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

Percentage Accurate: 88.1% → 98.5%

Time: 8.9s

Alternatives: 10

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: 98.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-281} \lor \neg \left(t\_0 \leq 5 \cdot 10^{-220}\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) - \frac{x \cdot z}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (+ x y) (- 1.0 (/ y z)))))
   (if (or (<= t_0 -2e-281) (not (<= t_0 5e-220)))
     t_0
     (- (- z) (/ (* x z) 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-281) || !(t_0 <= 5e-220)) {
		tmp = t_0;
	} else {
		tmp = -z - ((x * z) / 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-281)) .or. (.not. (t_0 <= 5d-220))) then
        tmp = t_0
    else
        tmp = -z - ((x * z) / 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-281) || !(t_0 <= 5e-220)) {
		tmp = t_0;
	} else {
		tmp = -z - ((x * z) / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x + y) / (1.0 - (y / z))
	tmp = 0
	if (t_0 <= -2e-281) or not (t_0 <= 5e-220):
		tmp = t_0
	else:
		tmp = -z - ((x * z) / 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-281) || !(t_0 <= 5e-220))
		tmp = t_0;
	else
		tmp = Float64(Float64(-z) - Float64(Float64(x * z) / 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-281) || ~((t_0 <= 5e-220)))
		tmp = t_0;
	else
		tmp = -z - ((x * z) / 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-281], N[Not[LessEqual[t$95$0, 5e-220]], $MachinePrecision]], t$95$0, N[((-z) - N[(N[(x * z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z))) < -2e-281 or 5.0000000000000002e-220 < (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z)))

   1. Initial program 99.9%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   if -2e-281 < (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z))) < 5.0000000000000002e-220

   1. Initial program 18.4%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0 93.8%

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

      1. mul-1-neg 93.8%

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

      2. associate-/l* 99.8%

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

      3. associate-/r/ 24.0%

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

      4. distribute-rgt-neg-in 24.0%

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

      5. +-commutative 24.0%

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

      6. distribute-neg-in 24.0%

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

      7. sub-neg 24.0%

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

   5. Simplified 24.0%

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

   6. Taylor expanded in y around 0 99.9%

      \[\leadsto \color{blue}{-1 \cdot z + -1 \cdot \frac{x \cdot z}{y}} \]
   7. Step-by-step derivation

      1. mul-1-neg 99.9%

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

      2. unsub-neg 99.9%

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

      3. neg-mul-1 99.9%

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

   8. Simplified 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \leq -2 \cdot 10^{-281} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \leq 5 \cdot 10^{-220}\right):\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) - \frac{x \cdot z}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 67.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{1 - \frac{y}{z}}\\ \mathbf{if}\;y \leq -9 \cdot 10^{+49}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq -9 \cdot 10^{-187}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 6.6 \cdot 10^{-156}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{-88}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{+33}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{+34}:\\ \;\;\;\;\frac{z}{y} \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ x (- 1.0 (/ y z)))))
   (if (<= y -9e+49)
     (- z)
     (if (<= y -9e-187)
       t_0
       (if (<= y 6.6e-156)
         (+ x y)
         (if (<= y 3.1e-88)
           t_0
           (if (<= y 1.6e+33)
             (+ x y)
             (if (<= y 2.3e+34) (* (/ z y) (- x)) (- z)))))))))

