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

Percentage Accurate: 88.2% → 99.7%

Time: 7.9s

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: 88.2% 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: 99.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{y}{z}\\ t_1 := \frac{x + y}{t\_0}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-268} \lor \neg \left(t\_1 \leq 0\right):\\ \;\;\;\;\frac{y}{t\_0} + \frac{x}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{\frac{y}{\left(-x\right) - y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ y z))) (t_1 (/ (+ x y) t_0)))
   (if (or (<= t_1 -2e-268) (not (<= t_1 0.0)))
     (+ (/ y t_0) (/ x t_0))
     (/ z (/ y (- (- x) y))))))

C

double code(double x, double y, double z) {
	double t_0 = 1.0 - (y / z);
	double t_1 = (x + y) / t_0;
	double tmp;
	if ((t_1 <= -2e-268) || !(t_1 <= 0.0)) {
		tmp = (y / t_0) + (x / 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) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 - (y / z)
    t_1 = (x + y) / t_0
    if ((t_1 <= (-2d-268)) .or. (.not. (t_1 <= 0.0d0))) then
        tmp = (y / t_0) + (x / 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 = 1.0 - (y / z);
	double t_1 = (x + y) / t_0;
	double tmp;
	if ((t_1 <= -2e-268) || !(t_1 <= 0.0)) {
		tmp = (y / t_0) + (x / t_0);
	} else {
		tmp = z / (y / (-x - y));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 1.0 - (y / z)
	t_1 = (x + y) / t_0
	tmp = 0
	if (t_1 <= -2e-268) or not (t_1 <= 0.0):
		tmp = (y / t_0) + (x / t_0)
	else:
		tmp = z / (y / (-x - y))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(1.0 - Float64(y / z))
	t_1 = Float64(Float64(x + y) / t_0)
	tmp = 0.0
	if ((t_1 <= -2e-268) || !(t_1 <= 0.0))
		tmp = Float64(Float64(y / t_0) + Float64(x / t_0));
	else
		tmp = Float64(z / Float64(y / Float64(Float64(-x) - y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 1.0 - (y / z);
	t_1 = (x + y) / t_0;
	tmp = 0.0;
	if ((t_1 <= -2e-268) || ~((t_1 <= 0.0)))
		tmp = (y / t_0) + (x / t_0);
	else
		tmp = z / (y / (-x - y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x + y), $MachinePrecision] / t$95$0), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -2e-268], N[Not[LessEqual[t$95$1, 0.0]], $MachinePrecision]], N[(N[(y / t$95$0), $MachinePrecision] + N[(x / t$95$0), $MachinePrecision]), $MachinePrecision], N[(z / N[(y / N[((-x) - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z))) < -1.99999999999999992e-268 or 0.0 < (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z)))

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 99.9%

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

      1. +-commutative 99.9%

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

   5. Simplified 99.9%

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

   if -1.99999999999999992e-268 < (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z))) < 0.0

   1. Initial program 8.4%

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

   3. Taylor expanded in z around 0 96.6%

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

      1. mul-1-neg 96.6%

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

      2. associate-/l* 99.9%

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

      3. distribute-rgt-neg-in 99.9%

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

      4. distribute-neg-frac2 99.9%

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

      5. +-commutative 99.9%

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

   5. Simplified 99.9%

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

      1. associate-*r/ 96.6%

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

      2. distribute-frac-neg2 96.6%

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

      3. add-sqr-sqrt 39.4%

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

      4. sqrt-unprod 8.3%

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

      5. sqr-neg 8.3%

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

      6. sqrt-unprod 2.0%

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

      7. add-sqr-sqrt 3.7%

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

      8. associate-*r/ 3.6%

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

      9. clear-num 3.6%

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

      10. un-div-inv 3.6%

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

      11. add-sqr-sqrt 2.0%

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

      12. sqrt-unprod 11.5%

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

      13. sqr-neg 11.5%

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

      14. sqrt-unprod 42.5%

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

      15. add-sqr-sqrt 100.0%

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

   7. Applied egg-rr 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \leq -2 \cdot 10^{-268} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \leq 0\right):\\ \;\;\;\;\frac{y}{1 - \frac{y}{z}} + \frac{x}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{\frac{y}{\left(-x\right) - y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 65.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{y}{z}\\ t_1 := \frac{y}{t\_0}\\ t_2 := \frac{x}{t\_0}\\ \mathbf{if}\;x \leq -2.25 \cdot 10^{-12}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq -3.8 \cdot 10^{-25}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-63}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;x \leq -5.2 \cdot 10^{-91}:\\ \;\;\;\;-z\\ \mathbf{elif}\;x \leq -6.3 \cdot 10^{-218}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-71}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ y z))) (t_1 (/ y t_0)) (t_2 (/ x t_0)))
   (if (<= x -2.25e-12)
     t_2
     (if (<= x -3.8e-25)
       t_1
       (if (<= x -5e-63)
         (+ x y)
         (if (<= x -5.2e-91)
           (- z)
           (if (<= x -6.3e-218) (+ x y) (if (<= x 2.6e-71) t_1 t_2))))))))

