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

Percentage Accurate: 88.1% → 99.7%

Time: 5.8s

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: 88.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{-271} \lor \neg \left(t_0 \leq 0\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) - \frac{z}{\frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (+ x y) (- 1.0 (/ y z)))))
   (if (or (<= t_0 -1e-271) (not (<= t_0 0.0))) t_0 (- (- z) (/ z (/ y x))))))

C

double code(double x, double y, double z) {
	double t_0 = (x + y) / (1.0 - (y / z));
	double tmp;
	if ((t_0 <= -1e-271) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = -z - (z / (y / 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) :: t_0
    real(8) :: tmp
    t_0 = (x + y) / (1.0d0 - (y / z))
    if ((t_0 <= (-1d-271)) .or. (.not. (t_0 <= 0.0d0))) then
        tmp = t_0
    else
        tmp = -z - (z / (y / x))
    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-271) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = -z - (z / (y / x));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x + y) / (1.0 - (y / z))
	tmp = 0
	if (t_0 <= -1e-271) or not (t_0 <= 0.0):
		tmp = t_0
	else:
		tmp = -z - (z / (y / x))
	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-271) || !(t_0 <= 0.0))
		tmp = t_0;
	else
		tmp = Float64(Float64(-z) - Float64(z / Float64(y / x)));
	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-271) || ~((t_0 <= 0.0)))
		tmp = t_0;
	else
		tmp = -z - (z / (y / x));
	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-271], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], t$95$0, N[((-z) - N[(z / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z))) < -9.99999999999999963e-272 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 -9.99999999999999963e-272 < (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z))) < 0.0

   1. Initial program 9.1%

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

   2. Taylor expanded in z around 0 96.8%

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

      1. associate-*r/ 96.8%

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

      2. +-commutative 96.8%

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

      3. *-commutative 96.8%

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

      4. associate-*r* 96.8%

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

      5. mul-1-neg 96.8%

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

      6. +-commutative 96.8%

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

   4. Simplified 96.8%

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

   5. Taylor expanded in y around 0 100.0%

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

      1. mul-1-neg 100.0%

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

      2. unsub-neg 100.0%

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

      3. mul-1-neg 100.0%

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

      4. associate-/l* 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \leq -1 \cdot 10^{-271} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \leq 0\right):\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) - \frac{z}{\frac{y}{x}}\\ \end{array} \]

Alternative 2: 70.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x + y\right) \cdot \left(1 + \frac{y}{z}\right)\\ t_1 := \left(-z\right) - \frac{z}{\frac{y}{x}}\\ \mathbf{if}\;z \leq -8 \cdot 10^{+128}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -2 \cdot 10^{+78}:\\ \;\;\;\;\left(-z\right) - z \cdot \frac{x}{y}\\ \mathbf{elif}\;z \leq -25000000000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -4 \cdot 10^{-294}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 9.4 \cdot 10^{-294}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{-31}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (+ x y) (+ 1.0 (/ y z)))) (t_1 (- (- z) (/ z (/ y x)))))
   (if (<= z -8e+128)
     t_0
     (if (<= z -2e+78)
       (- (- z) (* z (/ x y)))
       (if (<= z -25000000000000.0)
         t_0
         (if (<= z -4e-294)
           t_1
           (if (<= z 9.4e-294)
             (/ x (- 1.0 (/ y z)))
             (if (<= z 5.5e-31) t_1 t_0))))))))

C

double code(double x, double y, double z) {
	double t_0 = (x + y) * (1.0 + (y / z));
	double t_1 = -z - (z / (y / x));
	double tmp;
	if (z <= -8e+128) {
		tmp = t_0;
	} else if (z <= -2e+78) {
		tmp = -z - (z * (x / y));
	} else if (z <= -25000000000000.0) {
		tmp = t_0;
	} else if (z <= -4e-294) {
		tmp = t_1;
	} else if (z <= 9.4e-294) {
		tmp = x / (1.0 - (y / z));
	} else if (z <= 5.5e-31) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_0 = (x + y) * (1.0d0 + (y / z))
    t_1 = -z - (z / (y / x))
    if (z <= (-8d+128)) then
        tmp = t_0
    else if (z <= (-2d+78)) then
        tmp = -z - (z * (x / y))
    else if (z <= (-25000000000000.0d0)) then
        tmp = t_0
    else if (z <= (-4d-294)) then
        tmp = t_1
    else if (z <= 9.4d-294) then
        tmp = x / (1.0d0 - (y / z))
    else if (z <= 5.5d-31) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (x + y) * (1.0 + (y / z));
	double t_1 = -z - (z / (y / x));
	double tmp;
	if (z <= -8e+128) {
		tmp = t_0;
	} else if (z <= -2e+78) {
		tmp = -z - (z * (x / y));
	} else if (z <= -25000000000000.0) {
		tmp = t_0;
	} else if (z <= -4e-294) {
		tmp = t_1;
	} else if (z <= 9.4e-294) {
		tmp = x / (1.0 - (y / z));
	} else if (z <= 5.5e-31) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x + y) * (1.0 + (y / z))
	t_1 = -z - (z / (y / x))
	tmp = 0
	if z <= -8e+128:
		tmp = t_0
	elif z <= -2e+78:
		tmp = -z - (z * (x / y))
	elif z <= -25000000000000.0:
		tmp = t_0
	elif z <= -4e-294:
		tmp = t_1
	elif z <= 9.4e-294:
		tmp = x / (1.0 - (y / z))
	elif z <= 5.5e-31:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x + y) * Float64(1.0 + Float64(y / z)))
	t_1 = Float64(Float64(-z) - Float64(z / Float64(y / x)))
	tmp = 0.0
	if (z <= -8e+128)
		tmp = t_0;
	elseif (z <= -2e+78)
		tmp = Float64(Float64(-z) - Float64(z * Float64(x / y)));
	elseif (z <= -25000000000000.0)
		tmp = t_0;
	elseif (z <= -4e-294)
		tmp = t_1;
	elseif (z <= 9.4e-294)
		tmp = Float64(x / Float64(1.0 - Float64(y / z)));
	elseif (z <= 5.5e-31)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x + y) * (1.0 + (y / z));
	t_1 = -z - (z / (y / x));
	tmp = 0.0;
	if (z <= -8e+128)
		tmp = t_0;
	elseif (z <= -2e+78)
		tmp = -z - (z * (x / y));
	elseif (z <= -25000000000000.0)
		tmp = t_0;
	elseif (z <= -4e-294)
		tmp = t_1;
	elseif (z <= 9.4e-294)
		tmp = x / (1.0 - (y / z));
	elseif (z <= 5.5e-31)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x + y), $MachinePrecision] * N[(1.0 + N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[((-z) - N[(z / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -8e+128], t$95$0, If[LessEqual[z, -2e+78], N[((-z) - N[(z * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, -25000000000000.0], t$95$0, If[LessEqual[z, -4e-294], t$95$1, If[LessEqual[z, 9.4e-294], N[(x / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 5.5e-31], t$95$1, t$95$0]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -8.0000000000000006e128 or -2.00000000000000002e78 < z < -2.5e13 or 5.49999999999999958e-31 < z

