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

Percentage Accurate: 88.1% → 99.2%

Time: 7.8s

Alternatives: 14

Speedup: 0.3×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 88.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{if}\;t_0 \leq -4 \cdot 10^{-230} \lor \neg \left(t_0 \leq 0\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-\frac{z}{y}\right) - \left(z + \frac{z}{\frac{y}{x}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (+ x y) (- 1.0 (/ y z)))))
   (if (or (<= t_0 -4e-230) (not (<= t_0 0.0)))
     t_0
     (- (* z (- (/ z y))) (+ z (/ z (/ y x)))))))

C

double code(double x, double y, double z) {
	double t_0 = (x + y) / (1.0 - (y / z));
	double tmp;
	if ((t_0 <= -4e-230) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = (z * -(z / y)) - (z + (z / (y / x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x + y) / (1.0d0 - (y / z))
    if ((t_0 <= (-4d-230)) .or. (.not. (t_0 <= 0.0d0))) then
        tmp = t_0
    else
        tmp = (z * -(z / y)) - (z + (z / (y / x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (x + y) / (1.0 - (y / z));
	double tmp;
	if ((t_0 <= -4e-230) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = (z * -(z / y)) - (z + (z / (y / x)));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x + y) / (1.0 - (y / z))
	tmp = 0
	if (t_0 <= -4e-230) or not (t_0 <= 0.0):
		tmp = t_0
	else:
		tmp = (z * -(z / y)) - (z + (z / (y / x)))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x + y) / Float64(1.0 - Float64(y / z)))
	tmp = 0.0
	if ((t_0 <= -4e-230) || !(t_0 <= 0.0))
		tmp = t_0;
	else
		tmp = Float64(Float64(z * Float64(-Float64(z / y))) - Float64(z + Float64(z / Float64(y / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x + y) / (1.0 - (y / z));
	tmp = 0.0;
	if ((t_0 <= -4e-230) || ~((t_0 <= 0.0)))
		tmp = t_0;
	else
		tmp = (z * -(z / y)) - (z + (z / (y / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x + y), $MachinePrecision] / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -4e-230], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], t$95$0, N[(N[(z * (-N[(z / y), $MachinePrecision])), $MachinePrecision] - N[(z + N[(z / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z))) < -4.00000000000000019e-230 or 0.0 < (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z)))

   1. Initial program 99.9%

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

   if -4.00000000000000019e-230 < (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z))) < 0.0

   1. Initial program 20.8%

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

      1. clear-num 20.8%

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

      2. associate-/r/ 20.8%

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

   3. Applied egg-rr 20.8%

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

   4. Taylor expanded in y around inf 99.9%

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

      1. mul-1-neg 99.9%

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

      2. unsub-neg 99.9%

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

      3. neg-mul-1 99.9%

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

      4. associate-/l* 100.0%

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

      5. unpow2 100.0%

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

      6. associate-*r/ 100.0%

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

   6. Simplified 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \leq -4 \cdot 10^{-230} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \leq 0\right):\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-\frac{z}{y}\right) - \left(z + \frac{z}{\frac{y}{x}}\right)\\ \end{array} \]

Alternative 2: 99.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{if}\;t_0 \leq -4 \cdot 10^{-230} \lor \neg \left(t_0 \leq 0\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot \left(\left(-x\right) - z\right)}{y} - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (+ x y) (- 1.0 (/ y z)))))
   (if (or (<= t_0 -4e-230) (not (<= t_0 0.0)))
     t_0
     (- (/ (* z (- (- x) z)) y) z))))

C

double code(double x, double y, double z) {
	double t_0 = (x + y) / (1.0 - (y / z));
	double tmp;
	if ((t_0 <= -4e-230) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = ((z * (-x - z)) / y) - z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x + y) / (1.0d0 - (y / z))
    if ((t_0 <= (-4d-230)) .or. (.not. (t_0 <= 0.0d0))) then
        tmp = t_0
    else
        tmp = ((z * (-x - z)) / y) - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (x + y) / (1.0 - (y / z));
	double tmp;
	if ((t_0 <= -4e-230) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = ((z * (-x - z)) / y) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x + y) / (1.0 - (y / z))
	tmp = 0
	if (t_0 <= -4e-230) or not (t_0 <= 0.0):
		tmp = t_0
	else:
		tmp = ((z * (-x - z)) / y) - z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x + y) / Float64(1.0 - Float64(y / z)))
	tmp = 0.0
	if ((t_0 <= -4e-230) || !(t_0 <= 0.0))
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(z * Float64(Float64(-x) - z)) / y) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x + y) / (1.0 - (y / z));
	tmp = 0.0;
	if ((t_0 <= -4e-230) || ~((t_0 <= 0.0)))
		tmp = t_0;
	else
		tmp = ((z * (-x - z)) / y) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x + y), $MachinePrecision] / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -4e-230], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], t$95$0, N[(N[(N[(z * N[((-x) - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] - z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z))) < -4.00000000000000019e-230 or 0.0 < (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z)))