C

double code(double x, double y, double z) {
	double t_0 = x / (1.0 - (y / z));
	double tmp;
	if (y <= -9e+49) {
		tmp = -z;
	} else if (y <= -9e-187) {
		tmp = t_0;
	} else if (y <= 6.6e-156) {
		tmp = x + y;
	} else if (y <= 3.1e-88) {
		tmp = t_0;
	} else if (y <= 1.6e+33) {
		tmp = x + y;
	} else if (y <= 2.3e+34) {
		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) :: tmp
    t_0 = x / (1.0d0 - (y / z))
    if (y <= (-9d+49)) then
        tmp = -z
    else if (y <= (-9d-187)) then
        tmp = t_0
    else if (y <= 6.6d-156) then
        tmp = x + y
    else if (y <= 3.1d-88) then
        tmp = t_0
    else if (y <= 1.6d+33) then
        tmp = x + y
    else if (y <= 2.3d+34) 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 = x / (1.0 - (y / z));
	double tmp;
	if (y <= -9e+49) {
		tmp = -z;
	} else if (y <= -9e-187) {
		tmp = t_0;
	} else if (y <= 6.6e-156) {
		tmp = x + y;
	} else if (y <= 3.1e-88) {
		tmp = t_0;
	} else if (y <= 1.6e+33) {
		tmp = x + y;
	} else if (y <= 2.3e+34) {
		tmp = (z / y) * -x;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x / (1.0 - (y / z))
	tmp = 0
	if y <= -9e+49:
		tmp = -z
	elif y <= -9e-187:
		tmp = t_0
	elif y <= 6.6e-156:
		tmp = x + y
	elif y <= 3.1e-88:
		tmp = t_0
	elif y <= 1.6e+33:
		tmp = x + y
	elif y <= 2.3e+34:
		tmp = (z / y) * -x
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x / Float64(1.0 - Float64(y / z)))
	tmp = 0.0
	if (y <= -9e+49)
		tmp = Float64(-z);
	elseif (y <= -9e-187)
		tmp = t_0;
	elseif (y <= 6.6e-156)
		tmp = Float64(x + y);
	elseif (y <= 3.1e-88)
		tmp = t_0;
	elseif (y <= 1.6e+33)
		tmp = Float64(x + y);
	elseif (y <= 2.3e+34)
		tmp = Float64(Float64(z / y) * Float64(-x));
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x / (1.0 - (y / z));
	tmp = 0.0;
	if (y <= -9e+49)
		tmp = -z;
	elseif (y <= -9e-187)
		tmp = t_0;
	elseif (y <= 6.6e-156)
		tmp = x + y;
	elseif (y <= 3.1e-88)
		tmp = t_0;
	elseif (y <= 1.6e+33)
		tmp = x + y;
	elseif (y <= 2.3e+34)
		tmp = (z / y) * -x;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -9e+49], (-z), If[LessEqual[y, -9e-187], t$95$0, If[LessEqual[y, 6.6e-156], N[(x + y), $MachinePrecision], If[LessEqual[y, 3.1e-88], t$95$0, If[LessEqual[y, 1.6e+33], N[(x + y), $MachinePrecision], If[LessEqual[y, 2.3e+34], N[(N[(z / y), $MachinePrecision] * (-x)), $MachinePrecision], (-z)]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -8.99999999999999965e49 or 2.2999999999999998e34 < y

   1. Initial program 75.6%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 61.3%

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

      1. mul-1-neg 61.3%

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

   5. Simplified 61.3%

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

   if -8.99999999999999965e49 < y < -8.9999999999999996e-187 or 6.5999999999999997e-156 < y < 3.0999999999999998e-88

   1. Initial program 99.8%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 77.9%

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

   if -8.9999999999999996e-187 < y < 6.5999999999999997e-156 or 3.0999999999999998e-88 < y < 1.60000000000000009e33

   1. Initial program 100.0%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf 87.8%

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

      1. +-commutative 87.8%

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

   5. Simplified 87.8%

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

   if 1.60000000000000009e33 < y < 2.2999999999999998e34

   1. Initial program 100.0%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0 98.4%

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

      1. mul-1-neg 98.4%

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

      2. associate-/l* 100.0%

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

      3. associate-/r/ 100.0%

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

      4. distribute-rgt-neg-in 100.0%

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

      5. +-commutative 100.0%

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

      6. distribute-neg-in 100.0%

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

      7. sub-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around 0 98.4%

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot z}{y}} \]
   7. Step-by-step derivation

