - (z / (y / z));
	else
		tmp = t_1;
	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[(N[(x + y), $MachinePrecision] / t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-280], N[(N[(1.0 / t$95$0), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(N[((-z) - N[(z / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(z / N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z))) < -5.00000000000000028e-280

   1. Initial program 99.8%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Step-by-step derivation

      1. clear-num 99.7%

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

      2. associate-/r/ 99.9%

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

   3. Applied egg-rr 99.9%

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

   if -5.00000000000000028e-280 < (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z))) < 0.0

   1. Initial program 8.0%

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

   2. Taylor expanded in y around inf 99.9%

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

      1. mul-1-neg 99.9%

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

      2. unsub-neg 99.9%

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

      3. mul-1-neg 99.9%

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

      4. associate-/l* 99.9%

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

      5. unpow2 99.9%

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

      6. associate-/l* 99.9%

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

   4. Simplified 99.9%

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

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

   1. Initial program 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \leq -5 \cdot 10^{-280}:\\ \;\;\;\;\frac{1}{1 - \frac{y}{z}} \cdot \left(x + y\right)\\ \mathbf{elif}\;\frac{x + y}{1 - \frac{y}{z}} \leq 0:\\ \;\;\;\;\left(\left(-z\right) - \frac{z}{\frac{y}{x}}\right) - \frac{z}{\frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \end{array} \]

Alternative 2: 99.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{-280} \lor \neg \left(t_0 \leq 0\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{\left(-y\right) - x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (+ x y) (- 1.0 (/ y z)))))
   (if (or (<= t_0 -5e-280) (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 <= -5e-280) || !(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 <= (-5d-280)) .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 <= -5e-280) || !(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 <= -5e-280) 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 <= -5e-280) || !(t_0 <= 0.0))
		tmp = t_0;
	else
		tmp = Float64(z * Float64(Float64(Float64(-y) - 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 <= -5e-280) || ~((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, -5e-280], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], t$95$0, N[(z * N[(N[((-y) - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z))) < -5.00000000000000028e-280 or 0.0 < (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z)))

   1. Initial program 99.8%

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

   if -5.00000000000000028e-280 < (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z))) < 0.0

   1. Initial program 8.0%

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

   2. Taylor expanded in z around 0 97.9%

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

      1. mul-1-neg 97.9%

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

      2. associate-/l* 8.0%

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

      3. +-commutative 8.0%

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

      4. associate-/r/ 99.9%

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

      5. distribute-rgt-neg-in 99.9%

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

      6. +-commutative 99.9%

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

   4. Simplified 99.9%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \leq -5 \cdot 10^{-280} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \leq 0\right):\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{\left(-y\right) - x}{y}\\ \end{array} \]

Alternative 3: 99.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{y}{z}\\ t_1 := \frac{x + y}{t_0}\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{-280}:\\ \;\;\;\;\frac{1}{t_0} \cdot \left(x + y\right)\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;z \cdot \frac{\left(-y\right) - x}{y}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ y z))) (t_1 (/ (+ x y) t_0)))
   (if (<= t_1 -5e-280)
     (* (/ 1.0 t_0) (+ x y))
     (if (<= t_1 0.0) (* z (/ (- (- y) x) y)) t_1))))

