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

Percentage Accurate: 88.4% → 99.3%

Time: 5.6s

Alternatives: 11

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: 88.4% 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: 99.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{-252} \lor \neg \left(t_0 \leq 0\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot \left(\left(-x\right) - z\right)}{y} - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (+ x y) (- 1.0 (/ y z)))))
   (if (or (<= t_0 -1e-252) (not (<= t_0 0.0)))
     t_0
     (- (/ (* z (- (- x) z)) y) z))))

C

double code(double x, double y, double z) {
	double t_0 = (x + y) / (1.0 - (y / z));
	double tmp;
	if ((t_0 <= -1e-252) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = ((z * (-x - z)) / 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) :: t_0
    real(8) :: tmp
    t_0 = (x + y) / (1.0d0 - (y / z))
    if ((t_0 <= (-1d-252)) .or. (.not. (t_0 <= 0.0d0))) then
        tmp = t_0
    else
        tmp = ((z * (-x - z)) / y) - z
    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-252) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = ((z * (-x - z)) / y) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x + y) / (1.0 - (y / z))
	tmp = 0
	if (t_0 <= -1e-252) or not (t_0 <= 0.0):
		tmp = t_0
	else:
		tmp = ((z * (-x - z)) / y) - z
	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-252) || !(t_0 <= 0.0))
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(z * Float64(Float64(-x) - z)) / y) - z);
	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-252) || ~((t_0 <= 0.0)))
		tmp = t_0;
	else
		tmp = ((z * (-x - z)) / y) - z;
	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-252], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], t$95$0, N[(N[(N[(z * N[((-x) - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] - z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z))) < -9.99999999999999943e-253 or -0.0 < (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z)))

   1. Initial program 99.9%

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

   if -9.99999999999999943e-253 < (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z))) < -0.0

   1. Initial program 8.7%

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

   2. Taylor expanded in y around inf 100.0%

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

      1. associate--l+ 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-*r/ 100.0%

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

      4. div-sub 100.0%

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

      5. sub-neg 100.0%

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

      6. mul-1-neg 100.0%

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

      7. distribute-neg-out 100.0%

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

      8. *-lft-identity 100.0%

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

      9. metadata-eval 100.0%

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

      10. cancel-sign-sub-inv 100.0%

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

      11. distribute-neg-frac 100.0%

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

      12. mul-1-neg 100.0%

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

      13. mul-1-neg 100.0%

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

      14. unsub-neg 100.0%

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

   4. Simplified 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \leq -1 \cdot 10^{-252} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \leq 0\right):\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot \left(\left(-x\right) - z\right)}{y} - z\\ \end{array} \]

Alternative 2: 99.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{-233} \lor \neg \left(t_0 \leq 0\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-z}{\frac{y}{x + y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (+ x y) (- 1.0 (/ y z)))))
   (if (or (<= t_0 -1e-233) (not (<= t_0 0.0))) t_0 (/ (- z) (/ y (+ x y))))))

C

double code(double x, double y, double z) {
	double t_0 = (x + y) / (1.0 - (y / z));
	double tmp;
	if ((t_0 <= -1e-233) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = -z / (y / (x + y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x + y) / (1.0d0 - (y / z))
    if ((t_0 <= (-1d-233)) .or. (.not. (t_0 <= 0.0d0))) then
        tmp = t_0
    else
        tmp = -z / (y / (x + y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (x + y) / (1.0 - (y / z));
	double tmp;
	if ((t_0 <= -1e-233) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = -z / (y / (x + y));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x + y) / (1.0 - (y / z))
	tmp = 0
	if (t_0 <= -1e-233) or not (t_0 <= 0.0):
		tmp = t_0
	else:
		tmp = -z / (y / (x + y))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x + y) / Float64(1.0 - Float64(y / z)))
	tmp = 0.0
	if ((t_0 <= -1e-233) || !(t_0 <= 0.0))
		tmp = t_0;
	else
		tmp = Float64(Float64(-z) / Float64(y / Float64(x + y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x + y) / (1.0 - (y / z));
	tmp = 0.0;
	if ((t_0 <= -1e-233) || ~((t_0 <= 0.0)))
		tmp = t_0;
	else
		tmp = -z / (y / (x + y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x + y), $MachinePrecision] / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -1e-233], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], t$95$0, N[((-z) / N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z))) < -9.99999999999999958e-234 or -0.0 < (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z)))

