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

Percentage Accurate: 88.7% → 99.4%

Time: 6.3s

Alternatives: 11

Speedup: 0.3×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 88.7% 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: 99.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{y}{z}\\ t_1 := \frac{x + y}{t\_0}\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{-257}:\\ \;\;\;\;\frac{1}{t\_0} \cdot \left(x + y\right)\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\left(-z\right) - \frac{z \cdot \left(x + z\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ y z))) (t_1 (/ (+ x y) t_0)))
   (if (<= t_1 -4e-257)
     (* (/ 1.0 t_0) (+ x y))
     (if (<= t_1 0.0) (- (- z) (/ (* z (+ x z)) y)) t_1))))

C

double code(double x, double y, double z) {
	double t_0 = 1.0 - (y / z);
	double t_1 = (x + y) / t_0;
	double tmp;
	if (t_1 <= -4e-257) {
		tmp = (1.0 / t_0) * (x + y);
	} else if (t_1 <= 0.0) {
		tmp = -z - ((z * (x + z)) / y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 - (y / z)
    t_1 = (x + y) / t_0
    if (t_1 <= (-4d-257)) then
        tmp = (1.0d0 / t_0) * (x + y)
    else if (t_1 <= 0.0d0) then
        tmp = -z - ((z * (x + z)) / y)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = 1.0 - (y / z);
	double t_1 = (x + y) / t_0;
	double tmp;
	if (t_1 <= -4e-257) {
		tmp = (1.0 / t_0) * (x + y);
	} else if (t_1 <= 0.0) {
		tmp = -z - ((z * (x + z)) / y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 1.0 - (y / z)
	t_1 = (x + y) / t_0
	tmp = 0
	if t_1 <= -4e-257:
		tmp = (1.0 / t_0) * (x + y)
	elif t_1 <= 0.0:
		tmp = -z - ((z * (x + z)) / y)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(1.0 - Float64(y / z))
	t_1 = Float64(Float64(x + y) / t_0)
	tmp = 0.0
	if (t_1 <= -4e-257)
		tmp = Float64(Float64(1.0 / t_0) * Float64(x + y));
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(-z) - Float64(Float64(z * Float64(x + z)) / y));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 1.0 - (y / z);
	t_1 = (x + y) / t_0;
	tmp = 0.0;
	if (t_1 <= -4e-257)
		tmp = (1.0 / t_0) * (x + y);
	elseif (t_1 <= 0.0)
		tmp = -z - ((z * (x + z)) / y);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x + y), $MachinePrecision] / t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, -4e-257], N[(N[(1.0 / t$95$0), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[((-z) - N[(N[(z * N[(x + z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z))) < -3.9999999999999999e-257

   1. Initial program 99.9%

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

      1. clear-num 99.6%

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

      2. associate-/r/ 99.9%

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

   4. Applied egg-rr 99.9%

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

   if -3.9999999999999999e-257 < (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z))) < 0.0

   1. Initial program 6.1%

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

   3. Taylor expanded in y around inf 99.8%

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

      1. associate--l+ 99.8%

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

      2. associate-*r/ 99.8%

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

      3. div-sub 99.8%

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

      4. remove-double-neg 99.8%

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

      5. mul-1-neg 99.8%

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

      6. neg-mul-1 99.8%

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

      7. distribute-lft-out-- 99.8%

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

      8. mul-1-neg 99.8%

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

      9. distribute-neg-frac 99.8%

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

      10. unsub-neg 99.8%

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

      11. mul-1-neg 99.8%

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

      12. cancel-sign-sub-inv 99.8%

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

      13. metadata-eval 99.8%

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

      14. *-lft-identity 99.8%

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

      15. +-commutative 99.8%

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

      16. unpow2 99.8%

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

      17. distribute-rgt-out 99.8%

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

   5. Simplified 99.8%

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

   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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \leq -4 \cdot 10^{-257}:\\ \;\;\;\;\frac{1}{1 - \frac{y}{z}} \cdot \left(x + y\right)\\ \mathbf{elif}\;\frac{x + y}{1 - \frac{y}{z}} \leq 0:\\ \;\;\;\;\left(-z\right) - \frac{z \cdot \left(x + z\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{-257} \lor \neg \left(t\_0 \leq 0\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) - \frac{z \cdot \left(x + z\right)}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (+ x y) (- 1.0 (/ y z)))))
   (if (or (<= t_0 -4e-257) (not (<= t_0 0.0)))
     t_0
     (- (- z) (/ (* z (+ x z)) y)))))