C

double code(double x, double y, double z) {
	double t_0 = 1.0 - (y / z);
	double t_1 = y / t_0;
	double t_2 = x / t_0;
	double tmp;
	if (x <= -2.25e-12) {
		tmp = t_2;
	} else if (x <= -3.8e-25) {
		tmp = t_1;
	} else if (x <= -5e-63) {
		tmp = x + y;
	} else if (x <= -5.2e-91) {
		tmp = -z;
	} else if (x <= -6.3e-218) {
		tmp = x + y;
	} else if (x <= 2.6e-71) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	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) :: t_2
    real(8) :: tmp
    t_0 = 1.0d0 - (y / z)
    t_1 = y / t_0
    t_2 = x / t_0
    if (x <= (-2.25d-12)) then
        tmp = t_2
    else if (x <= (-3.8d-25)) then
        tmp = t_1
    else if (x <= (-5d-63)) then
        tmp = x + y
    else if (x <= (-5.2d-91)) then
        tmp = -z
    else if (x <= (-6.3d-218)) then
        tmp = x + y
    else if (x <= 2.6d-71) then
        tmp = t_1
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = 1.0 - (y / z);
	double t_1 = y / t_0;
	double t_2 = x / t_0;
	double tmp;
	if (x <= -2.25e-12) {
		tmp = t_2;
	} else if (x <= -3.8e-25) {
		tmp = t_1;
	} else if (x <= -5e-63) {
		tmp = x + y;
	} else if (x <= -5.2e-91) {
		tmp = -z;
	} else if (x <= -6.3e-218) {
		tmp = x + y;
	} else if (x <= 2.6e-71) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 1.0 - (y / z)
	t_1 = y / t_0
	t_2 = x / t_0
	tmp = 0
	if x <= -2.25e-12:
		tmp = t_2
	elif x <= -3.8e-25:
		tmp = t_1
	elif x <= -5e-63:
		tmp = x + y
	elif x <= -5.2e-91:
		tmp = -z
	elif x <= -6.3e-218:
		tmp = x + y
	elif x <= 2.6e-71:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(1.0 - Float64(y / z))
	t_1 = Float64(y / t_0)
	t_2 = Float64(x / t_0)
	tmp = 0.0
	if (x <= -2.25e-12)
		tmp = t_2;
	elseif (x <= -3.8e-25)
		tmp = t_1;
	elseif (x <= -5e-63)
		tmp = Float64(x + y);
	elseif (x <= -5.2e-91)
		tmp = Float64(-z);
	elseif (x <= -6.3e-218)
		tmp = Float64(x + y);
	elseif (x <= 2.6e-71)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 1.0 - (y / z);
	t_1 = y / t_0;
	t_2 = x / t_0;
	tmp = 0.0;
	if (x <= -2.25e-12)
		tmp = t_2;
	elseif (x <= -3.8e-25)
		tmp = t_1;
	elseif (x <= -5e-63)
		tmp = x + y;
	elseif (x <= -5.2e-91)
		tmp = -z;
	elseif (x <= -6.3e-218)
		tmp = x + y;
	elseif (x <= 2.6e-71)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(y / t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(x / t$95$0), $MachinePrecision]}, If[LessEqual[x, -2.25e-12], t$95$2, If[LessEqual[x, -3.8e-25], t$95$1, If[LessEqual[x, -5e-63], N[(x + y), $MachinePrecision], If[LessEqual[x, -5.2e-91], (-z), If[LessEqual[x, -6.3e-218], N[(x + y), $MachinePrecision], If[LessEqual[x, 2.6e-71], t$95$1, t$95$2]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -2.2499999999999999e-12 or 2.5999999999999999e-71 < x

   1. Initial program 89.2%

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

   3. Taylor expanded in x around inf 77.7%

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

   if -2.2499999999999999e-12 < x < -3.7999999999999998e-25 or -6.2999999999999996e-218 < x < 2.5999999999999999e-71