      1. mul-1-neg 98.4%

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

      2. distribute-neg-frac 98.4%

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

      3. distribute-lft-neg-out 98.4%

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

      4. associate-*r/ 100.0%

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

      5. *-commutative 100.0%

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

   8. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{z}{y} \cdot \left(-x\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9 \cdot 10^{+49}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq -9 \cdot 10^{-187}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{elif}\;y \leq 6.6 \cdot 10^{-156}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{-88}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{+33}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{+34}:\\ \;\;\;\;\frac{z}{y} \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 67.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{y}{z}\\ t_1 := \frac{x}{t\_0}\\ \mathbf{if}\;x \leq -5.5 \cdot 10^{-32}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 6 \cdot 10^{-98}:\\ \;\;\;\;\frac{y}{t\_0}\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{-75} \lor \neg \left(x \leq 9 \cdot 10^{+15}\right):\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ y z))) (t_1 (/ x t_0)))
   (if (<= x -5.5e-32)
     t_1
     (if (<= x 6e-98)
       (/ y t_0)
       (if (or (<= x 5.2e-75) (not (<= x 9e+15))) t_1 (- z))))))

C

double code(double x, double y, double z) {
	double t_0 = 1.0 - (y / z);
	double t_1 = x / t_0;
	double tmp;
	if (x <= -5.5e-32) {
		tmp = t_1;
	} else if (x <= 6e-98) {
		tmp = y / t_0;
	} else if ((x <= 5.2e-75) || !(x <= 9e+15)) {
		tmp = t_1;
	} 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 = x / t_0
    if (x <= (-5.5d-32)) then
        tmp = t_1
    else if (x <= 6d-98) then
        tmp = y / t_0
    else if ((x <= 5.2d-75) .or. (.not. (x <= 9d+15))) then
        tmp = t_1
    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 = x / t_0;
	double tmp;
	if (x <= -5.5e-32) {
		tmp = t_1;
	} else if (x <= 6e-98) {
		tmp = y / t_0;
	} else if ((x <= 5.2e-75) || !(x <= 9e+15)) {
		tmp = t_1;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 1.0 - (y / z)
	t_1 = x / t_0
	tmp = 0
	if x <= -5.5e-32:
		tmp = t_1
	elif x <= 6e-98:
		tmp = y / t_0
	elif (x <= 5.2e-75) or not (x <= 9e+15):
		tmp = t_1
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(1.0 - Float64(y / z))
	t_1 = Float64(x / t_0)
	tmp = 0.0
	if (x <= -5.5e-32)
		tmp = t_1;
	elseif (x <= 6e-98)
		tmp = Float64(y / t_0);
	elseif ((x <= 5.2e-75) || !(x <= 9e+15))
		tmp = t_1;
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 1.0 - (y / z);
	t_1 = x / t_0;
	tmp = 0.0;
	if (x <= -5.5e-32)
		tmp = t_1;
	elseif (x <= 6e-98)
		tmp = y / t_0;
	elseif ((x <= 5.2e-75) || ~((x <= 9e+15)))
		tmp = t_1;
	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[(x / t$95$0), $MachinePrecision]}, If[LessEqual[x, -5.5e-32], t$95$1, If[LessEqual[x, 6e-98], N[(y / t$95$0), $MachinePrecision], If[Or[LessEqual[x, 5.2e-75], N[Not[LessEqual[x, 9e+15]], $MachinePrecision]], t$95$1, (-z)]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.50000000000000024e-32 or 6e-98 < x < 5.2e-75 or 9e15 < x