C

double code(double x, double y, double z) {
	double t_0 = (x + y) / (1.0 - (y / z));
	double tmp;
	if ((t_0 <= -4e-257) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = -z - ((z * (x + 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) :: t_0
    real(8) :: tmp
    t_0 = (x + y) / (1.0d0 - (y / z))
    if ((t_0 <= (-4d-257)) .or. (.not. (t_0 <= 0.0d0))) then
        tmp = t_0
    else
        tmp = -z - ((z * (x + z)) / 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 <= -4e-257) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = -z - ((z * (x + z)) / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x + y) / (1.0 - (y / z))
	tmp = 0
	if (t_0 <= -4e-257) or not (t_0 <= 0.0):
		tmp = t_0
	else:
		tmp = -z - ((z * (x + z)) / 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 <= -4e-257) || !(t_0 <= 0.0))
		tmp = t_0;
	else
		tmp = Float64(Float64(-z) - Float64(Float64(z * Float64(x + z)) / 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 <= -4e-257) || ~((t_0 <= 0.0)))
		tmp = t_0;
	else
		tmp = -z - ((z * (x + z)) / 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, -4e-257], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], t$95$0, N[((-z) - N[(N[(z * N[(x + z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z))) < -3.9999999999999999e-257 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 -3.9999999999999999e-257 < (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z))) < 0.0

   1. Initial program 6.1%

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

   3. Taylor expanded in y around inf 99.8%

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

      1. associate--l+ 99.8%

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

      2. associate-*r/ 99.8%

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

      3. div-sub 99.8%

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

      4. remove-double-neg 99.8%

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

      5. mul-1-neg 99.8%

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

      6. neg-mul-1 99.8%

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

      7. distribute-lft-out-- 99.8%

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

      8. mul-1-neg 99.8%

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

      9. distribute-neg-frac 99.8%

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

      10. unsub-neg 99.8%

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

      11. mul-1-neg 99.8%

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

      12. cancel-sign-sub-inv 99.8%

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

      13. metadata-eval 99.8%

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

      14. *-lft-identity 99.8%

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

      15. +-commutative 99.8%

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

      16. unpow2 99.8%

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

      17. distribute-rgt-out 99.8%

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

   5. Simplified 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \leq -4 \cdot 10^{-257} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \leq 0\right):\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) - \frac{z \cdot \left(x + z\right)}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{-257} \lor \neg \left(t\_0 \leq 0\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{-\left(x + y\right)}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (+ x y) (- 1.0 (/ y z)))))
   (if (or (<= t_0 -4e-257) (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 <= -4e-257) || !(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 <= (-4d-257)) .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 <= -4e-257) || !(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 <= -4e-257) 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 <= -4e-257) || !(t_0 <= 0.0))
		tmp = t_0;
	else
		tmp = Float64(z * Float64(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 <= -4e-257) || ~((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, -4e-257], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], t$95$0, N[(z * N[((-N[(x + y), $MachinePrecision]) / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z))) < -3.9999999999999999e-257 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 -3.9999999999999999e-257 < (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z))) < 0.0

