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

Percentage Accurate: 87.7% → 99.8%

Time: 7.1s

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: 99.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-276} \lor \neg \left(t\_0 \leq 0\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{\left(-x\right) - y}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (+ x y) (- 1.0 (/ y z)))))
   (if (or (<= t_0 -2e-276) (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 <= -2e-276) || !(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 <= (-2d-276)) .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 <= -2e-276) || !(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 <= -2e-276) 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 <= -2e-276) || !(t_0 <= 0.0))
		tmp = t_0;
	else
		tmp = Float64(z * Float64(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 <= -2e-276) || ~((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, -2e-276], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], t$95$0, N[(z * N[(N[((-x) - y), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z))) < -2e-276 or -0.0 < (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z)))

   1. Initial program 99.9%

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

   if -2e-276 < (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z))) < -0.0

   1. Initial program 15.1%

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

   3. Taylor expanded in z around 0 91.5%

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

      1. mul-1-neg 91.5%

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

      2. associate-/l* 99.9%

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

      3. distribute-lft-neg-in 99.9%

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

      4. +-commutative 99.9%

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

   5. Simplified 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \leq -2 \cdot 10^{-276} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \leq 0\right):\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{\left(-x\right) - y}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 69.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{y}{z}\\ t_1 := \frac{y}{t\_0}\\ \mathbf{if}\;y \leq -3 \cdot 10^{+172}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq -4.8 \cdot 10^{-71}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 2.25 \cdot 10^{-79}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 7600000000000:\\ \;\;\;\;\frac{x}{t\_0}\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{+94} \lor \neg \left(y \leq 1.8 \cdot 10^{+111}\right) \land y \leq 8 \cdot 10^{+186}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ y z))) (t_1 (/ y t_0)))
   (if (<= y -3e+172)
     (- z)
     (if (<= y -4.8e-71)
       t_1
       (if (<= y 2.25e-79)
         (+ x y)
         (if (<= y 7600000000000.0)
           (/ x t_0)
           (if (or (<= y 5.5e+94) (and (not (<= y 1.8e+111)) (<= y 8e+186)))
             t_1
             (- z))))))))

