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

Percentage Accurate: 88.3% → 99.6%

Time: 6.9s

Alternatives: 9

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: 88.3% 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: 99.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-268} \lor \neg \left(t\_0 \leq 0\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) \cdot \frac{x + y}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (+ x y) (- 1.0 (/ y z)))))
   (if (or (<= t_0 -5e-268) (not (<= t_0 0.0))) t_0 (* (- z) (/ (+ x y) 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-268) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = -z * ((x + y) / 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-268)) .or. (.not. (t_0 <= 0.0d0))) then
        tmp = t_0
    else
        tmp = -z * ((x + y) / 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-268) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = -z * ((x + y) / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x + y) / (1.0 - (y / z))
	tmp = 0
	if (t_0 <= -5e-268) or not (t_0 <= 0.0):
		tmp = t_0
	else:
		tmp = -z * ((x + y) / 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-268) || !(t_0 <= 0.0))
		tmp = t_0;
	else
		tmp = Float64(Float64(-z) * Float64(Float64(x + y) / 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-268) || ~((t_0 <= 0.0)))
		tmp = t_0;
	else
		tmp = -z * ((x + y) / 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-268], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], t$95$0, N[((-z) * N[(N[(x + y), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z))) < -4.9999999999999999e-268 or -0.0 < (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z)))

   1. Initial program 99.9%

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

   if -4.9999999999999999e-268 < (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z))) < -0.0

   1. Initial program 5.4%

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

   3. Taylor expanded in z around 0 99.9%

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

      1. mul-1-neg 99.9%

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

      2. associate-/l* 100.0%

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

      3. distribute-rgt-neg-in 100.0%

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

      4. distribute-neg-frac2 100.0%

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

      5. +-commutative 100.0%

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

   5. Simplified 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \leq -5 \cdot 10^{-268} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \leq 0\right):\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) \cdot \frac{x + y}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 74.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.05 \cdot 10^{-20}:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{elif}\;y \leq 8.6 \cdot 10^{-127}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 1.16 \cdot 10^{-68}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{+44}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) \cdot \frac{x + y}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2.05e-20)
   (* z (- -1.0 (/ x y)))
   (if (<= y 8.6e-127)
     (+ x y)
     (if (<= y 1.16e-68)
       (/ x (- 1.0 (/ y z)))
       (if (<= y 1.8e+44) (+ x y) (* (- z) (/ (+ x y) y)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.05e-20) {
		tmp = z * (-1.0 - (x / y));
	} else if (y <= 8.6e-127) {
		tmp = x + y;
	} else if (y <= 1.16e-68) {
		tmp = x / (1.0 - (y / z));
	} else if (y <= 1.8e+44) {
		tmp = x + y;
	} else {
		tmp = -z * ((x + y) / 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 <= (-2.05d-20)) then
        tmp = z * ((-1.0d0) - (x / y))
    else if (y <= 8.6d-127) then
        tmp = x + y
    else if (y <= 1.16d-68) then
        tmp = x / (1.0d0 - (y / z))
    else if (y <= 1.8d+44) then
        tmp = x + y
    else
        tmp = -z * ((x + y) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.05e-20) {
		tmp = z * (-1.0 - (x / y));
	} else if (y <= 8.6e-127) {
		tmp = x + y;
	} else if (y <= 1.16e-68) {
		tmp = x / (1.0 - (y / z));
	} else if (y <= 1.8e+44) {
		tmp = x + y;
	} else {
		tmp = -z * ((x + y) / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -2.05e-20:
		tmp = z * (-1.0 - (x / y))
	elif y <= 8.6e-127:
		tmp = x + y
	elif y <= 1.16e-68:
		tmp = x / (1.0 - (y / z))
	elif y <= 1.8e+44:
		tmp = x + y
	else:
		tmp = -z * ((x + y) / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -2.05e-20)
		tmp = Float64(z * Float64(-1.0 - Float64(x / y)));
	elseif (y <= 8.6e-127)
		tmp = Float64(x + y);
	elseif (y <= 1.16e-68)
		tmp = Float64(x / Float64(1.0 - Float64(y / z)));
	elseif (y <= 1.8e+44)
		tmp = Float64(x + y);
	else
		tmp = Float64(Float64(-z) * Float64(Float64(x + y) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -2.05e-20)
		tmp = z * (-1.0 - (x / y));
	elseif (y <= 8.6e-127)
		tmp = x + y;
	elseif (y <= 1.16e-68)
		tmp = x / (1.0 - (y / z));
	elseif (y <= 1.8e+44)
		tmp = x + y;
	else
		tmp = -z * ((x + y) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -2.05e-20], N[(z * N[(-1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 8.6e-127], N[(x + y), $MachinePrecision], If[LessEqual[y, 1.16e-68], N[(x / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.8e+44], N[(x + y), $MachinePrecision], N[((-z) * N[(N[(x + y), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -2.05e-20