   1. Initial program 6.1%

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

   3. Taylor expanded in z around 0 99.8%

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

      1. mul-1-neg 99.8%

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

      2. associate-/l* 99.8%

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

      3. distribute-rgt-neg-in 99.8%

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

      4. distribute-neg-frac2 99.8%

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

      5. +-commutative 99.8%

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

   5. Simplified 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \leq -4 \cdot 10^{-257} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \leq 0\right):\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{-\left(x + y\right)}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 68.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.1 \cdot 10^{+238}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq -2.2 \cdot 10^{+84} \lor \neg \left(y \leq -130000000 \lor \neg \left(y \leq -6.7 \cdot 10^{-62}\right) \land y \leq 3.9 \cdot 10^{-30}\right):\\ \;\;\;\;\frac{z \cdot \left(x + y\right)}{-y}\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2.1e+238)
   (- z)
   (if (or (<= y -2.2e+84)
           (not
            (or (<= y -130000000.0)
                (and (not (<= y -6.7e-62)) (<= y 3.9e-30)))))
     (/ (* z (+ x y)) (- y))
     (+ x y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.1e+238) {
		tmp = -z;
	} else if ((y <= -2.2e+84) || !((y <= -130000000.0) || (!(y <= -6.7e-62) && (y <= 3.9e-30)))) {
		tmp = (z * (x + y)) / -y;
	} else {
		tmp = x + y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-2.1d+238)) then
        tmp = -z
    else if ((y <= (-2.2d+84)) .or. (.not. (y <= (-130000000.0d0)) .or. (.not. (y <= (-6.7d-62))) .and. (y <= 3.9d-30))) then
        tmp = (z * (x + y)) / -y
    else
        tmp = x + y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.1e+238) {
		tmp = -z;
	} else if ((y <= -2.2e+84) || !((y <= -130000000.0) || (!(y <= -6.7e-62) && (y <= 3.9e-30)))) {
		tmp = (z * (x + y)) / -y;
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -2.1e+238:
		tmp = -z
	elif (y <= -2.2e+84) or not ((y <= -130000000.0) or (not (y <= -6.7e-62) and (y <= 3.9e-30))):
		tmp = (z * (x + y)) / -y
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -2.1e+238)
		tmp = Float64(-z);
	elseif ((y <= -2.2e+84) || !((y <= -130000000.0) || (!(y <= -6.7e-62) && (y <= 3.9e-30))))
		tmp = Float64(Float64(z * Float64(x + y)) / Float64(-y));
	else
		tmp = Float64(x + y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -2.1e+238)
		tmp = -z;
	elseif ((y <= -2.2e+84) || ~(((y <= -130000000.0) || (~((y <= -6.7e-62)) && (y <= 3.9e-30)))))
		tmp = (z * (x + y)) / -y;
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -2.1e+238], (-z), If[Or[LessEqual[y, -2.2e+84], N[Not[Or[LessEqual[y, -130000000.0], And[N[Not[LessEqual[y, -6.7e-62]], $MachinePrecision], LessEqual[y, 3.9e-30]]]], $MachinePrecision]], N[(N[(z * N[(x + y), $MachinePrecision]), $MachinePrecision] / (-y)), $MachinePrecision], N[(x + y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.10000000000000007e238

   1. Initial program 73.0%

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

   3. Taylor expanded in y around inf 92.9%

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

      1. mul-1-neg 92.9%

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

   5. Simplified 92.9%

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

   if -2.10000000000000007e238 < y < -2.1999999999999998e84 or -1.3e8 < y < -6.69999999999999992e-62 or 3.9000000000000003e-30 < y

   1. Initial program 71.1%

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

   3. Taylor expanded in z around 0 75.6%

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

      1. mul-1-neg 75.6%

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

      2. +-commutative 75.6%

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

   5. Simplified 75.6%

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

   if -2.1999999999999998e84 < y < -1.3e8 or -6.69999999999999992e-62 < y < 3.9000000000000003e-30