C

double code(double x, double y, double z) {
	double t_0 = 1.0 - (y / z);
	double t_1 = (x + y) / t_0;
	double tmp;
	if (t_1 <= -5e-280) {
		tmp = (1.0 / t_0) * (x + y);
	} else if (t_1 <= 0.0) {
		tmp = z * ((-y - x) / y);
	} else {
		tmp = t_1;
	}
	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 = (x + y) / t_0
    if (t_1 <= (-5d-280)) then
        tmp = (1.0d0 / t_0) * (x + y)
    else if (t_1 <= 0.0d0) then
        tmp = z * ((-y - x) / y)
    else
        tmp = t_1
    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 = (x + y) / t_0;
	double tmp;
	if (t_1 <= -5e-280) {
		tmp = (1.0 / t_0) * (x + y);
	} else if (t_1 <= 0.0) {
		tmp = z * ((-y - x) / y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 1.0 - (y / z)
	t_1 = (x + y) / t_0
	tmp = 0
	if t_1 <= -5e-280:
		tmp = (1.0 / t_0) * (x + y)
	elif t_1 <= 0.0:
		tmp = z * ((-y - x) / y)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(1.0 - Float64(y / z))
	t_1 = Float64(Float64(x + y) / t_0)
	tmp = 0.0
	if (t_1 <= -5e-280)
		tmp = Float64(Float64(1.0 / t_0) * Float64(x + y));
	elseif (t_1 <= 0.0)
		tmp = Float64(z * Float64(Float64(Float64(-y) - x) / y));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 1.0 - (y / z);
	t_1 = (x + y) / t_0;
	tmp = 0.0;
	if (t_1 <= -5e-280)
		tmp = (1.0 / t_0) * (x + y);
	elseif (t_1 <= 0.0)
		tmp = z * ((-y - x) / y);
	else
		tmp = t_1;
	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[(N[(x + y), $MachinePrecision] / t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-280], N[(N[(1.0 / t$95$0), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(z * N[(N[((-y) - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z))) < -5.00000000000000028e-280

   1. Initial program 99.8%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Step-by-step derivation

      1. clear-num 99.7%

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

      2. associate-/r/ 99.9%

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

   3. Applied egg-rr 99.9%

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

   if -5.00000000000000028e-280 < (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z))) < 0.0

   1. Initial program 8.0%

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

   2. Taylor expanded in z around 0 97.9%

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

      1. mul-1-neg 97.9%

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

      2. associate-/l* 8.0%

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

      3. +-commutative 8.0%

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

      4. associate-/r/ 99.9%

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

      5. distribute-rgt-neg-in 99.9%

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

      6. +-commutative 99.9%

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

   4. Simplified 99.9%

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

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

   1. Initial program 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \leq -5 \cdot 10^{-280}:\\ \;\;\;\;\frac{1}{1 - \frac{y}{z}} \cdot \left(x + y\right)\\ \mathbf{elif}\;\frac{x + y}{1 - \frac{y}{z}} \leq 0:\\ \;\;\;\;z \cdot \frac{\left(-y\right) - x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \end{array} \]

Alternative 4: 67.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.35 \cdot 10^{+132}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{-233}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{+45}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.35e+132)
   (- z)
   (if (<= y 4.2e-233)
     (+ x y)
     (if (<= y 3.5e+45) (/ x (- 1.0 (/ y z))) (- z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.35e+132) {
		tmp = -z;
	} else if (y <= 4.2e-233) {
		tmp = x + y;
	} else if (y <= 3.5e+45) {
		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 <= (-1.35d+132)) then
        tmp = -z
    else if (y <= 4.2d-233) then
        tmp = x + y
    else if (y <= 3.5d+45) 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 <= -1.35e+132) {
		tmp = -z;
	} else if (y <= 4.2e-233) {
		tmp = x + y;
	} else if (y <= 3.5e+45) {
		tmp = x / (1.0 - (y / z));
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1.35e+132:
		tmp = -z
	elif y <= 4.2e-233:
		tmp = x + y
	elif y <= 3.5e+45:
		tmp = x / (1.0 - (y / z))
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.35e+132)
		tmp = Float64(-z);
	elseif (y <= 4.2e-233)
		tmp = Float64(x + y);
	elseif (y <= 3.5e+45)
		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 <= -1.35e+132)
		tmp = -z;
	elseif (y <= 4.2e-233)
		tmp = x + y;
	elseif (y <= 3.5e+45)
		tmp = x / (1.0 - (y / z));
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.35e+132], (-z), If[LessEqual[y, 4.2e-233], N[(x + y), $MachinePrecision], If[LessEqual[y, 3.5e+45], N[(x / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-z)]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.35e132 or 3.50000000000000023e45 < y

