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

Percentage Accurate: 88.6% → 99.5%

Time: 7.0s

Alternatives: 10

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: 88.6% 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: 99.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-250}:\\ \;\;\;\;\frac{x + y}{\frac{z - y}{z}}\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (+ x y) (- 1.0 (/ y z)))))
   (if (<= t_0 -2e-250)
     (/ (+ x y) (/ (- z y) z))
     (if (<= t_0 0.0) (* z (- -1.0 (/ x y))) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (x + y) / (1.0 - (y / z));
	double tmp;
	if (t_0 <= -2e-250) {
		tmp = (x + y) / ((z - y) / z);
	} else if (t_0 <= 0.0) {
		tmp = z * (-1.0 - (x / y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x + y) / (1.0d0 - (y / z))
    if (t_0 <= (-2d-250)) then
        tmp = (x + y) / ((z - y) / z)
    else if (t_0 <= 0.0d0) then
        tmp = z * ((-1.0d0) - (x / y))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (x + y) / (1.0 - (y / z));
	double tmp;
	if (t_0 <= -2e-250) {
		tmp = (x + y) / ((z - y) / z);
	} else if (t_0 <= 0.0) {
		tmp = z * (-1.0 - (x / y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x + y) / (1.0 - (y / z))
	tmp = 0
	if t_0 <= -2e-250:
		tmp = (x + y) / ((z - y) / z)
	elif t_0 <= 0.0:
		tmp = z * (-1.0 - (x / y))
	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 (t_0 <= -2e-250)
		tmp = Float64(Float64(x + y) / Float64(Float64(z - y) / z));
	elseif (t_0 <= 0.0)
		tmp = Float64(z * Float64(-1.0 - Float64(x / y)));
	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 (t_0 <= -2e-250)
		tmp = (x + y) / ((z - y) / z);
	elseif (t_0 <= 0.0)
		tmp = z * (-1.0 - (x / y));
	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[t$95$0, -2e-250], N[(N[(x + y), $MachinePrecision] / N[(N[(z - y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(z * N[(-1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z))) < -2.0000000000000001e-250

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0 99.9%

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

   if -2.0000000000000001e-250 < (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z))) < 0.0

   1. Initial program 15.8%

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

   3. Taylor expanded in y around inf 15.8%

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

      1. neg-mul-1 15.8%

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

      2. distribute-neg-frac2 15.8%

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

   5. Simplified 15.8%

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

      1. distribute-frac-neg2 15.8%

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

      2. distribute-frac-neg 15.8%

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

      3. associate-/r/ 99.9%

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

   7. Applied egg-rr 99.9%

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

   8. Taylor expanded in x around 0 99.9%

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

      1. sub-neg 99.9%

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

      2. *-commutative 99.9%

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

      3. metadata-eval 99.9%

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

      4. distribute-lft1-in 99.9%

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

      5. *-rgt-identity 99.9%

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

      6. associate-*r/ 99.9%

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

      7. rgt-mult-inverse 99.9%

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

      8. distribute-lft-in 99.8%

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

      9. +-commutative 99.8%

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

      10. *-commutative 99.8%

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

      11. mul-1-neg 99.8%

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

      12. neg-sub0 99.8%

          \[\leadsto \color{blue}{\left(0 - x \cdot \left(\frac{1}{x} + \frac{1}{y}\right)\right)} \cdot z \]

      13. distribute-rgt-in 99.9%

          \[\leadsto \left(0 - \color{blue}{\left(\frac{1}{x} \cdot x + \frac{1}{y} \cdot x\right)}\right) \cdot z \]

      14. lft-mult-inverse 99.9%

          \[\leadsto \left(0 - \left(\color{blue}{1} + \frac{1}{y} \cdot x\right)\right) \cdot z \]

      15. associate--r+ 99.9%

          \[\leadsto \color{blue}{\left(\left(0 - 1\right) - \frac{1}{y} \cdot x\right)} \cdot z \]

      16. metadata-eval 99.9%

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

      17. associate-*l/ 99.9%

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

      18. *-lft-identity 99.9%

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

   10. Simplified 99.9%

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

   if 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
3. Recombined 3 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \leq -2 \cdot 10^{-250}:\\ \;\;\;\;\frac{x + y}{\frac{z - y}{z}}\\ \mathbf{elif}\;\frac{x + y}{1 - \frac{y}{z}} \leq 0:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-250} \lor \neg \left(t\_0 \leq 0\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (+ x y) (- 1.0 (/ y z)))))
   (if (or (<= t_0 -2e-250) (not (<= t_0 0.0))) t_0 (* z (- -1.0 (/ x y))))))