   1. Initial program 99.2%

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

   3. Taylor expanded in z around inf 82.5%

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

      1. +-commutative 82.5%

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

   5. Simplified 82.5%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.1 \cdot 10^{+238}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq -2.2 \cdot 10^{+84} \lor \neg \left(y \leq -130000000 \lor \neg \left(y \leq -6.7 \cdot 10^{-62}\right) \land y \leq 3.9 \cdot 10^{-30}\right):\\ \;\;\;\;\frac{z \cdot \left(x + y\right)}{-y}\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \frac{-\left(x + y\right)}{y}\\ \mathbf{if}\;y \leq -2.2 \cdot 10^{+84}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq -22000000:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq -6.4 \cdot 10^{-62}:\\ \;\;\;\;\frac{z \cdot \left(x + y\right)}{-y}\\ \mathbf{elif}\;y \leq 2.95 \cdot 10^{-30}:\\ \;\;\;\;\left(x + y\right) \cdot \left(1 + \frac{y}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (/ (- (+ x y)) y))))
   (if (<= y -2.2e+84)
     t_0
     (if (<= y -22000000.0)
       (+ x y)
       (if (<= y -6.4e-62)
         (/ (* z (+ x y)) (- y))
         (if (<= y 2.95e-30) (* (+ x y) (+ 1.0 (/ y z))) t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = z * (-(x + y) / y);
	double tmp;
	if (y <= -2.2e+84) {
		tmp = t_0;
	} else if (y <= -22000000.0) {
		tmp = x + y;
	} else if (y <= -6.4e-62) {
		tmp = (z * (x + y)) / -y;
	} else if (y <= 2.95e-30) {
		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 = z * (-(x + y) / y)
    if (y <= (-2.2d+84)) then
        tmp = t_0
    else if (y <= (-22000000.0d0)) then
        tmp = x + y
    else if (y <= (-6.4d-62)) then
        tmp = (z * (x + y)) / -y
    else if (y <= 2.95d-30) 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 = z * (-(x + y) / y);
	double tmp;
	if (y <= -2.2e+84) {
		tmp = t_0;
	} else if (y <= -22000000.0) {
		tmp = x + y;
	} else if (y <= -6.4e-62) {
		tmp = (z * (x + y)) / -y;
	} else if (y <= 2.95e-30) {
		tmp = (x + y) * (1.0 + (y / z));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * (-(x + y) / y)
	tmp = 0
	if y <= -2.2e+84:
		tmp = t_0
	elif y <= -22000000.0:
		tmp = x + y
	elif y <= -6.4e-62:
		tmp = (z * (x + y)) / -y
	elif y <= 2.95e-30:
		tmp = (x + y) * (1.0 + (y / z))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(Float64(-Float64(x + y)) / y))
	tmp = 0.0
	if (y <= -2.2e+84)
		tmp = t_0;
	elseif (y <= -22000000.0)
		tmp = Float64(x + y);
	elseif (y <= -6.4e-62)
		tmp = Float64(Float64(z * Float64(x + y)) / Float64(-y));
	elseif (y <= 2.95e-30)
		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 = z * (-(x + y) / y);
	tmp = 0.0;
	if (y <= -2.2e+84)
		tmp = t_0;
	elseif (y <= -22000000.0)
		tmp = x + y;
	elseif (y <= -6.4e-62)
		tmp = (z * (x + y)) / -y;
	elseif (y <= 2.95e-30)
		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[(z * N[((-N[(x + y), $MachinePrecision]) / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.2e+84], t$95$0, If[LessEqual[y, -22000000.0], N[(x + y), $MachinePrecision], If[LessEqual[y, -6.4e-62], N[(N[(z * N[(x + y), $MachinePrecision]), $MachinePrecision] / (-y)), $MachinePrecision], If[LessEqual[y, 2.95e-30], N[(N[(x + y), $MachinePrecision] * N[(1.0 + N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -2.1999999999999998e84 or 2.9499999999999999e-30 < y