   1. Initial program 99.9%

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

   if -9.99999999999999958e-234 < (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z))) < -0.0

   1. Initial program 11.2%

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

   2. Taylor expanded in z around 0 94.7%

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

      1. mul-1-neg 94.7%

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

      2. associate-/l* 99.9%

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

      3. distribute-neg-frac 99.9%

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

      4. +-commutative 99.9%

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

   4. Simplified 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \leq -1 \cdot 10^{-233} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \leq 0\right):\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-z}{\frac{y}{x + y}}\\ \end{array} \]

Alternative 3: 73.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{y}{z}\\ t_1 := \left(-z\right) - z \cdot \frac{x}{y}\\ \mathbf{if}\;y \leq -2.8 \cdot 10^{+144}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -7.1 \cdot 10^{+59}:\\ \;\;\;\;\frac{y}{t_0}\\ \mathbf{elif}\;y \leq 6 \cdot 10^{-98}:\\ \;\;\;\;\frac{x}{t_0}\\ \mathbf{elif}\;y \leq 6.8 \cdot 10^{+57}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ y z))) (t_1 (- (- z) (* z (/ x y)))))
   (if (<= y -2.8e+144)
     t_1
     (if (<= y -7.1e+59)
       (/ y t_0)
       (if (<= y 6e-98) (/ x t_0) (if (<= y 6.8e+57) (+ x y) t_1))))))

C

double code(double x, double y, double z) {
	double t_0 = 1.0 - (y / z);
	double t_1 = -z - (z * (x / y));
	double tmp;
	if (y <= -2.8e+144) {
		tmp = t_1;
	} else if (y <= -7.1e+59) {
		tmp = y / t_0;
	} else if (y <= 6e-98) {
		tmp = x / t_0;
	} else if (y <= 6.8e+57) {
		tmp = x + y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 - (y / z)
    t_1 = -z - (z * (x / y))
    if (y <= (-2.8d+144)) then
        tmp = t_1
    else if (y <= (-7.1d+59)) then
        tmp = y / t_0
    else if (y <= 6d-98) then
        tmp = x / t_0
    else if (y <= 6.8d+57) then
        tmp = x + y
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = 1.0 - (y / z);
	double t_1 = -z - (z * (x / y));
	double tmp;
	if (y <= -2.8e+144) {
		tmp = t_1;
	} else if (y <= -7.1e+59) {
		tmp = y / t_0;
	} else if (y <= 6e-98) {
		tmp = x / t_0;
	} else if (y <= 6.8e+57) {
		tmp = x + y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 1.0 - (y / z)
	t_1 = -z - (z * (x / y))
	tmp = 0
	if y <= -2.8e+144:
		tmp = t_1
	elif y <= -7.1e+59:
		tmp = y / t_0
	elif y <= 6e-98:
		tmp = x / t_0
	elif y <= 6.8e+57:
		tmp = x + y
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(1.0 - Float64(y / z))
	t_1 = Float64(Float64(-z) - Float64(z * Float64(x / y)))
	tmp = 0.0
	if (y <= -2.8e+144)
		tmp = t_1;
	elseif (y <= -7.1e+59)
		tmp = Float64(y / t_0);
	elseif (y <= 6e-98)
		tmp = Float64(x / t_0);
	elseif (y <= 6.8e+57)
		tmp = Float64(x + y);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 1.0 - (y / z);
	t_1 = -z - (z * (x / y));
	tmp = 0.0;
	if (y <= -2.8e+144)
		tmp = t_1;
	elseif (y <= -7.1e+59)
		tmp = y / t_0;
	elseif (y <= 6e-98)
		tmp = x / t_0;
	elseif (y <= 6.8e+57)
		tmp = x + y;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[((-z) - N[(z * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.8e+144], t$95$1, If[LessEqual[y, -7.1e+59], N[(y / t$95$0), $MachinePrecision], If[LessEqual[y, 6e-98], N[(x / t$95$0), $MachinePrecision], If[LessEqual[y, 6.8e+57], N[(x + y), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -2.80000000000000007e144 or 6.79999999999999984e57 < y