   1. Initial program 99.9%

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

   if -4.00000000000000019e-230 < (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z))) < 0.0

   1. Initial program 20.8%

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

   2. Taylor expanded in y around inf 99.9%

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

      1. sub-neg 99.9%

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

      2. mul-1-neg 99.9%

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

      3. unsub-neg 99.9%

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

      4. associate-+l- 99.9%

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

      5. mul-1-neg 99.9%

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

      6. distribute-frac-neg 99.9%

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

      7. mul-1-neg 99.9%

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

      8. div-sub 99.9%

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

      9. sub-neg 99.9%

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

      10. mul-1-neg 99.9%

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

      11. remove-double-neg 99.9%

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

      12. unpow2 99.9%

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

      13. distribute-lft-out 99.9%

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

   4. Simplified 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \leq -4 \cdot 10^{-230} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \leq 0\right):\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot \left(\left(-x\right) - z\right)}{y} - z\\ \end{array} \]

Alternative 3: 99.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{-252} \lor \neg \left(t_0 \leq 0\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot \left(\left(-y\right) - x\right)}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (+ x y) (- 1.0 (/ y z)))))
   (if (or (<= t_0 -2e-252) (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 <= -2e-252) || !(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 <= (-2d-252)) .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 <= -2e-252) || !(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 <= -2e-252) 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 <= -2e-252) || !(t_0 <= 0.0))
		tmp = t_0;
	else
		tmp = Float64(Float64(z * Float64(Float64(-y) - x)) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x + y) / (1.0 - (y / z));
	tmp = 0.0;
	if ((t_0 <= -2e-252) || ~((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, -2e-252], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], t$95$0, N[(N[(z * N[((-y) - x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z))) < -1.99999999999999989e-252 or 0.0 < (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z)))

   1. Initial program 99.8%

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

   if -1.99999999999999989e-252 < (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z))) < 0.0

   1. Initial program 8.8%

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

   2. Taylor expanded in z around 0 99.8%

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

      1. associate-*r/ 99.8%

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

      2. +-commutative 99.8%

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

      3. *-commutative 99.8%

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

      4. associate-*r* 99.8%

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

      5. mul-1-neg 99.8%

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

      6. +-commutative 99.8%

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

   4. Simplified 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \leq -2 \cdot 10^{-252} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \leq 0\right):\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot \left(\left(-y\right) - x\right)}{y}\\ \end{array} \]

Alternative 4: 68.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x + y\right) \cdot \left(-\frac{z}{y}\right)\\ \mathbf{if}\;y \leq -4.4 \cdot 10^{+160}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq -1.85 \cdot 10^{-61}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 96000000:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{+92}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{+109}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{+143}:\\ \;\;\;\;\left(x \cdot z\right) \cdot \frac{1}{-y}\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (+ x y) (- (/ z y)))))
   (if (<= y -4.4e+160)
     (- z)
     (if (<= y -1.85e-61)
       t_0
       (if (<= y 96000000.0)
         (+ x y)
         (if (<= y 1.2e+92)
           t_0
           (if (<= y 5.2e+109)
             (+ x y)
             (if (<= y 1.5e+143) (* (* x z) (/ 1.0 (- y))) (- z)))))))))