   1. Initial program 95.4%

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

   3. Taylor expanded in x around 0 79.0%

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

   if -3.7999999999999998e-25 < x < -5.0000000000000002e-63 or -5.20000000000000028e-91 < x < -6.2999999999999996e-218

   1. Initial program 89.6%

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

   3. Taylor expanded in z around inf 81.5%

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

      1. +-commutative 81.5%

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

   5. Simplified 81.5%

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

   if -5.0000000000000002e-63 < x < -5.20000000000000028e-91

   1. Initial program 56.7%

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

   3. Taylor expanded in y around inf 71.4%

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

      1. mul-1-neg 71.4%

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

   5. Simplified 71.4%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.25 \cdot 10^{-12}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{elif}\;x \leq -3.8 \cdot 10^{-25}:\\ \;\;\;\;\frac{y}{1 - \frac{y}{z}}\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-63}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;x \leq -5.2 \cdot 10^{-91}:\\ \;\;\;\;-z\\ \mathbf{elif}\;x \leq -6.3 \cdot 10^{-218}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-71}:\\ \;\;\;\;\frac{y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-268} \lor \neg \left(t\_0 \leq 0\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{\frac{y}{\left(-x\right) - y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (+ x y) (- 1.0 (/ y z)))))
   (if (or (<= t_0 -2e-268) (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-268) || !(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-268)) .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-268) || !(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-268) 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-268) || !(t_0 <= 0.0))
		tmp = t_0;
	else
		tmp = Float64(z / Float64(y / Float64(Float64(-x) - y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x + y) / (1.0 - (y / z));
	tmp = 0.0;
	if ((t_0 <= -2e-268) || ~((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-268], 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 #s(literal 1 binary64) (/.f64 y z))) < -1.99999999999999992e-268 or 0.0 < (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z)))

   1. Initial program 99.9%

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

   if -1.99999999999999992e-268 < (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z))) < 0.0

   1. Initial program 8.4%

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

   3. Taylor expanded in z around 0 96.6%

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

      1. mul-1-neg 96.6%

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

      2. associate-/l* 99.9%

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

      3. distribute-rgt-neg-in 99.9%

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

      4. distribute-neg-frac2 99.9%

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

      5. +-commutative 99.9%

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

   5. Simplified 99.9%

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

      1. associate-*r/ 96.6%

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

      2. distribute-frac-neg2 96.6%

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

      3. add-sqr-sqrt 39.4%

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

      4. sqrt-unprod 8.3%

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

      5. sqr-neg 8.3%

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

      6. sqrt-unprod 2.0%

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

      7. add-sqr-sqrt 3.7%

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

      8. associate-*r/ 3.6%

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

      9. clear-num 3.6%

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

      10. un-div-inv 3.6%

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

      11. add-sqr-sqrt 2.0%

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

      12. sqrt-unprod 11.5%

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

      13. sqr-neg 11.5%

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

      14. sqrt-unprod 42.5%

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

      15. add-sqr-sqrt 100.0%

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

   7. Applied egg-rr 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \leq -2 \cdot 10^{-268} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \leq 0\right):\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{\frac{y}{\left(-x\right) - y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 59.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.3 \cdot 10^{-54}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq -1.5 \cdot 10^{-168}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{-137}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{-27}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -3.3e-54)
   (+ x y)
   (if (<= z -1.5e-168)
     (- z)
     (if (<= z 2.5e-137)
       (/ x (- 1.0 (/ y z)))
       (if (<= z 3.5e-27) (- z) (+ x y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -3.3e-54) {
		tmp = x + y;
	} else if (z <= -1.5e-168) {
		tmp = -z;
	} else if (z <= 2.5e-137) {
		tmp = x / (1.0 - (y / z));
	} else if (z <= 3.5e-27) {
		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 (z <= (-3.3d-54)) then
        tmp = x + y
    else if (z <= (-1.5d-168)) then
        tmp = -z
    else if (z <= 2.5d-137) then
        tmp = x / (1.0d0 - (y / z))
    else if (z <= 3.5d-27) 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 (z <= -3.3e-54) {
		tmp = x + y;
	} else if (z <= -1.5e-168) {
		tmp = -z;
	} else if (z <= 2.5e-137) {
		tmp = x / (1.0 - (y / z));
	} else if (z <= 3.5e-27) {
		tmp = -z;
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -3.3e-54:
		tmp = x + y
	elif z <= -1.5e-168:
		tmp = -z
	elif z <= 2.5e-137:
		tmp = x / (1.0 - (y / z))
	elif z <= 3.5e-27:
		tmp = -z
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -3.3e-54)
		tmp = Float64(x + y);
	elseif (z <= -1.5e-168)
		tmp = Float64(-z);
	elseif (z <= 2.5e-137)
		tmp = Float64(x / Float64(1.0 - Float64(y / z)));
	elseif (z <= 3.5e-27)
		tmp = Float64(-z);
	else
		tmp = Float64(x + y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -3.3e-54)
		tmp = x + y;
	elseif (z <= -1.5e-168)
		tmp = -z;
	elseif (z <= 2.5e-137)
		tmp = x / (1.0 - (y / z));
	elseif (z <= 3.5e-27)
		tmp = -z;
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -3.3e-54], N[(x + y), $MachinePrecision], If[LessEqual[z, -1.5e-168], (-z), If[LessEqual[z, 2.5e-137], N[(x / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 3.5e-27], (-z), N[(x + y), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.29999999999999993e-54 or 3.5000000000000001e-27 < z