   1. Initial program 63.1%

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

   2. Taylor expanded in y around inf 77.2%

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

      1. associate--l+ 77.2%

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

      2. +-commutative 77.2%

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

      3. associate-*r/ 77.2%

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

      4. div-sub 77.2%

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

      5. sub-neg 77.2%

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

      6. mul-1-neg 77.2%

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

      7. distribute-neg-out 77.2%

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

      8. *-lft-identity 77.2%

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

      9. metadata-eval 77.2%

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

      10. cancel-sign-sub-inv 77.2%

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

      11. distribute-neg-frac 77.2%

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

      12. mul-1-neg 77.2%

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

      13. mul-1-neg 77.2%

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

      14. unsub-neg 77.2%

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

   4. Simplified 77.2%

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

   5. Taylor expanded in z around 0 79.1%

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

      1. mul-1-neg 79.1%

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

      2. *-commutative 79.1%

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

      3. associate-*r/ 83.8%

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

      4. distribute-rgt-neg-in 83.8%

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

      5. distribute-neg-frac 83.8%

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

   7. Simplified 83.8%

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

   if -2.80000000000000007e144 < y < -7.10000000000000003e59

   1. Initial program 92.8%

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

   2. Taylor expanded in x around 0 91.5%

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

   if -7.10000000000000003e59 < y < 6e-98

   1. Initial program 99.2%

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

   2. Taylor expanded in x around inf 80.2%

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

   if 6e-98 < y < 6.79999999999999984e57

   1. Initial program 95.2%

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

   2. Taylor expanded in z around inf 68.1%

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

      1. +-commutative 68.1%

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

   4. Simplified 68.1%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.8 \cdot 10^{+144}:\\ \;\;\;\;\left(-z\right) - z \cdot \frac{x}{y}\\ \mathbf{elif}\;y \leq -7.1 \cdot 10^{+59}:\\ \;\;\;\;\frac{y}{1 - \frac{y}{z}}\\ \mathbf{elif}\;y \leq 6 \cdot 10^{-98}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{elif}\;y \leq 6.8 \cdot 10^{+57}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) - z \cdot \frac{x}{y}\\ \end{array} \]

Alternative 4: 68.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{y}{z}\\ \mathbf{if}\;y \leq -2.2 \cdot 10^{+174}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq -1.36 \cdot 10^{+57}:\\ \;\;\;\;\frac{y}{t_0}\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{-97}:\\ \;\;\;\;\frac{x}{t_0}\\ \mathbf{elif}\;y \leq 9.6 \cdot 10^{+120}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ y z))))
   (if (<= y -2.2e+174)
     (- z)
     (if (<= y -1.36e+57)
       (/ y t_0)
       (if (<= y 8.5e-97) (/ x t_0) (if (<= y 9.6e+120) (+ x y) (- z)))))))

C

double code(double x, double y, double z) {
	double t_0 = 1.0 - (y / z);
	double tmp;
	if (y <= -2.2e+174) {
		tmp = -z;
	} else if (y <= -1.36e+57) {
		tmp = y / t_0;
	} else if (y <= 8.5e-97) {
		tmp = x / t_0;
	} else if (y <= 9.6e+120) {
		tmp = x + y;
	} else {
		tmp = -z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - (y / z)
    if (y <= (-2.2d+174)) then
        tmp = -z
    else if (y <= (-1.36d+57)) then
        tmp = y / t_0
    else if (y <= 8.5d-97) then
        tmp = x / t_0
    else if (y <= 9.6d+120) then
        tmp = x + y
    else
        tmp = -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = 1.0 - (y / z);
	double tmp;
	if (y <= -2.2e+174) {
		tmp = -z;
	} else if (y <= -1.36e+57) {
		tmp = y / t_0;
	} else if (y <= 8.5e-97) {
		tmp = x / t_0;
	} else if (y <= 9.6e+120) {
		tmp = x + y;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 1.0 - (y / z)
	tmp = 0
	if y <= -2.2e+174:
		tmp = -z
	elif y <= -1.36e+57:
		tmp = y / t_0
	elif y <= 8.5e-97:
		tmp = x / t_0
	elif y <= 9.6e+120:
		tmp = x + y
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(1.0 - Float64(y / z))
	tmp = 0.0
	if (y <= -2.2e+174)
		tmp = Float64(-z);
	elseif (y <= -1.36e+57)
		tmp = Float64(y / t_0);
	elseif (y <= 8.5e-97)
		tmp = Float64(x / t_0);
	elseif (y <= 9.6e+120)
		tmp = Float64(x + y);
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 1.0 - (y / z);
	tmp = 0.0;
	if (y <= -2.2e+174)
		tmp = -z;
	elseif (y <= -1.36e+57)
		tmp = y / t_0;
	elseif (y <= 8.5e-97)
		tmp = x / t_0;
	elseif (y <= 9.6e+120)
		tmp = x + y;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.2e+174], (-z), If[LessEqual[y, -1.36e+57], N[(y / t$95$0), $MachinePrecision], If[LessEqual[y, 8.5e-97], N[(x / t$95$0), $MachinePrecision], If[LessEqual[y, 9.6e+120], N[(x + y), $MachinePrecision], (-z)]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -2.2000000000000002e174 or 9.60000000000000004e120 < y

