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

Percentage Accurate: 87.9% → 99.5%

Time: 7.6s

Alternatives: 10

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: 87.9% 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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-243} \lor \neg \left(t\_0 \leq 0\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-\frac{x + y}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (+ x y) (- 1.0 (/ y z)))))
   (if (or (<= t_0 -2e-243) (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-243) || !(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-243)) .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-243) || !(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-243) 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-243) || !(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-243) || ~((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-243], 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))) < -1.99999999999999999e-243 or 0.0 < (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z)))

   1. Initial program 99.8%

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

   if -1.99999999999999999e-243 < (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z))) < 0.0

   1. Initial program 8.4%

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

      1. clear-num 8.4%

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

      2. inv-pow 8.4%

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

   4. Applied egg-rr 8.4%

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

   5. Taylor expanded in z around 0 97.3%

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

      1. associate-*r/ 97.3%

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

      2. +-commutative 97.3%

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

      3. associate-*r* 97.3%

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

      4. associate-*r/ 99.8%

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

      5. mul-1-neg 99.8%

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

      6. +-commutative 99.8%

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

   7. Simplified 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \leq -2 \cdot 10^{-243} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \leq 0\right):\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-\frac{x + y}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 65.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{1 - \frac{y}{z}}\\ \mathbf{if}\;y \leq -1.7 \cdot 10^{+202}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq -9.5 \cdot 10^{+113}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq -1.35 \cdot 10^{+48}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{-81}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{-44}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 9 \cdot 10^{+28}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ x (- 1.0 (/ y z)))))
   (if (<= y -1.7e+202)
     (- z)
     (if (<= y -9.5e+113)
       t_0
       (if (<= y -1.35e+48)
         (- z)
         (if (<= y 3.5e-81)
           t_0
           (if (<= y 2.7e-44) (+ x y) (if (<= y 9e+28) t_0 (- z)))))))))

C

double code(double x, double y, double z) {
	double t_0 = x / (1.0 - (y / z));
	double tmp;
	if (y <= -1.7e+202) {
		tmp = -z;
	} else if (y <= -9.5e+113) {
		tmp = t_0;
	} else if (y <= -1.35e+48) {
		tmp = -z;
	} else if (y <= 3.5e-81) {
		tmp = t_0;
	} else if (y <= 2.7e-44) {
		tmp = x + y;
	} else if (y <= 9e+28) {
		tmp = t_0;
	} else {
		tmp = -z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / (1.0d0 - (y / z))
    if (y <= (-1.7d+202)) then
        tmp = -z
    else if (y <= (-9.5d+113)) then
        tmp = t_0
    else if (y <= (-1.35d+48)) then
        tmp = -z
    else if (y <= 3.5d-81) then
        tmp = t_0
    else if (y <= 2.7d-44) then
        tmp = x + y
    else if (y <= 9d+28) then
        tmp = t_0
    else
        tmp = -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x / (1.0 - (y / z));
	double tmp;
	if (y <= -1.7e+202) {
		tmp = -z;
	} else if (y <= -9.5e+113) {
		tmp = t_0;
	} else if (y <= -1.35e+48) {
		tmp = -z;
	} else if (y <= 3.5e-81) {
		tmp = t_0;
	} else if (y <= 2.7e-44) {
		tmp = x + y;
	} else if (y <= 9e+28) {
		tmp = t_0;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x / (1.0 - (y / z))
	tmp = 0
	if y <= -1.7e+202:
		tmp = -z
	elif y <= -9.5e+113:
		tmp = t_0
	elif y <= -1.35e+48:
		tmp = -z
	elif y <= 3.5e-81:
		tmp = t_0
	elif y <= 2.7e-44:
		tmp = x + y
	elif y <= 9e+28:
		tmp = t_0
	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 <= -1.7e+202)
		tmp = Float64(-z);
	elseif (y <= -9.5e+113)
		tmp = t_0;
	elseif (y <= -1.35e+48)
		tmp = Float64(-z);
	elseif (y <= 3.5e-81)
		tmp = t_0;
	elseif (y <= 2.7e-44)
		tmp = Float64(x + y);
	elseif (y <= 9e+28)
		tmp = t_0;
	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 <= -1.7e+202)
		tmp = -z;
	elseif (y <= -9.5e+113)
		tmp = t_0;
	elseif (y <= -1.35e+48)
		tmp = -z;
	elseif (y <= 3.5e-81)
		tmp = t_0;
	elseif (y <= 2.7e-44)
		tmp = x + y;
	elseif (y <= 9e+28)
		tmp = t_0;
	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, -1.7e+202], (-z), If[LessEqual[y, -9.5e+113], t$95$0, If[LessEqual[y, -1.35e+48], (-z), If[LessEqual[y, 3.5e-81], t$95$0, If[LessEqual[y, 2.7e-44], N[(x + y), $MachinePrecision], If[LessEqual[y, 9e+28], t$95$0, (-z)]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.7e202 or -9.5000000000000001e113 < y < -1.35000000000000002e48 or 8.9999999999999994e28 < y