   1. Initial program 67.2%

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

   3. Taylor expanded in z around 0 72.6%

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

      1. mul-1-neg 72.6%

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

      2. associate-/l* 84.4%

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

      3. distribute-rgt-neg-in 84.4%

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

      4. distribute-neg-frac2 84.4%

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

      5. +-commutative 84.4%

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

   5. Simplified 84.4%

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

   if -2.1999999999999998e84 < y < -2.2e7

   1. Initial program 94.4%

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

   3. Taylor expanded in z around inf 65.9%

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

      1. +-commutative 65.9%

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

   5. Simplified 65.9%

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

   if -2.2e7 < y < -6.40000000000000043e-62

   1. Initial program 99.7%

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

   3. Taylor expanded in z around 0 64.3%

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

      1. mul-1-neg 64.3%

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

      2. +-commutative 64.3%

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

   5. Simplified 64.3%

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

   if -6.40000000000000043e-62 < y < 2.9499999999999999e-30

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf 86.2%

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

      1. associate-+r+ 86.2%

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

      2. *-rgt-identity 86.2%

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

      3. *-commutative 86.2%

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

      4. associate-/l* 86.1%

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

      5. distribute-lft-in 86.1%

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

      6. +-commutative 86.1%

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

   5. Simplified 86.1%

      \[\leadsto \color{blue}{\left(y + x\right) \cdot \left(1 + \frac{y}{z}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{+84}:\\ \;\;\;\;z \cdot \frac{-\left(x + y\right)}{y}\\ \mathbf{elif}\;y \leq -22000000:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq -6.4 \cdot 10^{-62}:\\ \;\;\;\;\frac{z \cdot \left(x + y\right)}{-y}\\ \mathbf{elif}\;y \leq 2.95 \cdot 10^{-30}:\\ \;\;\;\;\left(x + y\right) \cdot \left(1 + \frac{y}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{-\left(x + y\right)}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \frac{-\left(x + y\right)}{y}\\ \mathbf{if}\;y \leq -2.2 \cdot 10^{+84}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq -105000000:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq -4.4 \cdot 10^{-62}:\\ \;\;\;\;\frac{z \cdot \left(x + y\right)}{-y}\\ \mathbf{elif}\;y \leq 9 \cdot 10^{-31}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (/ (- (+ x y)) y))))
   (if (<= y -2.2e+84)
     t_0
     (if (<= y -105000000.0)
       (+ x y)
       (if (<= y -4.4e-62)
         (/ (* z (+ x y)) (- y))
         (if (<= y 9e-31) (+ x y) t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = z * (-(x + y) / y);
	double tmp;
	if (y <= -2.2e+84) {
		tmp = t_0;
	} else if (y <= -105000000.0) {
		tmp = x + y;
	} else if (y <= -4.4e-62) {
		tmp = (z * (x + y)) / -y;
	} else if (y <= 9e-31) {
		tmp = 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 = z * (-(x + y) / y)
    if (y <= (-2.2d+84)) then
        tmp = t_0
    else if (y <= (-105000000.0d0)) then
        tmp = x + y
    else if (y <= (-4.4d-62)) then
        tmp = (z * (x + y)) / -y
    else if (y <= 9d-31) then
        tmp = 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 = z * (-(x + y) / y);
	double tmp;
	if (y <= -2.2e+84) {
		tmp = t_0;
	} else if (y <= -105000000.0) {
		tmp = x + y;
	} else if (y <= -4.4e-62) {
		tmp = (z * (x + y)) / -y;
	} else if (y <= 9e-31) {
		tmp = x + y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * (-(x + y) / y)
	tmp = 0
	if y <= -2.2e+84:
		tmp = t_0
	elif y <= -105000000.0:
		tmp = x + y
	elif y <= -4.4e-62:
		tmp = (z * (x + y)) / -y
	elif y <= 9e-31:
		tmp = x + y
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(Float64(-Float64(x + y)) / y))
	tmp = 0.0
	if (y <= -2.2e+84)
		tmp = t_0;
	elseif (y <= -105000000.0)
		tmp = Float64(x + y);
	elseif (y <= -4.4e-62)
		tmp = Float64(Float64(z * Float64(x + y)) / Float64(-y));
	elseif (y <= 9e-31)
		tmp = Float64(x + y);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * (-(x + y) / y);
	tmp = 0.0;
	if (y <= -2.2e+84)
		tmp = t_0;
	elseif (y <= -105000000.0)
		tmp = x + y;
	elseif (y <= -4.4e-62)
		tmp = (z * (x + y)) / -y;
	elseif (y <= 9e-31)
		tmp = x + y;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[((-N[(x + y), $MachinePrecision]) / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.2e+84], t$95$0, If[LessEqual[y, -105000000.0], N[(x + y), $MachinePrecision], If[LessEqual[y, -4.4e-62], N[(N[(z * N[(x + y), $MachinePrecision]), $MachinePrecision] / (-y)), $MachinePrecision], If[LessEqual[y, 9e-31], N[(x + y), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.1999999999999998e84 or 9.0000000000000008e-31 < y

   1. Initial program 67.2%

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

   3. Taylor expanded in z around 0 72.6%

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

      1. mul-1-neg 72.6%

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

      2. associate-/l* 84.4%

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

      3. distribute-rgt-neg-in 84.4%

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

      4. distribute-neg-frac2 84.4%

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

      5. +-commutative 84.4%

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

   5. Simplified 84.4%

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

   if -2.1999999999999998e84 < y < -1.05e8 or -4.40000000000000035e-62 < y < 9.0000000000000008e-31