   1. Initial program 76.9%

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

      1. add-cube-cbrt 75.7%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}\right) \cdot \sqrt[3]{x + y}}}{1 - \frac{y}{z}} \]

      2. associate-/l* 75.7%

         \[\leadsto \color{blue}{\left(\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}\right) \cdot \frac{\sqrt[3]{x + y}}{1 - \frac{y}{z}}} \]

      3. pow2 75.7%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{x + y}\right)}^{2}} \cdot \frac{\sqrt[3]{x + y}}{1 - \frac{y}{z}} \]

   4. Applied egg-rr 75.7%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{x + y}\right)}^{2} \cdot \frac{\sqrt[3]{x + y}}{1 - \frac{y}{z}}} \]

   5. Taylor expanded in z around 0 65.8%

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

      1. mul-1-neg 65.8%

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

      2. associate-/l* 79.0%

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

      3. distribute-lft-neg-in 79.0%

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

      4. +-commutative 79.0%

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

   7. Simplified 79.0%

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

   8. Taylor expanded in y around inf 74.1%

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

      1. mul-1-neg 74.1%

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

      2. *-commutative 74.1%

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

      3. associate-*r/ 79.0%

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

      4. *-commutative 79.0%

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

      5. distribute-lft-neg-out 79.0%

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

      6. distribute-rgt-in 79.0%

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

      7. sub-neg 79.0%

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

   10. Simplified 79.0%

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

   if -2.05e-20 < y < 8.59999999999999967e-127 or 1.1599999999999999e-68 < y < 1.8e44