   1. Initial program 67.0%

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

   3. Taylor expanded in y around inf 66.1%

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

      1. mul-1-neg 66.1%

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

   5. Simplified 66.1%

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

   if -1.7e202 < y < -9.5000000000000001e113 or -1.35000000000000002e48 < y < 3.49999999999999986e-81 or 2.6999999999999999e-44 < y < 8.9999999999999994e28

   1. Initial program 99.2%

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

   3. Taylor expanded in x around inf 71.2%

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

   if 3.49999999999999986e-81 < y < 2.6999999999999999e-44

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf 89.2%

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

      1. +-commutative 89.2%

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

   5. Simplified 89.2%

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

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{+202}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq -9.5 \cdot 10^{+113}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{elif}\;y \leq -1.35 \cdot 10^{+48}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{-81}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{-44}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 9 \cdot 10^{+28}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x + y\right) \cdot \left(1 + \frac{y}{z}\right)\\ \mathbf{if}\;z \leq -1.2 \cdot 10^{+79}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq -9.6 \cdot 10^{+64}:\\ \;\;\;\;\frac{-z}{\frac{y}{x + y}}\\ \mathbf{elif}\;z \leq -3 \cdot 10^{-14}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{-59}:\\ \;\;\;\;\frac{x \cdot \left(-z\right)}{y} - z\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (+ x y) (+ 1.0 (/ y z)))))
   (if (<= z -1.2e+79)
     t_0
     (if (<= z -9.6e+64)
       (/ (- z) (/ y (+ x y)))
       (if (<= z -3e-14)
         t_0
         (if (<= z 3.8e-59) (- (/ (* x (- z)) y) z) (+ x y)))))))

C

double code(double x, double y, double z) {
	double t_0 = (x + y) * (1.0 + (y / z));
	double tmp;
	if (z <= -1.2e+79) {
		tmp = t_0;
	} else if (z <= -9.6e+64) {
		tmp = -z / (y / (x + y));
	} else if (z <= -3e-14) {
		tmp = t_0;
	} else if (z <= 3.8e-59) {
		tmp = ((x * -z) / y) - 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) :: t_0
    real(8) :: tmp
    t_0 = (x + y) * (1.0d0 + (y / z))
    if (z <= (-1.2d+79)) then
        tmp = t_0
    else if (z <= (-9.6d+64)) then
        tmp = -z / (y / (x + y))
    else if (z <= (-3d-14)) then
        tmp = t_0
    else if (z <= 3.8d-59) then
        tmp = ((x * -z) / y) - z
    else
        tmp = 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 (z <= -1.2e+79) {
		tmp = t_0;
	} else if (z <= -9.6e+64) {
		tmp = -z / (y / (x + y));
	} else if (z <= -3e-14) {
		tmp = t_0;
	} else if (z <= 3.8e-59) {
		tmp = ((x * -z) / y) - z;
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x + y) * (1.0 + (y / z))
	tmp = 0
	if z <= -1.2e+79:
		tmp = t_0
	elif z <= -9.6e+64:
		tmp = -z / (y / (x + y))
	elif z <= -3e-14:
		tmp = t_0
	elif z <= 3.8e-59:
		tmp = ((x * -z) / y) - z
	else:
		tmp = 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 (z <= -1.2e+79)
		tmp = t_0;
	elseif (z <= -9.6e+64)
		tmp = Float64(Float64(-z) / Float64(y / Float64(x + y)));
	elseif (z <= -3e-14)
		tmp = t_0;
	elseif (z <= 3.8e-59)
		tmp = Float64(Float64(Float64(x * Float64(-z)) / y) - z);
	else
		tmp = 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 (z <= -1.2e+79)
		tmp = t_0;
	elseif (z <= -9.6e+64)
		tmp = -z / (y / (x + y));
	elseif (z <= -3e-14)
		tmp = t_0;
	elseif (z <= 3.8e-59)
		tmp = ((x * -z) / y) - z;
	else
		tmp = 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[LessEqual[z, -1.2e+79], t$95$0, If[LessEqual[z, -9.6e+64], N[((-z) / N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, -3e-14], t$95$0, If[LessEqual[z, 3.8e-59], N[(N[(N[(x * (-z)), $MachinePrecision] / y), $MachinePrecision] - z), $MachinePrecision], N[(x + y), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -1.19999999999999993e79 or -9.59999999999999997e64 < z < -2.9999999999999998e-14