C

double code(double x, double y, double z) {
	double t_0 = (x + y) / (1.0 - (y / z));
	double tmp;
	if ((t_0 <= -2e-250) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = (x + y) / (1.0d0 - (y / z))
    if ((t_0 <= (-2d-250)) .or. (.not. (t_0 <= 0.0d0))) then
        tmp = t_0
    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 t_0 = (x + y) / (1.0 - (y / z));
	double tmp;
	if ((t_0 <= -2e-250) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = z * (-1.0 - (x / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x + y) / (1.0 - (y / z))
	tmp = 0
	if (t_0 <= -2e-250) or not (t_0 <= 0.0):
		tmp = t_0
	else:
		tmp = z * (-1.0 - (x / y))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x + y) / Float64(1.0 - Float64(y / z)))
	tmp = 0.0
	if ((t_0 <= -2e-250) || !(t_0 <= 0.0))
		tmp = t_0;
	else
		tmp = Float64(z * Float64(-1.0 - Float64(x / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x + y) / (1.0 - (y / z));
	tmp = 0.0;
	if ((t_0 <= -2e-250) || ~((t_0 <= 0.0)))
		tmp = t_0;
	else
		tmp = z * (-1.0 - (x / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x + y), $MachinePrecision] / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -2e-250], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], t$95$0, N[(z * N[(-1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z))) < -2.0000000000000001e-250 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 -2.0000000000000001e-250 < (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z))) < 0.0

   1. Initial program 15.8%

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

   3. Taylor expanded in y around inf 15.8%

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

      1. neg-mul-1 15.8%

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

      2. distribute-neg-frac2 15.8%

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

   5. Simplified 15.8%

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

      1. distribute-frac-neg2 15.8%

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

      2. distribute-frac-neg 15.8%

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

      3. associate-/r/ 99.9%

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

   7. Applied egg-rr 99.9%

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

   8. Taylor expanded in x around 0 99.9%

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

      1. sub-neg 99.9%

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

      2. *-commutative 99.9%

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

      3. metadata-eval 99.9%

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

      4. distribute-lft1-in 99.9%

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

      5. *-rgt-identity 99.9%

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

      6. associate-*r/ 99.9%

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

      7. rgt-mult-inverse 99.9%

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

      8. distribute-lft-in 99.8%

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

      9. +-commutative 99.8%

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

      10. *-commutative 99.8%

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

      11. mul-1-neg 99.8%

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

      12. neg-sub0 99.8%

          \[\leadsto \color{blue}{\left(0 - x \cdot \left(\frac{1}{x} + \frac{1}{y}\right)\right)} \cdot z \]

      13. distribute-rgt-in 99.9%

          \[\leadsto \left(0 - \color{blue}{\left(\frac{1}{x} \cdot x + \frac{1}{y} \cdot x\right)}\right) \cdot z \]

      14. lft-mult-inverse 99.9%

          \[\leadsto \left(0 - \left(\color{blue}{1} + \frac{1}{y} \cdot x\right)\right) \cdot z \]

      15. associate--r+ 99.9%

          \[\leadsto \color{blue}{\left(\left(0 - 1\right) - \frac{1}{y} \cdot x\right)} \cdot z \]