   1. Initial program 92.3%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 77.9%

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

   if -5.50000000000000024e-32 < x < 6e-98

   1. Initial program 89.2%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 72.7%

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

   if 5.2e-75 < x < 9e15

   1. Initial program 78.7%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 58.1%

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

      1. mul-1-neg 58.1%

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

   5. Simplified 58.1%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{-32}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{elif}\;x \leq 6 \cdot 10^{-98}:\\ \;\;\;\;\frac{y}{1 - \frac{y}{z}}\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{-75} \lor \neg \left(x \leq 9 \cdot 10^{+15}\right):\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x + y\right) \cdot \left(1 + \frac{y}{z}\right)\\ \mathbf{if}\;z \leq -6.5 \cdot 10^{+56}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 3.7 \cdot 10^{-96}:\\ \;\;\;\;\left(-z\right) - \frac{x \cdot z}{y}\\ \mathbf{elif}\;z \leq 4.8 \cdot 10^{+14}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (+ x y) (+ 1.0 (/ y z)))))
   (if (<= z -6.5e+56)
     t_0
     (if (<= z 3.7e-96)
       (- (- z) (/ (* x z) y))
       (if (<= z 4.8e+14) (/ x (- 1.0 (/ y z))) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = (x + y) * (1.0 + (y / z));
	double tmp;
	if (z <= -6.5e+56) {
		tmp = t_0;
	} else if (z <= 3.7e-96) {
		tmp = -z - ((x * z) / y);
	} else if (z <= 4.8e+14) {
		tmp = x / (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 = (x + y) * (1.0d0 + (y / z))
    if (z <= (-6.5d+56)) then
        tmp = t_0
    else if (z <= 3.7d-96) then
        tmp = -z - ((x * z) / y)
    else if (z <= 4.8d+14) then
        tmp = x / (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 = (x + y) * (1.0 + (y / z));
	double tmp;
	if (z <= -6.5e+56) {
		tmp = t_0;
	} else if (z <= 3.7e-96) {
		tmp = -z - ((x * z) / y);
	} else if (z <= 4.8e+14) {
		tmp = x / (1.0 - (y / z));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x + y) * (1.0 + (y / z))
	tmp = 0
	if z <= -6.5e+56:
		tmp = t_0
	elif z <= 3.7e-96:
		tmp = -z - ((x * z) / y)
	elif z <= 4.8e+14:
		tmp = x / (1.0 - (y / z))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x + y) * Float64(1.0 + Float64(y / z)))
	tmp = 0.0
	if (z <= -6.5e+56)
		tmp = t_0;
	elseif (z <= 3.7e-96)
		tmp = Float64(Float64(-z) - Float64(Float64(x * z) / y));
	elseif (z <= 4.8e+14)
		tmp = Float64(x / Float64(1.0 - Float64(y / z)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x + y) * (1.0 + (y / z));
	tmp = 0.0;
	if (z <= -6.5e+56)
		tmp = t_0;
	elseif (z <= 3.7e-96)
		tmp = -z - ((x * z) / y);
	elseif (z <= 4.8e+14)
		tmp = x / (1.0 - (y / z));
	else
		tmp = t_0;
	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[LessEqual[z, -6.5e+56], t$95$0, If[LessEqual[z, 3.7e-96], N[((-z) - N[(N[(x * z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 4.8e+14], N[(x / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if z < -6.5000000000000001e56 or 4.8e14 < z

   1. Initial program 99.9%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf 75.6%

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

      1. associate-+r+ 75.6%

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

      2. associate-/l* 84.1%

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

      3. associate-/r/ 84.1%

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

      4. *-lft-identity 84.1%

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

      5. distribute-rgt-in 84.0%

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

      6. +-commutative 84.0%

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

   5. Simplified 84.0%

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

   if -6.5000000000000001e56 < z < 3.69999999999999986e-96

   1. Initial program 79.2%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0 75.0%

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

      1. mul-1-neg 75.0%

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

      2. associate-/l* 76.9%

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

      3. associate-/r/ 61.4%

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

      4. distribute-rgt-neg-in 61.4%

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

      5. +-commutative 61.4%

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

      6. distribute-neg-in 61.4%

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

      7. sub-neg 61.4%

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

   5. Simplified 61.4%

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

   6. Taylor expanded in y around 0 78.3%

      \[\leadsto \color{blue}{-1 \cdot z + -1 \cdot \frac{x \cdot z}{y}} \]
   7. Step-by-step derivation

