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

Percentage Accurate: 88.5% → 99.6%

Time: 6.4s

Alternatives: 6

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: 88.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{-263} \lor \neg \left(t_0 \leq 0\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-z}{\frac{y}{x + y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (+ x y) (- 1.0 (/ y z)))))
   (if (or (<= t_0 -2e-263) (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-263) || !(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-263)) .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-263) || !(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-263) 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-263) || !(t_0 <= 0.0))
		tmp = t_0;
	else
		tmp = Float64(Float64(-z) / Float64(y / Float64(x + y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x + y) / (1.0 - (y / z));
	tmp = 0.0;
	if ((t_0 <= -2e-263) || ~((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-263], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], t$95$0, N[((-z) / N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z))) < -2e-263 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 -2e-263 < (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z))) < 0.0

   1. Initial program 9.1%

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

   2. Taylor expanded in z around 0 96.5%

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

      1. mul-1-neg 96.5%

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

      2. associate-/l* 100.0%

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

      3. distribute-neg-frac 100.0%

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

      4. +-commutative 100.0%

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

   4. Simplified 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \leq -2 \cdot 10^{-263} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \leq 0\right):\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-z}{\frac{y}{x + y}}\\ \end{array} \]

Alternative 2: 65.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \frac{-x}{y}\\ \mathbf{if}\;y \leq -5.4 \cdot 10^{+135}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq -2.35 \cdot 10^{+67}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -3.2 \cdot 10^{+51}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 6.4 \cdot 10^{-12}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{+107}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (/ (- x) y))))
   (if (<= y -5.4e+135)
     (- z)
     (if (<= y -2.35e+67)
       t_0
       (if (<= y -3.2e+51)
         (- z)
         (if (<= y 6.4e-12) (+ x y) (if (<= y 8.2e+107) t_0 (- z))))))))

C

double code(double x, double y, double z) {
	double t_0 = z * (-x / y);
	double tmp;
	if (y <= -5.4e+135) {
		tmp = -z;
	} else if (y <= -2.35e+67) {
		tmp = t_0;
	} else if (y <= -3.2e+51) {
		tmp = -z;
	} else if (y <= 6.4e-12) {
		tmp = x + y;
	} else if (y <= 8.2e+107) {
		tmp = t_0;
	} else {
		tmp = -z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = z * (-x / y)
    if (y <= (-5.4d+135)) then
        tmp = -z
    else if (y <= (-2.35d+67)) then
        tmp = t_0
    else if (y <= (-3.2d+51)) then
        tmp = -z
    else if (y <= 6.4d-12) then
        tmp = x + y
    else if (y <= 8.2d+107) then
        tmp = t_0
    else
        tmp = -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = z * (-x / y);
	double tmp;
	if (y <= -5.4e+135) {
		tmp = -z;
	} else if (y <= -2.35e+67) {
		tmp = t_0;
	} else if (y <= -3.2e+51) {
		tmp = -z;
	} else if (y <= 6.4e-12) {
		tmp = x + y;
	} else if (y <= 8.2e+107) {
		tmp = t_0;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * (-x / y)
	tmp = 0
	if y <= -5.4e+135:
		tmp = -z
	elif y <= -2.35e+67:
		tmp = t_0
	elif y <= -3.2e+51:
		tmp = -z
	elif y <= 6.4e-12:
		tmp = x + y
	elif y <= 8.2e+107:
		tmp = t_0
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(Float64(-x) / y))
	tmp = 0.0
	if (y <= -5.4e+135)
		tmp = Float64(-z);
	elseif (y <= -2.35e+67)
		tmp = t_0;
	elseif (y <= -3.2e+51)
		tmp = Float64(-z);
	elseif (y <= 6.4e-12)
		tmp = Float64(x + y);
	elseif (y <= 8.2e+107)
		tmp = t_0;
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * (-x / y);
	tmp = 0.0;
	if (y <= -5.4e+135)
		tmp = -z;
	elseif (y <= -2.35e+67)
		tmp = t_0;
	elseif (y <= -3.2e+51)
		tmp = -z;
	elseif (y <= 6.4e-12)
		tmp = x + y;
	elseif (y <= 8.2e+107)
		tmp = t_0;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[((-x) / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -5.4e+135], (-z), If[LessEqual[y, -2.35e+67], t$95$0, If[LessEqual[y, -3.2e+51], (-z), If[LessEqual[y, 6.4e-12], N[(x + y), $MachinePrecision], If[LessEqual[y, 8.2e+107], t$95$0, (-z)]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -5.3999999999999997e135 or -2.35000000000000009e67 < y < -3.2000000000000002e51 or 8.1999999999999998e107 < y