C

double code(double x, double y, double z) {
	double t_0 = 1.0 - (y / z);
	double t_1 = y / t_0;
	double tmp;
	if (y <= -3e+172) {
		tmp = -z;
	} else if (y <= -4.8e-71) {
		tmp = t_1;
	} else if (y <= 2.25e-79) {
		tmp = x + y;
	} else if (y <= 7600000000000.0) {
		tmp = x / t_0;
	} else if ((y <= 5.5e+94) || (!(y <= 1.8e+111) && (y <= 8e+186))) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 - (y / z)
    t_1 = y / t_0
    if (y <= (-3d+172)) then
        tmp = -z
    else if (y <= (-4.8d-71)) then
        tmp = t_1
    else if (y <= 2.25d-79) then
        tmp = x + y
    else if (y <= 7600000000000.0d0) then
        tmp = x / t_0
    else if ((y <= 5.5d+94) .or. (.not. (y <= 1.8d+111)) .and. (y <= 8d+186)) then
        tmp = t_1
    else
        tmp = -z
    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 tmp;
	if (y <= -3e+172) {
		tmp = -z;
	} else if (y <= -4.8e-71) {
		tmp = t_1;
	} else if (y <= 2.25e-79) {
		tmp = x + y;
	} else if (y <= 7600000000000.0) {
		tmp = x / t_0;
	} else if ((y <= 5.5e+94) || (!(y <= 1.8e+111) && (y <= 8e+186))) {
		tmp = t_1;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 1.0 - (y / z)
	t_1 = y / t_0
	tmp = 0
	if y <= -3e+172:
		tmp = -z
	elif y <= -4.8e-71:
		tmp = t_1
	elif y <= 2.25e-79:
		tmp = x + y
	elif y <= 7600000000000.0:
		tmp = x / t_0
	elif (y <= 5.5e+94) or (not (y <= 1.8e+111) and (y <= 8e+186)):
		tmp = t_1
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(1.0 - Float64(y / z))
	t_1 = Float64(y / t_0)
	tmp = 0.0
	if (y <= -3e+172)
		tmp = Float64(-z);
	elseif (y <= -4.8e-71)
		tmp = t_1;
	elseif (y <= 2.25e-79)
		tmp = Float64(x + y);
	elseif (y <= 7600000000000.0)
		tmp = Float64(x / t_0);
	elseif ((y <= 5.5e+94) || (!(y <= 1.8e+111) && (y <= 8e+186)))
		tmp = t_1;
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 1.0 - (y / z);
	t_1 = y / t_0;
	tmp = 0.0;
	if (y <= -3e+172)
		tmp = -z;
	elseif (y <= -4.8e-71)
		tmp = t_1;
	elseif (y <= 2.25e-79)
		tmp = x + y;
	elseif (y <= 7600000000000.0)
		tmp = x / t_0;
	elseif ((y <= 5.5e+94) || (~((y <= 1.8e+111)) && (y <= 8e+186)))
		tmp = t_1;
	else
		tmp = -z;
	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]}, If[LessEqual[y, -3e+172], (-z), If[LessEqual[y, -4.8e-71], t$95$1, If[LessEqual[y, 2.25e-79], N[(x + y), $MachinePrecision], If[LessEqual[y, 7600000000000.0], N[(x / t$95$0), $MachinePrecision], If[Or[LessEqual[y, 5.5e+94], And[N[Not[LessEqual[y, 1.8e+111]], $MachinePrecision], LessEqual[y, 8e+186]]], t$95$1, (-z)]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -2.9999999999999999e172 or 5.4999999999999997e94 < y < 1.8000000000000001e111 or 7.99999999999999984e186 < y

   1. Initial program 56.5%

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

   3. Taylor expanded in y around inf 85.8%

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

      1. mul-1-neg 85.8%

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

   5. Simplified 85.8%

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

   if -2.9999999999999999e172 < y < -4.8e-71 or 7.6e12 < y < 5.4999999999999997e94 or 1.8000000000000001e111 < y < 7.99999999999999984e186

   1. Initial program 90.0%

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

   3. Taylor expanded in x around 0 66.7%

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

   if -4.8e-71 < y < 2.2500000000000001e-79

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf 86.1%

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

      1. +-commutative 86.1%

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

   5. Simplified 86.1%

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

   if 2.2500000000000001e-79 < y < 7.6e12

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf 63.5%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3 \cdot 10^{+172}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq -4.8 \cdot 10^{-71}:\\ \;\;\;\;\frac{y}{1 - \frac{y}{z}}\\ \mathbf{elif}\;y \leq 2.25 \cdot 10^{-79}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 7600000000000:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{+94} \lor \neg \left(y \leq 1.8 \cdot 10^{+111}\right) \land y \leq 8 \cdot 10^{+186}:\\ \;\;\;\;\frac{y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 66.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{1 - \frac{y}{z}}\\ \mathbf{if}\;y \leq -1.12 \cdot 10^{+42}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq -1.3 \cdot 10^{-14}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq -7.2 \cdot 10^{-44}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 6.3 \cdot 10^{-81}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{+81}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 7 \cdot 10^{+93}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ x (- 1.0 (/ y z)))))
   (if (<= y -1.12e+42)
     (- z)
     (if (<= y -1.3e-14)
       t_0
       (if (<= y -7.2e-44)
         (- z)
         (if (<= y 6.3e-81)
           (+ x y)
           (if (<= y 3.5e+81) t_0 (if (<= y 7e+93) y (- z)))))))))

