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

Percentage Accurate: 88.8% → 99.8%

Time: 7.1s

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: 88.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{-286} \lor \neg \left(t_0 \leq 5 \cdot 10^{-308}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) - z \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (+ x y) (- 1.0 (/ y z)))))
   (if (or (<= t_0 -5e-286) (not (<= t_0 5e-308)))
     t_0
     (- (- z) (* z (/ x y))))))

C

double code(double x, double y, double z) {
	double t_0 = (x + y) / (1.0 - (y / z));
	double tmp;
	if ((t_0 <= -5e-286) || !(t_0 <= 5e-308)) {
		tmp = t_0;
	} else {
		tmp = -z - (z * (x / y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x + y) / (1.0d0 - (y / z))
    if ((t_0 <= (-5d-286)) .or. (.not. (t_0 <= 5d-308))) then
        tmp = t_0
    else
        tmp = -z - (z * (x / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (x + y) / (1.0 - (y / z));
	double tmp;
	if ((t_0 <= -5e-286) || !(t_0 <= 5e-308)) {
		tmp = t_0;
	} else {
		tmp = -z - (z * (x / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x + y) / (1.0 - (y / z))
	tmp = 0
	if (t_0 <= -5e-286) or not (t_0 <= 5e-308):
		tmp = t_0
	else:
		tmp = -z - (z * (x / y))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x + y) / Float64(1.0 - Float64(y / z)))
	tmp = 0.0
	if ((t_0 <= -5e-286) || !(t_0 <= 5e-308))
		tmp = t_0;
	else
		tmp = Float64(Float64(-z) - Float64(z * 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 <= -5e-286) || ~((t_0 <= 5e-308)))
		tmp = t_0;
	else
		tmp = -z - (z * (x / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x + y), $MachinePrecision] / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -5e-286], N[Not[LessEqual[t$95$0, 5e-308]], $MachinePrecision]], t$95$0, N[((-z) - N[(z * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z))) < -5.00000000000000037e-286 or 4.99999999999999955e-308 < (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z)))

   1. Initial program 99.8%

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

   if -5.00000000000000037e-286 < (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z))) < 4.99999999999999955e-308

   1. Initial program 10.7%

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

   2. Taylor expanded in z around 0 97.1%

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

      1. mul-1-neg 97.1%

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

      2. associate-/l* 10.7%

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

      3. +-commutative 10.7%

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

      4. distribute-neg-frac 10.7%

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

      5. distribute-neg-in 10.7%

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

      6. +-commutative 10.7%

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

      7. sub-neg 10.7%

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

   4. Simplified 10.7%

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

   5. Taylor expanded in y around 0 99.9%

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

      1. distribute-lft-out 99.9%

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

      2. associate-*r/ 100.0%

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

      3. neg-mul-1 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \leq -5 \cdot 10^{-286} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \leq 5 \cdot 10^{-308}\right):\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) - z \cdot \frac{x}{y}\\ \end{array} \]

Alternative 2: 69.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{y}{z}\\ t_1 := \frac{x}{t_0}\\ \mathbf{if}\;y \leq -5.6 \cdot 10^{+182}:\\ \;\;\;\;z \cdot \left(-1 - \frac{z}{y}\right)\\ \mathbf{elif}\;y \leq -1.9 \cdot 10^{+48}:\\ \;\;\;\;\frac{y}{t_0}\\ \mathbf{elif}\;y \leq -6.5 \cdot 10^{-62}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{-60}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 4.1 \cdot 10^{+14}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ y z))) (t_1 (/ x t_0)))
   (if (<= y -5.6e+182)
     (* z (- -1.0 (/ z y)))
     (if (<= y -1.9e+48)
       (/ y t_0)
       (if (<= y -6.5e-62)
         t_1
         (if (<= y 1.15e-60) (+ x y) (if (<= y 4.1e+14) t_1 (- z))))))))

