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

Percentage Accurate: 87.6% → 99.6%

Time: 6.1s

Alternatives: 9

Speedup: 0.3×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 87.6% 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: 99.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{-250} \lor \neg \left(t_0 \leq 0\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-z}{\frac{y}{x + y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (+ x y) (- 1.0 (/ y z)))))
   (if (or (<= t_0 -5e-250) (not (<= t_0 0.0))) t_0 (/ (- z) (/ y (+ x y))))))

C

double code(double x, double y, double z) {
	double t_0 = (x + y) / (1.0 - (y / z));
	double tmp;
	if ((t_0 <= -5e-250) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = -z / (y / (x + y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x + y) / (1.0d0 - (y / z))
    if ((t_0 <= (-5d-250)) .or. (.not. (t_0 <= 0.0d0))) then
        tmp = t_0
    else
        tmp = -z / (y / (x + y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (x + y) / (1.0 - (y / z));
	double tmp;
	if ((t_0 <= -5e-250) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = -z / (y / (x + y));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x + y) / (1.0 - (y / z))
	tmp = 0
	if (t_0 <= -5e-250) or not (t_0 <= 0.0):
		tmp = t_0
	else:
		tmp = -z / (y / (x + y))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x + y) / Float64(1.0 - Float64(y / z)))
	tmp = 0.0
	if ((t_0 <= -5e-250) || !(t_0 <= 0.0))
		tmp = t_0;
	else
		tmp = Float64(Float64(-z) / Float64(y / Float64(x + y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x + y) / (1.0 - (y / z));
	tmp = 0.0;
	if ((t_0 <= -5e-250) || ~((t_0 <= 0.0)))
		tmp = t_0;
	else
		tmp = -z / (y / (x + y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x + y), $MachinePrecision] / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -5e-250], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], t$95$0, N[((-z) / N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z))) < -5.00000000000000027e-250 or 0.0 < (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z)))

   1. Initial program 99.9%

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

   if -5.00000000000000027e-250 < (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z))) < 0.0

   1. Initial program 14.8%

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

   3. Taylor expanded in z around 0 94.0%

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

      1. mul-1-neg 94.0%

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

      2. associate-/l* 100.0%

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

      3. distribute-neg-frac 100.0%

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

      4. +-commutative 100.0%

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

   5. Simplified 100.0%

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

4. Final simplification 99.9%

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

Alternative 2: 68.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{1 - \frac{y}{z}}\\ \mathbf{if}\;y \leq -4.8 \cdot 10^{+173}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq -2.5 \cdot 10^{+58}:\\ \;\;\;\;\left(x + y\right) \cdot \frac{-z}{y}\\ \mathbf{elif}\;y \leq -30000000000000:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq -5.4 \cdot 10^{-41}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -2.8 \cdot 10^{-170}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 2.35 \cdot 10^{+83}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ x (- 1.0 (/ y z)))))
   (if (<= y -4.8e+173)
     (- z)
     (if (<= y -2.5e+58)
       (* (+ x y) (/ (- z) y))
       (if (<= y -30000000000000.0)
         (+ x y)
         (if (<= y -5.4e-41)
           t_0
           (if (<= y -2.8e-170) (+ x y) (if (<= y 2.35e+83) t_0 (- z)))))))))