      16. metadata-eval 99.9%

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

      17. associate-*l/ 99.9%

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

      18. *-lft-identity 99.9%

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

   10. Simplified 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \leq -2 \cdot 10^{-250} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \leq 0\right):\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.4 \cdot 10^{-28} \lor \neg \left(y \leq 1.3 \cdot 10^{+22}\right):\\ \;\;\;\;z \cdot \frac{y}{z - y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{z}{z - y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -5.4e-28) (not (<= y 1.3e+22)))
   (* z (/ y (- z y)))
   (* x (/ z (- z y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -5.4e-28) || !(y <= 1.3e+22)) {
		tmp = z * (y / (z - y));
	} else {
		tmp = x * (z / (z - 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 <= (-5.4d-28)) .or. (.not. (y <= 1.3d+22))) then
        tmp = z * (y / (z - y))
    else
        tmp = x * (z / (z - y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -5.4e-28) || !(y <= 1.3e+22)) {
		tmp = z * (y / (z - y));
	} else {
		tmp = x * (z / (z - y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -5.4e-28) or not (y <= 1.3e+22):
		tmp = z * (y / (z - y))
	else:
		tmp = x * (z / (z - y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -5.4e-28) || !(y <= 1.3e+22))
		tmp = Float64(z * Float64(y / Float64(z - y)));
	else
		tmp = Float64(x * Float64(z / Float64(z - y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -5.4e-28) || ~((y <= 1.3e+22)))
		tmp = z * (y / (z - y));
	else
		tmp = x * (z / (z - y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -5.4e-28], N[Not[LessEqual[y, 1.3e+22]], $MachinePrecision]], N[(z * N[(y / N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(z / N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -5.3999999999999998e-28 or 1.3e22 < y

   1. Initial program 79.1%

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

   3. Taylor expanded in z around 0 79.1%

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

      1. *-un-lft-identity 79.1%

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

      2. div-inv 79.0%

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

      3. times-frac 72.4%

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

   5. Applied egg-rr 72.4%

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

   6. Taylor expanded in x around 0 55.1%

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

      1. *-commutative 55.1%

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

      2. associate-/l* 76.6%

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

   8. Simplified 76.6%

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

   if -5.3999999999999998e-28 < y < 1.3e22

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0 99.9%

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

   4. Taylor expanded in x around inf 60.9%

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

      1. associate-/l* 81.9%

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

   6. Simplified 81.9%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.4 \cdot 10^{-28} \lor \neg \left(y \leq 1.3 \cdot 10^{+22}\right):\\ \;\;\;\;z \cdot \frac{y}{z - y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{z}{z - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 66.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.4 \cdot 10^{-28} \lor \neg \left(y \leq 6.5 \cdot 10^{+53}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{z}{z - y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -6.4e-28) (not (<= y 6.5e+53))) (- z) (* x (/ z (- z y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -6.4e-28) || !(y <= 6.5e+53)) {
		tmp = -z;
	} else {
		tmp = x * (z / (z - 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 <= (-6.4d-28)) .or. (.not. (y <= 6.5d+53))) then
        tmp = -z
    else
        tmp = x * (z / (z - y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -6.4e-28) || !(y <= 6.5e+53)) {
		tmp = -z;
	} else {
		tmp = x * (z / (z - y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -6.4e-28) or not (y <= 6.5e+53):
		tmp = -z
	else:
		tmp = x * (z / (z - y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -6.4e-28) || !(y <= 6.5e+53))
		tmp = Float64(-z);
	else
		tmp = Float64(x * Float64(z / Float64(z - y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -6.4e-28) || ~((y <= 6.5e+53)))
		tmp = -z;
	else
		tmp = x * (z / (z - y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -6.4e-28], N[Not[LessEqual[y, 6.5e+53]], $MachinePrecision]], (-z), N[(x * N[(z / N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -6.39999999999999964e-28 or 6.50000000000000017e53 < y