C

double code(double x, double y, double z) {
	double t_0 = 1.0 - (y / z);
	double t_1 = x / t_0;
	double tmp;
	if (y <= -5.6e+182) {
		tmp = z * (-1.0 - (z / y));
	} else if (y <= -1.9e+48) {
		tmp = y / t_0;
	} else if (y <= -6.5e-62) {
		tmp = t_1;
	} else if (y <= 1.15e-60) {
		tmp = x + y;
	} else if (y <= 4.1e+14) {
		tmp = t_1;
	} else {
		tmp = -z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 - (y / z)
    t_1 = x / t_0
    if (y <= (-5.6d+182)) then
        tmp = z * ((-1.0d0) - (z / y))
    else if (y <= (-1.9d+48)) then
        tmp = y / t_0
    else if (y <= (-6.5d-62)) then
        tmp = t_1
    else if (y <= 1.15d-60) then
        tmp = x + y
    else if (y <= 4.1d+14) then
        tmp = t_1
    else
        tmp = -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = 1.0 - (y / z);
	double t_1 = x / t_0;
	double tmp;
	if (y <= -5.6e+182) {
		tmp = z * (-1.0 - (z / y));
	} else if (y <= -1.9e+48) {
		tmp = y / t_0;
	} else if (y <= -6.5e-62) {
		tmp = t_1;
	} else if (y <= 1.15e-60) {
		tmp = x + y;
	} else if (y <= 4.1e+14) {
		tmp = t_1;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 1.0 - (y / z)
	t_1 = x / t_0
	tmp = 0
	if y <= -5.6e+182:
		tmp = z * (-1.0 - (z / y))
	elif y <= -1.9e+48:
		tmp = y / t_0
	elif y <= -6.5e-62:
		tmp = t_1
	elif y <= 1.15e-60:
		tmp = x + y
	elif y <= 4.1e+14:
		tmp = t_1
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(1.0 - Float64(y / z))
	t_1 = Float64(x / t_0)
	tmp = 0.0
	if (y <= -5.6e+182)
		tmp = Float64(z * Float64(-1.0 - Float64(z / y)));
	elseif (y <= -1.9e+48)
		tmp = Float64(y / t_0);
	elseif (y <= -6.5e-62)
		tmp = t_1;
	elseif (y <= 1.15e-60)
		tmp = Float64(x + y);
	elseif (y <= 4.1e+14)
		tmp = t_1;
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 1.0 - (y / z);
	t_1 = x / t_0;
	tmp = 0.0;
	if (y <= -5.6e+182)
		tmp = z * (-1.0 - (z / y));
	elseif (y <= -1.9e+48)
		tmp = y / t_0;
	elseif (y <= -6.5e-62)
		tmp = t_1;
	elseif (y <= 1.15e-60)
		tmp = x + y;
	elseif (y <= 4.1e+14)
		tmp = t_1;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x / t$95$0), $MachinePrecision]}, If[LessEqual[y, -5.6e+182], N[(z * N[(-1.0 - N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -1.9e+48], N[(y / t$95$0), $MachinePrecision], If[LessEqual[y, -6.5e-62], t$95$1, If[LessEqual[y, 1.15e-60], N[(x + y), $MachinePrecision], If[LessEqual[y, 4.1e+14], t$95$1, (-z)]]]]]]]

Derivation

1. Split input into 5 regimes

2. if y < -5.60000000000000013e182

   1. Initial program 64.1%

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

   2. Taylor expanded in y around inf 63.0%

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

      1. mul-1-neg 63.0%

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

      2. unsub-neg 63.0%

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

      3. mul-1-neg 63.0%

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

      4. associate-/l* 63.2%

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

      5. unpow2 63.2%

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

      6. associate-/l* 90.6%

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

   4. Simplified 90.6%

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

      1. neg-mul-1 90.6%

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

      2. fma-neg 90.6%

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

      3. div-inv 90.6%

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

      4. clear-num 90.6%

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

      5. distribute-lft-neg-in 90.6%

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

      6. add-sqr-sqrt 58.0%

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

      7. sqrt-unprod 62.5%

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

      8. sqr-neg 62.5%

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

      9. sqrt-unprod 29.4%

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

      10. add-sqr-sqrt 84.5%

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

   6. Applied egg-rr 84.5%

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

      1. fma-udef 84.5%

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

      2. *-commutative 84.5%

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

      3. distribute-lft-out 84.5%

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

   8. Simplified 84.5%

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

   9. Taylor expanded in x around 0 60.6%

      \[\leadsto \color{blue}{-1 \cdot z - \frac{{z}^{2}}{y}} \]
   10. Step-by-step derivation