   1. Initial program 99.9%

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

   2. Taylor expanded in z around inf 72.3%

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

      1. associate-/l* 82.3%

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

      2. +-commutative 82.3%

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

      3. associate-/r/ 82.3%

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

      4. +-commutative 82.3%

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

      5. *-lft-identity 82.3%

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

      6. distribute-rgt-in 82.3%

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

      7. +-commutative 82.3%

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

      8. +-commutative 82.3%

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

   4. Simplified 82.3%

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

   if -8.0000000000000006e128 < z < -2.00000000000000002e78

   1. Initial program 99.7%

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

   2. Taylor expanded in z around 0 68.2%

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

      1. associate-*r/ 68.2%

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

      2. +-commutative 68.2%

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

      3. *-commutative 68.2%

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

      4. associate-*r* 68.2%

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

      5. mul-1-neg 68.2%

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

      6. +-commutative 68.2%

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

   4. Simplified 68.2%

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

   5. Taylor expanded in y around 0 67.5%

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

      1. mul-1-neg 67.5%

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

      2. unsub-neg 67.5%

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

      3. mul-1-neg 67.5%

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

      4. associate-/l* 100.0%

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

   7. Simplified 100.0%

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

      1. clear-num 100.0%

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

      2. associate-/r/ 100.0%

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

      3. clear-num 100.0%

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

   9. Applied egg-rr 100.0%

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

   if -2.5e13 < z < -4.00000000000000007e-294 or 9.4000000000000003e-294 < z < 5.49999999999999958e-31

   1. Initial program 74.9%

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

   2. Taylor expanded in z around 0 72.1%

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

      1. associate-*r/ 72.1%

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

      2. +-commutative 72.1%

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

      3. *-commutative 72.1%

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

      4. associate-*r* 72.1%

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

      5. mul-1-neg 72.1%

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

      6. +-commutative 72.1%

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

   4. Simplified 72.1%

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

   5. Taylor expanded in y around 0 75.4%

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

      1. mul-1-neg 75.4%

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

      2. unsub-neg 75.4%

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

      3. mul-1-neg 75.4%

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

      4. associate-/l* 75.4%

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

   7. Simplified 75.4%

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

   if -4.00000000000000007e-294 < z < 9.4000000000000003e-294

   1. Initial program 90.2%

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

   2. Taylor expanded in x around inf 90.2%

      \[\leadsto \color{blue}{\frac{x}{1 - \frac{y}{z}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8 \cdot 10^{+128}:\\ \;\;\;\;\left(x + y\right) \cdot \left(1 + \frac{y}{z}\right)\\ \mathbf{elif}\;z \leq -2 \cdot 10^{+78}:\\ \;\;\;\;\left(-z\right) - z \cdot \frac{x}{y}\\ \mathbf{elif}\;z \leq -25000000000000:\\ \;\;\;\;\left(x + y\right) \cdot \left(1 + \frac{y}{z}\right)\\ \mathbf{elif}\;z \leq -4 \cdot 10^{-294}:\\ \;\;\;\;\left(-z\right) - \frac{z}{\frac{y}{x}}\\ \mathbf{elif}\;z \leq 9.4 \cdot 10^{-294}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{-31}:\\ \;\;\;\;\left(-z\right) - \frac{z}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) \cdot \left(1 + \frac{y}{z}\right)\\ \end{array} \]

Alternative 3: 68.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{y}{z}\\ \mathbf{if}\;y \leq -2.9 \cdot 10^{+104}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq -1.2 \cdot 10^{-40}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq -5.4 \cdot 10^{-71}:\\ \;\;\;\;\frac{y}{t_0}\\ \mathbf{elif}\;y \leq 4.3 \cdot 10^{-162}:\\ \;\;\;\;\frac{x}{t_0}\\ \mathbf{elif}\;y \leq 7.3 \cdot 10^{-62}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{+154}:\\ \;\;\;\;\left(x + y\right) \cdot \left(-\frac{z}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ y z))))
   (if (<= y -2.9e+104)
     (- z)
     (if (<= y -1.2e-40)
       (+ x y)
       (if (<= y -5.4e-71)
         (/ y t_0)
         (if (<= y 4.3e-162)
           (/ x t_0)
           (if (<= y 7.3e-62)
             (+ x y)
             (if (<= y 7.5e+154) (* (+ x y) (- (/ z y))) (- z)))))))))

