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

Percentage Accurate: 87.9% → 99.6%

Time: 6.1s

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: 87.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-271} \lor \neg \left(t\_0 \leq 0\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot \left(x + z\right)}{-y} - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (+ x y) (- 1.0 (/ y z)))))
   (if (or (<= t_0 -5e-271) (not (<= t_0 0.0)))
     t_0
     (- (/ (* z (+ x z)) (- y)) z))))

C

double code(double x, double y, double z) {
	double t_0 = (x + y) / (1.0 - (y / z));
	double tmp;
	if ((t_0 <= -5e-271) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = ((z * (x + z)) / -y) - z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x + y) / (1.0d0 - (y / z))
    if ((t_0 <= (-5d-271)) .or. (.not. (t_0 <= 0.0d0))) then
        tmp = t_0
    else
        tmp = ((z * (x + z)) / -y) - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (x + y) / (1.0 - (y / z));
	double tmp;
	if ((t_0 <= -5e-271) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = ((z * (x + z)) / -y) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x + y) / (1.0 - (y / z))
	tmp = 0
	if (t_0 <= -5e-271) or not (t_0 <= 0.0):
		tmp = t_0
	else:
		tmp = ((z * (x + z)) / -y) - z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x + y) / Float64(1.0 - Float64(y / z)))
	tmp = 0.0
	if ((t_0 <= -5e-271) || !(t_0 <= 0.0))
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(z * Float64(x + z)) / Float64(-y)) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x + y) / (1.0 - (y / z));
	tmp = 0.0;
	if ((t_0 <= -5e-271) || ~((t_0 <= 0.0)))
		tmp = t_0;
	else
		tmp = ((z * (x + z)) / -y) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x + y), $MachinePrecision] / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -5e-271], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], t$95$0, N[(N[(N[(z * N[(x + z), $MachinePrecision]), $MachinePrecision] / (-y)), $MachinePrecision] - z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z))) < -5.0000000000000002e-271 or -0.0 < (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z)))

   1. Initial program 99.9%

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

   if -5.0000000000000002e-271 < (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z))) < -0.0

   1. Initial program 9.1%

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

   3. Taylor expanded in y around inf 100.0%

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

      1. associate--l+ 100.0%

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

      2. associate-*r/ 100.0%

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

      3. div-sub 100.0%

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

      4. remove-double-neg 100.0%

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

      5. mul-1-neg 100.0%

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

      6. neg-mul-1 100.0%

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

      7. distribute-lft-out-- 100.0%

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

      8. mul-1-neg 100.0%

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

      9. distribute-neg-frac 100.0%

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

      10. unsub-neg 100.0%

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

      11. mul-1-neg 100.0%

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

      12. cancel-sign-sub-inv 100.0%

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

      13. metadata-eval 100.0%

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

      14. *-lft-identity 100.0%

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

      15. +-commutative 100.0%

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

      16. unpow2 100.0%

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

      17. distribute-rgt-out 100.0%

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

   5. Simplified 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \leq -5 \cdot 10^{-271} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \leq 0\right):\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot \left(x + z\right)}{-y} - z\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-305} \lor \neg \left(t\_0 \leq 0\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot \left(\left(-x\right) - y\right)}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (+ x y) (- 1.0 (/ y z)))))
   (if (or (<= t_0 -1e-305) (not (<= t_0 0.0))) t_0 (/ (* z (- (- x) y)) y))))

C

double code(double x, double y, double z) {
	double t_0 = (x + y) / (1.0 - (y / z));
	double tmp;
	if ((t_0 <= -1e-305) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = (z * (-x - y)) / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x + y) / (1.0d0 - (y / z))
    if ((t_0 <= (-1d-305)) .or. (.not. (t_0 <= 0.0d0))) then
        tmp = t_0
    else
        tmp = (z * (-x - y)) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (x + y) / (1.0 - (y / z));
	double tmp;
	if ((t_0 <= -1e-305) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = (z * (-x - y)) / y;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x + y) / (1.0 - (y / z))
	tmp = 0
	if (t_0 <= -1e-305) or not (t_0 <= 0.0):
		tmp = t_0
	else:
		tmp = (z * (-x - y)) / y
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x + y) / Float64(1.0 - Float64(y / z)))
	tmp = 0.0
	if ((t_0 <= -1e-305) || !(t_0 <= 0.0))
		tmp = t_0;
	else
		tmp = Float64(Float64(z * Float64(Float64(-x) - y)) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x + y) / (1.0 - (y / z));
	tmp = 0.0;
	if ((t_0 <= -1e-305) || ~((t_0 <= 0.0)))
		tmp = t_0;
	else
		tmp = (z * (-x - y)) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x + y), $MachinePrecision] / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -1e-305], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], t$95$0, N[(N[(z * N[((-x) - y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z))) < -9.99999999999999996e-306 or -0.0 < (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z)))