       1. *-commutative 60.6%

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

       2. unpow2 60.6%

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

       3. associate-*r/ 85.1%

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

       4. distribute-lft-out-- 85.1%

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

   11. Simplified 85.1%

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

   if -5.60000000000000013e182 < y < -1.9e48

   1. Initial program 82.4%

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

   2. Taylor expanded in x around 0 62.0%

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

   if -1.9e48 < y < -6.50000000000000026e-62 or 1.1500000000000001e-60 < y < 4.1e14

   1. Initial program 97.0%

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

   2. Taylor expanded in x around inf 70.5%

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

   if -6.50000000000000026e-62 < y < 1.1500000000000001e-60

   1. Initial program 99.9%

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

   2. Taylor expanded in z around inf 89.1%

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

   if 4.1e14 < y

   1. Initial program 70.1%

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

   2. Taylor expanded in y around inf 57.2%

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

      1. mul-1-neg 57.2%

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

   4. Simplified 57.2%

      \[\leadsto \color{blue}{-z} \]
3. Recombined 5 regimes into one program.

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.6 \cdot 10^{+182}:\\ \;\;\;\;z \cdot \left(-1 - \frac{z}{y}\right)\\ \mathbf{elif}\;y \leq -1.9 \cdot 10^{+48}:\\ \;\;\;\;\frac{y}{1 - \frac{y}{z}}\\ \mathbf{elif}\;y \leq -6.5 \cdot 10^{-62}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{-60}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 4.1 \cdot 10^{+14}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 3: 68.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{1 - \frac{y}{z}}\\ \mathbf{if}\;y \leq -2.1 \cdot 10^{+48}:\\ \;\;\;\;z \cdot \left(-1 - \frac{z}{y}\right)\\ \mathbf{elif}\;y \leq -2.2 \cdot 10^{-44}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 7 \cdot 10^{-62}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 4.1 \cdot 10^{+14}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ x (- 1.0 (/ y z)))))
   (if (<= y -2.1e+48)
     (* z (- -1.0 (/ z y)))
     (if (<= y -2.2e-44)
       t_0
       (if (<= y 7e-62) (+ x y) (if (<= y 4.1e+14) t_0 (- z)))))))

C

double code(double x, double y, double z) {
	double t_0 = x / (1.0 - (y / z));
	double tmp;
	if (y <= -2.1e+48) {
		tmp = z * (-1.0 - (z / y));
	} else if (y <= -2.2e-44) {
		tmp = t_0;
	} else if (y <= 7e-62) {
		tmp = x + y;
	} else if (y <= 4.1e+14) {
		tmp = t_0;
	} else {
		tmp = -z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / (1.0d0 - (y / z))
    if (y <= (-2.1d+48)) then
        tmp = z * ((-1.0d0) - (z / y))
    else if (y <= (-2.2d-44)) then
        tmp = t_0
    else if (y <= 7d-62) then
        tmp = x + y
    else if (y <= 4.1d+14) then
        tmp = t_0
    else
        tmp = -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x / (1.0 - (y / z));
	double tmp;
	if (y <= -2.1e+48) {
		tmp = z * (-1.0 - (z / y));
	} else if (y <= -2.2e-44) {
		tmp = t_0;
	} else if (y <= 7e-62) {
		tmp = x + y;
	} else if (y <= 4.1e+14) {
		tmp = t_0;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x / (1.0 - (y / z))
	tmp = 0
	if y <= -2.1e+48:
		tmp = z * (-1.0 - (z / y))
	elif y <= -2.2e-44:
		tmp = t_0
	elif y <= 7e-62:
		tmp = x + y
	elif y <= 4.1e+14:
		tmp = t_0
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x / Float64(1.0 - Float64(y / z)))
	tmp = 0.0
	if (y <= -2.1e+48)
		tmp = Float64(z * Float64(-1.0 - Float64(z / y)));
	elseif (y <= -2.2e-44)
		tmp = t_0;
	elseif (y <= 7e-62)
		tmp = Float64(x + y);
	elseif (y <= 4.1e+14)
		tmp = t_0;
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x / (1.0 - (y / z));
	tmp = 0.0;
	if (y <= -2.1e+48)
		tmp = z * (-1.0 - (z / y));
	elseif (y <= -2.2e-44)
		tmp = t_0;
	elseif (y <= 7e-62)
		tmp = x + y;
	elseif (y <= 4.1e+14)
		tmp = t_0;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.1e+48], N[(z * N[(-1.0 - N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -2.2e-44], t$95$0, If[LessEqual[y, 7e-62], N[(x + y), $MachinePrecision], If[LessEqual[y, 4.1e+14], t$95$0, (-z)]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -2.0999999999999998e48