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf 75.5%

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

      1. associate-+r+ 75.5%

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

      2. *-lft-identity 75.5%

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

      3. associate-/l* 79.0%

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

      4. associate-/r/ 79.0%

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

      5. distribute-rgt-in 79.0%

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

      6. +-commutative 79.0%

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

   5. Simplified 79.0%

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

   if -1.19999999999999993e79 < z < -9.59999999999999997e64

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0 36.2%

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

      1. mul-1-neg 36.2%

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

      2. associate-/l* 84.3%

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

      3. distribute-neg-frac 84.3%

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

      4. +-commutative 84.3%

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

   5. Simplified 84.3%

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

   if -2.9999999999999998e-14 < z < 3.79999999999999983e-59

   1. Initial program 72.4%

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

   3. Taylor expanded in z around 0 77.7%

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

      1. mul-1-neg 77.7%

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

      2. associate-/l* 80.6%

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

      3. distribute-neg-frac 80.6%

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

      4. +-commutative 80.6%

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

   5. Simplified 80.6%

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

   6. Taylor expanded in y around 0 80.8%

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

      1. mul-1-neg 80.8%

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

      2. associate-*r/ 76.9%

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

      3. unsub-neg 76.9%

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

      4. mul-1-neg 76.9%

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

      5. associate-*r/ 80.8%

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

   8. Simplified 80.8%

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

   if 3.79999999999999983e-59 < z

   1. Initial program 98.6%

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

   3. Taylor expanded in z around inf 74.9%

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

      1. +-commutative 74.9%

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

   5. Simplified 74.9%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.2 \cdot 10^{+79}:\\ \;\;\;\;\left(x + y\right) \cdot \left(1 + \frac{y}{z}\right)\\ \mathbf{elif}\;z \leq -9.6 \cdot 10^{+64}:\\ \;\;\;\;\frac{-z}{\frac{y}{x + y}}\\ \mathbf{elif}\;z \leq -3 \cdot 10^{-14}:\\ \;\;\;\;\left(x + y\right) \cdot \left(1 + \frac{y}{z}\right)\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{-59}:\\ \;\;\;\;\frac{x \cdot \left(-z\right)}{y} - z\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 70.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{y}{z}\\ t_1 := \frac{y}{t\_0}\\ \mathbf{if}\;y \leq -2.9 \cdot 10^{+202}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq -4.8 \cdot 10^{+19}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{+33}:\\ \;\;\;\;\frac{x}{t\_0}\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{+157}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ y z))) (t_1 (/ y t_0)))
   (if (<= y -2.9e+202)
     (- z)
     (if (<= y -4.8e+19)
       t_1
       (if (<= y 1.7e+33) (/ x t_0) (if (<= y 4.5e+157) 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 <= -2.9e+202) {
		tmp = -z;
	} else if (y <= -4.8e+19) {
		tmp = t_1;
	} else if (y <= 1.7e+33) {
		tmp = x / t_0;
	} else if (y <= 4.5e+157) {
		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 <= (-2.9d+202)) then
        tmp = -z
    else if (y <= (-4.8d+19)) then
        tmp = t_1
    else if (y <= 1.7d+33) then
        tmp = x / t_0
    else if (y <= 4.5d+157) 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 <= -2.9e+202) {
		tmp = -z;
	} else if (y <= -4.8e+19) {
		tmp = t_1;
	} else if (y <= 1.7e+33) {
		tmp = x / t_0;
	} else if (y <= 4.5e+157) {
		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 <= -2.9e+202:
		tmp = -z
	elif y <= -4.8e+19:
		tmp = t_1
	elif y <= 1.7e+33:
		tmp = x / t_0
	elif y <= 4.5e+157:
		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 <= -2.9e+202)
		tmp = Float64(-z);
	elseif (y <= -4.8e+19)
		tmp = t_1;
	elseif (y <= 1.7e+33)
		tmp = Float64(x / t_0);
	elseif (y <= 4.5e+157)
		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 <= -2.9e+202)
		tmp = -z;
	elseif (y <= -4.8e+19)
		tmp = t_1;
	elseif (y <= 1.7e+33)
		tmp = x / t_0;
	elseif (y <= 4.5e+157)
		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, -2.9e+202], (-z), If[LessEqual[y, -4.8e+19], t$95$1, If[LessEqual[y, 1.7e+33], N[(x / t$95$0), $MachinePrecision], If[LessEqual[y, 4.5e+157], t$95$1, (-z)]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.8999999999999999e202 or 4.49999999999999985e157 < y