   1. Initial program 60.5%

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

   2. Taylor expanded in y around inf 74.6%

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

      1. mul-1-neg 74.6%

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

   4. Simplified 74.6%

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

   if -2.2000000000000002e174 < y < -1.36e57

   1. Initial program 85.2%

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

   2. Taylor expanded in x around 0 79.3%

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

   if -1.36e57 < y < 8.5000000000000002e-97

   1. Initial program 99.2%

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

   2. Taylor expanded in x around inf 80.2%

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

   if 8.5000000000000002e-97 < y < 9.60000000000000004e120

   1. Initial program 90.6%

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

   2. Taylor expanded in z around inf 61.7%

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

      1. +-commutative 61.7%

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

   4. Simplified 61.7%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{+174}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq -1.36 \cdot 10^{+57}:\\ \;\;\;\;\frac{y}{1 - \frac{y}{z}}\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{-97}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{elif}\;y \leq 9.6 \cdot 10^{+120}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 5: 69.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -8 \cdot 10^{-134} \lor \neg \left(z \leq 1.5 \cdot 10^{+97}\right):\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;\frac{-z}{\frac{y}{x + y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -8e-134) (not (<= z 1.5e+97))) (+ x y) (/ (- z) (/ y (+ x y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -8e-134) || !(z <= 1.5e+97)) {
		tmp = x + y;
	} else {
		tmp = -z / (y / (x + y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-8d-134)) .or. (.not. (z <= 1.5d+97))) then
        tmp = x + y
    else
        tmp = -z / (y / (x + y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -8e-134) || !(z <= 1.5e+97)) {
		tmp = x + y;
	} else {
		tmp = -z / (y / (x + y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -8e-134) or not (z <= 1.5e+97):
		tmp = x + y
	else:
		tmp = -z / (y / (x + y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -8e-134) || !(z <= 1.5e+97))
		tmp = Float64(x + y);
	else
		tmp = Float64(Float64(-z) / Float64(y / Float64(x + y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -8e-134) || ~((z <= 1.5e+97)))
		tmp = x + y;
	else
		tmp = -z / (y / (x + y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -8e-134], N[Not[LessEqual[z, 1.5e+97]], $MachinePrecision]], N[(x + y), $MachinePrecision], N[((-z) / N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -8.00000000000000032e-134 or 1.4999999999999999e97 < z

   1. Initial program 95.9%

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

   2. Taylor expanded in z around inf 78.7%

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

      1. +-commutative 78.7%

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

   4. Simplified 78.7%

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

   if -8.00000000000000032e-134 < z < 1.4999999999999999e97

   1. Initial program 76.9%

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

   2. Taylor expanded in z around 0 69.2%

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

      1. mul-1-neg 69.2%

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

      2. associate-/l* 75.2%

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

      3. distribute-neg-frac 75.2%

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

      4. +-commutative 75.2%

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

   4. Simplified 75.2%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8 \cdot 10^{-134} \lor \neg \left(z \leq 1.5 \cdot 10^{+97}\right):\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;\frac{-z}{\frac{y}{x + y}}\\ \end{array} \]