   1. Initial program 99.9%

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

   if -9.99999999999999996e-306 < (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z))) < -0.0

   1. Initial program 5.5%

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

   3. Taylor expanded in z around 0 99.9%

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

      1. mul-1-neg 99.9%

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

      2. +-commutative 99.9%

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

   5. Simplified 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \leq -1 \cdot 10^{-305} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \leq 0\right):\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot \left(\left(-x\right) - y\right)}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 56.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{+104}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq -8.6 \cdot 10^{+20}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -1.55 \cdot 10^{-13}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq -1.8 \cdot 10^{-48}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{-7}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 2.9 \cdot 10^{+86}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.7e+104)
   (- z)
   (if (<= y -8.6e+20)
     x
     (if (<= y -1.55e-13)
       (- z)
       (if (<= y -1.8e-48)
         y
         (if (<= y 1.3e-7) x (if (<= y 2.9e+86) y (- z))))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.7e+104) {
		tmp = -z;
	} else if (y <= -8.6e+20) {
		tmp = x;
	} else if (y <= -1.55e-13) {
		tmp = -z;
	} else if (y <= -1.8e-48) {
		tmp = y;
	} else if (y <= 1.3e-7) {
		tmp = x;
	} else if (y <= 2.9e+86) {
		tmp = y;
	} else {
		tmp = -z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-1.7d+104)) then
        tmp = -z
    else if (y <= (-8.6d+20)) then
        tmp = x
    else if (y <= (-1.55d-13)) then
        tmp = -z
    else if (y <= (-1.8d-48)) then
        tmp = y
    else if (y <= 1.3d-7) then
        tmp = x
    else if (y <= 2.9d+86) then
        tmp = y
    else
        tmp = -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.7e+104) {
		tmp = -z;
	} else if (y <= -8.6e+20) {
		tmp = x;
	} else if (y <= -1.55e-13) {
		tmp = -z;
	} else if (y <= -1.8e-48) {
		tmp = y;
	} else if (y <= 1.3e-7) {
		tmp = x;
	} else if (y <= 2.9e+86) {
		tmp = y;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1.7e+104:
		tmp = -z
	elif y <= -8.6e+20:
		tmp = x
	elif y <= -1.55e-13:
		tmp = -z
	elif y <= -1.8e-48:
		tmp = y
	elif y <= 1.3e-7:
		tmp = x
	elif y <= 2.9e+86:
		tmp = y
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.7e+104)
		tmp = Float64(-z);
	elseif (y <= -8.6e+20)
		tmp = x;
	elseif (y <= -1.55e-13)
		tmp = Float64(-z);
	elseif (y <= -1.8e-48)
		tmp = y;
	elseif (y <= 1.3e-7)
		tmp = x;
	elseif (y <= 2.9e+86)
		tmp = y;
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -1.7e+104)
		tmp = -z;
	elseif (y <= -8.6e+20)
		tmp = x;
	elseif (y <= -1.55e-13)
		tmp = -z;
	elseif (y <= -1.8e-48)
		tmp = y;
	elseif (y <= 1.3e-7)
		tmp = x;
	elseif (y <= 2.9e+86)
		tmp = y;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.7e+104], (-z), If[LessEqual[y, -8.6e+20], x, If[LessEqual[y, -1.55e-13], (-z), If[LessEqual[y, -1.8e-48], y, If[LessEqual[y, 1.3e-7], x, If[LessEqual[y, 2.9e+86], y, (-z)]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.6999999999999998e104 or -8.6e20 < y < -1.55e-13 or 2.8999999999999999e86 < y

   1. Initial program 76.0%

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

   3. Taylor expanded in y around inf 70.4%

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

      1. mul-1-neg 70.4%

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

   5. Simplified 70.4%

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

   if -1.6999999999999998e104 < y < -8.6e20 or -1.8000000000000001e-48 < y < 1.29999999999999999e-7

