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

Percentage Accurate: 87.7% → 99.5%

Time: 6.8s

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: 87.7% 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: 99.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-286}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\frac{z \cdot \left(\left(-x\right) - y\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y}{\frac{z - y}{z}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (+ x y) (- 1.0 (/ y z)))))
   (if (<= t_0 -1e-286)
     t_0
     (if (<= t_0 0.0) (/ (* z (- (- x) y)) y) (/ (+ x y) (/ (- z y) z))))))

C

double code(double x, double y, double z) {
	double t_0 = (x + y) / (1.0 - (y / z));
	double tmp;
	if (t_0 <= -1e-286) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = (z * (-x - y)) / y;
	} else {
		tmp = (x + y) / ((z - y) / z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x + y) / (1.0d0 - (y / z))
    if (t_0 <= (-1d-286)) then
        tmp = t_0
    else if (t_0 <= 0.0d0) then
        tmp = (z * (-x - y)) / y
    else
        tmp = (x + y) / ((z - y) / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (x + y) / (1.0 - (y / z));
	double tmp;
	if (t_0 <= -1e-286) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = (z * (-x - y)) / y;
	} else {
		tmp = (x + y) / ((z - y) / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x + y) / (1.0 - (y / z))
	tmp = 0
	if t_0 <= -1e-286:
		tmp = t_0
	elif t_0 <= 0.0:
		tmp = (z * (-x - y)) / y
	else:
		tmp = (x + y) / ((z - y) / z)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x + y) / Float64(1.0 - Float64(y / z)))
	tmp = 0.0
	if (t_0 <= -1e-286)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(z * Float64(Float64(-x) - y)) / y);
	else
		tmp = Float64(Float64(x + y) / Float64(Float64(z - y) / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x + y) / (1.0 - (y / z));
	tmp = 0.0;
	if (t_0 <= -1e-286)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = (z * (-x - y)) / y;
	else
		tmp = (x + y) / ((z - y) / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x + y), $MachinePrecision] / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-286], t$95$0, If[LessEqual[t$95$0, 0.0], N[(N[(z * N[((-x) - y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], N[(N[(x + y), $MachinePrecision] / N[(N[(z - y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z))) < -1.00000000000000005e-286

   1. Initial program 99.9%

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

   if -1.00000000000000005e-286 < (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z))) < -0.0

   1. Initial program 5.8%

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

   3. Taylor expanded in z around 0 99.9%

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

      1. associate-*r/ 99.9%

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

      2. *-commutative 99.9%

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

      3. associate-*r* 99.9%

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

      4. neg-mul-1 99.9%

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

      5. distribute-neg-in 99.9%

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

      6. unsub-neg 99.9%

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

   5. Simplified 99.9%

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

   if -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 z around 0 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \leq -1 \cdot 10^{-286}:\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{elif}\;\frac{x + y}{1 - \frac{y}{z}} \leq 0:\\ \;\;\;\;\frac{z \cdot \left(\left(-x\right) - y\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y}{\frac{z - y}{z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-286} \lor \neg \left(t\_0 \leq 0\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot \left(\left(-x\right) - y\right)}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (+ x y) (- 1.0 (/ y z)))))
   (if (or (<= t_0 -1e-286) (not (<= t_0 0.0))) t_0 (/ (* z (- (- x) y)) y))))

C

double code(double x, double y, double z) {
	double t_0 = (x + y) / (1.0 - (y / z));
	double tmp;
	if ((t_0 <= -1e-286) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = (z * (-x - y)) / 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 <= (-1d-286)) .or. (.not. (t_0 <= 0.0d0))) then
        tmp = t_0
    else
        tmp = (z * (-x - y)) / 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 <= -1e-286) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = (z * (-x - y)) / y;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x + y) / (1.0 - (y / z))
	tmp = 0
	if (t_0 <= -1e-286) or not (t_0 <= 0.0):
		tmp = t_0
	else:
		tmp = (z * (-x - y)) / 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 <= -1e-286) || !(t_0 <= 0.0))
		tmp = t_0;
	else
		tmp = Float64(Float64(z * Float64(Float64(-x) - y)) / 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 <= -1e-286) || ~((t_0 <= 0.0)))
		tmp = t_0;
	else
		tmp = (z * (-x - y)) / 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, -1e-286], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], t$95$0, N[(N[(z * N[((-x) - y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z))) < -1.00000000000000005e-286 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 -1.00000000000000005e-286 < (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z))) < -0.0