C

double code(double x, double y, double z) {
	double t_0 = x / (1.0 - (y / z));
	double tmp;
	if (y <= -4.8e+173) {
		tmp = -z;
	} else if (y <= -2.5e+58) {
		tmp = (x + y) * (-z / y);
	} else if (y <= -30000000000000.0) {
		tmp = x + y;
	} else if (y <= -5.4e-41) {
		tmp = t_0;
	} else if (y <= -2.8e-170) {
		tmp = x + y;
	} else if (y <= 2.35e+83) {
		tmp = t_0;
	} else {
		tmp = -z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / (1.0d0 - (y / z))
    if (y <= (-4.8d+173)) then
        tmp = -z
    else if (y <= (-2.5d+58)) then
        tmp = (x + y) * (-z / y)
    else if (y <= (-30000000000000.0d0)) then
        tmp = x + y
    else if (y <= (-5.4d-41)) then
        tmp = t_0
    else if (y <= (-2.8d-170)) then
        tmp = x + y
    else if (y <= 2.35d+83) then
        tmp = t_0
    else
        tmp = -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x / (1.0 - (y / z));
	double tmp;
	if (y <= -4.8e+173) {
		tmp = -z;
	} else if (y <= -2.5e+58) {
		tmp = (x + y) * (-z / y);
	} else if (y <= -30000000000000.0) {
		tmp = x + y;
	} else if (y <= -5.4e-41) {
		tmp = t_0;
	} else if (y <= -2.8e-170) {
		tmp = x + y;
	} else if (y <= 2.35e+83) {
		tmp = t_0;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x / (1.0 - (y / z))
	tmp = 0
	if y <= -4.8e+173:
		tmp = -z
	elif y <= -2.5e+58:
		tmp = (x + y) * (-z / y)
	elif y <= -30000000000000.0:
		tmp = x + y
	elif y <= -5.4e-41:
		tmp = t_0
	elif y <= -2.8e-170:
		tmp = x + y
	elif y <= 2.35e+83:
		tmp = t_0
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x / Float64(1.0 - Float64(y / z)))
	tmp = 0.0
	if (y <= -4.8e+173)
		tmp = Float64(-z);
	elseif (y <= -2.5e+58)
		tmp = Float64(Float64(x + y) * Float64(Float64(-z) / y));
	elseif (y <= -30000000000000.0)
		tmp = Float64(x + y);
	elseif (y <= -5.4e-41)
		tmp = t_0;
	elseif (y <= -2.8e-170)
		tmp = Float64(x + y);
	elseif (y <= 2.35e+83)
		tmp = t_0;
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x / (1.0 - (y / z));
	tmp = 0.0;
	if (y <= -4.8e+173)
		tmp = -z;
	elseif (y <= -2.5e+58)
		tmp = (x + y) * (-z / y);
	elseif (y <= -30000000000000.0)
		tmp = x + y;
	elseif (y <= -5.4e-41)
		tmp = t_0;
	elseif (y <= -2.8e-170)
		tmp = x + y;
	elseif (y <= 2.35e+83)
		tmp = t_0;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -4.8e+173], (-z), If[LessEqual[y, -2.5e+58], N[(N[(x + y), $MachinePrecision] * N[((-z) / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -30000000000000.0], N[(x + y), $MachinePrecision], If[LessEqual[y, -5.4e-41], t$95$0, If[LessEqual[y, -2.8e-170], N[(x + y), $MachinePrecision], If[LessEqual[y, 2.35e+83], t$95$0, (-z)]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -4.7999999999999998e173 or 2.3499999999999999e83 < y

   1. Initial program 69.4%

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

   3. Taylor expanded in y around inf 72.6%

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

      1. mul-1-neg 72.6%

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

   5. Simplified 72.6%

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

   if -4.7999999999999998e173 < y < -2.49999999999999993e58

   1. Initial program 82.3%

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

   3. Taylor expanded in z around 0 81.4%

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

      1. mul-1-neg 81.4%

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

      2. associate-/l* 85.2%

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

      3. associate-/r/ 70.0%

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

      4. distribute-rgt-neg-in 70.0%

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

      5. +-commutative 70.0%

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

      6. distribute-neg-in 70.0%

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

      7. sub-neg 70.0%

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

   5. Simplified 70.0%

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

   if -2.49999999999999993e58 < y < -3e13 or -5.4e-41 < y < -2.79999999999999995e-170

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf 77.0%

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

      1. +-commutative 77.0%

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

   5. Simplified 77.0%

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

   if -3e13 < y < -5.4e-41 or -2.79999999999999995e-170 < y < 2.3499999999999999e83

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 84.2%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{+173}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq -2.5 \cdot 10^{+58}:\\ \;\;\;\;\left(x + y\right) \cdot \frac{-z}{y}\\ \mathbf{elif}\;y \leq -30000000000000:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq -5.4 \cdot 10^{-41}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{elif}\;y \leq -2.8 \cdot 10^{-170}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 2.35 \cdot 10^{+83}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{1 - \frac{y}{z}}\\ t_1 := \frac{-z}{\frac{y}{x + y}}\\ \mathbf{if}\;y \leq -1.4 \cdot 10^{+58}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -580:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq -1.6 \cdot 10^{-45}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -2.7 \cdot 10^{-173}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 1.12 \cdot 10^{+82}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ x (- 1.0 (/ y z)))) (t_1 (/ (- z) (/ y (+ x y)))))
   (if (<= y -1.4e+58)
     t_1
     (if (<= y -580.0)
       (+ x y)
       (if (<= y -1.6e-45)
         t_0
         (if (<= y -2.7e-173) (+ x y) (if (<= y 1.12e+82) t_0 t_1)))))))