C

double code(double x, double y, double z) {
	double t_0 = x / (1.0 - (y / z));
	double tmp;
	if (y <= -1.12e+42) {
		tmp = -z;
	} else if (y <= -1.3e-14) {
		tmp = t_0;
	} else if (y <= -7.2e-44) {
		tmp = -z;
	} else if (y <= 6.3e-81) {
		tmp = x + y;
	} else if (y <= 3.5e+81) {
		tmp = t_0;
	} else if (y <= 7e+93) {
		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) :: t_0
    real(8) :: tmp
    t_0 = x / (1.0d0 - (y / z))
    if (y <= (-1.12d+42)) then
        tmp = -z
    else if (y <= (-1.3d-14)) then
        tmp = t_0
    else if (y <= (-7.2d-44)) then
        tmp = -z
    else if (y <= 6.3d-81) then
        tmp = x + y
    else if (y <= 3.5d+81) then
        tmp = t_0
    else if (y <= 7d+93) then
        tmp = y
    else
        tmp = -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x / (1.0 - (y / z));
	double tmp;
	if (y <= -1.12e+42) {
		tmp = -z;
	} else if (y <= -1.3e-14) {
		tmp = t_0;
	} else if (y <= -7.2e-44) {
		tmp = -z;
	} else if (y <= 6.3e-81) {
		tmp = x + y;
	} else if (y <= 3.5e+81) {
		tmp = t_0;
	} else if (y <= 7e+93) {
		tmp = y;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x / (1.0 - (y / z))
	tmp = 0
	if y <= -1.12e+42:
		tmp = -z
	elif y <= -1.3e-14:
		tmp = t_0
	elif y <= -7.2e-44:
		tmp = -z
	elif y <= 6.3e-81:
		tmp = x + y
	elif y <= 3.5e+81:
		tmp = t_0
	elif y <= 7e+93:
		tmp = y
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x / Float64(1.0 - Float64(y / z)))
	tmp = 0.0
	if (y <= -1.12e+42)
		tmp = Float64(-z);
	elseif (y <= -1.3e-14)
		tmp = t_0;
	elseif (y <= -7.2e-44)
		tmp = Float64(-z);
	elseif (y <= 6.3e-81)
		tmp = Float64(x + y);
	elseif (y <= 3.5e+81)
		tmp = t_0;
	elseif (y <= 7e+93)
		tmp = y;
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x / (1.0 - (y / z));
	tmp = 0.0;
	if (y <= -1.12e+42)
		tmp = -z;
	elseif (y <= -1.3e-14)
		tmp = t_0;
	elseif (y <= -7.2e-44)
		tmp = -z;
	elseif (y <= 6.3e-81)
		tmp = x + y;
	elseif (y <= 3.5e+81)
		tmp = t_0;
	elseif (y <= 7e+93)
		tmp = y;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.12e+42], (-z), If[LessEqual[y, -1.3e-14], t$95$0, If[LessEqual[y, -7.2e-44], (-z), If[LessEqual[y, 6.3e-81], N[(x + y), $MachinePrecision], If[LessEqual[y, 3.5e+81], t$95$0, If[LessEqual[y, 7e+93], y, (-z)]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -1.12e42 or -1.29999999999999998e-14 < y < -7.1999999999999998e-44 or 6.99999999999999996e93 < y

   1. Initial program 71.2%

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

   3. Taylor expanded in y around inf 75.6%

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

      1. mul-1-neg 75.6%

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

   5. Simplified 75.6%

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

   if -1.12e42 < y < -1.29999999999999998e-14 or 6.30000000000000023e-81 < y < 3.5e81