   1. Initial program 99.2%

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

   3. Taylor expanded in y around 0 60.3%

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

   if -1.55e-13 < y < -1.8000000000000001e-48 or 1.29999999999999999e-7 < y < 2.8999999999999999e86

   1. Initial program 96.7%

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

   3. Taylor expanded in x around 0 70.2%

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

   4. Taylor expanded in y around 0 52.7%

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

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{+104}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq -8.6 \cdot 10^{+20}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -1.55 \cdot 10^{-13}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq -1.8 \cdot 10^{-48}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{-7}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 2.9 \cdot 10^{+86}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 71.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x + y\right) \cdot \left(1 + \frac{y}{z}\right)\\ \mathbf{if}\;z \leq -1.72 \cdot 10^{+56}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{+16}:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{elif}\;z \leq 6.2 \cdot 10^{+83}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq 6.7 \cdot 10^{+125}:\\ \;\;\;\;\frac{y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (+ x y) (+ 1.0 (/ y z)))))
   (if (<= z -1.72e+56)
     t_0
     (if (<= z 3.5e+16)
       (* z (- -1.0 (/ x y)))
       (if (<= z 6.2e+83)
         (+ x y)
         (if (<= z 6.7e+125) (/ y (- 1.0 (/ y z))) t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = (x + y) * (1.0 + (y / z));
	double tmp;
	if (z <= -1.72e+56) {
		tmp = t_0;
	} else if (z <= 3.5e+16) {
		tmp = z * (-1.0 - (x / y));
	} else if (z <= 6.2e+83) {
		tmp = x + y;
	} else if (z <= 6.7e+125) {
		tmp = 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 = (x + y) * (1.0d0 + (y / z))
    if (z <= (-1.72d+56)) then
        tmp = t_0
    else if (z <= 3.5d+16) then
        tmp = z * ((-1.0d0) - (x / y))
    else if (z <= 6.2d+83) then
        tmp = x + y
    else if (z <= 6.7d+125) then
        tmp = 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 = (x + y) * (1.0 + (y / z));
	double tmp;
	if (z <= -1.72e+56) {
		tmp = t_0;
	} else if (z <= 3.5e+16) {
		tmp = z * (-1.0 - (x / y));
	} else if (z <= 6.2e+83) {
		tmp = x + y;
	} else if (z <= 6.7e+125) {
		tmp = y / (1.0 - (y / z));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x + y) * (1.0 + (y / z))
	tmp = 0
	if z <= -1.72e+56:
		tmp = t_0
	elif z <= 3.5e+16:
		tmp = z * (-1.0 - (x / y))
	elif z <= 6.2e+83:
		tmp = x + y
	elif z <= 6.7e+125:
		tmp = y / (1.0 - (y / z))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x + y) * Float64(1.0 + Float64(y / z)))
	tmp = 0.0
	if (z <= -1.72e+56)
		tmp = t_0;
	elseif (z <= 3.5e+16)
		tmp = Float64(z * Float64(-1.0 - Float64(x / y)));
	elseif (z <= 6.2e+83)
		tmp = Float64(x + y);
	elseif (z <= 6.7e+125)
		tmp = Float64(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 = (x + y) * (1.0 + (y / z));
	tmp = 0.0;
	if (z <= -1.72e+56)
		tmp = t_0;
	elseif (z <= 3.5e+16)
		tmp = z * (-1.0 - (x / y));
	elseif (z <= 6.2e+83)
		tmp = x + y;
	elseif (z <= 6.7e+125)
		tmp = 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[(x + y), $MachinePrecision] * N[(1.0 + N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.72e+56], t$95$0, If[LessEqual[z, 3.5e+16], N[(z * N[(-1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 6.2e+83], N[(x + y), $MachinePrecision], If[LessEqual[z, 6.7e+125], N[(y / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -1.72e56 or 6.7000000000000003e125 < z