C

double code(double x, double y, double z) {
	double t_0 = 1.0 - (y / z);
	double tmp;
	if (y <= -2.9e+104) {
		tmp = -z;
	} else if (y <= -1.2e-40) {
		tmp = x + y;
	} else if (y <= -5.4e-71) {
		tmp = y / t_0;
	} else if (y <= 4.3e-162) {
		tmp = x / t_0;
	} else if (y <= 7.3e-62) {
		tmp = x + y;
	} else if (y <= 7.5e+154) {
		tmp = (x + y) * -(z / 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 <= (-2.9d+104)) then
        tmp = -z
    else if (y <= (-1.2d-40)) then
        tmp = x + y
    else if (y <= (-5.4d-71)) then
        tmp = y / t_0
    else if (y <= 4.3d-162) then
        tmp = x / t_0
    else if (y <= 7.3d-62) then
        tmp = x + y
    else if (y <= 7.5d+154) then
        tmp = (x + y) * -(z / 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 <= -2.9e+104) {
		tmp = -z;
	} else if (y <= -1.2e-40) {
		tmp = x + y;
	} else if (y <= -5.4e-71) {
		tmp = y / t_0;
	} else if (y <= 4.3e-162) {
		tmp = x / t_0;
	} else if (y <= 7.3e-62) {
		tmp = x + y;
	} else if (y <= 7.5e+154) {
		tmp = (x + y) * -(z / y);
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 1.0 - (y / z)
	tmp = 0
	if y <= -2.9e+104:
		tmp = -z
	elif y <= -1.2e-40:
		tmp = x + y
	elif y <= -5.4e-71:
		tmp = y / t_0
	elif y <= 4.3e-162:
		tmp = x / t_0
	elif y <= 7.3e-62:
		tmp = x + y
	elif y <= 7.5e+154:
		tmp = (x + y) * -(z / 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 <= -2.9e+104)
		tmp = Float64(-z);
	elseif (y <= -1.2e-40)
		tmp = Float64(x + y);
	elseif (y <= -5.4e-71)
		tmp = Float64(y / t_0);
	elseif (y <= 4.3e-162)
		tmp = Float64(x / t_0);
	elseif (y <= 7.3e-62)
		tmp = Float64(x + y);
	elseif (y <= 7.5e+154)
		tmp = Float64(Float64(x + y) * Float64(-Float64(z / 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 <= -2.9e+104)
		tmp = -z;
	elseif (y <= -1.2e-40)
		tmp = x + y;
	elseif (y <= -5.4e-71)
		tmp = y / t_0;
	elseif (y <= 4.3e-162)
		tmp = x / t_0;
	elseif (y <= 7.3e-62)
		tmp = x + y;
	elseif (y <= 7.5e+154)
		tmp = (x + y) * -(z / 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, -2.9e+104], (-z), If[LessEqual[y, -1.2e-40], N[(x + y), $MachinePrecision], If[LessEqual[y, -5.4e-71], N[(y / t$95$0), $MachinePrecision], If[LessEqual[y, 4.3e-162], N[(x / t$95$0), $MachinePrecision], If[LessEqual[y, 7.3e-62], N[(x + y), $MachinePrecision], If[LessEqual[y, 7.5e+154], N[(N[(x + y), $MachinePrecision] * (-N[(z / y), $MachinePrecision])), $MachinePrecision], (-z)]]]]]]]

Derivation

1. Split input into 5 regimes

2. if y < -2.8999999999999998e104 or 7.5000000000000004e154 < y

   1. Initial program 64.0%

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

   2. Taylor expanded in y around inf 71.5%

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

      1. mul-1-neg 71.5%

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

   4. Simplified 71.5%

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

   if -2.8999999999999998e104 < y < -1.19999999999999996e-40 or 4.29999999999999996e-162 < y < 7.2999999999999998e-62

   1. Initial program 97.9%

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

   2. Taylor expanded in z around inf 68.8%

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

   if -1.19999999999999996e-40 < y < -5.4000000000000003e-71

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 70.3%

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

   if -5.4000000000000003e-71 < y < 4.29999999999999996e-162

   1. Initial program 99.9%

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

   2. Taylor expanded in x around inf 86.9%

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

   if 7.2999999999999998e-62 < y < 7.5000000000000004e154

   1. Initial program 94.7%

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

      1. clear-num 94.2%

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

      2. associate-/r/ 94.5%

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

   3. Applied egg-rr 94.5%

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

   4. Taylor expanded in y around inf 64.8%

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

      1. associate-*r/ 64.8%

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

      2. mul-1-neg 64.8%

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

   6. Simplified 64.8%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.9 \cdot 10^{+104}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq -1.2 \cdot 10^{-40}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq -5.4 \cdot 10^{-71}:\\ \;\;\;\;\frac{y}{1 - \frac{y}{z}}\\ \mathbf{elif}\;y \leq 4.3 \cdot 10^{-162}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{elif}\;y \leq 7.3 \cdot 10^{-62}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{+154}:\\ \;\;\;\;\left(x + y\right) \cdot \left(-\frac{z}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 4: 70.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-z\right) - z \cdot \frac{x}{y}\\ \mathbf{if}\;z \leq -8 \cdot 10^{+128}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq -9 \cdot 10^{+80}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -1380000000000:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq -8.8 \cdot 10^{-294}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 2.8 \cdot 10^{-294}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{elif}\;z \leq 5.1 \cdot 10^{-30}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (- z) (* z (/ x y)))))
   (if (<= z -8e+128)
     (+ x y)
     (if (<= z -9e+80)
       t_0
       (if (<= z -1380000000000.0)
         (+ x y)
         (if (<= z -8.8e-294)
           t_0
           (if (<= z 2.8e-294)
             (/ x (- 1.0 (/ y z)))
             (if (<= z 5.1e-30) t_0 (+ x y)))))))))