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf 85.7%

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

      1. +-commutative 85.7%

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

   5. Simplified 85.7%

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

   if 8.59999999999999967e-127 < y < 1.1599999999999999e-68

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 100.0%

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

   if 1.8e44 < y

   1. Initial program 74.8%

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

   3. Taylor expanded in z around 0 65.9%

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

      1. mul-1-neg 65.9%

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

      2. associate-/l* 80.8%

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

      3. distribute-rgt-neg-in 80.8%

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

      4. distribute-neg-frac2 80.8%

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

      5. +-commutative 80.8%

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

   5. Simplified 80.8%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.05 \cdot 10^{-20}:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{elif}\;y \leq 8.6 \cdot 10^{-127}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 1.16 \cdot 10^{-68}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{+44}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) \cdot \frac{x + y}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.1 \cdot 10^{+27}:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{elif}\;y \leq 8.6 \cdot 10^{-127}:\\ \;\;\;\;\left(x + y\right) \cdot \left(1 + \frac{y}{z}\right)\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{-69}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{elif}\;y \leq 2.9 \cdot 10^{+38}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) \cdot \frac{x + y}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -3.1e+27)
   (* z (- -1.0 (/ x y)))
   (if (<= y 8.6e-127)
     (* (+ x y) (+ 1.0 (/ y z)))
     (if (<= y 2.6e-69)
       (/ x (- 1.0 (/ y z)))
       (if (<= y 2.9e+38) (+ x y) (* (- z) (/ (+ x y) y)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -3.1e+27) {
		tmp = z * (-1.0 - (x / y));
	} else if (y <= 8.6e-127) {
		tmp = (x + y) * (1.0 + (y / z));
	} else if (y <= 2.6e-69) {
		tmp = x / (1.0 - (y / z));
	} else if (y <= 2.9e+38) {
		tmp = x + y;
	} else {
		tmp = -z * ((x + y) / 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 <= (-3.1d+27)) then
        tmp = z * ((-1.0d0) - (x / y))
    else if (y <= 8.6d-127) then
        tmp = (x + y) * (1.0d0 + (y / z))
    else if (y <= 2.6d-69) then
        tmp = x / (1.0d0 - (y / z))
    else if (y <= 2.9d+38) then
        tmp = x + y
    else
        tmp = -z * ((x + y) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -3.1e+27) {
		tmp = z * (-1.0 - (x / y));
	} else if (y <= 8.6e-127) {
		tmp = (x + y) * (1.0 + (y / z));
	} else if (y <= 2.6e-69) {
		tmp = x / (1.0 - (y / z));
	} else if (y <= 2.9e+38) {
		tmp = x + y;
	} else {
		tmp = -z * ((x + y) / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -3.1e+27:
		tmp = z * (-1.0 - (x / y))
	elif y <= 8.6e-127:
		tmp = (x + y) * (1.0 + (y / z))
	elif y <= 2.6e-69:
		tmp = x / (1.0 - (y / z))
	elif y <= 2.9e+38:
		tmp = x + y
	else:
		tmp = -z * ((x + y) / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -3.1e+27)
		tmp = Float64(z * Float64(-1.0 - Float64(x / y)));
	elseif (y <= 8.6e-127)
		tmp = Float64(Float64(x + y) * Float64(1.0 + Float64(y / z)));
	elseif (y <= 2.6e-69)
		tmp = Float64(x / Float64(1.0 - Float64(y / z)));
	elseif (y <= 2.9e+38)
		tmp = Float64(x + y);
	else
		tmp = Float64(Float64(-z) * Float64(Float64(x + y) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -3.1e+27)
		tmp = z * (-1.0 - (x / y));
	elseif (y <= 8.6e-127)
		tmp = (x + y) * (1.0 + (y / z));
	elseif (y <= 2.6e-69)
		tmp = x / (1.0 - (y / z));
	elseif (y <= 2.9e+38)
		tmp = x + y;
	else
		tmp = -z * ((x + y) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -3.1e+27], N[(z * N[(-1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 8.6e-127], N[(N[(x + y), $MachinePrecision] * N[(1.0 + N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.6e-69], N[(x / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.9e+38], N[(x + y), $MachinePrecision], N[((-z) * N[(N[(x + y), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if y < -3.09999999999999996e27

   1. Initial program 73.8%

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

      1. add-cube-cbrt 72.6%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}\right) \cdot \sqrt[3]{x + y}}}{1 - \frac{y}{z}} \]

      2. associate-/l* 72.6%

         \[\leadsto \color{blue}{\left(\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}\right) \cdot \frac{\sqrt[3]{x + y}}{1 - \frac{y}{z}}} \]

      3. pow2 72.6%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{x + y}\right)}^{2}} \cdot \frac{\sqrt[3]{x + y}}{1 - \frac{y}{z}} \]

   4. Applied egg-rr 72.6%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{x + y}\right)}^{2} \cdot \frac{\sqrt[3]{x + y}}{1 - \frac{y}{z}}} \]