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf 74.3%

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

      1. associate-+r+ 74.3%

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

      2. *-rgt-identity 74.3%

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

      3. *-commutative 74.3%

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

      4. associate-/l* 87.9%

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

      5. distribute-lft-in 87.9%

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

      6. +-commutative 87.9%

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

   5. Simplified 87.9%

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

   if -1.72e56 < z < 3.5e16

   1. Initial program 81.7%

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

      1. clear-num 81.5%

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

      2. inv-pow 81.5%

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

   4. Applied egg-rr 81.5%

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

   5. Taylor expanded in z around 0 70.2%

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

      1. mul-1-neg 70.2%

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

      2. +-commutative 70.2%

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

      3. distribute-frac-neg2 70.2%

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

      4. mul-1-neg 70.2%

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

      5. *-commutative 70.2%

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

      6. associate-/l/ 70.2%

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

      7. associate-/l* 70.2%

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

      8. metadata-eval 70.2%

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

      9. distribute-neg-frac2 70.2%

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

      10. /-rgt-identity 70.2%

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

      11. +-commutative 70.2%

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

   7. Simplified 70.2%

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

   8. Taylor expanded in x around 0 72.4%

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

      1. *-commutative 72.4%

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

      2. fma-define 72.4%

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

      3. mul-1-neg 72.4%

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

      4. *-commutative 72.4%

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

      5. associate-*r/ 70.9%

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

      6. fma-neg 70.9%

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

      7. distribute-lft-out-- 70.9%

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

   10. Simplified 70.9%

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

   if 3.5e16 < z < 6.19999999999999984e83

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf 91.0%

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

      1. +-commutative 91.0%

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

   5. Simplified 91.0%

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

   if 6.19999999999999984e83 < z < 6.7000000000000003e125

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 82.6%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.72 \cdot 10^{+56}:\\ \;\;\;\;\left(x + y\right) \cdot \left(1 + \frac{y}{z}\right)\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{+16}:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{elif}\;z \leq 6.2 \cdot 10^{+83}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq 6.7 \cdot 10^{+125}:\\ \;\;\;\;\frac{y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) \cdot \left(1 + \frac{y}{z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 71.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.25 \cdot 10^{+58}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{+22}:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{elif}\;z \leq 3.1 \cdot 10^{+83} \lor \neg \left(z \leq 1.5 \cdot 10^{+123}\right):\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{1 - \frac{y}{z}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.25e+58)
   (+ x y)
   (if (<= z 4.2e+22)
     (* z (- -1.0 (/ x y)))
     (if (or (<= z 3.1e+83) (not (<= z 1.5e+123)))
       (+ x y)
       (/ y (- 1.0 (/ y z)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.25e+58) {
		tmp = x + y;
	} else if (z <= 4.2e+22) {
		tmp = z * (-1.0 - (x / y));
	} else if ((z <= 3.1e+83) || !(z <= 1.5e+123)) {
		tmp = x + y;
	} else {
		tmp = y / (1.0 - (y / z));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-1.25d+58)) then
        tmp = x + y
    else if (z <= 4.2d+22) then
        tmp = z * ((-1.0d0) - (x / y))
    else if ((z <= 3.1d+83) .or. (.not. (z <= 1.5d+123))) then
        tmp = x + y
    else
        tmp = y / (1.0d0 - (y / z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.25e+58) {
		tmp = x + y;
	} else if (z <= 4.2e+22) {
		tmp = z * (-1.0 - (x / y));
	} else if ((z <= 3.1e+83) || !(z <= 1.5e+123)) {
		tmp = x + y;
	} else {
		tmp = y / (1.0 - (y / z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.25e+58:
		tmp = x + y
	elif z <= 4.2e+22:
		tmp = z * (-1.0 - (x / y))
	elif (z <= 3.1e+83) or not (z <= 1.5e+123):
		tmp = x + y
	else:
		tmp = y / (1.0 - (y / z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.25e+58)
		tmp = Float64(x + y);
	elseif (z <= 4.2e+22)
		tmp = Float64(z * Float64(-1.0 - Float64(x / y)));
	elseif ((z <= 3.1e+83) || !(z <= 1.5e+123))
		tmp = Float64(x + y);
	else
		tmp = Float64(y / Float64(1.0 - Float64(y / z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1.25e+58)
		tmp = x + y;
	elseif (z <= 4.2e+22)
		tmp = z * (-1.0 - (x / y));
	elseif ((z <= 3.1e+83) || ~((z <= 1.5e+123)))
		tmp = x + y;
	else
		tmp = y / (1.0 - (y / z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.25e+58], N[(x + y), $MachinePrecision], If[LessEqual[z, 4.2e+22], N[(z * N[(-1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[z, 3.1e+83], N[Not[LessEqual[z, 1.5e+123]], $MachinePrecision]], N[(x + y), $MachinePrecision], N[(y / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.24999999999999996e58 or 4.1999999999999996e22 < z < 3.09999999999999992e83 or 1.50000000000000004e123 < z