   1. Initial program 99.3%

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

   3. Taylor expanded in z around inf 78.2%

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

      1. +-commutative 78.2%

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

   5. Simplified 78.2%

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

   if -3.29999999999999993e-54 < z < -1.49999999999999996e-168 or 2.5e-137 < z < 3.5000000000000001e-27

   1. Initial program 73.0%

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

   3. Taylor expanded in y around inf 67.5%

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

      1. mul-1-neg 67.5%

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

   5. Simplified 67.5%

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

   if -1.49999999999999996e-168 < z < 2.5e-137

   1. Initial program 81.0%

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

   3. Taylor expanded in x around inf 64.9%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.3 \cdot 10^{-54}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq -1.5 \cdot 10^{-168}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{-137}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{-27}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -7.3 \cdot 10^{-31} \lor \neg \left(z \leq 1.85 \cdot 10^{-6}\right):\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{\frac{y}{\left(-x\right) - y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -7.3e-31) (not (<= z 1.85e-6)))
   (+ x y)
   (/ z (/ y (- (- x) y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -7.3e-31) || !(z <= 1.85e-6)) {
		tmp = x + y;
	} 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) :: tmp
    if ((z <= (-7.3d-31)) .or. (.not. (z <= 1.85d-6))) then
        tmp = x + y
    else
        tmp = z / (y / (-x - y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -7.3e-31) || !(z <= 1.85e-6)) {
		tmp = x + y;
	} else {
		tmp = z / (y / (-x - y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -7.3e-31) or not (z <= 1.85e-6):
		tmp = x + y
	else:
		tmp = z / (y / (-x - y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -7.3e-31) || !(z <= 1.85e-6))
		tmp = Float64(x + y);
	else
		tmp = Float64(z / Float64(y / Float64(Float64(-x) - y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -7.3e-31) || ~((z <= 1.85e-6)))
		tmp = x + y;
	else
		tmp = z / (y / (-x - y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -7.3e-31], N[Not[LessEqual[z, 1.85e-6]], $MachinePrecision]], N[(x + y), $MachinePrecision], N[(z / N[(y / N[((-x) - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -7.3000000000000003e-31 or 1.8500000000000001e-6 < z