   1. Initial program 5.8%

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

   3. Taylor expanded in z around 0 99.9%

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

      1. associate-*r/ 99.9%

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

      2. *-commutative 99.9%

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

      3. associate-*r* 99.9%

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

      4. neg-mul-1 99.9%

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

      5. distribute-neg-in 99.9%

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

      6. unsub-neg 99.9%

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

   5. Simplified 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \leq -1 \cdot 10^{-286} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \leq 0\right):\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot \left(\left(-x\right) - y\right)}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 68.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8.8 \cdot 10^{+82}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 8 \cdot 10^{-11}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 1.06 \cdot 10^{+49}:\\ \;\;\;\;x \cdot \frac{z}{z - y}\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -8.8e+82)
   (- z)
   (if (<= y 8e-11) (+ x y) (if (<= y 1.06e+49) (* x (/ z (- z y))) (- z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -8.8e+82) {
		tmp = -z;
	} else if (y <= 8e-11) {
		tmp = x + y;
	} else if (y <= 1.06e+49) {
		tmp = x * (z / (z - 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 <= (-8.8d+82)) then
        tmp = -z
    else if (y <= 8d-11) then
        tmp = x + y
    else if (y <= 1.06d+49) then
        tmp = x * (z / (z - 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 <= -8.8e+82) {
		tmp = -z;
	} else if (y <= 8e-11) {
		tmp = x + y;
	} else if (y <= 1.06e+49) {
		tmp = x * (z / (z - y));
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -8.8e+82:
		tmp = -z
	elif y <= 8e-11:
		tmp = x + y
	elif y <= 1.06e+49:
		tmp = x * (z / (z - y))
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -8.8e+82)
		tmp = Float64(-z);
	elseif (y <= 8e-11)
		tmp = Float64(x + y);
	elseif (y <= 1.06e+49)
		tmp = Float64(x * Float64(z / Float64(z - y)));
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -8.8e+82)
		tmp = -z;
	elseif (y <= 8e-11)
		tmp = x + y;
	elseif (y <= 1.06e+49)
		tmp = x * (z / (z - y));
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -8.8e+82], (-z), If[LessEqual[y, 8e-11], N[(x + y), $MachinePrecision], If[LessEqual[y, 1.06e+49], N[(x * N[(z / N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-z)]]]

Derivation

1. Split input into 3 regimes

2. if y < -8.8000000000000005e82 or 1.06e49 < y

   1. Initial program 74.8%

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

   3. Taylor expanded in y around inf 71.1%

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

      1. neg-mul-1 71.1%

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

   5. Simplified 71.1%

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

   if -8.8000000000000005e82 < y < 7.99999999999999952e-11

   1. Initial program 99.2%

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

   3. Taylor expanded in z around inf 79.0%

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

      1. +-commutative 79.0%

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

   5. Simplified 79.0%

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

   if 7.99999999999999952e-11 < y < 1.06e49

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0 99.9%

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

   4. Taylor expanded in x around inf 59.2%

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

      1. associate-/l* 72.7%

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

   6. Simplified 72.7%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.8 \cdot 10^{+82}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 8 \cdot 10^{-11}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 1.06 \cdot 10^{+49}:\\ \;\;\;\;x \cdot \frac{z}{z - y}\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.7 \cdot 10^{+63} \lor \neg \left(y \leq 6.1 \cdot 10^{-15}\right):\\ \;\;\;\;z \cdot \frac{\left(-x\right) - y}{y}\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -5.7e+63) (not (<= y 6.1e-15)))
   (* z (/ (- (- x) y) y))
   (+ x y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -5.7e+63) || !(y <= 6.1e-15)) {
		tmp = z * ((-x - y) / 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 <= (-5.7d+63)) .or. (.not. (y <= 6.1d-15))) then
        tmp = z * ((-x - y) / 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 <= -5.7e+63) || !(y <= 6.1e-15)) {
		tmp = z * ((-x - y) / y);
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -5.7e+63) or not (y <= 6.1e-15):
		tmp = z * ((-x - y) / y)
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -5.7e+63) || !(y <= 6.1e-15))
		tmp = Float64(z * Float64(Float64(Float64(-x) - y) / y));
	else
		tmp = Float64(x + y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -5.7e+63) || ~((y <= 6.1e-15)))
		tmp = z * ((-x - y) / y);
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -5.7e+63], N[Not[LessEqual[y, 6.1e-15]], $MachinePrecision]], N[(z * N[(N[((-x) - y), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(x + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -5.7000000000000002e63 or 6.09999999999999972e-15 < y