   1. Initial program 99.2%

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

   3. Taylor expanded in z around inf 82.5%

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

      1. +-commutative 82.5%

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

   5. Simplified 82.5%

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

   if -1.05e8 < y < -4.40000000000000035e-62

   1. Initial program 99.7%

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

   3. Taylor expanded in z around 0 64.3%

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

      1. mul-1-neg 64.3%

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

      2. +-commutative 64.3%

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

   5. Simplified 64.3%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{+84}:\\ \;\;\;\;z \cdot \frac{-\left(x + y\right)}{y}\\ \mathbf{elif}\;y \leq -105000000:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq -4.4 \cdot 10^{-62}:\\ \;\;\;\;\frac{z \cdot \left(x + y\right)}{-y}\\ \mathbf{elif}\;y \leq 9 \cdot 10^{-31}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{-\left(x + y\right)}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 69.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{y}{z}\\ \mathbf{if}\;y \leq -1.02 \cdot 10^{+138}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq -1.4 \cdot 10^{+38}:\\ \;\;\;\;\frac{y}{t\_0}\\ \mathbf{elif}\;y \leq -1.42 \cdot 10^{-98}:\\ \;\;\;\;\frac{x}{t\_0}\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{+72}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ y z))))
   (if (<= y -1.02e+138)
     (- z)
     (if (<= y -1.4e+38)
       (/ y t_0)
       (if (<= y -1.42e-98) (/ x t_0) (if (<= y 3.1e+72) (+ x y) (- z)))))))

C

double code(double x, double y, double z) {
	double t_0 = 1.0 - (y / z);
	double tmp;
	if (y <= -1.02e+138) {
		tmp = -z;
	} else if (y <= -1.4e+38) {
		tmp = y / t_0;
	} else if (y <= -1.42e-98) {
		tmp = x / t_0;
	} else if (y <= 3.1e+72) {
		tmp = x + y;
	} else {
		tmp = -z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - (y / z)
    if (y <= (-1.02d+138)) then
        tmp = -z
    else if (y <= (-1.4d+38)) then
        tmp = y / t_0
    else if (y <= (-1.42d-98)) then
        tmp = x / t_0
    else if (y <= 3.1d+72) then
        tmp = x + y
    else
        tmp = -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = 1.0 - (y / z);
	double tmp;
	if (y <= -1.02e+138) {
		tmp = -z;
	} else if (y <= -1.4e+38) {
		tmp = y / t_0;
	} else if (y <= -1.42e-98) {
		tmp = x / t_0;
	} else if (y <= 3.1e+72) {
		tmp = x + y;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 1.0 - (y / z)
	tmp = 0
	if y <= -1.02e+138:
		tmp = -z
	elif y <= -1.4e+38:
		tmp = y / t_0
	elif y <= -1.42e-98:
		tmp = x / t_0
	elif y <= 3.1e+72:
		tmp = x + y
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(1.0 - Float64(y / z))
	tmp = 0.0
	if (y <= -1.02e+138)
		tmp = Float64(-z);
	elseif (y <= -1.4e+38)
		tmp = Float64(y / t_0);
	elseif (y <= -1.42e-98)
		tmp = Float64(x / t_0);
	elseif (y <= 3.1e+72)
		tmp = Float64(x + y);
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 1.0 - (y / z);
	tmp = 0.0;
	if (y <= -1.02e+138)
		tmp = -z;
	elseif (y <= -1.4e+38)
		tmp = y / t_0;
	elseif (y <= -1.42e-98)
		tmp = x / t_0;
	elseif (y <= 3.1e+72)
		tmp = x + y;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.02e+138], (-z), If[LessEqual[y, -1.4e+38], N[(y / t$95$0), $MachinePrecision], If[LessEqual[y, -1.42e-98], N[(x / t$95$0), $MachinePrecision], If[LessEqual[y, 3.1e+72], N[(x + y), $MachinePrecision], (-z)]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -1.02e138 or 3.09999999999999988e72 < y