C

double code(double x, double y, double z) {
	double t_0 = -z - (z * (x / y));
	double tmp;
	if (z <= -8e+128) {
		tmp = x + y;
	} else if (z <= -9e+80) {
		tmp = t_0;
	} else if (z <= -1380000000000.0) {
		tmp = x + y;
	} else if (z <= -8.8e-294) {
		tmp = t_0;
	} else if (z <= 2.8e-294) {
		tmp = x / (1.0 - (y / z));
	} else if (z <= 5.1e-30) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = -z - (z * (x / y))
    if (z <= (-8d+128)) then
        tmp = x + y
    else if (z <= (-9d+80)) then
        tmp = t_0
    else if (z <= (-1380000000000.0d0)) then
        tmp = x + y
    else if (z <= (-8.8d-294)) then
        tmp = t_0
    else if (z <= 2.8d-294) then
        tmp = x / (1.0d0 - (y / z))
    else if (z <= 5.1d-30) then
        tmp = t_0
    else
        tmp = x + y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = -z - (z * (x / y));
	double tmp;
	if (z <= -8e+128) {
		tmp = x + y;
	} else if (z <= -9e+80) {
		tmp = t_0;
	} else if (z <= -1380000000000.0) {
		tmp = x + y;
	} else if (z <= -8.8e-294) {
		tmp = t_0;
	} else if (z <= 2.8e-294) {
		tmp = x / (1.0 - (y / z));
	} else if (z <= 5.1e-30) {
		tmp = t_0;
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = -z - (z * (x / y))
	tmp = 0
	if z <= -8e+128:
		tmp = x + y
	elif z <= -9e+80:
		tmp = t_0
	elif z <= -1380000000000.0:
		tmp = x + y
	elif z <= -8.8e-294:
		tmp = t_0
	elif z <= 2.8e-294:
		tmp = x / (1.0 - (y / z))
	elif z <= 5.1e-30:
		tmp = t_0
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(-z) - Float64(z * Float64(x / y)))
	tmp = 0.0
	if (z <= -8e+128)
		tmp = Float64(x + y);
	elseif (z <= -9e+80)
		tmp = t_0;
	elseif (z <= -1380000000000.0)
		tmp = Float64(x + y);
	elseif (z <= -8.8e-294)
		tmp = t_0;
	elseif (z <= 2.8e-294)
		tmp = Float64(x / Float64(1.0 - Float64(y / z)));
	elseif (z <= 5.1e-30)
		tmp = t_0;
	else
		tmp = Float64(x + y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = -z - (z * (x / y));
	tmp = 0.0;
	if (z <= -8e+128)
		tmp = x + y;
	elseif (z <= -9e+80)
		tmp = t_0;
	elseif (z <= -1380000000000.0)
		tmp = x + y;
	elseif (z <= -8.8e-294)
		tmp = t_0;
	elseif (z <= 2.8e-294)
		tmp = x / (1.0 - (y / z));
	elseif (z <= 5.1e-30)
		tmp = t_0;
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[((-z) - N[(z * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -8e+128], N[(x + y), $MachinePrecision], If[LessEqual[z, -9e+80], t$95$0, If[LessEqual[z, -1380000000000.0], N[(x + y), $MachinePrecision], If[LessEqual[z, -8.8e-294], t$95$0, If[LessEqual[z, 2.8e-294], N[(x / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 5.1e-30], t$95$0, N[(x + y), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -8.0000000000000006e128 or -9.00000000000000013e80 < z < -1.38e12 or 5.09999999999999972e-30 < z

   1. Initial program 99.9%

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

   2. Taylor expanded in z around inf 82.0%

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

   if -8.0000000000000006e128 < z < -9.00000000000000013e80 or -1.38e12 < z < -8.8e-294 or 2.79999999999999991e-294 < z < 5.09999999999999972e-30