   1. Initial program 56.5%

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

   2. Taylor expanded in y around inf 74.0%

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

      1. mul-1-neg 74.0%

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

   4. Simplified 74.0%

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

   if -1.35e132 < y < 4.1999999999999997e-233

   1. Initial program 95.3%

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

   2. Taylor expanded in z around inf 66.5%

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

   if 4.1999999999999997e-233 < y < 3.50000000000000023e45

   1. Initial program 98.3%

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

   2. Taylor expanded in x around inf 75.8%

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

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.35 \cdot 10^{+132}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{-233}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{+45}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 5: 68.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{y}{z}\\ \mathbf{if}\;y \leq -3.6 \cdot 10^{+134}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq -7 \cdot 10^{-35}:\\ \;\;\;\;\frac{y}{t_0}\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{+36}:\\ \;\;\;\;\frac{x}{t_0}\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ y z))))
   (if (<= y -3.6e+134)
     (- z)
     (if (<= y -7e-35) (/ y t_0) (if (<= y 1.9e+36) (/ x t_0) (- z))))))

C

double code(double x, double y, double z) {
	double t_0 = 1.0 - (y / z);
	double tmp;
	if (y <= -3.6e+134) {
		tmp = -z;
	} else if (y <= -7e-35) {
		tmp = y / t_0;
	} else if (y <= 1.9e+36) {
		tmp = x / 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 = 1.0d0 - (y / z)
    if (y <= (-3.6d+134)) then
        tmp = -z
    else if (y <= (-7d-35)) then
        tmp = y / t_0
    else if (y <= 1.9d+36) then
        tmp = x / 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 = 1.0 - (y / z);
	double tmp;
	if (y <= -3.6e+134) {
		tmp = -z;
	} else if (y <= -7e-35) {
		tmp = y / t_0;
	} else if (y <= 1.9e+36) {
		tmp = x / t_0;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 1.0 - (y / z)
	tmp = 0
	if y <= -3.6e+134:
		tmp = -z
	elif y <= -7e-35:
		tmp = y / t_0
	elif y <= 1.9e+36:
		tmp = x / t_0
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(1.0 - Float64(y / z))
	tmp = 0.0
	if (y <= -3.6e+134)
		tmp = Float64(-z);
	elseif (y <= -7e-35)
		tmp = Float64(y / t_0);
	elseif (y <= 1.9e+36)
		tmp = Float64(x / t_0);
	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 <= -3.6e+134)
		tmp = -z;
	elseif (y <= -7e-35)
		tmp = y / t_0;
	elseif (y <= 1.9e+36)
		tmp = x / t_0;
	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, -3.6e+134], (-z), If[LessEqual[y, -7e-35], N[(y / t$95$0), $MachinePrecision], If[LessEqual[y, 1.9e+36], N[(x / t$95$0), $MachinePrecision], (-z)]]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.59999999999999988e134 or 1.90000000000000012e36 < y