C

double code(double x, double y, double z) {
	double t_0 = (x + y) * -(z / y);
	double tmp;
	if (y <= -4.4e+160) {
		tmp = -z;
	} else if (y <= -1.85e-61) {
		tmp = t_0;
	} else if (y <= 96000000.0) {
		tmp = x + y;
	} else if (y <= 1.2e+92) {
		tmp = t_0;
	} else if (y <= 5.2e+109) {
		tmp = x + y;
	} else if (y <= 1.5e+143) {
		tmp = (x * z) * (1.0 / -y);
	} else {
		tmp = -z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x + y) * -(z / y)
    if (y <= (-4.4d+160)) then
        tmp = -z
    else if (y <= (-1.85d-61)) then
        tmp = t_0
    else if (y <= 96000000.0d0) then
        tmp = x + y
    else if (y <= 1.2d+92) then
        tmp = t_0
    else if (y <= 5.2d+109) then
        tmp = x + y
    else if (y <= 1.5d+143) then
        tmp = (x * z) * (1.0d0 / -y)
    else
        tmp = -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (x + y) * -(z / y);
	double tmp;
	if (y <= -4.4e+160) {
		tmp = -z;
	} else if (y <= -1.85e-61) {
		tmp = t_0;
	} else if (y <= 96000000.0) {
		tmp = x + y;
	} else if (y <= 1.2e+92) {
		tmp = t_0;
	} else if (y <= 5.2e+109) {
		tmp = x + y;
	} else if (y <= 1.5e+143) {
		tmp = (x * z) * (1.0 / -y);
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x + y) * -(z / y)
	tmp = 0
	if y <= -4.4e+160:
		tmp = -z
	elif y <= -1.85e-61:
		tmp = t_0
	elif y <= 96000000.0:
		tmp = x + y
	elif y <= 1.2e+92:
		tmp = t_0
	elif y <= 5.2e+109:
		tmp = x + y
	elif y <= 1.5e+143:
		tmp = (x * z) * (1.0 / -y)
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x + y) * Float64(-Float64(z / y)))
	tmp = 0.0
	if (y <= -4.4e+160)
		tmp = Float64(-z);
	elseif (y <= -1.85e-61)
		tmp = t_0;
	elseif (y <= 96000000.0)
		tmp = Float64(x + y);
	elseif (y <= 1.2e+92)
		tmp = t_0;
	elseif (y <= 5.2e+109)
		tmp = Float64(x + y);
	elseif (y <= 1.5e+143)
		tmp = Float64(Float64(x * z) * Float64(1.0 / Float64(-y)));
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x + y) * -(z / y);
	tmp = 0.0;
	if (y <= -4.4e+160)
		tmp = -z;
	elseif (y <= -1.85e-61)
		tmp = t_0;
	elseif (y <= 96000000.0)
		tmp = x + y;
	elseif (y <= 1.2e+92)
		tmp = t_0;
	elseif (y <= 5.2e+109)
		tmp = x + y;
	elseif (y <= 1.5e+143)
		tmp = (x * z) * (1.0 / -y);
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x + y), $MachinePrecision] * (-N[(z / y), $MachinePrecision])), $MachinePrecision]}, If[LessEqual[y, -4.4e+160], (-z), If[LessEqual[y, -1.85e-61], t$95$0, If[LessEqual[y, 96000000.0], N[(x + y), $MachinePrecision], If[LessEqual[y, 1.2e+92], t$95$0, If[LessEqual[y, 5.2e+109], N[(x + y), $MachinePrecision], If[LessEqual[y, 1.5e+143], N[(N[(x * z), $MachinePrecision] * N[(1.0 / (-y)), $MachinePrecision]), $MachinePrecision], (-z)]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -4.39999999999999984e160 or 1.5e143 < y

   1. Initial program 68.6%

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

   2. Taylor expanded in y around inf 78.6%

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

      1. mul-1-neg 78.6%

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

   4. Simplified 78.6%

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

   if -4.39999999999999984e160 < y < -1.85e-61 or 9.6e7 < y < 1.20000000000000002e92

   1. Initial program 94.3%

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

      1. clear-num 94.1%

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

      2. associate-/r/ 94.1%

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

   3. Applied egg-rr 94.1%

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

   4. Taylor expanded in y around inf 66.1%

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

      1. associate-*r/ 66.1%

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

      2. neg-mul-1 66.1%

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

   6. Simplified 66.1%

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

   if -1.85e-61 < y < 9.6e7 or 1.20000000000000002e92 < y < 5.1999999999999997e109

   1. Initial program 100.0%

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

   2. Taylor expanded in z around inf 85.0%

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

   if 5.1999999999999997e109 < y < 1.5e143

   1. Initial program 42.6%

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

   2. Taylor expanded in x around inf 23.0%

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

   3. Taylor expanded in y around inf 70.7%

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

      1. associate-*r/ 70.7%

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

      2. associate-*r* 70.7%

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

      3. mul-1-neg 70.7%

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

   5. Simplified 70.7%

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

      1. frac-2neg 70.7%

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

      2. div-inv 71.0%

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

      3. distribute-lft-neg-out 71.0%

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

      4. remove-double-neg 71.0%

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

      5. *-commutative 71.0%

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

   7. Applied egg-rr 71.0%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.4 \cdot 10^{+160}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq -1.85 \cdot 10^{-61}:\\ \;\;\;\;\left(x + y\right) \cdot \left(-\frac{z}{y}\right)\\ \mathbf{elif}\;y \leq 96000000:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{+92}:\\ \;\;\;\;\left(x + y\right) \cdot \left(-\frac{z}{y}\right)\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{+109}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{+143}:\\ \;\;\;\;\left(x \cdot z\right) \cdot \frac{1}{-y}\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 5: 67.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{1 - \frac{y}{z}}\\ \mathbf{if}\;y \leq -3.1 \cdot 10^{+33}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq -4.5 \cdot 10^{-77}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 16000000:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{+41}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{+125}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{+143}:\\ \;\;\;\;\frac{-z}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ x (- 1.0 (/ y z)))))
   (if (<= y -3.1e+33)
     (- z)
     (if (<= y -4.5e-77)
       t_0
       (if (<= y 16000000.0)
         (+ x y)
         (if (<= y 7.2e+41)
           t_0
           (if (<= y 2.5e+125)
             (+ x y)
             (if (<= y 1.5e+143) (/ (- z) (/ y x)) (- z)))))))))