Alternative 6: 68.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -8.5 \cdot 10^{-134}:\\ \;\;\;\;\left(x + y\right) \cdot \left(1 + \frac{y}{z}\right)\\ \mathbf{elif}\;z \leq 1.36 \cdot 10^{+97}:\\ \;\;\;\;\frac{-z}{\frac{y}{x + y}}\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -8.5e-134)
   (* (+ x y) (+ 1.0 (/ y z)))
   (if (<= z 1.36e+97) (/ (- z) (/ y (+ x y))) (+ x y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -8.5e-134) {
		tmp = (x + y) * (1.0 + (y / z));
	} else if (z <= 1.36e+97) {
		tmp = -z / (y / (x + y));
	} else {
		tmp = x + y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-8.5d-134)) then
        tmp = (x + y) * (1.0d0 + (y / z))
    else if (z <= 1.36d+97) then
        tmp = -z / (y / (x + y))
    else
        tmp = x + y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -8.5e-134) {
		tmp = (x + y) * (1.0 + (y / z));
	} else if (z <= 1.36e+97) {
		tmp = -z / (y / (x + y));
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -8.5e-134:
		tmp = (x + y) * (1.0 + (y / z))
	elif z <= 1.36e+97:
		tmp = -z / (y / (x + y))
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -8.5e-134)
		tmp = Float64(Float64(x + y) * Float64(1.0 + Float64(y / z)));
	elseif (z <= 1.36e+97)
		tmp = Float64(Float64(-z) / Float64(y / Float64(x + y)));
	else
		tmp = Float64(x + y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -8.5e-134)
		tmp = (x + y) * (1.0 + (y / z));
	elseif (z <= 1.36e+97)
		tmp = -z / (y / (x + y));
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -8.5e-134], N[(N[(x + y), $MachinePrecision] * N[(1.0 + N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.36e+97], N[((-z) / N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -8.50000000000000015e-134

   1. Initial program 94.3%

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

   2. Taylor expanded in z around inf 66.6%

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

      1. associate-+r+ 66.6%

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

      2. *-lft-identity 66.6%

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

      3. associate-/l* 70.4%

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

      4. associate-/r/ 73.3%

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

      5. distribute-rgt-in 73.2%

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

      6. +-commutative 73.2%

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

   4. Simplified 73.2%

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

   if -8.50000000000000015e-134 < z < 1.36e97

   1. Initial program 76.9%

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

   2. Taylor expanded in z around 0 69.2%

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

      1. mul-1-neg 69.2%

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

      2. associate-/l* 75.2%

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

      3. distribute-neg-frac 75.2%

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

      4. +-commutative 75.2%

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

   4. Simplified 75.2%

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

   if 1.36e97 < z

   1. Initial program 99.9%

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

   2. Taylor expanded in z around inf 92.7%

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

      1. +-commutative 92.7%

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

   4. Simplified 92.7%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8.5 \cdot 10^{-134}:\\ \;\;\;\;\left(x + y\right) \cdot \left(1 + \frac{y}{z}\right)\\ \mathbf{elif}\;z \leq 1.36 \cdot 10^{+97}:\\ \;\;\;\;\frac{-z}{\frac{y}{x + y}}\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 7: 67.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{+67}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{-97}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{+122}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -7.2e+67)
   (- z)
   (if (<= y 2.7e-97)
     (/ x (- 1.0 (/ y z)))
     (if (<= y 4.8e+122) (+ x y) (- z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -7.2e+67) {
		tmp = -z;
	} else if (y <= 2.7e-97) {
		tmp = x / (1.0 - (y / z));
	} else if (y <= 4.8e+122) {
		tmp = x + y;
	} else {
		tmp = -z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-7.2d+67)) then
        tmp = -z
    else if (y <= 2.7d-97) then
        tmp = x / (1.0d0 - (y / z))
    else if (y <= 4.8d+122) then
        tmp = x + y
    else
        tmp = -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -7.2e+67) {
		tmp = -z;
	} else if (y <= 2.7e-97) {
		tmp = x / (1.0 - (y / z));
	} else if (y <= 4.8e+122) {
		tmp = x + y;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -7.2e+67:
		tmp = -z
	elif y <= 2.7e-97:
		tmp = x / (1.0 - (y / z))
	elif y <= 4.8e+122:
		tmp = x + y
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -7.2e+67)
		tmp = Float64(-z);
	elseif (y <= 2.7e-97)
		tmp = Float64(x / Float64(1.0 - Float64(y / z)));
	elseif (y <= 4.8e+122)
		tmp = Float64(x + y);
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -7.2e+67)
		tmp = -z;
	elseif (y <= 2.7e-97)
		tmp = x / (1.0 - (y / z));
	elseif (y <= 4.8e+122)
		tmp = x + y;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -7.2e+67], (-z), If[LessEqual[y, 2.7e-97], N[(x / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.8e+122], N[(x + y), $MachinePrecision], (-z)]]]