   1. Initial program 76.1%

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

   2. Taylor expanded in z around 0 71.2%

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

      1. associate-*r/ 71.2%

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

      2. +-commutative 71.2%

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

      3. *-commutative 71.2%

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

      4. associate-*r* 71.2%

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

      5. mul-1-neg 71.2%

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

      6. +-commutative 71.2%

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

   4. Simplified 71.2%

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

   5. Taylor expanded in y around 0 74.3%

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

      1. mul-1-neg 74.3%

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

      2. unsub-neg 74.3%

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

      3. mul-1-neg 74.3%

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

      4. associate-/l* 76.6%

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

   7. Simplified 76.6%

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

      1. clear-num 76.4%

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

      2. associate-/r/ 76.5%

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

      3. clear-num 76.5%

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

   9. Applied egg-rr 76.5%

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

   if -8.8e-294 < z < 2.79999999999999991e-294

   1. Initial program 90.0%

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

   2. Taylor expanded in x around inf 90.0%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8 \cdot 10^{+128}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq -9 \cdot 10^{+80}:\\ \;\;\;\;\left(-z\right) - z \cdot \frac{x}{y}\\ \mathbf{elif}\;z \leq -1380000000000:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq -8.8 \cdot 10^{-294}:\\ \;\;\;\;\left(-z\right) - z \cdot \frac{x}{y}\\ \mathbf{elif}\;z \leq 2.8 \cdot 10^{-294}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{elif}\;z \leq 5.1 \cdot 10^{-30}:\\ \;\;\;\;\left(-z\right) - z \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 5: 71.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-z\right) - \frac{z}{\frac{y}{x}}\\ \mathbf{if}\;z \leq -8 \cdot 10^{+128}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq -3.7 \cdot 10^{+79}:\\ \;\;\;\;\left(-z\right) - z \cdot \frac{x}{y}\\ \mathbf{elif}\;z \leq -15200000:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq -4 \cdot 10^{-294}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 9.4 \cdot 10^{-294}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{elif}\;z \leq 2 \cdot 10^{-32}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (- z) (/ z (/ y x)))))
   (if (<= z -8e+128)
     (+ x y)
     (if (<= z -3.7e+79)
       (- (- z) (* z (/ x y)))
       (if (<= z -15200000.0)
         (+ x y)
         (if (<= z -4e-294)
           t_0
           (if (<= z 9.4e-294)
             (/ x (- 1.0 (/ y z)))
             (if (<= z 2e-32) t_0 (+ x y)))))))))

C

double code(double x, double y, double z) {
	double t_0 = -z - (z / (y / x));
	double tmp;
	if (z <= -8e+128) {
		tmp = x + y;
	} else if (z <= -3.7e+79) {
		tmp = -z - (z * (x / y));
	} else if (z <= -15200000.0) {
		tmp = x + y;
	} else if (z <= -4e-294) {
		tmp = t_0;
	} else if (z <= 9.4e-294) {
		tmp = x / (1.0 - (y / z));
	} else if (z <= 2e-32) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = -z - (z / (y / x))
    if (z <= (-8d+128)) then
        tmp = x + y
    else if (z <= (-3.7d+79)) then
        tmp = -z - (z * (x / y))
    else if (z <= (-15200000.0d0)) then
        tmp = x + y
    else if (z <= (-4d-294)) then
        tmp = t_0
    else if (z <= 9.4d-294) then
        tmp = x / (1.0d0 - (y / z))
    else if (z <= 2d-32) then
        tmp = t_0
    else
        tmp = x + y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = -z - (z / (y / x));
	double tmp;
	if (z <= -8e+128) {
		tmp = x + y;
	} else if (z <= -3.7e+79) {
		tmp = -z - (z * (x / y));
	} else if (z <= -15200000.0) {
		tmp = x + y;
	} else if (z <= -4e-294) {
		tmp = t_0;
	} else if (z <= 9.4e-294) {
		tmp = x / (1.0 - (y / z));
	} else if (z <= 2e-32) {
		tmp = t_0;
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = -z - (z / (y / x))
	tmp = 0
	if z <= -8e+128:
		tmp = x + y
	elif z <= -3.7e+79:
		tmp = -z - (z * (x / y))
	elif z <= -15200000.0:
		tmp = x + y
	elif z <= -4e-294:
		tmp = t_0
	elif z <= 9.4e-294:
		tmp = x / (1.0 - (y / z))
	elif z <= 2e-32:
		tmp = t_0
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(-z) - Float64(z / Float64(y / x)))
	tmp = 0.0
	if (z <= -8e+128)
		tmp = Float64(x + y);
	elseif (z <= -3.7e+79)
		tmp = Float64(Float64(-z) - Float64(z * Float64(x / y)));
	elseif (z <= -15200000.0)
		tmp = Float64(x + y);
	elseif (z <= -4e-294)
		tmp = t_0;
	elseif (z <= 9.4e-294)
		tmp = Float64(x / Float64(1.0 - Float64(y / z)));
	elseif (z <= 2e-32)
		tmp = t_0;
	else
		tmp = Float64(x + y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = -z - (z / (y / x));
	tmp = 0.0;
	if (z <= -8e+128)
		tmp = x + y;
	elseif (z <= -3.7e+79)
		tmp = -z - (z * (x / y));
	elseif (z <= -15200000.0)
		tmp = x + y;
	elseif (z <= -4e-294)
		tmp = t_0;
	elseif (z <= 9.4e-294)
		tmp = x / (1.0 - (y / z));
	elseif (z <= 2e-32)
		tmp = t_0;
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[((-z) - N[(z / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -8e+128], N[(x + y), $MachinePrecision], If[LessEqual[z, -3.7e+79], N[((-z) - N[(z * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, -15200000.0], N[(x + y), $MachinePrecision], If[LessEqual[z, -4e-294], t$95$0, If[LessEqual[z, 9.4e-294], N[(x / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2e-32], t$95$0, N[(x + y), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -8.0000000000000006e128 or -3.70000000000000009e79 < z < -1.52e7 or 2.00000000000000011e-32 < z

   1. Initial program 99.9%

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

   2. Taylor expanded in z around inf 82.0%

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

   if -8.0000000000000006e128 < z < -3.70000000000000009e79

   1. Initial program 99.7%

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

   2. Taylor expanded in z around 0 68.2%

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

      1. associate-*r/ 68.2%

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

      2. +-commutative 68.2%

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

      3. *-commutative 68.2%

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

      4. associate-*r* 68.2%

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

      5. mul-1-neg 68.2%

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

      6. +-commutative 68.2%

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

   4. Simplified 68.2%

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

   5. Taylor expanded in y around 0 67.5%

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

      1. mul-1-neg 67.5%

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

      2. unsub-neg 67.5%

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

      3. mul-1-neg 67.5%

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

      4. associate-/l* 100.0%

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

   7. Simplified 100.0%

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

      1. clear-num 100.0%

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

      2. associate-/r/ 100.0%

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

      3. clear-num 100.0%

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

   9. Applied egg-rr 100.0%

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

   if -1.52e7 < z < -4.00000000000000007e-294 or 9.4000000000000003e-294 < z < 2.00000000000000011e-32