C

double code(double x, double y, double z) {
	double t_0 = x / (1.0 - (y / z));
	double t_1 = -z / (y / (x + y));
	double tmp;
	if (y <= -1.4e+58) {
		tmp = t_1;
	} else if (y <= -580.0) {
		tmp = x + y;
	} else if (y <= -1.6e-45) {
		tmp = t_0;
	} else if (y <= -2.7e-173) {
		tmp = x + y;
	} else if (y <= 1.12e+82) {
		tmp = t_0;
	} 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 = x / (1.0d0 - (y / z))
    t_1 = -z / (y / (x + y))
    if (y <= (-1.4d+58)) then
        tmp = t_1
    else if (y <= (-580.0d0)) then
        tmp = x + y
    else if (y <= (-1.6d-45)) then
        tmp = t_0
    else if (y <= (-2.7d-173)) then
        tmp = x + y
    else if (y <= 1.12d+82) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x / (1.0 - (y / z));
	double t_1 = -z / (y / (x + y));
	double tmp;
	if (y <= -1.4e+58) {
		tmp = t_1;
	} else if (y <= -580.0) {
		tmp = x + y;
	} else if (y <= -1.6e-45) {
		tmp = t_0;
	} else if (y <= -2.7e-173) {
		tmp = x + y;
	} else if (y <= 1.12e+82) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x / (1.0 - (y / z))
	t_1 = -z / (y / (x + y))
	tmp = 0
	if y <= -1.4e+58:
		tmp = t_1
	elif y <= -580.0:
		tmp = x + y
	elif y <= -1.6e-45:
		tmp = t_0
	elif y <= -2.7e-173:
		tmp = x + y
	elif y <= 1.12e+82:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x / Float64(1.0 - Float64(y / z)))
	t_1 = Float64(Float64(-z) / Float64(y / Float64(x + y)))
	tmp = 0.0
	if (y <= -1.4e+58)
		tmp = t_1;
	elseif (y <= -580.0)
		tmp = Float64(x + y);
	elseif (y <= -1.6e-45)
		tmp = t_0;
	elseif (y <= -2.7e-173)
		tmp = Float64(x + y);
	elseif (y <= 1.12e+82)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x / (1.0 - (y / z));
	t_1 = -z / (y / (x + y));
	tmp = 0.0;
	if (y <= -1.4e+58)
		tmp = t_1;
	elseif (y <= -580.0)
		tmp = x + y;
	elseif (y <= -1.6e-45)
		tmp = t_0;
	elseif (y <= -2.7e-173)
		tmp = x + y;
	elseif (y <= 1.12e+82)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[((-z) / N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.4e+58], t$95$1, If[LessEqual[y, -580.0], N[(x + y), $MachinePrecision], If[LessEqual[y, -1.6e-45], t$95$0, If[LessEqual[y, -2.7e-173], N[(x + y), $MachinePrecision], If[LessEqual[y, 1.12e+82], t$95$0, t$95$1]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.3999999999999999e58 or 1.11999999999999998e82 < y