C

double code(double x, double y, double z) {
	double t_0 = x / (1.0 - (y / z));
	double tmp;
	if (y <= -3.1e+33) {
		tmp = -z;
	} else if (y <= -4.5e-77) {
		tmp = t_0;
	} else if (y <= 16000000.0) {
		tmp = x + y;
	} else if (y <= 7.2e+41) {
		tmp = t_0;
	} else if (y <= 2.5e+125) {
		tmp = x + y;
	} else if (y <= 1.5e+143) {
		tmp = -z / (y / x);
	} else {
		tmp = -z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / (1.0d0 - (y / z))
    if (y <= (-3.1d+33)) then
        tmp = -z
    else if (y <= (-4.5d-77)) then
        tmp = t_0
    else if (y <= 16000000.0d0) then
        tmp = x + y
    else if (y <= 7.2d+41) then
        tmp = t_0
    else if (y <= 2.5d+125) then
        tmp = x + y
    else if (y <= 1.5d+143) then
        tmp = -z / (y / x)
    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 <= -3.1e+33) {
		tmp = -z;
	} else if (y <= -4.5e-77) {
		tmp = t_0;
	} else if (y <= 16000000.0) {
		tmp = x + y;
	} else if (y <= 7.2e+41) {
		tmp = t_0;
	} else if (y <= 2.5e+125) {
		tmp = x + y;
	} else if (y <= 1.5e+143) {
		tmp = -z / (y / x);
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x / (1.0 - (y / z))
	tmp = 0
	if y <= -3.1e+33:
		tmp = -z
	elif y <= -4.5e-77:
		tmp = t_0
	elif y <= 16000000.0:
		tmp = x + y
	elif y <= 7.2e+41:
		tmp = t_0
	elif y <= 2.5e+125:
		tmp = x + y
	elif y <= 1.5e+143:
		tmp = -z / (y / x)
	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 <= -3.1e+33)
		tmp = Float64(-z);
	elseif (y <= -4.5e-77)
		tmp = t_0;
	elseif (y <= 16000000.0)
		tmp = Float64(x + y);
	elseif (y <= 7.2e+41)
		tmp = t_0;
	elseif (y <= 2.5e+125)
		tmp = Float64(x + y);
	elseif (y <= 1.5e+143)
		tmp = Float64(Float64(-z) / Float64(y / x));
	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 <= -3.1e+33)
		tmp = -z;
	elseif (y <= -4.5e-77)
		tmp = t_0;
	elseif (y <= 16000000.0)
		tmp = x + y;
	elseif (y <= 7.2e+41)
		tmp = t_0;
	elseif (y <= 2.5e+125)
		tmp = x + y;
	elseif (y <= 1.5e+143)
		tmp = -z / (y / x);
	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, -3.1e+33], (-z), If[LessEqual[y, -4.5e-77], t$95$0, If[LessEqual[y, 16000000.0], N[(x + y), $MachinePrecision], If[LessEqual[y, 7.2e+41], t$95$0, If[LessEqual[y, 2.5e+125], N[(x + y), $MachinePrecision], If[LessEqual[y, 1.5e+143], N[((-z) / N[(y / x), $MachinePrecision]), $MachinePrecision], (-z)]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -3.1e33 or 1.5e143 < y

   1. Initial program 75.2%

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

   2. Taylor expanded in y around inf 67.2%

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

      1. mul-1-neg 67.2%

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

   4. Simplified 67.2%

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

   if -3.1e33 < y < -4.5000000000000001e-77 or 1.6e7 < y < 7.20000000000000051e41

   1. Initial program 97.4%

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

   2. Taylor expanded in x around inf 58.1%

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

   if -4.5000000000000001e-77 < y < 1.6e7 or 7.20000000000000051e41 < y < 2.49999999999999981e125