   1. Initial program 78.1%

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

   3. Taylor expanded in y around inf 61.4%

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

      1. neg-mul-1 61.4%

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

   5. Simplified 61.4%

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

   if -6.39999999999999964e-28 < y < 6.50000000000000017e53

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0 99.9%

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

   4. Taylor expanded in x around inf 60.5%

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

      1. associate-/l* 80.6%

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

   6. Simplified 80.6%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.4 \cdot 10^{-28} \lor \neg \left(y \leq 6.5 \cdot 10^{+53}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{z}{z - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{-28}:\\ \;\;\;\;z \cdot \frac{y}{z - y}\\ \mathbf{elif}\;y \leq 1.76 \cdot 10^{+34}:\\ \;\;\;\;\frac{x}{\frac{z - y}{z}}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2.2e-28)
   (* z (/ y (- z y)))
   (if (<= y 1.76e+34) (/ x (/ (- z y) z)) (* z (- -1.0 (/ x y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.2e-28) {
		tmp = z * (y / (z - y));
	} else if (y <= 1.76e+34) {
		tmp = x / ((z - y) / z);
	} 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 (y <= (-2.2d-28)) then
        tmp = z * (y / (z - y))
    else if (y <= 1.76d+34) then
        tmp = x / ((z - y) / z)
    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 (y <= -2.2e-28) {
		tmp = z * (y / (z - y));
	} else if (y <= 1.76e+34) {
		tmp = x / ((z - y) / z);
	} else {
		tmp = z * (-1.0 - (x / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -2.2e-28:
		tmp = z * (y / (z - y))
	elif y <= 1.76e+34:
		tmp = x / ((z - y) / z)
	else:
		tmp = z * (-1.0 - (x / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -2.2e-28)
		tmp = Float64(z * Float64(y / Float64(z - y)));
	elseif (y <= 1.76e+34)
		tmp = Float64(x / Float64(Float64(z - y) / z));
	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 (y <= -2.2e-28)
		tmp = z * (y / (z - y));
	elseif (y <= 1.76e+34)
		tmp = x / ((z - y) / z);
	else
		tmp = z * (-1.0 - (x / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -2.2e-28], N[(z * N[(y / N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.76e+34], N[(x / N[(N[(z - y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(z * N[(-1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.19999999999999996e-28

   1. Initial program 81.4%

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

   3. Taylor expanded in z around 0 81.4%

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

      1. *-un-lft-identity 81.4%

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

      2. div-inv 81.1%

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

      3. times-frac 75.4%

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

   5. Applied egg-rr 75.4%

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

   6. Taylor expanded in x around 0 59.2%

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

      1. *-commutative 59.2%

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

      2. associate-/l* 76.4%

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

   8. Simplified 76.4%

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

   if -2.19999999999999996e-28 < y < 1.75999999999999995e34

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 80.5%

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

   4. Taylor expanded in z around 0 80.5%

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

   if 1.75999999999999995e34 < y

   1. Initial program 73.8%

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

   3. Taylor expanded in y around inf 57.2%

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

      1. neg-mul-1 57.2%

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

      2. distribute-neg-frac2 57.2%

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

   5. Simplified 57.2%

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

      1. distribute-frac-neg2 57.2%

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

      2. distribute-frac-neg 57.2%

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

      3. associate-/r/ 83.3%

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

   7. Applied egg-rr 83.3%

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

   8. Taylor expanded in x around 0 83.3%

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

      1. sub-neg 83.3%

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

      2. *-commutative 83.3%

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

      3. metadata-eval 83.3%

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

      4. distribute-lft1-in 83.3%

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

      5. *-rgt-identity 83.3%

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

      6. associate-*r/ 83.3%

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

      7. rgt-mult-inverse 83.2%

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

      8. distribute-lft-in 83.2%

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

      9. +-commutative 83.2%

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

      10. *-commutative 83.2%

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

      11. mul-1-neg 83.2%

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

      12. neg-sub0 83.2%

          \[\leadsto \color{blue}{\left(0 - x \cdot \left(\frac{1}{x} + \frac{1}{y}\right)\right)} \cdot z \]

      13. distribute-rgt-in 83.2%

          \[\leadsto \left(0 - \color{blue}{\left(\frac{1}{x} \cdot x + \frac{1}{y} \cdot x\right)}\right) \cdot z \]

      14. lft-mult-inverse 83.3%

          \[\leadsto \left(0 - \left(\color{blue}{1} + \frac{1}{y} \cdot x\right)\right) \cdot z \]

      15. associate--r+ 83.3%

          \[\leadsto \color{blue}{\left(\left(0 - 1\right) - \frac{1}{y} \cdot x\right)} \cdot z \]