   1. Initial program 72.8%

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

   3. Taylor expanded in z around 0 67.5%

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

      1. mul-1-neg 67.5%

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

      2. associate-/l* 83.5%

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

      3. distribute-neg-frac 83.5%

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

      4. +-commutative 83.5%

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

   5. Simplified 83.5%

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

   if -1.3999999999999999e58 < y < -580 or -1.60000000000000004e-45 < y < -2.7e-173

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf 78.5%

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

      1. +-commutative 78.5%

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

   5. Simplified 78.5%

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

   if -580 < y < -1.60000000000000004e-45 or -2.7e-173 < y < 1.11999999999999998e82

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 82.9%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.4 \cdot 10^{+58}:\\ \;\;\;\;\frac{-z}{\frac{y}{x + y}}\\ \mathbf{elif}\;y \leq -580:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq -1.6 \cdot 10^{-45}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{elif}\;y \leq -2.7 \cdot 10^{-173}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 1.12 \cdot 10^{+82}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-z}{\frac{y}{x + y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{1 - \frac{y}{z}}\\ t_1 := \frac{-z}{\frac{y}{x + y}}\\ \mathbf{if}\;y \leq -9 \cdot 10^{+59}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -2.1 \cdot 10^{+16}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq -2.95 \cdot 10^{-46}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -3.3 \cdot 10^{-172}:\\ \;\;\;\;x + \left(y + \frac{x \cdot y}{z}\right)\\ \mathbf{elif}\;y \leq 1.12 \cdot 10^{+82}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ x (- 1.0 (/ y z)))) (t_1 (/ (- z) (/ y (+ x y)))))
   (if (<= y -9e+59)
     t_1
     (if (<= y -2.1e+16)
       (+ x y)
       (if (<= y -2.95e-46)
         t_0
         (if (<= y -3.3e-172)
           (+ x (+ y (/ (* x y) z)))
           (if (<= y 1.12e+82) t_0 t_1)))))))

C

double code(double x, double y, double z) {
	double t_0 = x / (1.0 - (y / z));
	double t_1 = -z / (y / (x + y));
	double tmp;
	if (y <= -9e+59) {
		tmp = t_1;
	} else if (y <= -2.1e+16) {
		tmp = x + y;
	} else if (y <= -2.95e-46) {
		tmp = t_0;
	} else if (y <= -3.3e-172) {
		tmp = x + (y + ((x * y) / z));
	} else if (y <= 1.12e+82) {
		tmp = t_0;
	} 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 = x / (1.0d0 - (y / z))
    t_1 = -z / (y / (x + y))
    if (y <= (-9d+59)) then
        tmp = t_1
    else if (y <= (-2.1d+16)) then
        tmp = x + y
    else if (y <= (-2.95d-46)) then
        tmp = t_0
    else if (y <= (-3.3d-172)) then
        tmp = x + (y + ((x * y) / z))
    else if (y <= 1.12d+82) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x / (1.0 - (y / z));
	double t_1 = -z / (y / (x + y));
	double tmp;
	if (y <= -9e+59) {
		tmp = t_1;
	} else if (y <= -2.1e+16) {
		tmp = x + y;
	} else if (y <= -2.95e-46) {
		tmp = t_0;
	} else if (y <= -3.3e-172) {
		tmp = x + (y + ((x * y) / z));
	} else if (y <= 1.12e+82) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x / (1.0 - (y / z))
	t_1 = -z / (y / (x + y))
	tmp = 0
	if y <= -9e+59:
		tmp = t_1
	elif y <= -2.1e+16:
		tmp = x + y
	elif y <= -2.95e-46:
		tmp = t_0
	elif y <= -3.3e-172:
		tmp = x + (y + ((x * y) / z))
	elif y <= 1.12e+82:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x / Float64(1.0 - Float64(y / z)))
	t_1 = Float64(Float64(-z) / Float64(y / Float64(x + y)))
	tmp = 0.0
	if (y <= -9e+59)
		tmp = t_1;
	elseif (y <= -2.1e+16)
		tmp = Float64(x + y);
	elseif (y <= -2.95e-46)
		tmp = t_0;
	elseif (y <= -3.3e-172)
		tmp = Float64(x + Float64(y + Float64(Float64(x * y) / z)));
	elseif (y <= 1.12e+82)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x / (1.0 - (y / z));
	t_1 = -z / (y / (x + y));
	tmp = 0.0;
	if (y <= -9e+59)
		tmp = t_1;
	elseif (y <= -2.1e+16)
		tmp = x + y;
	elseif (y <= -2.95e-46)
		tmp = t_0;
	elseif (y <= -3.3e-172)
		tmp = x + (y + ((x * y) / z));
	elseif (y <= 1.12e+82)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[((-z) / N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -9e+59], t$95$1, If[LessEqual[y, -2.1e+16], N[(x + y), $MachinePrecision], If[LessEqual[y, -2.95e-46], t$95$0, If[LessEqual[y, -3.3e-172], N[(x + N[(y + N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.12e+82], t$95$0, t$95$1]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -8.99999999999999919e59 or 1.11999999999999998e82 < y