Derivation

1. Split input into 3 regimes

2. if y < -7.1999999999999998e67 or 4.8000000000000004e122 < y

   1. Initial program 66.4%

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

   2. Taylor expanded in y around inf 69.5%

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

      1. mul-1-neg 69.5%

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

   4. Simplified 69.5%

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

   if -7.1999999999999998e67 < y < 2.69999999999999985e-97

   1. Initial program 99.2%

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

   2. Taylor expanded in x around inf 80.2%

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

   if 2.69999999999999985e-97 < y < 4.8000000000000004e122

   1. Initial program 90.6%

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

   2. Taylor expanded in z around inf 61.7%

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

      1. +-commutative 61.7%

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

   4. Simplified 61.7%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{+67}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{-97}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{+122}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 8: 56.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.1 \cdot 10^{+62}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 5600000000:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{+120}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2.1e+62)
   (- z)
   (if (<= y 5600000000.0) x (if (<= y 2.6e+120) y (- z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.1e+62) {
		tmp = -z;
	} else if (y <= 5600000000.0) {
		tmp = x;
	} else if (y <= 2.6e+120) {
		tmp = y;
	} else {
		tmp = -z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-2.1d+62)) then
        tmp = -z
    else if (y <= 5600000000.0d0) then
        tmp = x
    else if (y <= 2.6d+120) then
        tmp = y
    else
        tmp = -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.1e+62) {
		tmp = -z;
	} else if (y <= 5600000000.0) {
		tmp = x;
	} else if (y <= 2.6e+120) {
		tmp = y;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -2.1e+62:
		tmp = -z
	elif y <= 5600000000.0:
		tmp = x
	elif y <= 2.6e+120:
		tmp = y
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -2.1e+62)
		tmp = Float64(-z);
	elseif (y <= 5600000000.0)
		tmp = x;
	elseif (y <= 2.6e+120)
		tmp = y;
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -2.1e+62)
		tmp = -z;
	elseif (y <= 5600000000.0)
		tmp = x;
	elseif (y <= 2.6e+120)
		tmp = y;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -2.1e+62], (-z), If[LessEqual[y, 5600000000.0], x, If[LessEqual[y, 2.6e+120], y, (-z)]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.1e62 or 2.5999999999999999e120 < y

   1. Initial program 66.4%

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

   2. Taylor expanded in y around inf 69.5%

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

      1. mul-1-neg 69.5%

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

   4. Simplified 69.5%

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

   if -2.1e62 < y < 5.6e9

   1. Initial program 99.3%

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

   2. Taylor expanded in y around 0 58.4%

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

   if 5.6e9 < y < 2.5999999999999999e120

   1. Initial program 77.0%

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

   2. Taylor expanded in z around inf 50.5%

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

      1. associate-+r+ 50.5%

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

      2. *-lft-identity 50.5%

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

      3. associate-/l* 55.5%

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

      4. associate-/r/ 55.5%

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

      5. distribute-rgt-in 55.5%

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

      6. +-commutative 55.5%

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

   4. Simplified 55.5%

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

   5. Taylor expanded in x around 0 45.7%

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

   6. Taylor expanded in y around 0 47.1%

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

4. Final simplification 60.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.1 \cdot 10^{+62}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 5600000000:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{+120}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 9: 67.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.8 \cdot 10^{+76} \lor \neg \left(y \leq 4.3 \cdot 10^{+120}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1.8e+76) (not (<= y 4.3e+120))) (- z) (+ x y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.8e+76) || !(y <= 4.3e+120)) {
		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 <= (-1.8d+76)) .or. (.not. (y <= 4.3d+120))) 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 <= -1.8e+76) || !(y <= 4.3e+120)) {
		tmp = -z;
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -1.8e+76) or not (y <= 4.3e+120):
		tmp = -z
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -1.8e+76) || !(y <= 4.3e+120))
		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 <= -1.8e+76) || ~((y <= 4.3e+120)))
		tmp = -z;
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -1.8e+76], N[Not[LessEqual[y, 4.3e+120]], $MachinePrecision]], (-z), N[(x + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.8000000000000001e76 or 4.3000000000000002e120 < y