   1. Initial program 70.6%

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

   2. Taylor expanded in y around inf 71.6%

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

      1. mul-1-neg 71.6%

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

   4. Simplified 71.6%

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

   if -5.3999999999999997e135 < y < -2.35000000000000009e67 or 6.4000000000000002e-12 < y < 8.1999999999999998e107

   1. Initial program 92.2%

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

   2. Taylor expanded in z around 0 65.0%

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

      1. mul-1-neg 65.0%

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

      2. associate-/l* 65.1%

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

      3. associate-/r/ 60.0%

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

      4. distribute-rgt-neg-in 60.0%

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

      5. +-commutative 60.0%

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

      6. distribute-neg-in 60.0%

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

      7. sub-neg 60.0%

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

   4. Simplified 60.0%

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

      1. sub-neg 60.0%

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

      2. distribute-lft-in 60.0%

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

      3. add-sqr-sqrt 16.8%

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

      4. sqrt-unprod 52.5%

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

      5. sqr-neg 52.5%

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

      6. sqrt-unprod 35.7%

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

      7. add-sqr-sqrt 49.2%

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

   6. Applied egg-rr 49.2%

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

      1. distribute-lft-out 49.2%

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

      2. sub-neg 49.2%

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

      3. associate-*l/ 54.2%

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

      4. associate-*r/ 54.3%

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

      5. div-sub 54.3%

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

      6. *-inverses 54.3%

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

   8. Simplified 54.3%

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

   9. Taylor expanded in x around inf 54.7%

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

       1. mul-1-neg 54.7%

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

       2. associate-*l/ 54.8%

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

       3. *-commutative 54.8%

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

       4. distribute-rgt-neg-in 54.8%

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

   11. Simplified 54.8%

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

   if -3.2000000000000002e51 < y < 6.4000000000000002e-12

   1. Initial program 99.9%

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

   2. Taylor expanded in z around inf 81.3%

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

      1. +-commutative 81.3%

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

   4. Simplified 81.3%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.4 \cdot 10^{+135}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq -2.35 \cdot 10^{+67}:\\ \;\;\;\;z \cdot \frac{-x}{y}\\ \mathbf{elif}\;y \leq -3.2 \cdot 10^{+51}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 6.4 \cdot 10^{-12}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{+107}:\\ \;\;\;\;z \cdot \frac{-x}{y}\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 3: 74.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.2 \cdot 10^{-39} \lor \neg \left(y \leq 1.45 \cdot 10^{-13}\right):\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -6.2e-39) (not (<= y 1.45e-13)))
   (* z (- -1.0 (/ x y)))
   (+ x y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -6.2e-39) || !(y <= 1.45e-13)) {
		tmp = z * (-1.0 - (x / 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 <= (-6.2d-39)) .or. (.not. (y <= 1.45d-13))) then
        tmp = z * ((-1.0d0) - (x / 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 <= -6.2e-39) || !(y <= 1.45e-13)) {
		tmp = z * (-1.0 - (x / y));
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -6.2e-39) or not (y <= 1.45e-13):
		tmp = z * (-1.0 - (x / y))
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -6.2e-39) || !(y <= 1.45e-13))
		tmp = Float64(z * Float64(-1.0 - Float64(x / y)));
	else
		tmp = Float64(x + y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -6.2e-39) || ~((y <= 1.45e-13)))
		tmp = z * (-1.0 - (x / y));
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -6.2e-39], N[Not[LessEqual[y, 1.45e-13]], $MachinePrecision]], N[(z * N[(-1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -6.1999999999999994e-39 or 1.4499999999999999e-13 < y