   1. Initial program 72.8%

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

   3. Taylor expanded in z around 0 67.5%

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

      1. mul-1-neg 67.5%

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

      2. associate-/l* 83.5%

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

      3. distribute-neg-frac 83.5%

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

      4. +-commutative 83.5%

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

   5. Simplified 83.5%

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

   if -8.99999999999999919e59 < y < -2.1e16

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf 71.8%

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

      1. +-commutative 71.8%

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

   5. Simplified 71.8%

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

   if -2.1e16 < y < -2.95e-46 or -3.3e-172 < y < 1.11999999999999998e82

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 82.9%

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

   if -2.95e-46 < y < -3.3e-172

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 83.0%

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

      1. mul-1-neg 83.0%

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

   5. Simplified 83.0%

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

   6. Taylor expanded in x around 0 83.1%

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

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9 \cdot 10^{+59}:\\ \;\;\;\;\frac{-z}{\frac{y}{x + y}}\\ \mathbf{elif}\;y \leq -2.1 \cdot 10^{+16}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq -2.95 \cdot 10^{-46}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{elif}\;y \leq -3.3 \cdot 10^{-172}:\\ \;\;\;\;x + \left(y + \frac{x \cdot y}{z}\right)\\ \mathbf{elif}\;y \leq 1.12 \cdot 10^{+82}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-z}{\frac{y}{x + y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 68.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{1 - \frac{y}{z}}\\ \mathbf{if}\;y \leq -2.8 \cdot 10^{+85}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq -3 \cdot 10^{-41}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -1.2 \cdot 10^{-170}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{+84}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ x (- 1.0 (/ y z)))))
   (if (<= y -2.8e+85)
     (- z)
     (if (<= y -3e-41)
       t_0
       (if (<= y -1.2e-170) (+ x y) (if (<= y 8.5e+84) t_0 (- z)))))))

C

double code(double x, double y, double z) {
	double t_0 = x / (1.0 - (y / z));
	double tmp;
	if (y <= -2.8e+85) {
		tmp = -z;
	} else if (y <= -3e-41) {
		tmp = t_0;
	} else if (y <= -1.2e-170) {
		tmp = x + y;
	} else if (y <= 8.5e+84) {
		tmp = t_0;
	} else {
		tmp = -z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / (1.0d0 - (y / z))
    if (y <= (-2.8d+85)) then
        tmp = -z
    else if (y <= (-3d-41)) then
        tmp = t_0
    else if (y <= (-1.2d-170)) then
        tmp = x + y
    else if (y <= 8.5d+84) then
        tmp = t_0
    else
        tmp = -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x / (1.0 - (y / z));
	double tmp;
	if (y <= -2.8e+85) {
		tmp = -z;
	} else if (y <= -3e-41) {
		tmp = t_0;
	} else if (y <= -1.2e-170) {
		tmp = x + y;
	} else if (y <= 8.5e+84) {
		tmp = t_0;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x / (1.0 - (y / z))
	tmp = 0
	if y <= -2.8e+85:
		tmp = -z
	elif y <= -3e-41:
		tmp = t_0
	elif y <= -1.2e-170:
		tmp = x + y
	elif y <= 8.5e+84:
		tmp = t_0
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x / Float64(1.0 - Float64(y / z)))
	tmp = 0.0
	if (y <= -2.8e+85)
		tmp = Float64(-z);
	elseif (y <= -3e-41)
		tmp = t_0;
	elseif (y <= -1.2e-170)
		tmp = Float64(x + y);
	elseif (y <= 8.5e+84)
		tmp = t_0;
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x / (1.0 - (y / z));
	tmp = 0.0;
	if (y <= -2.8e+85)
		tmp = -z;
	elseif (y <= -3e-41)
		tmp = t_0;
	elseif (y <= -1.2e-170)
		tmp = x + y;
	elseif (y <= 8.5e+84)
		tmp = t_0;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.8e+85], (-z), If[LessEqual[y, -3e-41], t$95$0, If[LessEqual[y, -1.2e-170], N[(x + y), $MachinePrecision], If[LessEqual[y, 8.5e+84], t$95$0, (-z)]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.7999999999999999e85 or 8.5000000000000008e84 < y

   1. Initial program 71.2%

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

   3. Taylor expanded in y around inf 69.7%

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

      1. mul-1-neg 69.7%

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

   5. Simplified 69.7%

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

   if -2.7999999999999999e85 < y < -2.99999999999999989e-41 or -1.2e-170 < y < 8.5000000000000008e84