      16. metadata-eval 83.3%

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

      17. associate-*l/ 83.3%

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

      18. *-lft-identity 83.3%

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

   10. Simplified 83.3%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{-28}:\\ \;\;\;\;z \cdot \frac{y}{z - y}\\ \mathbf{elif}\;y \leq 1.76 \cdot 10^{+34}:\\ \;\;\;\;\frac{x}{\frac{z - y}{z}}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8.5 \cdot 10^{-29}:\\ \;\;\;\;z \cdot \frac{y}{z - y}\\ \mathbf{elif}\;y \leq 10^{+32}:\\ \;\;\;\;x \cdot \frac{z}{z - y}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -8.5e-29)
   (* z (/ y (- z y)))
   (if (<= y 1e+32) (* x (/ z (- z y))) (* z (- -1.0 (/ x y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -8.5e-29) {
		tmp = z * (y / (z - y));
	} else if (y <= 1e+32) {
		tmp = x * (z / (z - 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 (y <= (-8.5d-29)) then
        tmp = z * (y / (z - y))
    else if (y <= 1d+32) then
        tmp = x * (z / (z - 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 (y <= -8.5e-29) {
		tmp = z * (y / (z - y));
	} else if (y <= 1e+32) {
		tmp = x * (z / (z - y));
	} else {
		tmp = z * (-1.0 - (x / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -8.5e-29:
		tmp = z * (y / (z - y))
	elif y <= 1e+32:
		tmp = x * (z / (z - y))
	else:
		tmp = z * (-1.0 - (x / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -8.5e-29)
		tmp = Float64(z * Float64(y / Float64(z - y)));
	elseif (y <= 1e+32)
		tmp = Float64(x * Float64(z / Float64(z - 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 (y <= -8.5e-29)
		tmp = z * (y / (z - y));
	elseif (y <= 1e+32)
		tmp = x * (z / (z - y));
	else
		tmp = z * (-1.0 - (x / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -8.5e-29], N[(z * N[(y / N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1e+32], N[(x * N[(z / N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z * N[(-1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -8.5000000000000001e-29

   1. Initial program 81.4%

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

   3. Taylor expanded in z around 0 81.4%

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

      1. *-un-lft-identity 81.4%

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

      2. div-inv 81.1%

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

      3. times-frac 75.4%

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

   5. Applied egg-rr 75.4%

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

   6. Taylor expanded in x around 0 59.2%

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

      1. *-commutative 59.2%

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

      2. associate-/l* 76.4%

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

   8. Simplified 76.4%

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

   if -8.5000000000000001e-29 < y < 1.00000000000000005e32

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0 99.9%

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

   4. Taylor expanded in x around inf 60.2%

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

      1. associate-/l* 80.5%

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

   6. Simplified 80.5%

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

   if 1.00000000000000005e32 < y

   1. Initial program 73.8%

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

   3. Taylor expanded in y around inf 57.2%

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

      1. neg-mul-1 57.2%

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

      2. distribute-neg-frac2 57.2%

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

   5. Simplified 57.2%

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

      1. distribute-frac-neg2 57.2%

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

      2. distribute-frac-neg 57.2%

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

      3. associate-/r/ 83.3%

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

   7. Applied egg-rr 83.3%

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

   8. Taylor expanded in x around 0 83.3%

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

      1. sub-neg 83.3%

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

      2. *-commutative 83.3%

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

      3. metadata-eval 83.3%

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

      4. distribute-lft1-in 83.3%

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

      5. *-rgt-identity 83.3%

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

      6. associate-*r/ 83.3%

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

      7. rgt-mult-inverse 83.2%

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

      8. distribute-lft-in 83.2%

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

      9. +-commutative 83.2%

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

      10. *-commutative 83.2%

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

      11. mul-1-neg 83.2%

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

      12. neg-sub0 83.2%

          \[\leadsto \color{blue}{\left(0 - x \cdot \left(\frac{1}{x} + \frac{1}{y}\right)\right)} \cdot z \]

      13. distribute-rgt-in 83.2%

          \[\leadsto \left(0 - \color{blue}{\left(\frac{1}{x} \cdot x + \frac{1}{y} \cdot x\right)}\right) \cdot z \]

      14. lft-mult-inverse 83.3%

          \[\leadsto \left(0 - \left(\color{blue}{1} + \frac{1}{y} \cdot x\right)\right) \cdot z \]

      15. associate--r+ 83.3%

          \[\leadsto \color{blue}{\left(\left(0 - 1\right) - \frac{1}{y} \cdot x\right)} \cdot z \]