      1. mul-1-neg 78.3%

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

      2. unsub-neg 78.3%

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

      3. neg-mul-1 78.3%

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

   8. Simplified 78.3%

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

   if 3.69999999999999986e-96 < z < 4.8e14

   1. Initial program 96.0%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 80.7%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.5 \cdot 10^{+56}:\\ \;\;\;\;\left(x + y\right) \cdot \left(1 + \frac{y}{z}\right)\\ \mathbf{elif}\;z \leq 3.7 \cdot 10^{-96}:\\ \;\;\;\;\left(-z\right) - \frac{x \cdot z}{y}\\ \mathbf{elif}\;z \leq 4.8 \cdot 10^{+14}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) \cdot \left(1 + \frac{y}{z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6.3 \cdot 10^{+56}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq 5.8 \cdot 10^{-100}:\\ \;\;\;\;\frac{z}{y} \cdot \left(-x\right) - z\\ \mathbf{elif}\;z \leq 5200000000000:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -6.3e+56)
   (+ x y)
   (if (<= z 5.8e-100)
     (- (* (/ z y) (- x)) z)
     (if (<= z 5200000000000.0) (/ x (- 1.0 (/ y z))) (+ x y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -6.3e+56) {
		tmp = x + y;
	} else if (z <= 5.8e-100) {
		tmp = ((z / y) * -x) - z;
	} else if (z <= 5200000000000.0) {
		tmp = x / (1.0 - (y / z));
	} else {
		tmp = x + y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-6.3d+56)) then
        tmp = x + y
    else if (z <= 5.8d-100) then
        tmp = ((z / y) * -x) - z
    else if (z <= 5200000000000.0d0) then
        tmp = x / (1.0d0 - (y / z))
    else
        tmp = x + y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -6.3e+56) {
		tmp = x + y;
	} else if (z <= 5.8e-100) {
		tmp = ((z / y) * -x) - z;
	} else if (z <= 5200000000000.0) {
		tmp = x / (1.0 - (y / z));
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -6.3e+56:
		tmp = x + y
	elif z <= 5.8e-100:
		tmp = ((z / y) * -x) - z
	elif z <= 5200000000000.0:
		tmp = x / (1.0 - (y / z))
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -6.3e+56)
		tmp = Float64(x + y);
	elseif (z <= 5.8e-100)
		tmp = Float64(Float64(Float64(z / y) * Float64(-x)) - z);
	elseif (z <= 5200000000000.0)
		tmp = Float64(x / Float64(1.0 - Float64(y / z)));
	else
		tmp = Float64(x + y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -6.3e+56)
		tmp = x + y;
	elseif (z <= 5.8e-100)
		tmp = ((z / y) * -x) - z;
	elseif (z <= 5200000000000.0)
		tmp = x / (1.0 - (y / z));
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -6.3e+56], N[(x + y), $MachinePrecision], If[LessEqual[z, 5.8e-100], N[(N[(N[(z / y), $MachinePrecision] * (-x)), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[z, 5200000000000.0], N[(x / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -6.3000000000000001e56 or 5.2e12 < z