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf 87.8%

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

      1. +-commutative 87.8%

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

   5. Simplified 87.8%

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

   if -1.24999999999999996e58 < z < 4.1999999999999996e22

   1. Initial program 81.7%

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

      1. clear-num 81.5%

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

      2. inv-pow 81.5%

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

   4. Applied egg-rr 81.5%

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

   5. Taylor expanded in z around 0 70.2%

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

      1. mul-1-neg 70.2%

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

      2. +-commutative 70.2%

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

      3. distribute-frac-neg2 70.2%

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

      4. mul-1-neg 70.2%

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

      5. *-commutative 70.2%

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

      6. associate-/l/ 70.2%

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

      7. associate-/l* 70.2%

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

      8. metadata-eval 70.2%

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

      9. distribute-neg-frac2 70.2%

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

      10. /-rgt-identity 70.2%

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

      11. +-commutative 70.2%

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

   7. Simplified 70.2%

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

   8. Taylor expanded in x around 0 72.4%

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

      1. *-commutative 72.4%

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

      2. fma-define 72.4%

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

      3. mul-1-neg 72.4%

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

      4. *-commutative 72.4%

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

      5. associate-*r/ 70.9%

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

      6. fma-neg 70.9%

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

      7. distribute-lft-out-- 70.9%

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

   10. Simplified 70.9%

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

   if 3.09999999999999992e83 < z < 1.50000000000000004e123

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 82.6%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.25 \cdot 10^{+58}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{+22}:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{elif}\;z \leq 3.1 \cdot 10^{+83} \lor \neg \left(z \leq 1.5 \cdot 10^{+123}\right):\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{1 - \frac{y}{z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 72.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.9 \cdot 10^{+52} \lor \neg \left(z \leq 1.72 \cdot 10^{+16}\right):\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.9e+52) (not (<= z 1.72e+16)))
   (+ x y)
   (* z (- -1.0 (/ x y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.9e+52) || !(z <= 1.72e+16)) {
		tmp = x + y;
	} else {
		tmp = z * (-1.0 - (x / y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-1.9d+52)) .or. (.not. (z <= 1.72d+16))) then
        tmp = x + y
    else
        tmp = z * ((-1.0d0) - (x / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.9e+52) || !(z <= 1.72e+16)) {
		tmp = x + y;
	} else {
		tmp = z * (-1.0 - (x / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1.9e+52) or not (z <= 1.72e+16):
		tmp = x + y
	else:
		tmp = z * (-1.0 - (x / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.9e+52) || !(z <= 1.72e+16))
		tmp = Float64(x + y);
	else
		tmp = Float64(z * Float64(-1.0 - Float64(x / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1.9e+52) || ~((z <= 1.72e+16)))
		tmp = x + y;
	else
		tmp = z * (-1.0 - (x / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.9e+52], N[Not[LessEqual[z, 1.72e+16]], $MachinePrecision]], N[(x + y), $MachinePrecision], N[(z * N[(-1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.9e52 or 1.72e16 < z