   1. Initial program 99.9%

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

   2. Taylor expanded in z around inf 82.6%

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

   if 2.49999999999999981e125 < y < 1.5e143

   1. Initial program 18.0%

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

   2. Taylor expanded in x around inf 18.0%

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

   3. Taylor expanded in y around inf 86.1%

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

      1. mul-1-neg 86.1%

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

      2. associate-/l* 86.3%

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

      3. distribute-neg-frac 86.3%

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

   5. Simplified 86.3%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.1 \cdot 10^{+33}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq -4.5 \cdot 10^{-77}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{elif}\;y \leq 16000000:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{+41}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{+125}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{+143}:\\ \;\;\;\;\frac{-z}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 6: 73.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.25 \cdot 10^{-61} \lor \neg \left(y \leq 1.8 \cdot 10^{-35}\right):\\ \;\;\;\;\left(-z\right) - x \cdot \frac{z}{y}\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) \cdot \left(1 + \frac{y}{z}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -2.25e-61) (not (<= y 1.8e-35)))
   (- (- z) (* x (/ z y)))
   (* (+ x y) (+ 1.0 (/ y z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -2.25e-61) || !(y <= 1.8e-35)) {
		tmp = -z - (x * (z / y));
	} else {
		tmp = (x + y) * (1.0 + (y / z));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((y <= (-2.25d-61)) .or. (.not. (y <= 1.8d-35))) then
        tmp = -z - (x * (z / y))
    else
        tmp = (x + y) * (1.0d0 + (y / z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -2.25e-61) || !(y <= 1.8e-35)) {
		tmp = -z - (x * (z / y));
	} else {
		tmp = (x + y) * (1.0 + (y / z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -2.25e-61) or not (y <= 1.8e-35):
		tmp = -z - (x * (z / y))
	else:
		tmp = (x + y) * (1.0 + (y / z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -2.25e-61) || !(y <= 1.8e-35))
		tmp = Float64(Float64(-z) - Float64(x * Float64(z / y)));
	else
		tmp = Float64(Float64(x + y) * Float64(1.0 + Float64(y / z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -2.25e-61) || ~((y <= 1.8e-35)))
		tmp = -z - (x * (z / y));
	else
		tmp = (x + y) * (1.0 + (y / z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -2.25e-61], N[Not[LessEqual[y, 1.8e-35]], $MachinePrecision]], N[((-z) - N[(x * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x + y), $MachinePrecision] * N[(1.0 + N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.25e-61 or 1.80000000000000009e-35 < y

   1. Initial program 80.1%

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

   2. Taylor expanded in z around 0 63.9%

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

      1. associate-*r/ 63.9%

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

      2. +-commutative 63.9%

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

      3. *-commutative 63.9%

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

      4. associate-*r* 63.9%

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

      5. mul-1-neg 63.9%

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

      6. +-commutative 63.9%

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

   4. Simplified 63.9%

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

   5. Taylor expanded in y around 0 73.1%

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

      1. mul-1-neg 73.1%

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

      2. unsub-neg 73.1%

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

      3. mul-1-neg 73.1%

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

      4. *-commutative 73.1%

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

      5. associate-*r/ 73.3%

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

   7. Simplified 73.3%

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

   if -2.25e-61 < y < 1.80000000000000009e-35

   1. Initial program 100.0%

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

   2. Taylor expanded in z around inf 86.7%

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

      1. associate-/l* 80.2%

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

      2. +-commutative 80.2%

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

      3. associate-/r/ 86.6%

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

      4. +-commutative 86.6%

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

      5. *-lft-identity 86.6%

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

      6. distribute-rgt-in 86.6%

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

      7. +-commutative 86.6%

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

      8. +-commutative 86.6%

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

   4. Simplified 86.6%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.25 \cdot 10^{-61} \lor \neg \left(y \leq 1.8 \cdot 10^{-35}\right):\\ \;\;\;\;\left(-z\right) - x \cdot \frac{z}{y}\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) \cdot \left(1 + \frac{y}{z}\right)\\ \end{array} \]

Alternative 7: 73.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.3 \cdot 10^{-61} \lor \neg \left(y \leq 2200000\right):\\ \;\;\;\;\left(-z\right) - x \cdot \frac{z}{y}\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -2.3e-61) (not (<= y 2200000.0)))
   (- (- z) (* x (/ z y)))
   (+ x y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -2.3e-61) || !(y <= 2200000.0)) {
		tmp = -z - (x * (z / 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 <= (-2.3d-61)) .or. (.not. (y <= 2200000.0d0))) then
        tmp = -z - (x * (z / 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 <= -2.3e-61) || !(y <= 2200000.0)) {
		tmp = -z - (x * (z / y));
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -2.3e-61) or not (y <= 2200000.0):
		tmp = -z - (x * (z / y))
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -2.3e-61) || !(y <= 2200000.0))
		tmp = Float64(Float64(-z) - Float64(x * Float64(z / y)));
	else
		tmp = Float64(x + y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -2.3e-61) || ~((y <= 2200000.0)))
		tmp = -z - (x * (z / y));
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -2.3e-61], N[Not[LessEqual[y, 2200000.0]], $MachinePrecision]], N[((-z) - N[(x * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.29999999999999992e-61 or 2.2e6 < y