   1. Initial program 99.9%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf 82.5%

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

      1. +-commutative 82.5%

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

   5. Simplified 82.5%

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

   if -6.3000000000000001e56 < z < 5.79999999999999951e-100

   1. Initial program 79.2%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0 75.0%

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

      1. mul-1-neg 75.0%

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

      2. +-commutative 75.0%

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

   5. Simplified 75.0%

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

   6. Taylor expanded in y around 0 78.3%

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

      1. *-commutative 78.3%

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

      2. associate-/l* 76.9%

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

      3. associate-/r/ 76.7%

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

   8. Simplified 76.7%

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

   if 5.79999999999999951e-100 < z < 5.2e12

   1. Initial program 95.8%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 84.0%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.3 \cdot 10^{+56}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq 5.8 \cdot 10^{-100}:\\ \;\;\;\;\frac{z}{y} \cdot \left(-x\right) - z\\ \mathbf{elif}\;z \leq 5200000000000:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6.3 \cdot 10^{+56}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq 4.3 \cdot 10^{-94}:\\ \;\;\;\;\left(-z\right) - \frac{x \cdot z}{y}\\ \mathbf{elif}\;z \leq 3850000000000:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -6.3e+56)
   (+ x y)
   (if (<= z 4.3e-94)
     (- (- z) (/ (* x z) y))
     (if (<= z 3850000000000.0) (/ x (- 1.0 (/ y z))) (+ x y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -6.3e+56) {
		tmp = x + y;
	} else if (z <= 4.3e-94) {
		tmp = -z - ((x * z) / y);
	} else if (z <= 3850000000000.0) {
		tmp = x / (1.0 - (y / z));
	} else {
		tmp = x + y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-6.3d+56)) then
        tmp = x + y
    else if (z <= 4.3d-94) then
        tmp = -z - ((x * z) / y)
    else if (z <= 3850000000000.0d0) then
        tmp = x / (1.0d0 - (y / z))
    else
        tmp = x + y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -6.3e+56) {
		tmp = x + y;
	} else if (z <= 4.3e-94) {
		tmp = -z - ((x * z) / y);
	} else if (z <= 3850000000000.0) {
		tmp = x / (1.0 - (y / z));
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -6.3e+56:
		tmp = x + y
	elif z <= 4.3e-94:
		tmp = -z - ((x * z) / y)
	elif z <= 3850000000000.0:
		tmp = x / (1.0 - (y / z))
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -6.3e+56)
		tmp = Float64(x + y);
	elseif (z <= 4.3e-94)
		tmp = Float64(Float64(-z) - Float64(Float64(x * z) / y));
	elseif (z <= 3850000000000.0)
		tmp = Float64(x / Float64(1.0 - Float64(y / z)));
	else
		tmp = Float64(x + y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -6.3e+56)
		tmp = x + y;
	elseif (z <= 4.3e-94)
		tmp = -z - ((x * z) / y);
	elseif (z <= 3850000000000.0)
		tmp = x / (1.0 - (y / z));
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -6.3e+56], N[(x + y), $MachinePrecision], If[LessEqual[z, 4.3e-94], N[((-z) - N[(N[(x * z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 3850000000000.0], N[(x / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -6.3000000000000001e56 or 3.85e12 < z

   1. Initial program 99.9%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf 82.5%

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

      1. +-commutative 82.5%

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

   5. Simplified 82.5%

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

   if -6.3000000000000001e56 < z < 4.2999999999999998e-94

   1. Initial program 79.2%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0 75.0%

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

      1. mul-1-neg 75.0%

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

      2. associate-/l* 76.9%

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

      3. associate-/r/ 61.4%

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

      4. distribute-rgt-neg-in 61.4%

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

      5. +-commutative 61.4%

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

      6. distribute-neg-in 61.4%

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

      7. sub-neg 61.4%

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

   5. Simplified 61.4%

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

   6. Taylor expanded in y around 0 78.3%

      \[\leadsto \color{blue}{-1 \cdot z + -1 \cdot \frac{x \cdot z}{y}} \]
   7. Step-by-step derivation

      1. mul-1-neg 78.3%

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

      2. unsub-neg 78.3%

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

      3. neg-mul-1 78.3%

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

   8. Simplified 78.3%

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

   if 4.2999999999999998e-94 < z < 3.85e12

   1. Initial program 95.8%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 84.0%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.3 \cdot 10^{+56}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq 4.3 \cdot 10^{-94}:\\ \;\;\;\;\left(-z\right) - \frac{x \cdot z}{y}\\ \mathbf{elif}\;z \leq 3850000000000:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.4 \cdot 10^{+124} \lor \neg \left(y \leq 2.1 \cdot 10^{+33}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -3.4e+124) (not (<= y 2.1e+33))) (- z) (+ x y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -3.4e+124) || !(y <= 2.1e+33)) {
		tmp = -z;
	} else {
		tmp = x + y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((y <= (-3.4d+124)) .or. (.not. (y <= 2.1d+33))) then
        tmp = -z
    else
        tmp = x + y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -3.4e+124) || !(y <= 2.1e+33)) {
		tmp = -z;
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -3.4e+124) or not (y <= 2.1e+33):
		tmp = -z
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -3.4e+124) || !(y <= 2.1e+33))
		tmp = Float64(-z);
	else
		tmp = Float64(x + y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -3.4e+124) || ~((y <= 2.1e+33)))
		tmp = -z;
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -3.4e+124], N[Not[LessEqual[y, 2.1e+33]], $MachinePrecision]], (-z), N[(x + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.4e124 or 2.1000000000000001e33 < y

