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

Percentage Accurate: 87.7% → 99.1%

Time: 6.2s

Alternatives: 7

Speedup: 0.3×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 87.7% 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: 99.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-223} \lor \neg \left(t\_0 \leq 0\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (+ x y) (- 1.0 (/ y z)))))
   (if (or (<= t_0 -1e-223) (not (<= t_0 0.0))) t_0 (* z (- -1.0 (/ x y))))))

C

double code(double x, double y, double z) {
	double t_0 = (x + y) / (1.0 - (y / z));
	double tmp;
	if ((t_0 <= -1e-223) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = z * (-1.0 - (x / y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x + y) / (1.0d0 - (y / z))
    if ((t_0 <= (-1d-223)) .or. (.not. (t_0 <= 0.0d0))) then
        tmp = t_0
    else
        tmp = z * ((-1.0d0) - (x / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (x + y) / (1.0 - (y / z));
	double tmp;
	if ((t_0 <= -1e-223) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = z * (-1.0 - (x / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x + y) / (1.0 - (y / z))
	tmp = 0
	if (t_0 <= -1e-223) or not (t_0 <= 0.0):
		tmp = t_0
	else:
		tmp = z * (-1.0 - (x / y))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x + y) / Float64(1.0 - Float64(y / z)))
	tmp = 0.0
	if ((t_0 <= -1e-223) || !(t_0 <= 0.0))
		tmp = t_0;
	else
		tmp = Float64(z * Float64(-1.0 - Float64(x / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x + y) / (1.0 - (y / z));
	tmp = 0.0;
	if ((t_0 <= -1e-223) || ~((t_0 <= 0.0)))
		tmp = t_0;
	else
		tmp = z * (-1.0 - (x / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x + y), $MachinePrecision] / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -1e-223], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], t$95$0, N[(z * N[(-1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z))) < -9.9999999999999997e-224 or 0.0 < (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z)))

   1. Initial program 99.8%

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

   if -9.9999999999999997e-224 < (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z))) < 0.0

   1. Initial program 18.6%

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

   3. Taylor expanded in y around inf 18.6%

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

      1. neg-mul-1 18.6%

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

      2. distribute-neg-frac 18.6%

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

   5. Simplified 18.6%

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

      1. distribute-frac-neg 18.6%

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

      2. distribute-frac-neg2 18.6%

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

      3. associate-/r/ 99.9%

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

      4. +-commutative 99.9%

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

   7. Applied egg-rr 99.9%

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

   8. Taylor expanded in y around 0 87.7%

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

      1. distribute-lft-out 87.7%

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

      2. distribute-rgt-in 87.7%

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

      3. associate-*r/ 87.7%

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

      4. mul-1-neg 87.7%

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

      5. associate-/l* 99.9%

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

      6. +-commutative 99.9%

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

      7. distribute-rgt-neg-in 99.9%

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

      8. *-lft-identity 99.9%

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

      9. associate-*l/ 99.7%

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

      10. distribute-lft-in 99.8%

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

      11. lft-mult-inverse 99.9%

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

      12. distribute-neg-in 99.9%

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

      13. metadata-eval 99.9%

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

      14. associate-*l/ 99.9%

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

      15. associate-*r/ 99.9%

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

      16. *-lft-identity 99.9%

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

      17. unsub-neg 99.9%

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

   10. Simplified 99.9%

       \[\leadsto \color{blue}{z \cdot \left(-1 - \frac{x}{y}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \leq -1 \cdot 10^{-223} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \leq 0\right):\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 72.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.4 \cdot 10^{+39}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq 9.2 \cdot 10^{+33}:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) \cdot \left(1 + \frac{y}{z}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -5.4e+39)
   (+ x y)
   (if (<= z 9.2e+33) (* z (- -1.0 (/ x y))) (* (+ x y) (+ 1.0 (/ y z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -5.4e+39) {
		tmp = x + y;
	} else if (z <= 9.2e+33) {
		tmp = z * (-1.0 - (x / y));
	} else {
		tmp = (x + y) * (1.0 + (y / 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 (z <= (-5.4d+39)) then
        tmp = x + y
    else if (z <= 9.2d+33) then
        tmp = z * ((-1.0d0) - (x / y))
    else
        tmp = (x + y) * (1.0d0 + (y / z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -5.4e+39) {
		tmp = x + y;
	} else if (z <= 9.2e+33) {
		tmp = z * (-1.0 - (x / y));
	} else {
		tmp = (x + y) * (1.0 + (y / z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -5.4e+39:
		tmp = x + y
	elif z <= 9.2e+33:
		tmp = z * (-1.0 - (x / y))
	else:
		tmp = (x + y) * (1.0 + (y / z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -5.4e+39)
		tmp = Float64(x + y);
	elseif (z <= 9.2e+33)
		tmp = Float64(z * Float64(-1.0 - Float64(x / y)));
	else
		tmp = Float64(Float64(x + y) * Float64(1.0 + Float64(y / z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -5.4e+39)
		tmp = x + y;
	elseif (z <= 9.2e+33)
		tmp = z * (-1.0 - (x / y));
	else
		tmp = (x + y) * (1.0 + (y / z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -5.4e+39], N[(x + y), $MachinePrecision], If[LessEqual[z, 9.2e+33], N[(z * N[(-1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x + y), $MachinePrecision] * N[(1.0 + N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.40000000000000007e39

