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

Percentage Accurate: 88.4% → 99.1%

Time: 6.4s

Alternatives: 8

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: 88.4% 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: 99.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-232} \lor \neg \left(t\_0 \leq 0\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (+ x y) (- 1.0 (/ y z)))))
   (if (or (<= t_0 -2e-232) (not (<= t_0 0.0))) t_0 (* z (- -1.0 (/ 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-232) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = z * (-1.0 - (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-232)) .or. (.not. (t_0 <= 0.0d0))) then
        tmp = t_0
    else
        tmp = z * ((-1.0d0) - (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-232) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = z * (-1.0 - (x / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x + y) / (1.0 - (y / z))
	tmp = 0
	if (t_0 <= -2e-232) or not (t_0 <= 0.0):
		tmp = t_0
	else:
		tmp = z * (-1.0 - (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-232) || !(t_0 <= 0.0))
		tmp = t_0;
	else
		tmp = Float64(z * Float64(-1.0 - 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-232) || ~((t_0 <= 0.0)))
		tmp = t_0;
	else
		tmp = z * (-1.0 - (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-232], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], t$95$0, N[(z * N[(-1.0 - 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))) < -2.00000000000000005e-232 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 -2.00000000000000005e-232 < (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z))) < -0.0

   1. Initial program 26.1%

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

   3. Taylor expanded in y around inf 26.1%

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

      1. neg-mul-1 26.1%

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

      2. distribute-neg-frac 26.1%

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

   5. Simplified 26.1%

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

   6. Taylor expanded in x around 0 99.9%

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

      1. associate-*l/ 99.9%

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

      2. *-commutative 99.9%

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

      3. distribute-lft-in 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-lft-out 99.9%

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

      6. neg-mul-1 99.9%

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

      7. distribute-rgt-neg-in 99.9%

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

      8. mul-1-neg 99.9%

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

      9. *-commutative 99.9%

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

      10. distribute-lft-out 100.0%

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

      11. mul-1-neg 100.0%

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

      12. distribute-neg-frac2 100.0%

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

   8. Simplified 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \leq -2 \cdot 10^{-232} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \leq 0\right):\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 73.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{if}\;y \leq -5.8 \cdot 10^{+31}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq -5.7 \cdot 10^{-95}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{elif}\;y \leq 10^{+122}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (- -1.0 (/ x y)))))
   (if (<= y -5.8e+31)
     t_0
     (if (<= y -5.7e-95)
       (/ x (- 1.0 (/ y z)))
       (if (<= y 1e+122) (+ x y) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = z * (-1.0 - (x / y));
	double tmp;
	if (y <= -5.8e+31) {
		tmp = t_0;
	} else if (y <= -5.7e-95) {
		tmp = x / (1.0 - (y / z));
	} else if (y <= 1e+122) {
		tmp = x + y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = z * ((-1.0d0) - (x / y))
    if (y <= (-5.8d+31)) then
        tmp = t_0
    else if (y <= (-5.7d-95)) then
        tmp = x / (1.0d0 - (y / z))
    else if (y <= 1d+122) then
        tmp = x + y
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = z * (-1.0 - (x / y));
	double tmp;
	if (y <= -5.8e+31) {
		tmp = t_0;
	} else if (y <= -5.7e-95) {
		tmp = x / (1.0 - (y / z));
	} else if (y <= 1e+122) {
		tmp = x + y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * (-1.0 - (x / y))
	tmp = 0
	if y <= -5.8e+31:
		tmp = t_0
	elif y <= -5.7e-95:
		tmp = x / (1.0 - (y / z))
	elif y <= 1e+122:
		tmp = x + y
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(-1.0 - Float64(x / y)))
	tmp = 0.0
	if (y <= -5.8e+31)
		tmp = t_0;
	elseif (y <= -5.7e-95)
		tmp = Float64(x / Float64(1.0 - Float64(y / z)));
	elseif (y <= 1e+122)
		tmp = Float64(x + y);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * (-1.0 - (x / y));
	tmp = 0.0;
	if (y <= -5.8e+31)
		tmp = t_0;
	elseif (y <= -5.7e-95)
		tmp = x / (1.0 - (y / z));
	elseif (y <= 1e+122)
		tmp = x + y;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[(-1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -5.8e+31], t$95$0, If[LessEqual[y, -5.7e-95], N[(x / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1e+122], N[(x + y), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -5.8000000000000001e31 or 1.00000000000000001e122 < y