   1. Initial program 74.9%

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

   2. Taylor expanded in z around 0 72.1%

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

      1. associate-*r/ 72.1%

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

      2. +-commutative 72.1%

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

      3. *-commutative 72.1%

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

      4. associate-*r* 72.1%

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

      5. mul-1-neg 72.1%

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

      6. +-commutative 72.1%

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

   4. Simplified 72.1%

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

   5. Taylor expanded in y around 0 75.4%

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

      1. mul-1-neg 75.4%

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

      2. unsub-neg 75.4%

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

      3. mul-1-neg 75.4%

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

      4. associate-/l* 75.4%

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

   7. Simplified 75.4%

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

   if -4.00000000000000007e-294 < z < 9.4000000000000003e-294

   1. Initial program 90.2%

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

   2. Taylor expanded in x around inf 90.2%

      \[\leadsto \color{blue}{\frac{x}{1 - \frac{y}{z}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8 \cdot 10^{+128}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq -3.7 \cdot 10^{+79}:\\ \;\;\;\;\left(-z\right) - z \cdot \frac{x}{y}\\ \mathbf{elif}\;z \leq -15200000:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq -4 \cdot 10^{-294}:\\ \;\;\;\;\left(-z\right) - \frac{z}{\frac{y}{x}}\\ \mathbf{elif}\;z \leq 9.4 \cdot 10^{-294}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{elif}\;z \leq 2 \cdot 10^{-32}:\\ \;\;\;\;\left(-z\right) - \frac{z}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 6: 67.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{y}{z}\\ t_1 := \frac{x}{t_0}\\ \mathbf{if}\;y \leq -6 \cdot 10^{+104}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq -2.15 \cdot 10^{-40}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq -5.8 \cdot 10^{-69}:\\ \;\;\;\;\frac{y}{t_0}\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{-160}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 9.2 \cdot 10^{-43}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 2.95 \cdot 10^{+26}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ y z))) (t_1 (/ x t_0)))
   (if (<= y -6e+104)
     (- z)
     (if (<= y -2.15e-40)
       (+ x y)
       (if (<= y -5.8e-69)
         (/ y t_0)
         (if (<= y 1.15e-160)
           t_1
           (if (<= y 9.2e-43) (+ x y) (if (<= y 2.95e+26) 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 <= -6e+104) {
		tmp = -z;
	} else if (y <= -2.15e-40) {
		tmp = x + y;
	} else if (y <= -5.8e-69) {
		tmp = y / t_0;
	} else if (y <= 1.15e-160) {
		tmp = t_1;
	} else if (y <= 9.2e-43) {
		tmp = x + y;
	} else if (y <= 2.95e+26) {
		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 <= (-6d+104)) then
        tmp = -z
    else if (y <= (-2.15d-40)) then
        tmp = x + y
    else if (y <= (-5.8d-69)) then
        tmp = y / t_0
    else if (y <= 1.15d-160) then
        tmp = t_1
    else if (y <= 9.2d-43) then
        tmp = x + y
    else if (y <= 2.95d+26) 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 <= -6e+104) {
		tmp = -z;
	} else if (y <= -2.15e-40) {
		tmp = x + y;
	} else if (y <= -5.8e-69) {
		tmp = y / t_0;
	} else if (y <= 1.15e-160) {
		tmp = t_1;
	} else if (y <= 9.2e-43) {
		tmp = x + y;
	} else if (y <= 2.95e+26) {
		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 <= -6e+104:
		tmp = -z
	elif y <= -2.15e-40:
		tmp = x + y
	elif y <= -5.8e-69:
		tmp = y / t_0
	elif y <= 1.15e-160:
		tmp = t_1
	elif y <= 9.2e-43:
		tmp = x + y
	elif y <= 2.95e+26:
		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 <= -6e+104)
		tmp = Float64(-z);
	elseif (y <= -2.15e-40)
		tmp = Float64(x + y);
	elseif (y <= -5.8e-69)
		tmp = Float64(y / t_0);
	elseif (y <= 1.15e-160)
		tmp = t_1;
	elseif (y <= 9.2e-43)
		tmp = Float64(x + y);
	elseif (y <= 2.95e+26)
		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 <= -6e+104)
		tmp = -z;
	elseif (y <= -2.15e-40)
		tmp = x + y;
	elseif (y <= -5.8e-69)
		tmp = y / t_0;
	elseif (y <= 1.15e-160)
		tmp = t_1;
	elseif (y <= 9.2e-43)
		tmp = x + y;
	elseif (y <= 2.95e+26)
		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, -6e+104], (-z), If[LessEqual[y, -2.15e-40], N[(x + y), $MachinePrecision], If[LessEqual[y, -5.8e-69], N[(y / t$95$0), $MachinePrecision], If[LessEqual[y, 1.15e-160], t$95$1, If[LessEqual[y, 9.2e-43], N[(x + y), $MachinePrecision], If[LessEqual[y, 2.95e+26], t$95$1, (-z)]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -5.99999999999999937e104 or 2.95000000000000015e26 < y

   1. Initial program 69.9%

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

   2. Taylor expanded in y around inf 66.7%

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

      1. mul-1-neg 66.7%

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

   4. Simplified 66.7%

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

   if -5.99999999999999937e104 < y < -2.1500000000000001e-40 or 1.14999999999999992e-160 < y < 9.1999999999999995e-43