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 79.9%

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

   if -2.99999999999999989e-41 < y < -1.2e-170

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf 79.3%

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

      1. +-commutative 79.3%

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

   5. Simplified 79.3%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.8 \cdot 10^{+85}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq -3 \cdot 10^{-41}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{elif}\;y \leq -1.2 \cdot 10^{-170}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{+84}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.25 \cdot 10^{+68} \lor \neg \left(y \leq 1.26 \cdot 10^{+82}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1.25e+68) (not (<= y 1.26e+82))) (- z) (+ x y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.25e+68) || !(y <= 1.26e+82)) {
		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 <= (-1.25d+68)) .or. (.not. (y <= 1.26d+82))) 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 <= -1.25e+68) || !(y <= 1.26e+82)) {
		tmp = -z;
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -1.25e+68) or not (y <= 1.26e+82):
		tmp = -z
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -1.25e+68) || !(y <= 1.26e+82))
		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 <= -1.25e+68) || ~((y <= 1.26e+82)))
		tmp = -z;
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -1.25e+68], N[Not[LessEqual[y, 1.26e+82]], $MachinePrecision]], (-z), N[(x + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.2500000000000001e68 or 1.2600000000000001e82 < y

   1. Initial program 72.0%

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

   3. Taylor expanded in y around inf 68.7%

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

      1. mul-1-neg 68.7%

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

   5. Simplified 68.7%

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

   if -1.2500000000000001e68 < y < 1.2600000000000001e82

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf 75.0%

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

      1. +-commutative 75.0%

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

   5. Simplified 75.0%

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

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.25 \cdot 10^{+68} \lor \neg \left(y \leq 1.26 \cdot 10^{+82}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 56.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.95 \cdot 10^{+59} \lor \neg \left(y \leq 1.12 \cdot 10^{+82}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1.95e+59) (not (<= y 1.12e+82))) (- z) x))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.95e+59) || !(y <= 1.12e+82)) {
		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 <= (-1.95d+59)) .or. (.not. (y <= 1.12d+82))) 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 <= -1.95e+59) || !(y <= 1.12e+82)) {
		tmp = -z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -1.95e+59) or not (y <= 1.12e+82):
		tmp = -z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -1.95e+59) || !(y <= 1.12e+82))
		tmp = Float64(-z);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -1.95e+59) || ~((y <= 1.12e+82)))
		tmp = -z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -1.95e+59], N[Not[LessEqual[y, 1.12e+82]], $MachinePrecision]], (-z), x]

Derivation

1. Split input into 2 regimes

2. if y < -1.95000000000000011e59 or 1.11999999999999998e82 < y

   1. Initial program 72.8%

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

   3. Taylor expanded in y around inf 67.7%

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

      1. mul-1-neg 67.7%

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

   5. Simplified 67.7%

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

   if -1.95000000000000011e59 < y < 1.11999999999999998e82

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 59.7%

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

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.95 \cdot 10^{+59} \lor \neg \left(y \leq 1.12 \cdot 10^{+82}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 39.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{-196}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{-161}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.45e-196) x (if (<= x 1.5e-161) y x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.45e-196) {
		tmp = x;
	} else if (x <= 1.5e-161) {
		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 <= (-1.45d-196)) then
        tmp = x
    else if (x <= 1.5d-161) 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 <= -1.45e-196) {
		tmp = x;
	} else if (x <= 1.5e-161) {
		tmp = y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.45e-196:
		tmp = x
	elif x <= 1.5e-161:
		tmp = y
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.45e-196)
		tmp = x;
	elseif (x <= 1.5e-161)
		tmp = y;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.45e-196)
		tmp = x;
	elseif (x <= 1.5e-161)
		tmp = y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.45e-196], x, If[LessEqual[x, 1.5e-161], y, x]]

Derivation

1. Split input into 2 regimes

2. if x < -1.44999999999999994e-196 or 1.49999999999999994e-161 < x

   1. Initial program 90.3%

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

   3. Taylor expanded in y around 0 45.2%

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

   if -1.44999999999999994e-196 < x < 1.49999999999999994e-161

   1. Initial program 84.7%

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

   3. Taylor expanded in y around 0 45.9%

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

      1. mul-1-neg 45.9%

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

   5. Simplified 45.9%

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

   6. Taylor expanded in x around 0 46.0%

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

   7. Taylor expanded in x around 0 36.5%

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

4. Final simplification 43.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{-196}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{-161}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 33.5% accurate, 9.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := x

Derivation

1. Initial program 89.2%

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

3. Taylor expanded in y around 0 39.2%

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

4. Final simplification 39.2%

   \[\leadsto x \]
5. Add Preprocessing

Developer target: 93.6% 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 2024024 
(FPCore (x y z)
  :name "Graphics.Rendering.Chart.Backend.Diagrams:calcFontMetrics from Chart-diagrams-1.5.1, A"
  :precision binary64

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

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