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf 81.9%

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

      1. +-commutative 81.9%

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

   5. Simplified 81.9%

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

   if -1.9e52 < z < 1.72e16

   1. Initial program 81.7%

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

      1. clear-num 81.5%

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

      2. inv-pow 81.5%

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

   4. Applied egg-rr 81.5%

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

   5. Taylor expanded in z around 0 70.2%

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

      1. mul-1-neg 70.2%

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

      2. +-commutative 70.2%

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

      3. distribute-frac-neg2 70.2%

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

      4. mul-1-neg 70.2%

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

      5. *-commutative 70.2%

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

      6. associate-/l/ 70.2%

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

      7. associate-/l* 70.2%

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

      8. metadata-eval 70.2%

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

      9. distribute-neg-frac2 70.2%

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

      10. /-rgt-identity 70.2%

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

      11. +-commutative 70.2%

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

   7. Simplified 70.2%

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

   8. Taylor expanded in x around 0 72.4%

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

      1. *-commutative 72.4%

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

      2. fma-define 72.4%

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

      3. mul-1-neg 72.4%

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

      4. *-commutative 72.4%

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

      5. associate-*r/ 70.9%

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

      6. fma-neg 70.9%

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

      7. distribute-lft-out-- 70.9%

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

   10. Simplified 70.9%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.9 \cdot 10^{+52} \lor \neg \left(z \leq 1.72 \cdot 10^{+16}\right):\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.2 \cdot 10^{+105} \lor \neg \left(y \leq 5.9 \cdot 10^{+85}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -4.2e+105) (not (<= y 5.9e+85))) (- z) (+ x y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -4.2e+105) || !(y <= 5.9e+85)) {
		tmp = -z;
	} else {
		tmp = x + y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((y <= (-4.2d+105)) .or. (.not. (y <= 5.9d+85))) then
        tmp = -z
    else
        tmp = x + y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -4.2e+105) || !(y <= 5.9e+85)) {
		tmp = -z;
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -4.2e+105) or not (y <= 5.9e+85):
		tmp = -z
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -4.2e+105) || !(y <= 5.9e+85))
		tmp = Float64(-z);
	else
		tmp = Float64(x + y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -4.2e+105) || ~((y <= 5.9e+85)))
		tmp = -z;
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -4.2e+105], N[Not[LessEqual[y, 5.9e+85]], $MachinePrecision]], (-z), N[(x + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.2000000000000002e105 or 5.9e85 < y

   1. Initial program 74.6%

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

   3. Taylor expanded in y around inf 69.8%

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

      1. mul-1-neg 69.8%

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

   5. Simplified 69.8%

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

   if -4.2000000000000002e105 < y < 5.9e85

   1. Initial program 98.8%

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

   3. Taylor expanded in z around inf 71.3%

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

      1. +-commutative 71.3%

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

   5. Simplified 71.3%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.2 \cdot 10^{+105} \lor \neg \left(y \leq 5.9 \cdot 10^{+85}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 40.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.8 \cdot 10^{-155}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{-149}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -3.8e-155) x (if (<= x 1.7e-149) y x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -3.8e-155) {
		tmp = x;
	} else if (x <= 1.7e-149) {
		tmp = y;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-3.8d-155)) then
        tmp = x
    else if (x <= 1.7d-149) then
        tmp = y
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -3.8e-155) {
		tmp = x;
	} else if (x <= 1.7e-149) {
		tmp = y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -3.8e-155:
		tmp = x
	elif x <= 1.7e-149:
		tmp = y
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -3.8e-155)
		tmp = x;
	elseif (x <= 1.7e-149)
		tmp = y;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -3.8e-155)
		tmp = x;
	elseif (x <= 1.7e-149)
		tmp = y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -3.8e-155], x, If[LessEqual[x, 1.7e-149], y, x]]

Derivation

1. Split input into 2 regimes

2. if x < -3.7999999999999998e-155 or 1.6999999999999999e-149 < x

   1. Initial program 91.6%

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

   3. Taylor expanded in y around 0 45.6%

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

   if -3.7999999999999998e-155 < x < 1.6999999999999999e-149

   1. Initial program 88.4%

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

   3. Taylor expanded in x around 0 78.2%

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

   4. Taylor expanded in y around 0 46.4%

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

4. Final simplification 45.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.8 \cdot 10^{-155}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{-149}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 34.6% accurate, 9.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := x

Derivation

1. Initial program 90.7%

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

3. Taylor expanded in y around 0 35.8%

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

4. Final simplification 35.8%

   \[\leadsto x \]
5. Add Preprocessing

Developer target: 93.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y + x}{-y} \cdot z\\ \mathbf{if}\;y < -3.7429310762689856 \cdot 10^{+171}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y < 3.5534662456086734 \cdot 10^{+168}:\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024081 
(FPCore (x y z)
  :name "Graphics.Rendering.Chart.Backend.Diagrams:calcFontMetrics from Chart-diagrams-1.5.1, A"
  :precision binary64

  :alt
  (if (< y -3.7429310762689856e+171) (* (/ (+ y x) (- y)) z) (if (< y 3.5534662456086734e+168) (/ (+ x y) (- 1.0 (/ y z))) (* (/ (+ y x) (- y)) z)))

  (/ (+ x y) (- 1.0 (/ y z))))