   1. Initial program 73.7%

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

   2. Taylor expanded in y around inf 67.5%

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

      1. mul-1-neg 67.5%

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

      2. unsub-neg 67.5%

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

      3. mul-1-neg 67.5%

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

      4. associate-/l* 67.7%

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

      5. unpow2 67.7%

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

      6. associate-/l* 80.6%

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

   4. Simplified 80.6%

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

      1. neg-mul-1 80.6%

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

      2. fma-neg 80.6%

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

      3. div-inv 80.6%

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

      4. clear-num 80.6%

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

      5. distribute-lft-neg-in 80.6%

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

      6. add-sqr-sqrt 51.2%

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

      7. sqrt-unprod 60.5%

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

      8. sqr-neg 60.5%

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

      9. sqrt-unprod 24.8%

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

      10. add-sqr-sqrt 62.9%

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

   6. Applied egg-rr 62.9%

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

      1. fma-udef 62.9%

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

      2. *-commutative 62.9%

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

      3. distribute-lft-out 62.9%

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

   8. Simplified 62.9%

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

   9. Taylor expanded in x around 0 51.8%

      \[\leadsto \color{blue}{-1 \cdot z - \frac{{z}^{2}}{y}} \]
   10. Step-by-step derivation

       1. *-commutative 51.8%

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

       2. unpow2 51.8%

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

       3. associate-*r/ 63.4%

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

       4. distribute-lft-out-- 63.4%

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

   11. Simplified 63.4%

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

   if -2.0999999999999998e48 < y < -2.20000000000000012e-44 or 7.0000000000000003e-62 < y < 4.1e14

   1. Initial program 97.0%

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

   2. Taylor expanded in x around inf 70.5%

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

   if -2.20000000000000012e-44 < y < 7.0000000000000003e-62

   1. Initial program 99.9%

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

   2. Taylor expanded in z around inf 89.1%

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

   if 4.1e14 < y

   1. Initial program 70.1%

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

   2. Taylor expanded in y around inf 57.2%

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

      1. mul-1-neg 57.2%

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

   4. Simplified 57.2%

      \[\leadsto \color{blue}{-z} \]
3. Recombined 4 regimes into one program.

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.1 \cdot 10^{+48}:\\ \;\;\;\;z \cdot \left(-1 - \frac{z}{y}\right)\\ \mathbf{elif}\;y \leq -2.2 \cdot 10^{-44}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{elif}\;y \leq 7 \cdot 10^{-62}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 4.1 \cdot 10^{+14}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 4: 74.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{-22} \lor \neg \left(y \leq 4.4 \cdot 10^{-26}\right):\\ \;\;\;\;\left(-z\right) - z \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -4.8e-22) (not (<= y 4.4e-26)))
   (- (- z) (* z (/ x y)))
   (+ x y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -4.8e-22) || !(y <= 4.4e-26)) {
		tmp = -z - (z * (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 <= (-4.8d-22)) .or. (.not. (y <= 4.4d-26))) then
        tmp = -z - (z * (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 <= -4.8e-22) || !(y <= 4.4e-26)) {
		tmp = -z - (z * (x / y));
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -4.8e-22) or not (y <= 4.4e-26):
		tmp = -z - (z * (x / y))
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -4.8e-22) || !(y <= 4.4e-26))
		tmp = Float64(Float64(-z) - Float64(z * 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 <= -4.8e-22) || ~((y <= 4.4e-26)))
		tmp = -z - (z * (x / y));
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -4.8e-22], N[Not[LessEqual[y, 4.4e-26]], $MachinePrecision]], N[((-z) - N[(z * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.80000000000000005e-22 or 4.4000000000000002e-26 < y