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf 84.9%

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

      1. +-commutative 84.9%

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

   5. Simplified 84.9%

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

   if -5.40000000000000007e39 < z < 9.20000000000000042e33

   1. Initial program 82.7%

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

   3. Taylor expanded in y around inf 61.3%

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

      1. neg-mul-1 61.3%

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

      2. distribute-neg-frac 61.3%

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

   5. Simplified 61.3%

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

      1. distribute-frac-neg 61.3%

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

      2. distribute-frac-neg2 61.3%

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

      3. associate-/r/ 74.5%

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

      4. +-commutative 74.5%

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

   7. Applied egg-rr 74.5%

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

   8. Taylor expanded in y around 0 72.6%

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

      1. distribute-lft-out 72.6%

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

      2. distribute-rgt-in 72.6%

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

      3. associate-*r/ 72.6%

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

      4. mul-1-neg 72.6%

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

      5. associate-/l* 74.5%

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

      6. +-commutative 74.5%

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

      7. distribute-rgt-neg-in 74.5%

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

      8. *-lft-identity 74.5%

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

      9. associate-*l/ 74.4%

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

      10. distribute-lft-in 74.4%

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

      11. lft-mult-inverse 74.4%

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

      12. distribute-neg-in 74.4%

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

      13. metadata-eval 74.4%

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

      14. associate-*l/ 74.5%

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

      15. associate-*r/ 74.5%

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

      16. *-lft-identity 74.5%

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

      17. unsub-neg 74.5%

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

   10. Simplified 74.5%

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

   if 9.20000000000000042e33 < z

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf 73.1%

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

      1. associate-+r+ 73.1%

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

      2. *-rgt-identity 73.1%

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

      3. *-commutative 73.1%

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

      4. associate-/l* 79.5%

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

      5. distribute-lft-in 79.5%

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

      6. +-commutative 79.5%

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

   5. Simplified 79.5%

      \[\leadsto \color{blue}{\left(y + x\right) \cdot \left(1 + \frac{y}{z}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.4 \cdot 10^{+39}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq 9.2 \cdot 10^{+33}:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) \cdot \left(1 + \frac{y}{z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 72.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.15 \cdot 10^{+40} \lor \neg \left(z \leq 4.2 \cdot 10^{+33}\right):\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -2.15e+40) (not (<= z 4.2e+33)))
   (+ x y)
   (* z (- -1.0 (/ x y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -2.15e+40) || !(z <= 4.2e+33)) {
		tmp = x + y;
	} else {
		tmp = z * (-1.0 - (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 <= (-2.15d+40)) .or. (.not. (z <= 4.2d+33))) then
        tmp = x + y
    else
        tmp = z * ((-1.0d0) - (x / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -2.15e+40) || !(z <= 4.2e+33)) {
		tmp = x + y;
	} else {
		tmp = z * (-1.0 - (x / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -2.15e+40) or not (z <= 4.2e+33):
		tmp = x + y
	else:
		tmp = z * (-1.0 - (x / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -2.15e+40) || !(z <= 4.2e+33))
		tmp = Float64(x + y);
	else
		tmp = Float64(z * Float64(-1.0 - Float64(x / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -2.15e+40) || ~((z <= 4.2e+33)))
		tmp = x + y;
	else
		tmp = z * (-1.0 - (x / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -2.15e+40], N[Not[LessEqual[z, 4.2e+33]], $MachinePrecision]], N[(x + y), $MachinePrecision], N[(z * N[(-1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.1500000000000001e40 or 4.2000000000000001e33 < z