   1. Initial program 55.5%

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

   2. Taylor expanded in y around inf 74.5%

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

      1. mul-1-neg 74.5%

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

   4. Simplified 74.5%

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

   if -3.59999999999999988e134 < y < -6.99999999999999992e-35

   1. Initial program 88.2%

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

   2. Taylor expanded in x around 0 57.0%

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

   if -6.99999999999999992e-35 < y < 1.90000000000000012e36

   1. Initial program 99.1%

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

   2. Taylor expanded in x around inf 76.8%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.6 \cdot 10^{+134}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq -7 \cdot 10^{-35}:\\ \;\;\;\;\frac{y}{1 - \frac{y}{z}}\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{+36}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 6: 71.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.85 \cdot 10^{+46}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{+35}:\\ \;\;\;\;\frac{z \cdot \left(\left(-y\right) - x\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.85e+46)
   (+ x y)
   (if (<= z 1.35e+35) (/ (* z (- (- y) x)) y) (+ x y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.85e+46) {
		tmp = x + y;
	} else if (z <= 1.35e+35) {
		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 (z <= (-1.85d+46)) then
        tmp = x + y
    else if (z <= 1.35d+35) 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 (z <= -1.85e+46) {
		tmp = x + y;
	} else if (z <= 1.35e+35) {
		tmp = (z * (-y - x)) / y;
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.85e+46:
		tmp = x + y
	elif z <= 1.35e+35:
		tmp = (z * (-y - x)) / y
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.85e+46)
		tmp = Float64(x + y);
	elseif (z <= 1.35e+35)
		tmp = Float64(Float64(z * Float64(Float64(-y) - x)) / y);
	else
		tmp = Float64(x + y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1.85e+46)
		tmp = x + y;
	elseif (z <= 1.35e+35)
		tmp = (z * (-y - x)) / y;
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.85e+46], N[(x + y), $MachinePrecision], If[LessEqual[z, 1.35e+35], N[(N[(z * N[((-y) - x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], N[(x + y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.84999999999999995e46 or 1.35000000000000001e35 < z

   1. Initial program 99.9%

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

   2. Taylor expanded in z around inf 84.2%

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

   if -1.84999999999999995e46 < z < 1.35000000000000001e35

   1. Initial program 69.1%

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

   2. Taylor expanded in z around 0 73.3%

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

      1. mul-1-neg 73.3%

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

      2. +-commutative 73.3%

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

      3. *-commutative 73.3%

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

      4. +-commutative 73.3%

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

   4. Simplified 73.3%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.85 \cdot 10^{+46}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{+35}:\\ \;\;\;\;\frac{z \cdot \left(\left(-y\right) - x\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 7: 72.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{+38}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq 2.8 \cdot 10^{+32}:\\ \;\;\;\;z \cdot \frac{\left(-y\right) - x}{y}\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1e+38)
   (+ x y)
   (if (<= z 2.8e+32) (* z (/ (- (- y) x) y)) (+ x y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1e+38) {
		tmp = x + y;
	} else if (z <= 2.8e+32) {
		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 (z <= (-1d+38)) then
        tmp = x + y
    else if (z <= 2.8d+32) 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 (z <= -1e+38) {
		tmp = x + y;
	} else if (z <= 2.8e+32) {
		tmp = z * ((-y - x) / y);
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1e+38:
		tmp = x + y
	elif z <= 2.8e+32:
		tmp = z * ((-y - x) / y)
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1e+38)
		tmp = Float64(x + y);
	elseif (z <= 2.8e+32)
		tmp = Float64(z * Float64(Float64(Float64(-y) - x) / y));
	else
		tmp = Float64(x + y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1e+38)
		tmp = x + y;
	elseif (z <= 2.8e+32)
		tmp = z * ((-y - x) / y);
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1e+38], N[(x + y), $MachinePrecision], If[LessEqual[z, 2.8e+32], N[(z * N[(N[((-y) - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(x + y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -9.99999999999999977e37 or 2.8e32 < z