   1. Initial program 99.2%

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

   3. Taylor expanded in z around inf 80.6%

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

      1. +-commutative 80.6%

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

   5. Simplified 80.6%

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

   if -7.3000000000000003e-31 < z < 1.8500000000000001e-6

   1. Initial program 79.4%

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

   3. Taylor expanded in z around 0 69.6%

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

      1. mul-1-neg 69.6%

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

      2. associate-/l* 72.4%

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

      3. distribute-rgt-neg-in 72.4%

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

      4. distribute-neg-frac2 72.4%

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

      5. +-commutative 72.4%

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

   5. Simplified 72.4%

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

      1. associate-*r/ 69.6%

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

      2. distribute-frac-neg2 69.6%

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

      3. add-sqr-sqrt 34.5%

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

      4. sqrt-unprod 23.7%

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

      5. sqr-neg 23.7%

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

      6. sqrt-unprod 1.7%

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

      7. add-sqr-sqrt 3.3%

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

      8. associate-*r/ 3.1%

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

      9. clear-num 3.1%

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

      10. un-div-inv 3.1%

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

      11. add-sqr-sqrt 1.6%

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

      12. sqrt-unprod 26.2%

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

      13. sqr-neg 26.2%

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

      14. sqrt-unprod 37.6%

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

      15. add-sqr-sqrt 73.1%

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

   7. Applied egg-rr 73.1%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.3 \cdot 10^{-31} \lor \neg \left(z \leq 1.85 \cdot 10^{-6}\right):\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{\frac{y}{\left(-x\right) - y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.5 \cdot 10^{+52} \lor \neg \left(y \leq 5.2 \cdot 10^{+67}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -2.5e+52) (not (<= y 5.2e+67))) (- z) (+ x y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -2.5e+52) || !(y <= 5.2e+67)) {
		tmp = -z;
	} else {
		tmp = x + y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((y <= (-2.5d+52)) .or. (.not. (y <= 5.2d+67))) then
        tmp = -z
    else
        tmp = x + y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -2.5e+52) || !(y <= 5.2e+67)) {
		tmp = -z;
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -2.5e+52) or not (y <= 5.2e+67):
		tmp = -z
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -2.5e+52) || !(y <= 5.2e+67))
		tmp = Float64(-z);
	else
		tmp = Float64(x + y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -2.5e+52) || ~((y <= 5.2e+67)))
		tmp = -z;
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -2.5e+52], N[Not[LessEqual[y, 5.2e+67]], $MachinePrecision]], (-z), N[(x + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.5e52 or 5.2000000000000001e67 < y

   1. Initial program 72.0%

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

   3. Taylor expanded in y around inf 65.6%

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

      1. mul-1-neg 65.6%

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

   5. Simplified 65.6%

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

   if -2.5e52 < y < 5.2000000000000001e67

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf 70.7%

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

      1. +-commutative 70.7%

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

   5. Simplified 70.7%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.5 \cdot 10^{+52} \lor \neg \left(y \leq 5.2 \cdot 10^{+67}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 58.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.5 \cdot 10^{-10} \lor \neg \left(y \leq 1.05 \cdot 10^{-22}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -3.5e-10) (not (<= y 1.05e-22))) (- z) x))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -3.5e-10) || !(y <= 1.05e-22)) {
		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 <= (-3.5d-10)) .or. (.not. (y <= 1.05d-22))) 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 <= -3.5e-10) || !(y <= 1.05e-22)) {
		tmp = -z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -3.5e-10) or not (y <= 1.05e-22):
		tmp = -z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -3.5e-10) || !(y <= 1.05e-22))
		tmp = Float64(-z);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -3.5e-10) || ~((y <= 1.05e-22)))
		tmp = -z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -3.5e-10], N[Not[LessEqual[y, 1.05e-22]], $MachinePrecision]], (-z), x]

Derivation

1. Split input into 2 regimes

2. if y < -3.4999999999999998e-10 or 1.05000000000000004e-22 < y

   1. Initial program 79.0%

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

   3. Taylor expanded in y around inf 56.4%

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

      1. mul-1-neg 56.4%

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

   5. Simplified 56.4%

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

   if -3.4999999999999998e-10 < y < 1.05000000000000004e-22

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 59.4%

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

4. Final simplification 58.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.5 \cdot 10^{-10} \lor \neg \left(y \leq 1.05 \cdot 10^{-22}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 39.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{-203}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{-77}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -5.5e-203) x (if (<= x 1.7e-77) y x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -5.5e-203) {
		tmp = x;
	} else if (x <= 1.7e-77) {
		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 <= (-5.5d-203)) then
        tmp = x
    else if (x <= 1.7d-77) 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 <= -5.5e-203) {
		tmp = x;
	} else if (x <= 1.7e-77) {
		tmp = y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -5.5e-203:
		tmp = x
	elif x <= 1.7e-77:
		tmp = y
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -5.5e-203)
		tmp = x;
	elseif (x <= 1.7e-77)
		tmp = y;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -5.5e-203)
		tmp = x;
	elseif (x <= 1.7e-77)
		tmp = y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -5.5e-203], x, If[LessEqual[x, 1.7e-77], y, x]]

Derivation

1. Split input into 2 regimes

2. if x < -5.5000000000000002e-203 or 1.69999999999999991e-77 < x

   1. Initial program 87.8%

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

   3. Taylor expanded in y around 0 46.5%

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

   if -5.5000000000000002e-203 < x < 1.69999999999999991e-77

   1. Initial program 94.2%

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

   3. Taylor expanded in x around 0 77.1%

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

   4. Taylor expanded in y around 0 39.1%

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

Alternative 9: 34.2% 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 89.9%

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

3. Taylor expanded in y around 0 36.1%

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

Developer target: 93.7% 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 2024084 
(FPCore (x y z)
  :name "Graphics.Rendering.Chart.Backend.Diagrams:calcFontMetrics from Chart-diagrams-1.5.1, A"
  :precision binary64

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

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