   1. Initial program 75.4%

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

   2. Taylor expanded in z around 0 64.9%

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

      1. mul-1-neg 64.9%

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

      2. associate-/l* 52.0%

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

      3. +-commutative 52.0%

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

      4. distribute-neg-frac 52.0%

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

      5. distribute-neg-in 52.0%

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

      6. +-commutative 52.0%

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

      7. sub-neg 52.0%

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

   4. Simplified 52.0%

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

   5. Taylor expanded in y around 0 70.5%

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

      1. distribute-lft-out 70.5%

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

      2. associate-*r/ 76.5%

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

      3. neg-mul-1 76.5%

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

   7. Simplified 76.5%

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

   if -4.80000000000000005e-22 < y < 4.4000000000000002e-26

   1. Initial program 99.9%

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

   2. Taylor expanded in z around inf 85.7%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{-22} \lor \neg \left(y \leq 4.4 \cdot 10^{-26}\right):\\ \;\;\;\;\left(-z\right) - z \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 5: 68.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6 \cdot 10^{+47}:\\ \;\;\;\;z \cdot \left(-1 - \frac{z}{y}\right)\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{+24}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -6e+47) (* z (- -1.0 (/ z y))) (if (<= y 1.4e+24) (+ x y) (- z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -6e+47) {
		tmp = z * (-1.0 - (z / y));
	} else if (y <= 1.4e+24) {
		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 <= (-6d+47)) then
        tmp = z * ((-1.0d0) - (z / y))
    else if (y <= 1.4d+24) 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 <= -6e+47) {
		tmp = z * (-1.0 - (z / y));
	} else if (y <= 1.4e+24) {
		tmp = x + y;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -6e+47:
		tmp = z * (-1.0 - (z / y))
	elif y <= 1.4e+24:
		tmp = x + y
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -6e+47)
		tmp = Float64(z * Float64(-1.0 - Float64(z / y)));
	elseif (y <= 1.4e+24)
		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 <= -6e+47)
		tmp = z * (-1.0 - (z / y));
	elseif (y <= 1.4e+24)
		tmp = x + y;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -6e+47], N[(z * N[(-1.0 - N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.4e+24], N[(x + y), $MachinePrecision], (-z)]]

Derivation

1. Split input into 3 regimes

2. if y < -6.0000000000000003e47

   1. Initial program 74.1%

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

   2. Taylor expanded in y around inf 68.0%

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

      1. mul-1-neg 68.0%

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

      2. unsub-neg 68.0%

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

      3. mul-1-neg 68.0%

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

      4. associate-/l* 68.1%

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

      5. unpow2 68.1%

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

      6. associate-/l* 80.9%

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

   4. Simplified 80.9%

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

      1. neg-mul-1 80.9%

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

      2. fma-neg 80.9%

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

      3. div-inv 80.8%

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

      4. clear-num 80.9%

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

      5. distribute-lft-neg-in 80.9%

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

      6. add-sqr-sqrt 50.5%

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

      7. sqrt-unprod 59.7%

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

      8. sqr-neg 59.7%

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

      9. sqrt-unprod 24.4%

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

      10. add-sqr-sqrt 62.0%

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

   6. Applied egg-rr 62.0%

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

      1. fma-udef 62.0%

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

      2. *-commutative 62.0%

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

      3. distribute-lft-out 62.0%

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

   8. Simplified 62.0%

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

   9. Taylor expanded in x around 0 51.1%

      \[\leadsto \color{blue}{-1 \cdot z - \frac{{z}^{2}}{y}} \]
   10. Step-by-step derivation

       1. *-commutative 51.1%

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

       2. unpow2 51.1%

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

       3. associate-*r/ 62.6%

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

       4. distribute-lft-out-- 62.6%

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

   11. Simplified 62.6%

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

   if -6.0000000000000003e47 < y < 1.4000000000000001e24

   1. Initial program 99.1%

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

   2. Taylor expanded in z around inf 77.6%

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

   if 1.4000000000000001e24 < y

   1. Initial program 69.1%

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

   2. Taylor expanded in y around inf 59.1%

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

      1. mul-1-neg 59.1%

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

   4. Simplified 59.1%

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

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6 \cdot 10^{+47}:\\ \;\;\;\;z \cdot \left(-1 - \frac{z}{y}\right)\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{+24}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 6: 68.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.6 \cdot 10^{+37}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 5.7 \cdot 10^{+20}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -7.6e+37) (- z) (if (<= y 5.7e+20) (+ x y) (- z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -7.6e+37) {
		tmp = -z;
	} else if (y <= 5.7e+20) {
		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 <= (-7.6d+37)) then
        tmp = -z
    else if (y <= 5.7d+20) 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 <= -7.6e+37) {
		tmp = -z;
	} else if (y <= 5.7e+20) {
		tmp = x + y;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -7.6e+37:
		tmp = -z
	elif y <= 5.7e+20:
		tmp = x + y
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -7.6e+37)
		tmp = Float64(-z);
	elseif (y <= 5.7e+20)
		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 <= -7.6e+37)
		tmp = -z;
	elseif (y <= 5.7e+20)
		tmp = x + y;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -7.6e+37], (-z), If[LessEqual[y, 5.7e+20], N[(x + y), $MachinePrecision], (-z)]]