   1. Initial program 99.9%

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

   2. Taylor expanded in z around inf 84.2%

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

   if -9.99999999999999977e37 < z < 2.8e32

   1. Initial program 69.1%

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

   2. Taylor expanded in z around 0 73.3%

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

      1. mul-1-neg 73.3%

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

      2. associate-/l* 48.4%

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

      3. +-commutative 48.4%

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

      4. associate-/r/ 77.9%

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

      5. distribute-rgt-neg-in 77.9%

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

      6. +-commutative 77.9%

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

   4. Simplified 77.9%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{+38}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq 2.8 \cdot 10^{+32}:\\ \;\;\;\;z \cdot \frac{\left(-y\right) - x}{y}\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 8: 72.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.4 \cdot 10^{+45}:\\ \;\;\;\;\left(x + y\right) \cdot \left(1 + \frac{y}{z}\right)\\ \mathbf{elif}\;z \leq 4.3 \cdot 10^{+33}:\\ \;\;\;\;z \cdot \frac{\left(-y\right) - x}{y}\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.4e+45)
   (* (+ x y) (+ 1.0 (/ y z)))
   (if (<= z 4.3e+33) (* z (/ (- (- y) x) y)) (+ x y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.4e+45) {
		tmp = (x + y) * (1.0 + (y / z));
	} else if (z <= 4.3e+33) {
		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 (z <= (-1.4d+45)) then
        tmp = (x + y) * (1.0d0 + (y / z))
    else if (z <= 4.3d+33) 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 (z <= -1.4e+45) {
		tmp = (x + y) * (1.0 + (y / z));
	} else if (z <= 4.3e+33) {
		tmp = z * ((-y - x) / y);
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.4e+45:
		tmp = (x + y) * (1.0 + (y / z))
	elif z <= 4.3e+33:
		tmp = z * ((-y - x) / y)
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.4e+45)
		tmp = Float64(Float64(x + y) * Float64(1.0 + Float64(y / z)));
	elseif (z <= 4.3e+33)
		tmp = Float64(z * Float64(Float64(Float64(-y) - x) / y));
	else
		tmp = Float64(x + y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1.4e+45)
		tmp = (x + y) * (1.0 + (y / z));
	elseif (z <= 4.3e+33)
		tmp = z * ((-y - x) / y);
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.4e+45], N[(N[(x + y), $MachinePrecision] * N[(1.0 + N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 4.3e+33], N[(z * N[(N[((-y) - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(x + y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.4e45

   1. Initial program 99.9%

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

   2. Taylor expanded in z around inf 80.6%

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

      1. associate-/l* 87.9%

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

      2. +-commutative 87.9%

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

      3. associate-/r/ 87.9%

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

      4. +-commutative 87.9%

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

      5. *-lft-identity 87.9%

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

      6. distribute-rgt-in 87.9%

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

      7. +-commutative 87.9%

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

      8. +-commutative 87.9%

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

   4. Simplified 87.9%

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

   if -1.4e45 < z < 4.30000000000000028e33

   1. Initial program 69.1%

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

   2. Taylor expanded in z around 0 73.3%

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

      1. mul-1-neg 73.3%

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

      2. associate-/l* 48.4%

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

      3. +-commutative 48.4%

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

      4. associate-/r/ 77.9%

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

      5. distribute-rgt-neg-in 77.9%

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

      6. +-commutative 77.9%

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

   4. Simplified 77.9%

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

   if 4.30000000000000028e33 < z

   1. Initial program 99.9%

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

   2. Taylor expanded in z around inf 81.7%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.4 \cdot 10^{+45}:\\ \;\;\;\;\left(x + y\right) \cdot \left(1 + \frac{y}{z}\right)\\ \mathbf{elif}\;z \leq 4.3 \cdot 10^{+33}:\\ \;\;\;\;z \cdot \frac{\left(-y\right) - x}{y}\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 9: 57.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.2 \cdot 10^{+62}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq -1.45 \cdot 10^{-36}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{+35}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -4.2e+62)
   (- z)
   (if (<= y -1.45e-36) y (if (<= y 3.5e+35) x (- z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -4.2e+62) {
		tmp = -z;
	} else if (y <= -1.45e-36) {
		tmp = y;
	} else if (y <= 3.5e+35) {
		tmp = x;
	} else {
		tmp = -z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-4.2d+62)) then
        tmp = -z
    else if (y <= (-1.45d-36)) then
        tmp = y
    else if (y <= 3.5d+35) then
        tmp = x
    else
        tmp = -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -4.2e+62) {
		tmp = -z;
	} else if (y <= -1.45e-36) {
		tmp = y;
	} else if (y <= 3.5e+35) {
		tmp = x;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -4.2e+62:
		tmp = -z
	elif y <= -1.45e-36:
		tmp = y
	elif y <= 3.5e+35:
		tmp = x
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -4.2e+62)
		tmp = Float64(-z);
	elseif (y <= -1.45e-36)
		tmp = y;
	elseif (y <= 3.5e+35)
		tmp = x;
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -4.2e+62)
		tmp = -z;
	elseif (y <= -1.45e-36)
		tmp = y;
	elseif (y <= 3.5e+35)
		tmp = x;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -4.2e+62], (-z), If[LessEqual[y, -1.45e-36], y, If[LessEqual[y, 3.5e+35], x, (-z)]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.2e62 or 3.5000000000000001e35 < y