   5. Taylor expanded in z around 0 68.6%

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

      1. mul-1-neg 68.6%

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

      2. associate-/l* 83.6%

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

      3. distribute-lft-neg-in 83.6%

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

      4. +-commutative 83.6%

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

   7. Simplified 83.6%

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

   8. Taylor expanded in y around inf 78.0%

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

      1. mul-1-neg 78.0%

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

      2. *-commutative 78.0%

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

      3. associate-*r/ 83.6%

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

      4. *-commutative 83.6%

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

      5. distribute-lft-neg-out 83.6%

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

      6. distribute-rgt-in 83.6%

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

      7. sub-neg 83.6%

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

   10. Simplified 83.6%

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

   if -3.09999999999999996e27 < y < 8.59999999999999967e-127

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf 86.2%

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

      1. associate-+r+ 86.2%

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

      2. *-rgt-identity 86.2%

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

      3. *-commutative 86.2%

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

      4. associate-/l* 86.2%

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

      5. distribute-lft-in 86.2%

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

      6. +-commutative 86.2%

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

   5. Simplified 86.2%

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

   if 8.59999999999999967e-127 < y < 2.6000000000000002e-69

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 100.0%

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

   if 2.6000000000000002e-69 < y < 2.90000000000000007e38

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf 73.8%

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

      1. +-commutative 73.8%

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

   5. Simplified 73.8%

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

   if 2.90000000000000007e38 < y

   1. Initial program 74.8%

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

   3. Taylor expanded in z around 0 65.9%

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

      1. mul-1-neg 65.9%

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

      2. associate-/l* 80.8%

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

      3. distribute-rgt-neg-in 80.8%

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

      4. distribute-neg-frac2 80.8%

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

      5. +-commutative 80.8%

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

   5. Simplified 80.8%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.1 \cdot 10^{+27}:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{elif}\;y \leq 8.6 \cdot 10^{-127}:\\ \;\;\;\;\left(x + y\right) \cdot \left(1 + \frac{y}{z}\right)\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{-69}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{elif}\;y \leq 2.9 \cdot 10^{+38}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) \cdot \frac{x + y}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{if}\;y \leq -7.1 \cdot 10^{-20}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 8 \cdot 10^{-127}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 7.8 \cdot 10^{-69}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{+37}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (- -1.0 (/ x y)))))
   (if (<= y -7.1e-20)
     t_0
     (if (<= y 8e-127)
       (+ x y)
       (if (<= y 7.8e-69)
         (/ x (- 1.0 (/ y z)))
         (if (<= y 1.9e+37) (+ x y) t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = z * (-1.0 - (x / y));
	double tmp;
	if (y <= -7.1e-20) {
		tmp = t_0;
	} else if (y <= 8e-127) {
		tmp = x + y;
	} else if (y <= 7.8e-69) {
		tmp = x / (1.0 - (y / z));
	} else if (y <= 1.9e+37) {
		tmp = x + y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = z * ((-1.0d0) - (x / y))
    if (y <= (-7.1d-20)) then
        tmp = t_0
    else if (y <= 8d-127) then
        tmp = x + y
    else if (y <= 7.8d-69) then
        tmp = x / (1.0d0 - (y / z))
    else if (y <= 1.9d+37) then
        tmp = x + y
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = z * (-1.0 - (x / y));
	double tmp;
	if (y <= -7.1e-20) {
		tmp = t_0;
	} else if (y <= 8e-127) {
		tmp = x + y;
	} else if (y <= 7.8e-69) {
		tmp = x / (1.0 - (y / z));
	} else if (y <= 1.9e+37) {
		tmp = x + y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * (-1.0 - (x / y))
	tmp = 0
	if y <= -7.1e-20:
		tmp = t_0
	elif y <= 8e-127:
		tmp = x + y
	elif y <= 7.8e-69:
		tmp = x / (1.0 - (y / z))
	elif y <= 1.9e+37:
		tmp = x + y
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(-1.0 - Float64(x / y)))
	tmp = 0.0
	if (y <= -7.1e-20)
		tmp = t_0;
	elseif (y <= 8e-127)
		tmp = Float64(x + y);
	elseif (y <= 7.8e-69)
		tmp = Float64(x / Float64(1.0 - Float64(y / z)));
	elseif (y <= 1.9e+37)
		tmp = Float64(x + y);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * (-1.0 - (x / y));
	tmp = 0.0;
	if (y <= -7.1e-20)
		tmp = t_0;
	elseif (y <= 8e-127)
		tmp = x + y;
	elseif (y <= 7.8e-69)
		tmp = x / (1.0 - (y / z));
	elseif (y <= 1.9e+37)
		tmp = x + y;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[(-1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -7.1e-20], t$95$0, If[LessEqual[y, 8e-127], N[(x + y), $MachinePrecision], If[LessEqual[y, 7.8e-69], N[(x / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.9e+37], N[(x + y), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -7.0999999999999998e-20 or 1.89999999999999995e37 < y

   1. Initial program 75.9%

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

      1. add-cube-cbrt 74.6%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}\right) \cdot \sqrt[3]{x + y}}}{1 - \frac{y}{z}} \]

      2. associate-/l* 74.6%

         \[\leadsto \color{blue}{\left(\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}\right) \cdot \frac{\sqrt[3]{x + y}}{1 - \frac{y}{z}}} \]

      3. pow2 74.6%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{x + y}\right)}^{2}} \cdot \frac{\sqrt[3]{x + y}}{1 - \frac{y}{z}} \]