   1. Initial program 77.8%

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

   3. Taylor expanded in z around 0 65.2%

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

      1. mul-1-neg 65.2%

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

      2. associate-/l* 81.3%

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

      3. distribute-rgt-neg-in 81.3%

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

      4. distribute-neg-frac2 81.3%

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

      5. +-commutative 81.3%

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

   5. Simplified 81.3%

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

   if -5.7000000000000002e63 < y < 6.09999999999999972e-15

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf 80.5%

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

      1. +-commutative 80.5%

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

   5. Simplified 80.5%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.7 \cdot 10^{+63} \lor \neg \left(y \leq 6.1 \cdot 10^{-15}\right):\\ \;\;\;\;z \cdot \frac{\left(-x\right) - y}{y}\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 68.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.5 \cdot 10^{+86} \lor \neg \left(y \leq 10^{+55}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -3.5e+86) (not (<= y 1e+55))) (- z) (+ x y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -3.5e+86) || !(y <= 1e+55)) {
		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 <= (-3.5d+86)) .or. (.not. (y <= 1d+55))) 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 <= -3.5e+86) || !(y <= 1e+55)) {
		tmp = -z;
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -3.5e+86) or not (y <= 1e+55):
		tmp = -z
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -3.5e+86) || !(y <= 1e+55))
		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 <= -3.5e+86) || ~((y <= 1e+55)))
		tmp = -z;
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -3.5e+86], N[Not[LessEqual[y, 1e+55]], $MachinePrecision]], (-z), N[(x + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.50000000000000019e86 or 1.00000000000000001e55 < y

   1. Initial program 74.5%

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

   3. Taylor expanded in y around inf 71.8%

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

      1. neg-mul-1 71.8%

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

   5. Simplified 71.8%

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

   if -3.50000000000000019e86 < y < 1.00000000000000001e55

   1. Initial program 99.3%

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

   3. Taylor expanded in z around inf 74.5%

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

      1. +-commutative 74.5%

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

   5. Simplified 74.5%

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

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.5 \cdot 10^{+86} \lor \neg \left(y \leq 10^{+55}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 57.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.4 \cdot 10^{+74} \lor \neg \left(y \leq 3.3 \cdot 10^{-15}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -2.4e+74) (not (<= y 3.3e-15))) (- z) x))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -2.4e+74) || !(y <= 3.3e-15)) {
		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 <= (-2.4d+74)) .or. (.not. (y <= 3.3d-15))) 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 <= -2.4e+74) || !(y <= 3.3e-15)) {
		tmp = -z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -2.4e+74) or not (y <= 3.3e-15):
		tmp = -z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -2.4e+74) || !(y <= 3.3e-15))
		tmp = Float64(-z);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -2.4e+74) || ~((y <= 3.3e-15)))
		tmp = -z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -2.4e+74], N[Not[LessEqual[y, 3.3e-15]], $MachinePrecision]], (-z), x]

Derivation

1. Split input into 2 regimes

2. if y < -2.40000000000000008e74 or 3.3e-15 < y

   1. Initial program 78.3%

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

   3. Taylor expanded in y around inf 65.6%

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

      1. neg-mul-1 65.6%

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

   5. Simplified 65.6%

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

   if -2.40000000000000008e74 < y < 3.3e-15

   1. Initial program 99.2%

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

   3. Taylor expanded in y around 0 62.2%

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

4. Final simplification 63.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.4 \cdot 10^{+74} \lor \neg \left(y \leq 3.3 \cdot 10^{-15}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 39.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-30}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{-218}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -2e-30) x (if (<= x 4.6e-218) y x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -2e-30) {
		tmp = x;
	} else if (x <= 4.6e-218) {
		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-30)) then
        tmp = x
    else if (x <= 4.6d-218) 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-30) {
		tmp = x;
	} else if (x <= 4.6e-218) {
		tmp = y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -2e-30:
		tmp = x
	elif x <= 4.6e-218:
		tmp = y
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -2e-30)
		tmp = x;
	elseif (x <= 4.6e-218)
		tmp = y;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -2e-30)
		tmp = x;
	elseif (x <= 4.6e-218)
		tmp = y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -2e-30], x, If[LessEqual[x, 4.6e-218], y, x]]

Derivation

1. Split input into 2 regimes

2. if x < -2e-30 or 4.59999999999999989e-218 < x

   1. Initial program 88.3%

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

   3. Taylor expanded in y around 0 45.4%

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

   if -2e-30 < x < 4.59999999999999989e-218

   1. Initial program 91.4%

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

   3. Taylor expanded in z around inf 51.7%

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

      1. +-commutative 51.7%

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

   5. Simplified 51.7%

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

   6. Taylor expanded in y around inf 37.5%

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

Alternative 8: 34.6% accurate, 9.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := x

Derivation

1. Initial program 89.2%

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

3. Taylor expanded in y around 0 36.8%

   \[\leadsto \color{blue}{x} \]
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 2024146 
(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))))