   1. Initial program 98.0%

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

   2. Taylor expanded in z around inf 67.0%

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

   if -2.1500000000000001e-40 < y < -5.7999999999999997e-69

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 70.3%

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

   if -5.7999999999999997e-69 < y < 1.14999999999999992e-160 or 9.1999999999999995e-43 < y < 2.95000000000000015e26

   1. Initial program 99.9%

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

   2. Taylor expanded in x around inf 84.1%

      \[\leadsto \color{blue}{\frac{x}{1 - \frac{y}{z}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6 \cdot 10^{+104}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq -2.15 \cdot 10^{-40}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq -5.8 \cdot 10^{-69}:\\ \;\;\;\;\frac{y}{1 - \frac{y}{z}}\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{-160}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{elif}\;y \leq 9.2 \cdot 10^{-43}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 2.95 \cdot 10^{+26}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 7: 67.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{1 - \frac{y}{z}}\\ \mathbf{if}\;y \leq -2.4 \cdot 10^{+104}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq -1.6 \cdot 10^{-205}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{-161}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 8 \cdot 10^{-40}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{+22}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 5.7 \cdot 10^{+72}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 1.95 \cdot 10^{+120}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ x (- 1.0 (/ y z)))))
   (if (<= y -2.4e+104)
     (- z)
     (if (<= y -1.6e-205)
       (+ x y)
       (if (<= y 4.2e-161)
         t_0
         (if (<= y 8e-40)
           (+ x y)
           (if (<= y 7.2e+22)
             t_0
             (if (<= y 5.7e+72)
               (- z)
               (if (<= y 1.95e+120) (+ x y) (- z))))))))))

C

double code(double x, double y, double z) {
	double t_0 = x / (1.0 - (y / z));
	double tmp;
	if (y <= -2.4e+104) {
		tmp = -z;
	} else if (y <= -1.6e-205) {
		tmp = x + y;
	} else if (y <= 4.2e-161) {
		tmp = t_0;
	} else if (y <= 8e-40) {
		tmp = x + y;
	} else if (y <= 7.2e+22) {
		tmp = t_0;
	} else if (y <= 5.7e+72) {
		tmp = -z;
	} else if (y <= 1.95e+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 = x / (1.0d0 - (y / z))
    if (y <= (-2.4d+104)) then
        tmp = -z
    else if (y <= (-1.6d-205)) then
        tmp = x + y
    else if (y <= 4.2d-161) then
        tmp = t_0
    else if (y <= 8d-40) then
        tmp = x + y
    else if (y <= 7.2d+22) then
        tmp = t_0
    else if (y <= 5.7d+72) then
        tmp = -z
    else if (y <= 1.95d+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 = x / (1.0 - (y / z));
	double tmp;
	if (y <= -2.4e+104) {
		tmp = -z;
	} else if (y <= -1.6e-205) {
		tmp = x + y;
	} else if (y <= 4.2e-161) {
		tmp = t_0;
	} else if (y <= 8e-40) {
		tmp = x + y;
	} else if (y <= 7.2e+22) {
		tmp = t_0;
	} else if (y <= 5.7e+72) {
		tmp = -z;
	} else if (y <= 1.95e+120) {
		tmp = x + y;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x / (1.0 - (y / z))
	tmp = 0
	if y <= -2.4e+104:
		tmp = -z
	elif y <= -1.6e-205:
		tmp = x + y
	elif y <= 4.2e-161:
		tmp = t_0
	elif y <= 8e-40:
		tmp = x + y
	elif y <= 7.2e+22:
		tmp = t_0
	elif y <= 5.7e+72:
		tmp = -z
	elif y <= 1.95e+120:
		tmp = x + y
	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 <= -2.4e+104)
		tmp = Float64(-z);
	elseif (y <= -1.6e-205)
		tmp = Float64(x + y);
	elseif (y <= 4.2e-161)
		tmp = t_0;
	elseif (y <= 8e-40)
		tmp = Float64(x + y);
	elseif (y <= 7.2e+22)
		tmp = t_0;
	elseif (y <= 5.7e+72)
		tmp = Float64(-z);
	elseif (y <= 1.95e+120)
		tmp = Float64(x + y);
	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 <= -2.4e+104)
		tmp = -z;
	elseif (y <= -1.6e-205)
		tmp = x + y;
	elseif (y <= 4.2e-161)
		tmp = t_0;
	elseif (y <= 8e-40)
		tmp = x + y;
	elseif (y <= 7.2e+22)
		tmp = t_0;
	elseif (y <= 5.7e+72)
		tmp = -z;
	elseif (y <= 1.95e+120)
		tmp = x + y;
	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, -2.4e+104], (-z), If[LessEqual[y, -1.6e-205], N[(x + y), $MachinePrecision], If[LessEqual[y, 4.2e-161], t$95$0, If[LessEqual[y, 8e-40], N[(x + y), $MachinePrecision], If[LessEqual[y, 7.2e+22], t$95$0, If[LessEqual[y, 5.7e+72], (-z), If[LessEqual[y, 1.95e+120], N[(x + y), $MachinePrecision], (-z)]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.4e104 or 7.2e22 < y < 5.6999999999999997e72 or 1.9499999999999999e120 < y

   1. Initial program 67.2%

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

   2. Taylor expanded in y around inf 71.4%

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

      1. mul-1-neg 71.4%

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

   4. Simplified 71.4%

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

   if -2.4e104 < y < -1.60000000000000005e-205 or 4.2000000000000001e-161 < y < 7.9999999999999994e-40 or 5.6999999999999997e72 < y < 1.9499999999999999e120

   1. Initial program 98.9%

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

   2. Taylor expanded in z around inf 66.7%

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

   if -1.60000000000000005e-205 < y < 4.2000000000000001e-161 or 7.9999999999999994e-40 < y < 7.2e22