   1. Initial program 59.6%

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

   3. Taylor expanded in y around inf 73.5%

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

      1. mul-1-neg 73.5%

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

   5. Simplified 73.5%

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

   if -1.02e138 < y < -1.4e38

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 73.4%

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

   if -1.4e38 < y < -1.41999999999999999e-98

   1. Initial program 96.8%

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

   3. Taylor expanded in x around inf 61.1%

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

   if -1.41999999999999999e-98 < y < 3.09999999999999988e72

   1. Initial program 98.4%

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

   3. Taylor expanded in z around inf 78.0%

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

      1. +-commutative 78.0%

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

   5. Simplified 78.0%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.02 \cdot 10^{+138}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq -1.4 \cdot 10^{+38}:\\ \;\;\;\;\frac{y}{1 - \frac{y}{z}}\\ \mathbf{elif}\;y \leq -1.42 \cdot 10^{-98}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{+72}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 68.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.9 \cdot 10^{+113} \lor \neg \left(y \leq 1.85 \cdot 10^{+76}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -5.9e+113) (not (<= y 1.85e+76))) (- z) (+ x y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -5.9e+113) || !(y <= 1.85e+76)) {
		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 <= (-5.9d+113)) .or. (.not. (y <= 1.85d+76))) 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 <= -5.9e+113) || !(y <= 1.85e+76)) {
		tmp = -z;
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -5.9e+113) or not (y <= 1.85e+76):
		tmp = -z
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -5.9e+113) || !(y <= 1.85e+76))
		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 <= -5.9e+113) || ~((y <= 1.85e+76)))
		tmp = -z;
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -5.9e+113], N[Not[LessEqual[y, 1.85e+76]], $MachinePrecision]], (-z), N[(x + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -5.90000000000000023e113 or 1.85e76 < y

   1. Initial program 60.5%

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

   3. Taylor expanded in y around inf 73.2%

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

      1. mul-1-neg 73.2%

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

   5. Simplified 73.2%

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

   if -5.90000000000000023e113 < y < 1.85e76

   1. Initial program 98.2%

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

   3. Taylor expanded in z around inf 71.2%

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

      1. +-commutative 71.2%

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

   5. Simplified 71.2%

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

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.9 \cdot 10^{+113} \lor \neg \left(y \leq 1.85 \cdot 10^{+76}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 58.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.05 \cdot 10^{+39} \lor \neg \left(y \leq 5.2 \cdot 10^{-39}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -2.05e+39) (not (<= y 5.2e-39))) (- z) x))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -2.05e+39) || !(y <= 5.2e-39)) {
		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 <= (-2.05d+39)) .or. (.not. (y <= 5.2d-39))) 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 <= -2.05e+39) || !(y <= 5.2e-39)) {
		tmp = -z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -2.05e+39) or not (y <= 5.2e-39):
		tmp = -z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -2.05e+39) || !(y <= 5.2e-39))
		tmp = Float64(-z);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -2.05e+39) || ~((y <= 5.2e-39)))
		tmp = -z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -2.05e+39], N[Not[LessEqual[y, 5.2e-39]], $MachinePrecision]], (-z), x]

Derivation

1. Split input into 2 regimes

2. if y < -2.05000000000000002e39 or 5.2e-39 < y

   1. Initial program 69.7%

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

   3. Taylor expanded in y around inf 61.4%

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

      1. mul-1-neg 61.4%

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

   5. Simplified 61.4%

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

   if -2.05000000000000002e39 < y < 5.2e-39

   1. Initial program 99.2%

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

   3. Taylor expanded in y around 0 62.2%

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

4. Final simplification 61.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.05 \cdot 10^{+39} \lor \neg \left(y \leq 5.2 \cdot 10^{-39}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 40.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{-202}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{-138}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.5e-202) x (if (<= x 1.8e-138) y x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.5e-202) {
		tmp = x;
	} else if (x <= 1.8e-138) {
		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 <= (-1.5d-202)) then
        tmp = x
    else if (x <= 1.8d-138) 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 <= -1.5e-202) {
		tmp = x;
	} else if (x <= 1.8e-138) {
		tmp = y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.5e-202:
		tmp = x
	elif x <= 1.8e-138:
		tmp = y
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.5e-202)
		tmp = x;
	elseif (x <= 1.8e-138)
		tmp = y;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.5e-202)
		tmp = x;
	elseif (x <= 1.8e-138)
		tmp = y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.5e-202], x, If[LessEqual[x, 1.8e-138], y, x]]

Derivation

1. Split input into 2 regimes

2. if x < -1.50000000000000005e-202 or 1.80000000000000009e-138 < x

   1. Initial program 84.0%

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

   3. Taylor expanded in y around 0 44.5%

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

   if -1.50000000000000005e-202 < x < 1.80000000000000009e-138

   1. Initial program 88.7%

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

   3. Taylor expanded in x around 0 75.7%

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

   4. Taylor expanded in y around 0 43.4%

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

Alternative 11: 35.6% 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 85.3%

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

3. Taylor expanded in y around 0 37.0%

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

Developer target: 94.3% 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 2024085 
(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))))