   1. Initial program 60.8%

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

   2. Taylor expanded in y around inf 66.7%

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

      1. mul-1-neg 66.7%

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

   4. Simplified 66.7%

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

   if -4.2e62 < y < -1.45000000000000006e-36

   1. Initial program 94.9%

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

   2. Taylor expanded in x around 0 53.9%

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

   3. Taylor expanded in y around 0 36.7%

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

   if -1.45000000000000006e-36 < y < 3.5000000000000001e35

   1. Initial program 99.1%

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

   2. Taylor expanded in y around 0 59.5%

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

4. Final simplification 60.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.2 \cdot 10^{+62}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq -1.45 \cdot 10^{-36}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{+35}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 10: 67.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.35 \cdot 10^{+132}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 5.4 \cdot 10^{+42}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.35e+132) (- z) (if (<= y 5.4e+42) (+ x y) (- z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.35e+132) {
		tmp = -z;
	} else if (y <= 5.4e+42) {
		tmp = x + y;
	} else {
		tmp = -z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-1.35d+132)) then
        tmp = -z
    else if (y <= 5.4d+42) then
        tmp = x + y
    else
        tmp = -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.35e+132) {
		tmp = -z;
	} else if (y <= 5.4e+42) {
		tmp = x + y;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1.35e+132:
		tmp = -z
	elif y <= 5.4e+42:
		tmp = x + y
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.35e+132)
		tmp = Float64(-z);
	elseif (y <= 5.4e+42)
		tmp = Float64(x + y);
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -1.35e+132)
		tmp = -z;
	elseif (y <= 5.4e+42)
		tmp = x + y;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.35e+132], (-z), If[LessEqual[y, 5.4e+42], N[(x + y), $MachinePrecision], (-z)]]

Derivation

1. Split input into 2 regimes

2. if y < -1.35e132 or 5.4000000000000001e42 < y

   1. Initial program 56.5%

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

   2. Taylor expanded in y around inf 74.0%

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

      1. mul-1-neg 74.0%

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

   4. Simplified 74.0%

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

   if -1.35e132 < y < 5.4000000000000001e42

   1. Initial program 96.4%

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

   2. Taylor expanded in z around inf 64.3%

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

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.35 \cdot 10^{+132}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 5.4 \cdot 10^{+42}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 11: 37.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.52 \cdot 10^{-37}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 1.22 \cdot 10^{+23}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.52e-37) y (if (<= y 1.22e+23) x y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.52e-37) {
		tmp = y;
	} else if (y <= 1.22e+23) {
		tmp = x;
	} else {
		tmp = y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-1.52d-37)) then
        tmp = y
    else if (y <= 1.22d+23) then
        tmp = x
    else
        tmp = y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.52e-37) {
		tmp = y;
	} else if (y <= 1.22e+23) {
		tmp = x;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1.52e-37:
		tmp = y
	elif y <= 1.22e+23:
		tmp = x
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.52e-37)
		tmp = y;
	elseif (y <= 1.22e+23)
		tmp = x;
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -1.52e-37)
		tmp = y;
	elseif (y <= 1.22e+23)
		tmp = x;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.52e-37], y, If[LessEqual[y, 1.22e+23], x, y]]

Derivation

1. Split input into 2 regimes

2. if y < -1.52e-37 or 1.22e23 < y

   1. Initial program 66.3%

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

   2. Taylor expanded in x around 0 51.3%

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

   3. Taylor expanded in y around 0 21.5%

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

   if -1.52e-37 < y < 1.22e23

   1. Initial program 99.8%

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

   2. Taylor expanded in y around 0 60.5%

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

4. Final simplification 39.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.52 \cdot 10^{-37}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 1.22 \cdot 10^{+23}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 12: 34.7% 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 81.9%

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

2. Taylor expanded in y around 0 31.7%

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

3. Final simplification 31.7%

   \[\leadsto x \]

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 2023229 
(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))))