   1. Initial program 79.3%

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

   2. Taylor expanded in z around 0 64.4%

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

      1. associate-*r/ 64.4%

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

      2. +-commutative 64.4%

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

      3. *-commutative 64.4%

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

      4. associate-*r* 64.4%

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

      5. mul-1-neg 64.4%

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

      6. +-commutative 64.4%

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

   4. Simplified 64.4%

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

   5. Taylor expanded in y around 0 74.0%

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

      1. mul-1-neg 74.0%

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

      2. unsub-neg 74.0%

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

      3. mul-1-neg 74.0%

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

      4. *-commutative 74.0%

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

      5. associate-*r/ 74.2%

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

   7. Simplified 74.2%

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

   if -2.29999999999999992e-61 < y < 2.2e6

   1. Initial program 100.0%

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

   2. Taylor expanded in z around inf 84.5%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.3 \cdot 10^{-61} \lor \neg \left(y \leq 2200000\right):\\ \;\;\;\;\left(-z\right) - x \cdot \frac{z}{y}\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 8: 66.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.8 \cdot 10^{+32}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq -9 \cdot 10^{-37}:\\ \;\;\;\;x \cdot \left(-\frac{z}{y}\right)\\ \mathbf{elif}\;y \leq -2.7 \cdot 10^{-61}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 95000000:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.8e+32)
   (- z)
   (if (<= y -9e-37)
     (* x (- (/ z y)))
     (if (<= y -2.7e-61) (- z) (if (<= y 95000000.0) (+ x y) (- z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.8e+32) {
		tmp = -z;
	} else if (y <= -9e-37) {
		tmp = x * -(z / y);
	} else if (y <= -2.7e-61) {
		tmp = -z;
	} else if (y <= 95000000.0) {
		tmp = x + y;
	} else {
		tmp = -z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-1.8d+32)) then
        tmp = -z
    else if (y <= (-9d-37)) then
        tmp = x * -(z / y)
    else if (y <= (-2.7d-61)) then
        tmp = -z
    else if (y <= 95000000.0d0) then
        tmp = x + y
    else
        tmp = -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.8e+32) {
		tmp = -z;
	} else if (y <= -9e-37) {
		tmp = x * -(z / y);
	} else if (y <= -2.7e-61) {
		tmp = -z;
	} else if (y <= 95000000.0) {
		tmp = x + y;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1.8e+32:
		tmp = -z
	elif y <= -9e-37:
		tmp = x * -(z / y)
	elif y <= -2.7e-61:
		tmp = -z
	elif y <= 95000000.0:
		tmp = x + y
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.8e+32)
		tmp = Float64(-z);
	elseif (y <= -9e-37)
		tmp = Float64(x * Float64(-Float64(z / y)));
	elseif (y <= -2.7e-61)
		tmp = Float64(-z);
	elseif (y <= 95000000.0)
		tmp = Float64(x + y);
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -1.8e+32)
		tmp = -z;
	elseif (y <= -9e-37)
		tmp = x * -(z / y);
	elseif (y <= -2.7e-61)
		tmp = -z;
	elseif (y <= 95000000.0)
		tmp = x + y;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.8e+32], (-z), If[LessEqual[y, -9e-37], N[(x * (-N[(z / y), $MachinePrecision])), $MachinePrecision], If[LessEqual[y, -2.7e-61], (-z), If[LessEqual[y, 95000000.0], N[(x + y), $MachinePrecision], (-z)]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.7999999999999998e32 or -9.00000000000000081e-37 < y < -2.69999999999999993e-61 or 9.5e7 < y