   1. Initial program 76.3%

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

   3. Taylor expanded in y around inf 60.0%

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

      1. neg-mul-1 60.0%

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

      2. distribute-neg-frac 60.0%

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

   5. Simplified 60.0%

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

   6. Taylor expanded in x around 0 74.2%

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

      1. associate-*l/ 83.6%

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

      2. *-commutative 83.6%

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

      3. distribute-lft-in 83.6%

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

      4. +-commutative 83.6%

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

      5. distribute-lft-out 83.6%

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

      6. neg-mul-1 83.6%

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

      7. distribute-rgt-neg-in 83.6%

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

      8. mul-1-neg 83.6%

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

      9. *-commutative 83.6%

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

      10. distribute-lft-out 83.6%

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

      11. mul-1-neg 83.6%

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

      12. distribute-neg-frac2 83.6%

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

   8. Simplified 83.6%

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

   if -5.8000000000000001e31 < y < -5.7e-95

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 74.1%

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

   if -5.7e-95 < y < 1.00000000000000001e122

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf 79.6%

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

      1. +-commutative 79.6%

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

   5. Simplified 79.6%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.8 \cdot 10^{+31}:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{elif}\;y \leq -5.7 \cdot 10^{-95}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{elif}\;y \leq 10^{+122}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 68.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.1 \cdot 10^{+32}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq -5.5 \cdot 10^{-95}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{+126}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -3.1e+32)
   (- z)
   (if (<= y -5.5e-95)
     (/ x (- 1.0 (/ y z)))
     (if (<= y 1.1e+126) (+ x y) (- z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -3.1e+32) {
		tmp = -z;
	} else if (y <= -5.5e-95) {
		tmp = x / (1.0 - (y / z));
	} else if (y <= 1.1e+126) {
		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 <= (-3.1d+32)) then
        tmp = -z
    else if (y <= (-5.5d-95)) then
        tmp = x / (1.0d0 - (y / z))
    else if (y <= 1.1d+126) 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 <= -3.1e+32) {
		tmp = -z;
	} else if (y <= -5.5e-95) {
		tmp = x / (1.0 - (y / z));
	} else if (y <= 1.1e+126) {
		tmp = x + y;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -3.1e+32:
		tmp = -z
	elif y <= -5.5e-95:
		tmp = x / (1.0 - (y / z))
	elif y <= 1.1e+126:
		tmp = x + y
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -3.1e+32)
		tmp = Float64(-z);
	elseif (y <= -5.5e-95)
		tmp = Float64(x / Float64(1.0 - Float64(y / z)));
	elseif (y <= 1.1e+126)
		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 <= -3.1e+32)
		tmp = -z;
	elseif (y <= -5.5e-95)
		tmp = x / (1.0 - (y / z));
	elseif (y <= 1.1e+126)
		tmp = x + y;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -3.1e+32], (-z), If[LessEqual[y, -5.5e-95], N[(x / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.1e+126], N[(x + y), $MachinePrecision], (-z)]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.09999999999999993e32 or 1.09999999999999999e126 < y

   1. Initial program 76.3%

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

   3. Taylor expanded in y around inf 64.7%

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

      1. neg-mul-1 64.7%

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

   5. Simplified 64.7%

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

   if -3.09999999999999993e32 < y < -5.50000000000000003e-95

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 74.1%

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

   if -5.50000000000000003e-95 < y < 1.09999999999999999e126

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf 79.6%

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

      1. +-commutative 79.6%

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

   5. Simplified 79.6%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.1 \cdot 10^{+32}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq -5.5 \cdot 10^{-95}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{+126}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 66.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{y}{z}\\ \mathbf{if}\;x \leq -3.5 \cdot 10^{-48} \lor \neg \left(x \leq 1.7 \cdot 10^{-145}\right):\\ \;\;\;\;\frac{x}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ y z))))
   (if (or (<= x -3.5e-48) (not (<= x 1.7e-145))) (/ 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 <= -3.5e-48) || !(x <= 1.7e-145)) {
		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 <= (-3.5d-48)) .or. (.not. (x <= 1.7d-145))) 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 <= -3.5e-48) || !(x <= 1.7e-145)) {
		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 <= -3.5e-48) or not (x <= 1.7e-145):
		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 <= -3.5e-48) || !(x <= 1.7e-145))
		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 <= -3.5e-48) || ~((x <= 1.7e-145)))
		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, -3.5e-48], N[Not[LessEqual[x, 1.7e-145]], $MachinePrecision]], N[(x / t$95$0), $MachinePrecision], N[(y / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.49999999999999991e-48 or 1.6999999999999999e-145 < x