   4. Applied egg-rr 74.6%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{x + y}\right)}^{2} \cdot \frac{\sqrt[3]{x + y}}{1 - \frac{y}{z}}} \]

   5. Taylor expanded in z around 0 65.8%

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

      1. mul-1-neg 65.8%

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

      2. associate-/l* 79.9%

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

      3. distribute-lft-neg-in 79.9%

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

      4. +-commutative 79.9%

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

   7. Simplified 79.9%

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

   8. Taylor expanded in y around inf 74.9%

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

      1. mul-1-neg 74.9%

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

      2. *-commutative 74.9%

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

      3. associate-*r/ 79.9%

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

      4. *-commutative 79.9%

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

      5. distribute-lft-neg-out 79.9%

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

      6. distribute-rgt-in 79.9%

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

      7. sub-neg 79.9%

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

   10. Simplified 79.9%

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

   if -7.0999999999999998e-20 < y < 8.0000000000000002e-127 or 7.79999999999999961e-69 < y < 1.89999999999999995e37

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf 85.7%

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

      1. +-commutative 85.7%

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

   5. Simplified 85.7%

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

   if 8.0000000000000002e-127 < y < 7.79999999999999961e-69

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 100.0%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.1 \cdot 10^{-20}:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{elif}\;y \leq 8 \cdot 10^{-127}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 7.8 \cdot 10^{-69}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{+37}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.5 \cdot 10^{-20} \lor \neg \left(y \leq 7 \cdot 10^{+37}\right):\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -3.5e-20) (not (<= y 7e+37))) (* z (- -1.0 (/ x y))) (+ x y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -3.5e-20) || !(y <= 7e+37)) {
		tmp = z * (-1.0 - (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 ((y <= (-3.5d-20)) .or. (.not. (y <= 7d+37))) then
        tmp = z * ((-1.0d0) - (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 ((y <= -3.5e-20) || !(y <= 7e+37)) {
		tmp = z * (-1.0 - (x / y));
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -3.5e-20) or not (y <= 7e+37):
		tmp = z * (-1.0 - (x / y))
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -3.5e-20) || !(y <= 7e+37))
		tmp = Float64(z * Float64(-1.0 - Float64(x / y)));
	else
		tmp = Float64(x + y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -3.5e-20) || ~((y <= 7e+37)))
		tmp = z * (-1.0 - (x / y));
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -3.5e-20], N[Not[LessEqual[y, 7e+37]], $MachinePrecision]], N[(z * N[(-1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.50000000000000003e-20 or 7e37 < y

   1. Initial program 75.9%

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

      1. add-cube-cbrt 74.6%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}\right) \cdot \sqrt[3]{x + y}}}{1 - \frac{y}{z}} \]

      2. associate-/l* 74.6%

         \[\leadsto \color{blue}{\left(\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}\right) \cdot \frac{\sqrt[3]{x + y}}{1 - \frac{y}{z}}} \]

      3. pow2 74.6%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{x + y}\right)}^{2}} \cdot \frac{\sqrt[3]{x + y}}{1 - \frac{y}{z}} \]

   4. Applied egg-rr 74.6%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{x + y}\right)}^{2} \cdot \frac{\sqrt[3]{x + y}}{1 - \frac{y}{z}}} \]