   1. Initial program 76.3%

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

   2. Taylor expanded in y around inf 59.8%

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

      1. mul-1-neg 59.8%

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

   4. Simplified 59.8%

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

   if -1.7999999999999998e32 < y < -9.00000000000000081e-37

   1. Initial program 99.7%

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

   2. Taylor expanded in x around inf 59.7%

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

   3. Taylor expanded in y around inf 54.0%

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

      1. associate-*r/ 54.0%

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

      2. associate-*r* 54.0%

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

      3. mul-1-neg 54.0%

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

   5. Simplified 54.0%

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

   6. Taylor expanded in z around 0 54.0%

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

      1. mul-1-neg 54.0%

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

      2. associate-*l/ 54.1%

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

      3. *-commutative 54.1%

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

      4. distribute-rgt-neg-in 54.1%

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

      5. distribute-frac-neg 54.1%

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

   8. Simplified 54.1%

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

   if -2.69999999999999993e-61 < y < 9.5e7

   1. Initial program 100.0%

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

   2. Taylor expanded in z around inf 84.5%

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

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.8 \cdot 10^{+32}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq -9 \cdot 10^{-37}:\\ \;\;\;\;x \cdot \left(-\frac{z}{y}\right)\\ \mathbf{elif}\;y \leq -2.7 \cdot 10^{-61}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 95000000:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 9: 66.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.8 \cdot 10^{+37}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq -1.6 \cdot 10^{-38}:\\ \;\;\;\;\frac{-z}{\frac{y}{x}}\\ \mathbf{elif}\;y \leq -2.7 \cdot 10^{-61}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 150000000:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -3.8e+37)
   (- z)
   (if (<= y -1.6e-38)
     (/ (- z) (/ y x))
     (if (<= y -2.7e-61) (- z) (if (<= y 150000000.0) (+ x y) (- z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -3.8e+37) {
		tmp = -z;
	} else if (y <= -1.6e-38) {
		tmp = -z / (y / x);
	} else if (y <= -2.7e-61) {
		tmp = -z;
	} else if (y <= 150000000.0) {
		tmp = x + y;
	} else {
		tmp = -z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-3.8d+37)) then
        tmp = -z
    else if (y <= (-1.6d-38)) then
        tmp = -z / (y / x)
    else if (y <= (-2.7d-61)) then
        tmp = -z
    else if (y <= 150000000.0d0) then
        tmp = x + y
    else
        tmp = -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -3.8e+37) {
		tmp = -z;
	} else if (y <= -1.6e-38) {
		tmp = -z / (y / x);
	} else if (y <= -2.7e-61) {
		tmp = -z;
	} else if (y <= 150000000.0) {
		tmp = x + y;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -3.8e+37:
		tmp = -z
	elif y <= -1.6e-38:
		tmp = -z / (y / x)
	elif y <= -2.7e-61:
		tmp = -z
	elif y <= 150000000.0:
		tmp = x + y
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -3.8e+37)
		tmp = Float64(-z);
	elseif (y <= -1.6e-38)
		tmp = Float64(Float64(-z) / Float64(y / x));
	elseif (y <= -2.7e-61)
		tmp = Float64(-z);
	elseif (y <= 150000000.0)
		tmp = Float64(x + y);
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -3.8e+37)
		tmp = -z;
	elseif (y <= -1.6e-38)
		tmp = -z / (y / x);
	elseif (y <= -2.7e-61)
		tmp = -z;
	elseif (y <= 150000000.0)
		tmp = x + y;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -3.8e+37], (-z), If[LessEqual[y, -1.6e-38], N[((-z) / N[(y / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -2.7e-61], (-z), If[LessEqual[y, 150000000.0], N[(x + y), $MachinePrecision], (-z)]]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.7999999999999999e37 or -1.59999999999999989e-38 < y < -2.69999999999999993e-61 or 1.5e8 < y

   1. Initial program 76.1%

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

   2. Taylor expanded in y around inf 60.3%

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

      1. mul-1-neg 60.3%

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

   4. Simplified 60.3%

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

   if -3.7999999999999999e37 < y < -1.59999999999999989e-38

   1. Initial program 99.7%

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

   2. Taylor expanded in x around inf 56.9%

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

   3. Taylor expanded in y around inf 51.4%

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

      1. mul-1-neg 51.4%

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

      2. associate-/l* 51.7%

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

      3. distribute-neg-frac 51.7%

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

   5. Simplified 51.7%

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

   if -2.69999999999999993e-61 < y < 1.5e8

   1. Initial program 100.0%

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

   2. Taylor expanded in z around inf 84.5%

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

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.8 \cdot 10^{+37}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq -1.6 \cdot 10^{-38}:\\ \;\;\;\;\frac{-z}{\frac{y}{x}}\\ \mathbf{elif}\;y \leq -2.7 \cdot 10^{-61}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 150000000:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 10: 67.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{y}{z}\\ \mathbf{if}\;x \leq -1.46 \cdot 10^{-37} \lor \neg \left(x \leq 1.4 \cdot 10^{-32}\right):\\ \;\;\;\;\frac{x}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ y z))))
   (if (or (<= x -1.46e-37) (not (<= x 1.4e-32))) (/ x t_0) (/ y t_0))))

C

double code(double x, double y, double z) {
	double t_0 = 1.0 - (y / z);
	double tmp;
	if ((x <= -1.46e-37) || !(x <= 1.4e-32)) {
		tmp = x / t_0;
	} else {
		tmp = y / 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 = 1.0d0 - (y / z)
    if ((x <= (-1.46d-37)) .or. (.not. (x <= 1.4d-32))) then
        tmp = x / t_0
    else
        tmp = y / t_0
    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 ((x <= -1.46e-37) || !(x <= 1.4e-32)) {
		tmp = x / t_0;
	} else {
		tmp = y / t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 1.0 - (y / z)
	tmp = 0
	if (x <= -1.46e-37) or not (x <= 1.4e-32):
		tmp = x / t_0
	else:
		tmp = y / t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(1.0 - Float64(y / z))
	tmp = 0.0
	if ((x <= -1.46e-37) || !(x <= 1.4e-32))
		tmp = Float64(x / t_0);
	else
		tmp = Float64(y / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 1.0 - (y / z);
	tmp = 0.0;
	if ((x <= -1.46e-37) || ~((x <= 1.4e-32)))
		tmp = x / t_0;
	else
		tmp = y / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[x, -1.46e-37], N[Not[LessEqual[x, 1.4e-32]], $MachinePrecision]], N[(x / t$95$0), $MachinePrecision], N[(y / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.46e-37 or 1.3999999999999999e-32 < x