   1. Initial program 81.9%

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

      1. clear-num 81.7%

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

      2. associate-/r/ 81.8%

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

   3. Applied egg-rr 81.8%

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

   4. Taylor expanded in z around 0 62.4%

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

      1. mul-1-neg 62.4%

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

      2. associate-/l* 76.2%

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

      3. +-commutative 76.2%

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

      4. associate-/r/ 60.8%

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

      5. distribute-rgt-in 60.8%

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

      6. distribute-neg-in 60.8%

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

      7. *-commutative 60.8%

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

      8. associate-*l/ 60.7%

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

      9. associate-/l* 73.2%

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

      10. *-inverses 73.2%

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

      11. /-rgt-identity 73.2%

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

      12. unsub-neg 73.2%

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

      13. associate-*r/ 72.6%

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

      14. unsub-neg 72.6%

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

      15. mul-1-neg 72.6%

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

      16. +-commutative 72.6%

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

      17. unsub-neg 72.6%

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

      18. mul-1-neg 72.6%

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

      19. associate-/l* 72.5%

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

      20. distribute-neg-frac 72.5%

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

   6. Simplified 72.5%

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

   7. Taylor expanded in x around 0 72.6%

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

      1. neg-mul-1 72.6%

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

      2. mul-1-neg 72.6%

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

      3. associate-*l/ 76.2%

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

      4. distribute-lft-neg-in 76.2%

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

      5. cancel-sign-sub-inv 76.2%

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

      6. neg-mul-1 76.2%

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

      7. distribute-rgt-out-- 76.2%

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

   9. Simplified 76.2%

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

   if -6.1999999999999994e-39 < y < 1.4499999999999999e-13

   1. Initial program 99.9%

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

   2. Taylor expanded in z around inf 88.7%

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

      1. +-commutative 88.7%

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

   4. Simplified 88.7%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.2 \cdot 10^{-39} \lor \neg \left(y \leq 1.45 \cdot 10^{-13}\right):\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 4: 68.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.5 \cdot 10^{+51} \lor \neg \left(y \leq 5 \cdot 10^{+77}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -2.5e+51) (not (<= y 5e+77))) (- z) (+ x y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -2.5e+51) || !(y <= 5e+77)) {
		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+51)) .or. (.not. (y <= 5d+77))) 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+51) || !(y <= 5e+77)) {
		tmp = -z;
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -2.5e+51) or not (y <= 5e+77):
		tmp = -z
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -2.5e+51) || !(y <= 5e+77))
		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+51) || ~((y <= 5e+77)))
		tmp = -z;
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -2.5e+51], N[Not[LessEqual[y, 5e+77]], $MachinePrecision]], (-z), N[(x + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.5e51 or 5.00000000000000004e77 < y

   1. Initial program 74.9%

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

   2. Taylor expanded in y around inf 60.2%

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

      1. mul-1-neg 60.2%

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

   4. Simplified 60.2%

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

   if -2.5e51 < y < 5.00000000000000004e77

   1. Initial program 99.3%

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

   2. Taylor expanded in z around inf 77.5%

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

      1. +-commutative 77.5%

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

   4. Simplified 77.5%

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

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.5 \cdot 10^{+51} \lor \neg \left(y \leq 5 \cdot 10^{+77}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 5: 57.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.8 \cdot 10^{-40} \lor \neg \left(y \leq 1.6 \cdot 10^{+77}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -6.8e-40) (not (<= y 1.6e+77))) (- z) x))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -6.8e-40) || !(y <= 1.6e+77)) {
		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 <= (-6.8d-40)) .or. (.not. (y <= 1.6d+77))) 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 <= -6.8e-40) || !(y <= 1.6e+77)) {
		tmp = -z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -6.8e-40) or not (y <= 1.6e+77):
		tmp = -z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -6.8e-40) || !(y <= 1.6e+77))
		tmp = Float64(-z);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -6.8e-40) || ~((y <= 1.6e+77)))
		tmp = -z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -6.8e-40], N[Not[LessEqual[y, 1.6e+77]], $MachinePrecision]], (-z), x]

Derivation

1. Split input into 2 regimes

2. if y < -6.79999999999999968e-40 or 1.6000000000000001e77 < y

   1. Initial program 79.9%

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

   2. Taylor expanded in y around inf 55.6%

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

      1. mul-1-neg 55.6%

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

   4. Simplified 55.6%

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

   if -6.79999999999999968e-40 < y < 1.6000000000000001e77

   1. Initial program 99.3%

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

   2. Taylor expanded in y around 0 60.8%

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

4. Final simplification 58.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.8 \cdot 10^{-40} \lor \neg \left(y \leq 1.6 \cdot 10^{+77}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6: 35.5% accurate, 9.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := x

Derivation

1. Initial program 90.6%

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

2. Taylor expanded in y around 0 38.7%

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

3. Final simplification 38.7%

   \[\leadsto x \]

Developer target: 93.7% 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 2023336 
(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))))