   1. Initial program 99.9%

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

   2. Taylor expanded in x around inf 90.1%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.4 \cdot 10^{+104}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq -1.6 \cdot 10^{-205}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{-161}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{elif}\;y \leq 8 \cdot 10^{-40}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{+22}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{elif}\;y \leq 5.7 \cdot 10^{+72}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 1.95 \cdot 10^{+120}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 8: 67.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{+104}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+23} \lor \neg \left(y \leq 3.9 \cdot 10^{+70}\right) \land y \leq 2 \cdot 10^{+138}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -7e+104)
   (- z)
   (if (or (<= y 2e+23) (and (not (<= y 3.9e+70)) (<= y 2e+138)))
     (+ x y)
     (- z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -7e+104) {
		tmp = -z;
	} else if ((y <= 2e+23) || (!(y <= 3.9e+70) && (y <= 2e+138))) {
		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 <= (-7d+104)) then
        tmp = -z
    else if ((y <= 2d+23) .or. (.not. (y <= 3.9d+70)) .and. (y <= 2d+138)) 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 <= -7e+104) {
		tmp = -z;
	} else if ((y <= 2e+23) || (!(y <= 3.9e+70) && (y <= 2e+138))) {
		tmp = x + y;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -7e+104:
		tmp = -z
	elif (y <= 2e+23) or (not (y <= 3.9e+70) and (y <= 2e+138)):
		tmp = x + y
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -7e+104)
		tmp = Float64(-z);
	elseif ((y <= 2e+23) || (!(y <= 3.9e+70) && (y <= 2e+138)))
		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 <= -7e+104)
		tmp = -z;
	elseif ((y <= 2e+23) || (~((y <= 3.9e+70)) && (y <= 2e+138)))
		tmp = x + y;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -7e+104], (-z), If[Or[LessEqual[y, 2e+23], And[N[Not[LessEqual[y, 3.9e+70]], $MachinePrecision], LessEqual[y, 2e+138]]], N[(x + y), $MachinePrecision], (-z)]]

Derivation

1. Split input into 2 regimes

2. if y < -7.0000000000000003e104 or 1.9999999999999998e23 < y < 3.89999999999999975e70 or 2.0000000000000001e138 < y

   1. Initial program 67.2%

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

   2. Taylor expanded in y around inf 71.4%

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

      1. mul-1-neg 71.4%

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

   4. Simplified 71.4%

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

   if -7.0000000000000003e104 < y < 1.9999999999999998e23 or 3.89999999999999975e70 < y < 2.0000000000000001e138

   1. Initial program 99.3%

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

   2. Taylor expanded in z around inf 71.1%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{+104}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+23} \lor \neg \left(y \leq 3.9 \cdot 10^{+70}\right) \land y \leq 2 \cdot 10^{+138}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 9: 58.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8.5 \cdot 10^{+45}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 7.3 \cdot 10^{-62}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -8.5e+45) (- z) (if (<= y 7.3e-62) x (- z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -8.5e+45) {
		tmp = -z;
	} else if (y <= 7.3e-62) {
		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 <= (-8.5d+45)) then
        tmp = -z
    else if (y <= 7.3d-62) 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 <= -8.5e+45) {
		tmp = -z;
	} else if (y <= 7.3e-62) {
		tmp = x;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -8.5e+45:
		tmp = -z
	elif y <= 7.3e-62:
		tmp = x
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -8.5e+45)
		tmp = Float64(-z);
	elseif (y <= 7.3e-62)
		tmp = x;
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -8.5e+45)
		tmp = -z;
	elseif (y <= 7.3e-62)
		tmp = x;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -8.5e+45], (-z), If[LessEqual[y, 7.3e-62], x, (-z)]]

Derivation

1. Split input into 2 regimes

2. if y < -8.4999999999999996e45 or 7.2999999999999998e-62 < y

   1. Initial program 75.4%

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

   2. Taylor expanded in y around inf 58.0%

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

      1. mul-1-neg 58.0%

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

   4. Simplified 58.0%

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

   if -8.4999999999999996e45 < y < 7.2999999999999998e-62

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 59.8%

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

4. Final simplification 59.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.5 \cdot 10^{+45}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 7.3 \cdot 10^{-62}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 10: 38.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{-79}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{-69}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2.2e-79) y (if (<= y 2.5e-69) x y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.2e-79) {
		tmp = y;
	} else if (y <= 2.5e-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 <= (-2.2d-79)) then
        tmp = y
    else if (y <= 2.5d-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 <= -2.2e-79) {
		tmp = y;
	} else if (y <= 2.5e-69) {
		tmp = x;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -2.2e-79:
		tmp = y
	elif y <= 2.5e-69:
		tmp = x
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -2.2e-79)
		tmp = y;
	elseif (y <= 2.5e-69)
		tmp = x;
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -2.2e-79)
		tmp = y;
	elseif (y <= 2.5e-69)
		tmp = x;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -2.2e-79], y, If[LessEqual[y, 2.5e-69], x, y]]

Derivation

1. Split input into 2 regimes

2. if y < -2.1999999999999999e-79 or 2.50000000000000017e-69 < y

   1. Initial program 80.0%

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

   2. Taylor expanded in x around 0 56.1%

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

   3. Taylor expanded in y around 0 22.5%

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

   if -2.1999999999999999e-79 < y < 2.50000000000000017e-69

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 69.1%

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

4. Final simplification 41.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{-79}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{-69}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 11: 35.3% 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 88.2%

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

2. Taylor expanded in y around 0 35.0%

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

3. Final simplification 35.0%

   \[\leadsto x \]

Developer target: 94.1% 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 2023200 
(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))))