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf 82.2%

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

      1. +-commutative 82.2%

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

   5. Simplified 82.2%

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

   if -2.1500000000000001e40 < z < 4.2000000000000001e33

   1. Initial program 82.7%

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

   3. Taylor expanded in y around inf 61.3%

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

      1. neg-mul-1 61.3%

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

      2. distribute-neg-frac 61.3%

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

   5. Simplified 61.3%

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

      1. distribute-frac-neg 61.3%

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

      2. distribute-frac-neg2 61.3%

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

      3. associate-/r/ 74.5%

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

      4. +-commutative 74.5%

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

   7. Applied egg-rr 74.5%

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

   8. Taylor expanded in y around 0 72.6%

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

      1. distribute-lft-out 72.6%

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

      2. distribute-rgt-in 72.6%

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

      3. associate-*r/ 72.6%

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

      4. mul-1-neg 72.6%

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

      5. associate-/l* 74.5%

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

      6. +-commutative 74.5%

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

      7. distribute-rgt-neg-in 74.5%

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

      8. *-lft-identity 74.5%

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

      9. associate-*l/ 74.4%

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

      10. distribute-lft-in 74.4%

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

      11. lft-mult-inverse 74.4%

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

      12. distribute-neg-in 74.4%

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

      13. metadata-eval 74.4%

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

      14. associate-*l/ 74.5%

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

      15. associate-*r/ 74.5%

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

      16. *-lft-identity 74.5%

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

      17. unsub-neg 74.5%

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

   10. Simplified 74.5%

       \[\leadsto \color{blue}{z \cdot \left(-1 - \frac{x}{y}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.15 \cdot 10^{+40} \lor \neg \left(z \leq 4.2 \cdot 10^{+33}\right):\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 67.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.5 \cdot 10^{+17} \lor \neg \left(y \leq 4.8 \cdot 10^{+95}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -7.5e+17) (not (<= y 4.8e+95))) (- z) (+ x y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -7.5e+17) || !(y <= 4.8e+95)) {
		tmp = -z;
	} else {
		tmp = x + y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((y <= (-7.5d+17)) .or. (.not. (y <= 4.8d+95))) then
        tmp = -z
    else
        tmp = x + y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -7.5e+17) || !(y <= 4.8e+95)) {
		tmp = -z;
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -7.5e+17) or not (y <= 4.8e+95):
		tmp = -z
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -7.5e+17) || !(y <= 4.8e+95))
		tmp = Float64(-z);
	else
		tmp = Float64(x + y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -7.5e+17) || ~((y <= 4.8e+95)))
		tmp = -z;
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -7.5e+17], N[Not[LessEqual[y, 4.8e+95]], $MachinePrecision]], (-z), N[(x + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -7.5e17 or 4.8000000000000001e95 < y

   1. Initial program 76.8%

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

   3. Taylor expanded in y around inf 61.6%

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

      1. mul-1-neg 61.6%

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

   5. Simplified 61.6%

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

   if -7.5e17 < y < 4.8000000000000001e95

   1. Initial program 99.3%

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

   3. Taylor expanded in z around inf 70.1%

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

      1. +-commutative 70.1%

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

   5. Simplified 70.1%

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

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.5 \cdot 10^{+17} \lor \neg \left(y \leq 4.8 \cdot 10^{+95}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 57.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.3 \cdot 10^{-43} \lor \neg \left(y \leq 1.05 \cdot 10^{+83}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1.3e-43) (not (<= y 1.05e+83))) (- z) x))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.3e-43) || !(y <= 1.05e+83)) {
		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 <= (-1.3d-43)) .or. (.not. (y <= 1.05d+83))) 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 <= -1.3e-43) || !(y <= 1.05e+83)) {
		tmp = -z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -1.3e-43) or not (y <= 1.05e+83):
		tmp = -z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -1.3e-43) || !(y <= 1.05e+83))
		tmp = Float64(-z);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -1.3e-43) || ~((y <= 1.05e+83)))
		tmp = -z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -1.3e-43], N[Not[LessEqual[y, 1.05e+83]], $MachinePrecision]], (-z), x]

Derivation

1. Split input into 2 regimes

2. if y < -1.3e-43 or 1.05000000000000001e83 < y

   1. Initial program 78.7%

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

   3. Taylor expanded in y around inf 57.3%

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

      1. mul-1-neg 57.3%

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

   5. Simplified 57.3%

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

   if -1.3e-43 < y < 1.05000000000000001e83

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 59.9%

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

4. Final simplification 58.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.3 \cdot 10^{-43} \lor \neg \left(y \leq 1.05 \cdot 10^{+83}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 35.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.8 \cdot 10^{-33}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= y -3.8e-33) y x))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -3.8e-33) {
		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 (y <= (-3.8d-33)) 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 (y <= -3.8e-33) {
		tmp = y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -3.8e-33:
		tmp = y
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -3.8e-33)
		tmp = y;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -3.8e-33)
		tmp = y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -3.8e-33], y, x]

Derivation

1. Split input into 2 regimes

2. if y < -3.79999999999999994e-33

   1. Initial program 81.8%

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

   3. Taylor expanded in x around 0 61.2%

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

   4. Taylor expanded in y around 0 18.5%

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

   if -3.79999999999999994e-33 < y

   1. Initial program 93.4%

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

   3. Taylor expanded in y around 0 46.6%

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

4. Final simplification 39.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.8 \cdot 10^{-33}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 34.0% 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 36.2%

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

4. Final simplification 36.2%

   \[\leadsto x \]
5. Add Preprocessing

Developer target: 93.4% 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 2024055 
(FPCore (x y z)
  :name "Graphics.Rendering.Chart.Backend.Diagrams:calcFontMetrics from Chart-diagrams-1.5.1, A"
  :precision binary64

  :alt
  (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))))