   1. Initial program 93.4%

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

   3. Taylor expanded in x around inf 77.6%

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

   if -3.49999999999999991e-48 < x < 1.6999999999999999e-145

   1. Initial program 89.2%

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

   3. Taylor expanded in x around 0 77.0%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.5 \cdot 10^{-48} \lor \neg \left(x \leq 1.7 \cdot 10^{-145}\right):\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{1 - \frac{y}{z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 57.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.9 \cdot 10^{+30}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{-41}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{+126}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -3.9e+30) (- z) (if (<= y 8.2e-41) x (if (<= y 1.1e+126) y (- z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -3.9e+30) {
		tmp = -z;
	} else if (y <= 8.2e-41) {
		tmp = x;
	} else if (y <= 1.1e+126) {
		tmp = 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 <= (-3.9d+30)) then
        tmp = -z
    else if (y <= 8.2d-41) then
        tmp = x
    else if (y <= 1.1d+126) then
        tmp = 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 <= -3.9e+30) {
		tmp = -z;
	} else if (y <= 8.2e-41) {
		tmp = x;
	} else if (y <= 1.1e+126) {
		tmp = y;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -3.9e+30:
		tmp = -z
	elif y <= 8.2e-41:
		tmp = x
	elif y <= 1.1e+126:
		tmp = y
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -3.9e+30)
		tmp = Float64(-z);
	elseif (y <= 8.2e-41)
		tmp = x;
	elseif (y <= 1.1e+126)
		tmp = y;
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -3.9e+30)
		tmp = -z;
	elseif (y <= 8.2e-41)
		tmp = x;
	elseif (y <= 1.1e+126)
		tmp = y;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -3.9e+30], (-z), If[LessEqual[y, 8.2e-41], x, If[LessEqual[y, 1.1e+126], y, (-z)]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.90000000000000011e30 or 1.09999999999999999e126 < y

   1. Initial program 76.3%

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

   3. Taylor expanded in y around inf 64.7%

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

      1. neg-mul-1 64.7%

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

   5. Simplified 64.7%

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

   if -3.90000000000000011e30 < y < 8.20000000000000028e-41

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 63.3%

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

   if 8.20000000000000028e-41 < y < 1.09999999999999999e126

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf 62.6%

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

      1. +-commutative 62.6%

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

   5. Simplified 62.6%

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

   6. Taylor expanded in y around inf 50.0%

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

Alternative 6: 67.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9 \cdot 10^{+35} \lor \neg \left(y \leq 1.05 \cdot 10^{+123}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -9e+35) (not (<= y 1.05e+123))) (- z) (+ x y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -9e+35) || !(y <= 1.05e+123)) {
		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 <= (-9d+35)) .or. (.not. (y <= 1.05d+123))) 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 <= -9e+35) || !(y <= 1.05e+123)) {
		tmp = -z;
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -9e+35) or not (y <= 1.05e+123):
		tmp = -z
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -9e+35) || !(y <= 1.05e+123))
		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 <= -9e+35) || ~((y <= 1.05e+123)))
		tmp = -z;
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -9e+35], N[Not[LessEqual[y, 1.05e+123]], $MachinePrecision]], (-z), N[(x + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -8.9999999999999993e35 or 1.04999999999999997e123 < y

   1. Initial program 76.3%

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

   3. Taylor expanded in y around inf 64.7%

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

      1. neg-mul-1 64.7%

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

   5. Simplified 64.7%

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

   if -8.9999999999999993e35 < y < 1.04999999999999997e123

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf 75.6%

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

      1. +-commutative 75.6%

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

   5. Simplified 75.6%

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

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9 \cdot 10^{+35} \lor \neg \left(y \leq 1.05 \cdot 10^{+123}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 40.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-55}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-150}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -2e-55) x (if (<= x 2.5e-150) y x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -2e-55) {
		tmp = x;
	} else if (x <= 2.5e-150) {
		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 <= (-2d-55)) then
        tmp = x
    else if (x <= 2.5d-150) 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 <= -2e-55) {
		tmp = x;
	} else if (x <= 2.5e-150) {
		tmp = y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -2e-55:
		tmp = x
	elif x <= 2.5e-150:
		tmp = y
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -2e-55)
		tmp = x;
	elseif (x <= 2.5e-150)
		tmp = y;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -2e-55)
		tmp = x;
	elseif (x <= 2.5e-150)
		tmp = y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -2e-55], x, If[LessEqual[x, 2.5e-150], y, x]]

Derivation

1. Split input into 2 regimes

2. if x < -1.99999999999999999e-55 or 2.49999999999999995e-150 < x

   1. Initial program 93.0%

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

   3. Taylor expanded in y around 0 53.4%

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

   if -1.99999999999999999e-55 < x < 2.49999999999999995e-150

   1. Initial program 89.7%

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

   3. Taylor expanded in z around inf 53.8%

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

      1. +-commutative 53.8%

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

   5. Simplified 53.8%

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

   6. Taylor expanded in y around inf 44.3%

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

Alternative 8: 35.1% 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 91.8%

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

3. Taylor expanded in y around 0 38.7%

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

Developer Target 1: 93.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y + x}{-y} \cdot z\\ \mathbf{if}\;y < -3.7429310762689856 \cdot 10^{+171}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y < 3.5534662456086734 \cdot 10^{+168}:\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024160 
(FPCore (x y z)
  :name "Graphics.Rendering.Chart.Backend.Diagrams:calcFontMetrics from Chart-diagrams-1.5.1, A"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< y -3742931076268985600000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (* (/ (+ y x) (- y)) z) (if (< y 3553466245608673400000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (/ (+ x y) (- 1 (/ y z))) (* (/ (+ y x) (- y)) z))))

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