   1. Initial program 95.2%

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

   3. Taylor expanded in x around inf 56.5%

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

   if -7.1999999999999998e-44 < y < 6.30000000000000023e-81

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf 84.7%

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

      1. +-commutative 84.7%

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

   5. Simplified 84.7%

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

   if 3.5e81 < y < 6.99999999999999996e93

   1. Initial program 81.8%

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

   3. Taylor expanded in x around 0 81.8%

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

   4. Taylor expanded in y around 0 80.5%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.12 \cdot 10^{+42}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq -1.3 \cdot 10^{-14}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{elif}\;y \leq -7.2 \cdot 10^{-44}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 6.3 \cdot 10^{-81}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{+81}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{elif}\;y \leq 7 \cdot 10^{+93}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \frac{\left(-x\right) - y}{y}\\ \mathbf{if}\;y \leq -550000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq -4 \cdot 10^{-71}:\\ \;\;\;\;\frac{y}{1 - \frac{y}{z}}\\ \mathbf{elif}\;y \leq 2.12 \cdot 10^{-78}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (/ (- (- x) y) y))))
   (if (<= y -550000.0)
     t_0
     (if (<= y -4e-71)
       (/ y (- 1.0 (/ y z)))
       (if (<= y 2.12e-78) (+ x y) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = z * ((-x - y) / y);
	double tmp;
	if (y <= -550000.0) {
		tmp = t_0;
	} else if (y <= -4e-71) {
		tmp = y / (1.0 - (y / z));
	} else if (y <= 2.12e-78) {
		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 * ((-x - y) / y)
    if (y <= (-550000.0d0)) then
        tmp = t_0
    else if (y <= (-4d-71)) then
        tmp = y / (1.0d0 - (y / z))
    else if (y <= 2.12d-78) 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 * ((-x - y) / y);
	double tmp;
	if (y <= -550000.0) {
		tmp = t_0;
	} else if (y <= -4e-71) {
		tmp = y / (1.0 - (y / z));
	} else if (y <= 2.12e-78) {
		tmp = x + y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * ((-x - y) / y)
	tmp = 0
	if y <= -550000.0:
		tmp = t_0
	elif y <= -4e-71:
		tmp = y / (1.0 - (y / z))
	elif y <= 2.12e-78:
		tmp = x + y
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(Float64(Float64(-x) - y) / y))
	tmp = 0.0
	if (y <= -550000.0)
		tmp = t_0;
	elseif (y <= -4e-71)
		tmp = Float64(y / Float64(1.0 - Float64(y / z)));
	elseif (y <= 2.12e-78)
		tmp = Float64(x + y);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * ((-x - y) / y);
	tmp = 0.0;
	if (y <= -550000.0)
		tmp = t_0;
	elseif (y <= -4e-71)
		tmp = y / (1.0 - (y / z));
	elseif (y <= 2.12e-78)
		tmp = x + y;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[(N[((-x) - y), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -550000.0], t$95$0, If[LessEqual[y, -4e-71], N[(y / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.12e-78], N[(x + y), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -5.5e5 or 2.1199999999999999e-78 < y

   1. Initial program 76.1%

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

   3. Taylor expanded in z around 0 68.1%

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

      1. mul-1-neg 68.1%

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

      2. associate-/l* 78.4%

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

      3. distribute-lft-neg-in 78.4%

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

      4. +-commutative 78.4%

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

   5. Simplified 78.4%

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

   if -5.5e5 < y < -3.9999999999999997e-71

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 83.1%

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

   if -3.9999999999999997e-71 < y < 2.1199999999999999e-78

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf 86.1%

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

      1. +-commutative 86.1%

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

   5. Simplified 86.1%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -550000:\\ \;\;\;\;z \cdot \frac{\left(-x\right) - y}{y}\\ \mathbf{elif}\;y \leq -4 \cdot 10^{-71}:\\ \;\;\;\;\frac{y}{1 - \frac{y}{z}}\\ \mathbf{elif}\;y \leq 2.12 \cdot 10^{-78}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{\left(-x\right) - y}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3 \cdot 10^{+51} \lor \neg \left(y \leq 2.6 \cdot 10^{+94}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -3e+51) (not (<= y 2.6e+94))) (- z) (+ x y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -3e+51) || !(y <= 2.6e+94)) {
		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 <= (-3d+51)) .or. (.not. (y <= 2.6d+94))) 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 <= -3e+51) || !(y <= 2.6e+94)) {
		tmp = -z;
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -3e+51) or not (y <= 2.6e+94):
		tmp = -z
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -3e+51) || !(y <= 2.6e+94))
		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 <= -3e+51) || ~((y <= 2.6e+94)))
		tmp = -z;
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -3e+51], N[Not[LessEqual[y, 2.6e+94]], $MachinePrecision]], (-z), N[(x + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3e51 or 2.5999999999999999e94 < y