   1. Initial program 86.7%

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

   2. Taylor expanded in x around inf 69.5%

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

   if -1.46e-37 < x < 1.3999999999999999e-32

   1. Initial program 90.2%

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

   2. Taylor expanded in x around 0 76.0%

      \[\leadsto \color{blue}{\frac{y}{1 - \frac{y}{z}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.46 \cdot 10^{-37} \lor \neg \left(x \leq 1.4 \cdot 10^{-32}\right):\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{1 - \frac{y}{z}}\\ \end{array} \]

Alternative 11: 66.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.32 \cdot 10^{-61}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 70000000:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.32e-61) (- z) (if (<= y 70000000.0) (+ x y) (- z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.32e-61) {
		tmp = -z;
	} else if (y <= 70000000.0) {
		tmp = x + y;
	} else {
		tmp = -z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-1.32d-61)) then
        tmp = -z
    else if (y <= 70000000.0d0) then
        tmp = x + y
    else
        tmp = -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.32e-61) {
		tmp = -z;
	} else if (y <= 70000000.0) {
		tmp = x + y;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1.32e-61:
		tmp = -z
	elif y <= 70000000.0:
		tmp = x + y
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.32e-61)
		tmp = Float64(-z);
	elseif (y <= 70000000.0)
		tmp = Float64(x + y);
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -1.32e-61)
		tmp = -z;
	elseif (y <= 70000000.0)
		tmp = x + y;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.32e-61], (-z), If[LessEqual[y, 70000000.0], N[(x + y), $MachinePrecision], (-z)]]

Derivation

1. Split input into 2 regimes

2. if y < -1.32000000000000002e-61 or 7e7 < y

   1. Initial program 79.3%

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

   2. Taylor expanded in y around inf 55.8%

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

      1. mul-1-neg 55.8%

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

   4. Simplified 55.8%

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

   if -1.32000000000000002e-61 < y < 7e7

   1. Initial program 100.0%

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

   2. Taylor expanded in z around inf 84.5%

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

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.32 \cdot 10^{-61}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 70000000:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 12: 59.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.9 \cdot 10^{-61}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{-33}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.9e-61) (- z) (if (<= y 1.4e-33) x (- z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.9e-61) {
		tmp = -z;
	} else if (y <= 1.4e-33) {
		tmp = x;
	} else {
		tmp = -z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-1.9d-61)) then
        tmp = -z
    else if (y <= 1.4d-33) then
        tmp = x
    else
        tmp = -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.9e-61) {
		tmp = -z;
	} else if (y <= 1.4e-33) {
		tmp = x;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1.9e-61:
		tmp = -z
	elif y <= 1.4e-33:
		tmp = x
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.9e-61)
		tmp = Float64(-z);
	elseif (y <= 1.4e-33)
		tmp = x;
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -1.9e-61)
		tmp = -z;
	elseif (y <= 1.4e-33)
		tmp = x;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.9e-61], (-z), If[LessEqual[y, 1.4e-33], x, (-z)]]

Derivation

1. Split input into 2 regimes

2. if y < -1.8999999999999999e-61 or 1.4e-33 < y

   1. Initial program 80.0%

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

   2. Taylor expanded in y around inf 55.4%

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

      1. mul-1-neg 55.4%

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

   4. Simplified 55.4%

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

   if -1.8999999999999999e-61 < y < 1.4e-33

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 69.5%

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

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.9 \cdot 10^{-61}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{-33}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 13: 40.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{-102}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.06 \cdot 10^{-148}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.5e-102) x (if (<= x 1.06e-148) y x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.5e-102) {
		tmp = x;
	} else if (x <= 1.06e-148) {
		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-102)) then
        tmp = x
    else if (x <= 1.06d-148) 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-102) {
		tmp = x;
	} else if (x <= 1.06e-148) {
		tmp = y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.5e-102:
		tmp = x
	elif x <= 1.06e-148:
		tmp = y
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.5e-102)
		tmp = x;
	elseif (x <= 1.06e-148)
		tmp = y;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.5e-102)
		tmp = x;
	elseif (x <= 1.06e-148)
		tmp = y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.5e-102], x, If[LessEqual[x, 1.06e-148], y, x]]

Derivation

1. Split input into 2 regimes

2. if x < -1.5e-102 or 1.06000000000000003e-148 < x

   1. Initial program 87.6%

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

   2. Taylor expanded in y around 0 43.5%

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

   if -1.5e-102 < x < 1.06000000000000003e-148

   1. Initial program 89.4%

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

   2. Taylor expanded in x around 0 82.9%

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

   3. Taylor expanded in y around 0 45.6%

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

4. Final simplification 44.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{-102}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.06 \cdot 10^{-148}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 14: 34.6% accurate, 9.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := x

Derivation

1. Initial program 88.1%

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

2. Taylor expanded in y around 0 33.9%

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

3. Final simplification 33.9%

   \[\leadsto x \]

Developer target: 93.9% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023252 
(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))))