Derivation

1. Split input into 2 regimes

2. if y < -7.59999999999999979e37 or 5.7e20 < y

   1. Initial program 72.5%

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

   2. Taylor expanded in y around inf 60.2%

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

      1. mul-1-neg 60.2%

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

   4. Simplified 60.2%

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

   if -7.59999999999999979e37 < y < 5.7e20

   1. Initial program 99.1%

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

   2. Taylor expanded in z around inf 78.6%

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

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.6 \cdot 10^{+37}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 5.7 \cdot 10^{+20}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 7: 59.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8.2 \cdot 10^{+30}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{-24}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -8.2e+30) (- z) (if (<= y 2.8e-24) x (- z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -8.2e+30) {
		tmp = -z;
	} else if (y <= 2.8e-24) {
		tmp = x;
	} else {
		tmp = -z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-8.2d+30)) then
        tmp = -z
    else if (y <= 2.8d-24) then
        tmp = x
    else
        tmp = -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -8.2e+30) {
		tmp = -z;
	} else if (y <= 2.8e-24) {
		tmp = x;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -8.2e+30:
		tmp = -z
	elif y <= 2.8e-24:
		tmp = x
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -8.2e+30)
		tmp = Float64(-z);
	elseif (y <= 2.8e-24)
		tmp = x;
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -8.2e+30)
		tmp = -z;
	elseif (y <= 2.8e-24)
		tmp = x;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -8.2e+30], (-z), If[LessEqual[y, 2.8e-24], x, (-z)]]

Derivation

1. Split input into 2 regimes

2. if y < -8.20000000000000011e30 or 2.8000000000000002e-24 < y

   1. Initial program 74.2%

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

   2. Taylor expanded in y around inf 57.9%

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

      1. mul-1-neg 57.9%

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

   4. Simplified 57.9%

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

   if -8.20000000000000011e30 < y < 2.8000000000000002e-24

   1. Initial program 99.0%

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

   2. Taylor expanded in y around 0 66.7%

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

4. Final simplification 61.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.2 \cdot 10^{+30}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{-24}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 8: 38.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.15 \cdot 10^{+43}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 1.95 \cdot 10^{-69}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -3.15e+43) y (if (<= y 1.95e-69) x y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -3.15e+43) {
		tmp = y;
	} else if (y <= 1.95e-69) {
		tmp = x;
	} else {
		tmp = y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-3.15d+43)) then
        tmp = y
    else if (y <= 1.95d-69) then
        tmp = x
    else
        tmp = y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -3.15e+43) {
		tmp = y;
	} else if (y <= 1.95e-69) {
		tmp = x;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -3.15e+43:
		tmp = y
	elif y <= 1.95e-69:
		tmp = x
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -3.15e+43)
		tmp = y;
	elseif (y <= 1.95e-69)
		tmp = x;
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -3.15e+43)
		tmp = y;
	elseif (y <= 1.95e-69)
		tmp = x;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -3.15e+43], y, If[LessEqual[y, 1.95e-69], x, y]]

Derivation

1. Split input into 2 regimes

2. if y < -3.1499999999999999e43 or 1.9499999999999999e-69 < y

   1. Initial program 75.4%

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

   2. Taylor expanded in x around 0 53.5%

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

   3. Taylor expanded in y around 0 17.3%

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

   if -3.1499999999999999e43 < y < 1.9499999999999999e-69

   1. Initial program 99.0%

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

   2. Taylor expanded in y around 0 69.9%

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

4. Final simplification 39.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.15 \cdot 10^{+43}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 1.95 \cdot 10^{-69}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

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 85.2%

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

2. Taylor expanded in y around 0 34.3%

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

3. Final simplification 34.3%

   \[\leadsto x \]

Developer target: 94.0% 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 2023221 
(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))))