      16. metadata-eval 83.3%

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

      17. associate-*l/ 83.3%

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

      18. *-lft-identity 83.3%

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

   10. Simplified 83.3%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.5 \cdot 10^{-29}:\\ \;\;\;\;z \cdot \frac{y}{z - y}\\ \mathbf{elif}\;y \leq 10^{+32}:\\ \;\;\;\;x \cdot \frac{z}{z - y}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 66.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.5 \cdot 10^{-28} \lor \neg \left(y \leq 4.7 \cdot 10^{+67}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -2.5e-28) (not (<= y 4.7e+67))) (- z) (+ x y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -2.5e-28) || !(y <= 4.7e+67)) {
		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 <= (-2.5d-28)) .or. (.not. (y <= 4.7d+67))) 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 <= -2.5e-28) || !(y <= 4.7e+67)) {
		tmp = -z;
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -2.5e-28) or not (y <= 4.7e+67):
		tmp = -z
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -2.5e-28) || !(y <= 4.7e+67))
		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 <= -2.5e-28) || ~((y <= 4.7e+67)))
		tmp = -z;
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -2.5e-28], N[Not[LessEqual[y, 4.7e+67]], $MachinePrecision]], (-z), N[(x + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.5000000000000001e-28 or 4.70000000000000017e67 < y

   1. Initial program 78.1%

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

   3. Taylor expanded in y around inf 61.4%

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

      1. neg-mul-1 61.4%

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

   5. Simplified 61.4%

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

   if -2.5000000000000001e-28 < y < 4.70000000000000017e67

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf 76.9%

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

      1. +-commutative 76.9%

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

   5. Simplified 76.9%

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

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.5 \cdot 10^{-28} \lor \neg \left(y \leq 4.7 \cdot 10^{+67}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 58.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-30} \lor \neg \left(y \leq 8 \cdot 10^{+41}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -5e-30) (not (<= y 8e+41))) (- z) x))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -5e-30) || !(y <= 8e+41)) {
		tmp = -z;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((y <= (-5d-30)) .or. (.not. (y <= 8d+41))) then
        tmp = -z
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -5e-30) || !(y <= 8e+41)) {
		tmp = -z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -5e-30) or not (y <= 8e+41):
		tmp = -z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -5e-30) || !(y <= 8e+41))
		tmp = Float64(-z);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -5e-30) || ~((y <= 8e+41)))
		tmp = -z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -5e-30], N[Not[LessEqual[y, 8e+41]], $MachinePrecision]], (-z), x]

Derivation

1. Split input into 2 regimes

2. if y < -4.99999999999999972e-30 or 8.00000000000000005e41 < y

   1. Initial program 78.1%

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

   3. Taylor expanded in y around inf 61.4%

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

      1. neg-mul-1 61.4%

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

   5. Simplified 61.4%

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

   if -4.99999999999999972e-30 < y < 8.00000000000000005e41

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 66.1%

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

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-30} \lor \neg \left(y \leq 8 \cdot 10^{+41}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 38.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.3 \cdot 10^{-24}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{+89}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.3e-24) y (if (<= y 1.9e+89) x y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.3e-24) {
		tmp = y;
	} else if (y <= 1.9e+89) {
		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 <= (-1.3d-24)) then
        tmp = y
    else if (y <= 1.9d+89) 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 <= -1.3e-24) {
		tmp = y;
	} else if (y <= 1.9e+89) {
		tmp = x;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1.3e-24:
		tmp = y
	elif y <= 1.9e+89:
		tmp = x
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.3e-24)
		tmp = y;
	elseif (y <= 1.9e+89)
		tmp = x;
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -1.3e-24)
		tmp = y;
	elseif (y <= 1.9e+89)
		tmp = x;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.3e-24], y, If[LessEqual[y, 1.9e+89], x, y]]

Derivation

1. Split input into 2 regimes

2. if y < -1.3e-24 or 1.90000000000000012e89 < y

   1. Initial program 77.0%

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

   3. Taylor expanded in z around inf 24.5%

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

      1. +-commutative 24.5%

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

   5. Simplified 24.5%

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

   6. Taylor expanded in y around inf 19.3%

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

   if -1.3e-24 < y < 1.90000000000000012e89

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 64.6%

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

Alternative 10: 35.7% 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 90.3%

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

3. Taylor expanded in y around 0 41.1%

   \[\leadsto \color{blue}{x} \]
4. Add Preprocessing

Developer Target 1: 93.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024135 
(FPCore (x y z)
  :name "Graphics.Rendering.Chart.Backend.Diagrams:calcFontMetrics from Chart-diagrams-1.5.1, A"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< y -3742931076268985600000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (* (/ (+ y x) (- y)) z) (if (< y 3553466245608673400000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (/ (+ x y) (- 1 (/ y z))) (* (/ (+ y x) (- y)) z))))

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