   1. Initial program 66.4%

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

   2. Taylor expanded in y around inf 69.5%

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

      1. mul-1-neg 69.5%

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

   4. Simplified 69.5%

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

   if -1.8000000000000001e76 < y < 4.3000000000000002e120

   1. Initial program 96.8%

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

   2. Taylor expanded in z around inf 69.3%

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

      1. +-commutative 69.3%

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

   4. Simplified 69.3%

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

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.8 \cdot 10^{+76} \lor \neg \left(y \leq 4.3 \cdot 10^{+120}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 10: 40.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.6 \cdot 10^{-154}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{-67}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -7.6e-154) x (if (<= x 2.4e-67) y x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -7.6e-154) {
		tmp = x;
	} else if (x <= 2.4e-67) {
		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 <= (-7.6d-154)) then
        tmp = x
    else if (x <= 2.4d-67) 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 <= -7.6e-154) {
		tmp = x;
	} else if (x <= 2.4e-67) {
		tmp = y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -7.6e-154:
		tmp = x
	elif x <= 2.4e-67:
		tmp = y
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -7.6e-154)
		tmp = x;
	elseif (x <= 2.4e-67)
		tmp = y;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -7.6e-154)
		tmp = x;
	elseif (x <= 2.4e-67)
		tmp = y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -7.6e-154], x, If[LessEqual[x, 2.4e-67], y, x]]

Derivation

1. Split input into 2 regimes

2. if x < -7.60000000000000019e-154 or 2.4e-67 < x

   1. Initial program 86.2%

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

   2. Taylor expanded in y around 0 47.8%

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

   if -7.60000000000000019e-154 < x < 2.4e-67

   1. Initial program 90.7%

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

   2. Taylor expanded in z around inf 47.3%

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

      1. associate-+r+ 47.3%

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

      2. *-lft-identity 47.3%

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

      3. associate-/l* 49.9%

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

      4. associate-/r/ 49.9%

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

      5. distribute-rgt-in 49.9%

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

      6. +-commutative 49.9%

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

   4. Simplified 49.9%

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

   5. Taylor expanded in x around 0 39.6%

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

   6. Taylor expanded in y around 0 40.2%

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

4. Final simplification 45.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.6 \cdot 10^{-154}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{-67}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 11: 34.6% accurate, 9.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := x

Derivation

1. Initial program 87.4%

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

2. Taylor expanded in y around 0 38.2%

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

3. Final simplification 38.2%

   \[\leadsto x \]

Developer target: 93.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y + x}{-y} \cdot z\\ \mathbf{if}\;y < -3.7429310762689856 \cdot 10^{+171}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y < 3.5534662456086734 \cdot 10^{+168}:\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (/ (+ y x) (- y)) z)))
   (if (< y -3.7429310762689856e+171)
     t_0
     (if (< y 3.5534662456086734e+168) (/ (+ x y) (- 1.0 (/ y z))) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = ((y + x) / -y) * z;
	double tmp;
	if (y < -3.7429310762689856e+171) {
		tmp = t_0;
	} else if (y < 3.5534662456086734e+168) {
		tmp = (x + y) / (1.0 - (y / z));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((y + x) / -y) * z
    if (y < (-3.7429310762689856d+171)) then
        tmp = t_0
    else if (y < 3.5534662456086734d+168) then
        tmp = (x + y) / (1.0d0 - (y / z))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = ((y + x) / -y) * z;
	double tmp;
	if (y < -3.7429310762689856e+171) {
		tmp = t_0;
	} else if (y < 3.5534662456086734e+168) {
		tmp = (x + y) / (1.0 - (y / z));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = ((y + x) / -y) * z
	tmp = 0
	if y < -3.7429310762689856e+171:
		tmp = t_0
	elif y < 3.5534662456086734e+168:
		tmp = (x + y) / (1.0 - (y / z))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(y + x) / Float64(-y)) * z)
	tmp = 0.0
	if (y < -3.7429310762689856e+171)
		tmp = t_0;
	elseif (y < 3.5534662456086734e+168)
		tmp = Float64(Float64(x + y) / Float64(1.0 - Float64(y / z)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = ((y + x) / -y) * z;
	tmp = 0.0;
	if (y < -3.7429310762689856e+171)
		tmp = t_0;
	elseif (y < 3.5534662456086734e+168)
		tmp = (x + y) / (1.0 - (y / z));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(y + x), $MachinePrecision] / (-y)), $MachinePrecision] * z), $MachinePrecision]}, If[Less[y, -3.7429310762689856e+171], t$95$0, If[Less[y, 3.5534662456086734e+168], N[(N[(x + y), $MachinePrecision] / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Reproduce

herbie shell --seed 2023318 
(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))))