   5. Taylor expanded in z around 0 65.8%

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

      1. mul-1-neg 65.8%

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

      2. associate-/l* 79.9%

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

      3. distribute-lft-neg-in 79.9%

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

      4. +-commutative 79.9%

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

   7. Simplified 79.9%

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

   8. Taylor expanded in y around inf 74.9%

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

      1. mul-1-neg 74.9%

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

      2. *-commutative 74.9%

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

      3. associate-*r/ 79.9%

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

      4. *-commutative 79.9%

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

      5. distribute-lft-neg-out 79.9%

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

      6. distribute-rgt-in 79.9%

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

      7. sub-neg 79.9%

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

   10. Simplified 79.9%

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

   if -3.50000000000000003e-20 < y < 7e37

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf 84.2%

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

      1. +-commutative 84.2%

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

   5. Simplified 84.2%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.5 \cdot 10^{-20} \lor \neg \left(y \leq 7 \cdot 10^{+37}\right):\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 68.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.2 \cdot 10^{+66} \lor \neg \left(y \leq 7.7 \cdot 10^{+62}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -4.2e+66) (not (<= y 7.7e+62))) (- z) (+ x y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -4.2e+66) || !(y <= 7.7e+62)) {
		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 <= (-4.2d+66)) .or. (.not. (y <= 7.7d+62))) 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 <= -4.2e+66) || !(y <= 7.7e+62)) {
		tmp = -z;
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -4.2e+66) or not (y <= 7.7e+62):
		tmp = -z
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -4.2e+66) || !(y <= 7.7e+62))
		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 <= -4.2e+66) || ~((y <= 7.7e+62)))
		tmp = -z;
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -4.2e+66], N[Not[LessEqual[y, 7.7e+62]], $MachinePrecision]], (-z), N[(x + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.20000000000000011e66 or 7.7000000000000003e62 < y

   1. Initial program 72.3%

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

   3. Taylor expanded in y around inf 69.1%

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

      1. mul-1-neg 69.1%

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

   5. Simplified 69.1%

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

   if -4.20000000000000011e66 < y < 7.7000000000000003e62

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf 80.1%

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

      1. +-commutative 80.1%

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

   5. Simplified 80.1%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.2 \cdot 10^{+66} \lor \neg \left(y \leq 7.7 \cdot 10^{+62}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 59.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.5 \cdot 10^{-20} \lor \neg \left(y \leq 3.9 \cdot 10^{+34}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -3.5e-20) (not (<= y 3.9e+34))) (- z) x))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -3.5e-20) || !(y <= 3.9e+34)) {
		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 <= (-3.5d-20)) .or. (.not. (y <= 3.9d+34))) 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 <= -3.5e-20) || !(y <= 3.9e+34)) {
		tmp = -z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -3.5e-20) or not (y <= 3.9e+34):
		tmp = -z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -3.5e-20) || !(y <= 3.9e+34))
		tmp = Float64(-z);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -3.5e-20) || ~((y <= 3.9e+34)))
		tmp = -z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -3.5e-20], N[Not[LessEqual[y, 3.9e+34]], $MachinePrecision]], (-z), x]

Derivation

1. Split input into 2 regimes

2. if y < -3.50000000000000003e-20 or 3.90000000000000019e34 < y

   1. Initial program 76.3%

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

   3. Taylor expanded in y around inf 63.8%

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

      1. mul-1-neg 63.8%

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

   5. Simplified 63.8%

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

   if -3.50000000000000003e-20 < y < 3.90000000000000019e34

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 66.3%

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

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.5 \cdot 10^{-20} \lor \neg \left(y \leq 3.9 \cdot 10^{+34}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 41.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{-129}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{-169}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -4.8e-129) x (if (<= x 3.8e-169) y x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -4.8e-129) {
		tmp = x;
	} else if (x <= 3.8e-169) {
		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 <= (-4.8d-129)) then
        tmp = x
    else if (x <= 3.8d-169) 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 <= -4.8e-129) {
		tmp = x;
	} else if (x <= 3.8e-169) {
		tmp = y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -4.8e-129:
		tmp = x
	elif x <= 3.8e-169:
		tmp = y
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -4.8e-129)
		tmp = x;
	elseif (x <= 3.8e-169)
		tmp = y;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -4.8e-129)
		tmp = x;
	elseif (x <= 3.8e-169)
		tmp = y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -4.8e-129], x, If[LessEqual[x, 3.8e-169], y, x]]

Derivation

1. Split input into 2 regimes

2. if x < -4.79999999999999977e-129 or 3.8e-169 < x

   1. Initial program 88.5%

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

   3. Taylor expanded in y around 0 47.4%

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

   if -4.79999999999999977e-129 < x < 3.8e-169

   1. Initial program 91.5%

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

   3. Taylor expanded in x around 0 78.3%

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

   4. Taylor expanded in y around 0 43.4%

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

4. Final simplification 46.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{-129}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{-169}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 35.5% 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 89.2%

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

3. Taylor expanded in y around 0 40.5%

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

4. Final simplification 40.5%

   \[\leadsto x \]
5. Add Preprocessing

Developer target: 93.9% 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 2024130 
(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))))