   1. Initial program 75.8%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 68.7%

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

      1. mul-1-neg 68.7%

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

   5. Simplified 68.7%

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

   if -3.4e124 < y < 2.1000000000000001e33

   1. Initial program 97.2%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf 67.6%

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

      1. +-commutative 67.6%

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

   5. Simplified 67.6%

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

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.4 \cdot 10^{+124} \lor \neg \left(y \leq 2.1 \cdot 10^{+33}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 58.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.2 \cdot 10^{+48} \lor \neg \left(y \leq 2.3 \cdot 10^{+32}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -5.2e+48) (not (<= y 2.3e+32))) (- z) x))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -5.2e+48) || !(y <= 2.3e+32)) {
		tmp = -z;
	} 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 ((y <= (-5.2d+48)) .or. (.not. (y <= 2.3d+32))) then
        tmp = -z
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -5.2e+48) || !(y <= 2.3e+32)) {
		tmp = -z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -5.2e+48) or not (y <= 2.3e+32):
		tmp = -z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -5.2e+48) || !(y <= 2.3e+32))
		tmp = Float64(-z);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -5.2e+48) || ~((y <= 2.3e+32)))
		tmp = -z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -5.2e+48], N[Not[LessEqual[y, 2.3e+32]], $MachinePrecision]], (-z), x]

Derivation

1. Split input into 2 regimes

2. if y < -5.1999999999999999e48 or 2.3e32 < y

   1. Initial program 76.1%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 60.2%

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

      1. mul-1-neg 60.2%

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

   5. Simplified 60.2%

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

   if -5.1999999999999999e48 < y < 2.3e32

   1. Initial program 99.9%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 56.6%

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

4. Final simplification 58.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.2 \cdot 10^{+48} \lor \neg \left(y \leq 2.3 \cdot 10^{+32}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 40.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{-162}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{-147}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.05e-162) x (if (<= x 4.6e-147) y x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.05e-162) {
		tmp = x;
	} else if (x <= 4.6e-147) {
		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 <= (-1.05d-162)) then
        tmp = x
    else if (x <= 4.6d-147) 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 <= -1.05e-162) {
		tmp = x;
	} else if (x <= 4.6e-147) {
		tmp = y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.05e-162:
		tmp = x
	elif x <= 4.6e-147:
		tmp = y
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.05e-162)
		tmp = x;
	elseif (x <= 4.6e-147)
		tmp = y;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.05e-162)
		tmp = x;
	elseif (x <= 4.6e-147)
		tmp = y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.05e-162], x, If[LessEqual[x, 4.6e-147], y, x]]

Derivation

1. Split input into 2 regimes

2. if x < -1.05e-162 or 4.59999999999999981e-147 < x

   1. Initial program 91.0%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 45.8%

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

   if -1.05e-162 < x < 4.59999999999999981e-147

   1. Initial program 87.9%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 81.0%

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

   4. Taylor expanded in y around 0 42.6%

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

4. Final simplification 45.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{-162}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{-147}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 34.4% accurate, 9.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := x

Derivation

1. Initial program 90.3%

   \[\frac{x + y}{1 - \frac{y}{z}} \]
2. Add Preprocessing

3. Taylor expanded in y around 0 37.6%

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

4. Final simplification 37.6%

   \[\leadsto x \]
5. Add Preprocessing

Developer target: 93.5% accurate, 0.5× 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 2024034 
(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))))