   1. Initial program 68.9%

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

   3. Taylor expanded in y around inf 77.3%

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

      1. mul-1-neg 77.3%

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

   5. Simplified 77.3%

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

   if -3e51 < y < 2.5999999999999999e94

   1. Initial program 98.0%

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

   3. Taylor expanded in z around inf 69.0%

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

      1. +-commutative 69.0%

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

   5. Simplified 69.0%

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

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3 \cdot 10^{+51} \lor \neg \left(y \leq 2.6 \cdot 10^{+94}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 55.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{-71} \lor \neg \left(y \leq 3.15 \cdot 10^{-105}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -4e-71) (not (<= y 3.15e-105))) (- z) x))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -4e-71) || !(y <= 3.15e-105)) {
		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 <= (-4d-71)) .or. (.not. (y <= 3.15d-105))) 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 <= -4e-71) || !(y <= 3.15e-105)) {
		tmp = -z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -4e-71) or not (y <= 3.15e-105):
		tmp = -z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -4e-71) || !(y <= 3.15e-105))
		tmp = Float64(-z);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -4e-71) || ~((y <= 3.15e-105)))
		tmp = -z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -4e-71], N[Not[LessEqual[y, 3.15e-105]], $MachinePrecision]], (-z), x]

Derivation

1. Split input into 2 regimes

2. if y < -3.9999999999999997e-71 or 3.15e-105 < y

   1. Initial program 78.8%

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

   3. Taylor expanded in y around inf 62.2%

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

      1. mul-1-neg 62.2%

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

   5. Simplified 62.2%

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

   if -3.9999999999999997e-71 < y < 3.15e-105

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 71.8%

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

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{-71} \lor \neg \left(y \leq 3.15 \cdot 10^{-105}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 40.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.2 \cdot 10^{-104}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{-207}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -9.2e-104) x (if (<= x 6.2e-207) y x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -9.2e-104) {
		tmp = x;
	} else if (x <= 6.2e-207) {
		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 <= (-9.2d-104)) then
        tmp = x
    else if (x <= 6.2d-207) 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 <= -9.2e-104) {
		tmp = x;
	} else if (x <= 6.2e-207) {
		tmp = y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -9.2e-104:
		tmp = x
	elif x <= 6.2e-207:
		tmp = y
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -9.2e-104)
		tmp = x;
	elseif (x <= 6.2e-207)
		tmp = y;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -9.2e-104)
		tmp = x;
	elseif (x <= 6.2e-207)
		tmp = y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -9.2e-104], x, If[LessEqual[x, 6.2e-207], y, x]]

Derivation

1. Split input into 2 regimes

2. if x < -9.1999999999999998e-104 or 6.2000000000000003e-207 < x

   1. Initial program 85.3%

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

   3. Taylor expanded in y around 0 38.1%

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

   if -9.1999999999999998e-104 < x < 6.2000000000000003e-207

   1. Initial program 86.5%

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

   3. Taylor expanded in x around 0 74.9%

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

   4. Taylor expanded in y around 0 35.3%

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

4. Final simplification 37.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.2 \cdot 10^{-104}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{-207}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. 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 85.6%

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

3. Taylor expanded in y around 0 32.0%

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

4. Final simplification 32.0%

   \[\leadsto x \]
5. Add Preprocessing

Developer target: 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 2024040 
(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))))