   1. Initial program 54.4%

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

   3. Taylor expanded in y around inf 81.2%

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

      1. mul-1-neg 81.2%

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

   5. Simplified 81.2%

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

   if -2.8999999999999999e202 < y < -4.8e19 or 1.7e33 < y < 4.49999999999999985e157

   1. Initial program 87.0%

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

   3. Taylor expanded in x around 0 63.2%

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

   if -4.8e19 < y < 1.7e33

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf 72.3%

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

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.9 \cdot 10^{+202}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq -4.8 \cdot 10^{+19}:\\ \;\;\;\;\frac{y}{1 - \frac{y}{z}}\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{+33}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{+157}:\\ \;\;\;\;\frac{y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 65.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.1 \cdot 10^{+163} \lor \neg \left(y \leq -4.3 \cdot 10^{+108}\right) \land \left(y \leq -2.8 \cdot 10^{-38} \lor \neg \left(y \leq 1.9 \cdot 10^{+129}\right)\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -4.1e+163)
         (and (not (<= y -4.3e+108))
              (or (<= y -2.8e-38) (not (<= y 1.9e+129)))))
   (- z)
   (+ x y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -4.1e+163) || (!(y <= -4.3e+108) && ((y <= -2.8e-38) || !(y <= 1.9e+129)))) {
		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 <= (-4.1d+163)) .or. (.not. (y <= (-4.3d+108))) .and. (y <= (-2.8d-38)) .or. (.not. (y <= 1.9d+129))) 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 <= -4.1e+163) || (!(y <= -4.3e+108) && ((y <= -2.8e-38) || !(y <= 1.9e+129)))) {
		tmp = -z;
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -4.1e+163) or (not (y <= -4.3e+108) and ((y <= -2.8e-38) or not (y <= 1.9e+129))):
		tmp = -z
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -4.1e+163) || (!(y <= -4.3e+108) && ((y <= -2.8e-38) || !(y <= 1.9e+129))))
		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 <= -4.1e+163) || (~((y <= -4.3e+108)) && ((y <= -2.8e-38) || ~((y <= 1.9e+129)))))
		tmp = -z;
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -4.1e+163], And[N[Not[LessEqual[y, -4.3e+108]], $MachinePrecision], Or[LessEqual[y, -2.8e-38], N[Not[LessEqual[y, 1.9e+129]], $MachinePrecision]]]], (-z), N[(x + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.0999999999999999e163 or -4.29999999999999996e108 < y < -2.8e-38 or 1.90000000000000003e129 < y

   1. Initial program 66.7%

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

   3. Taylor expanded in y around inf 63.9%

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

      1. mul-1-neg 63.9%

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

   5. Simplified 63.9%

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

   if -4.0999999999999999e163 < y < -4.29999999999999996e108 or -2.8e-38 < y < 1.90000000000000003e129

   1. Initial program 99.8%

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

   3. Taylor expanded in z around inf 69.7%

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

      1. +-commutative 69.7%

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

   5. Simplified 69.7%

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

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.1 \cdot 10^{+163} \lor \neg \left(y \leq -4.3 \cdot 10^{+108}\right) \land \left(y \leq -2.8 \cdot 10^{-38} \lor \neg \left(y \leq 1.9 \cdot 10^{+129}\right)\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 72.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.8 \cdot 10^{-21} \lor \neg \left(z \leq 2.3 \cdot 10^{-59}\right):\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-\frac{x + y}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -4.8e-21) (not (<= z 2.3e-59)))
   (+ x y)
   (* z (- (/ (+ x y) y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -4.8e-21) || !(z <= 2.3e-59)) {
		tmp = x + y;
	} 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) :: tmp
    if ((z <= (-4.8d-21)) .or. (.not. (z <= 2.3d-59))) then
        tmp = x + y
    else
        tmp = z * -((x + y) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -4.8e-21) || !(z <= 2.3e-59)) {
		tmp = x + y;
	} else {
		tmp = z * -((x + y) / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -4.8e-21) or not (z <= 2.3e-59):
		tmp = x + y
	else:
		tmp = z * -((x + y) / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -4.8e-21) || !(z <= 2.3e-59))
		tmp = Float64(x + y);
	else
		tmp = Float64(z * Float64(-Float64(Float64(x + y) / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -4.8e-21) || ~((z <= 2.3e-59)))
		tmp = x + y;
	else
		tmp = z * -((x + y) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -4.8e-21], N[Not[LessEqual[z, 2.3e-59]], $MachinePrecision]], N[(x + y), $MachinePrecision], N[(z * (-N[(N[(x + y), $MachinePrecision] / y), $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4.7999999999999999e-21 or 2.29999999999999979e-59 < z

   1. Initial program 99.2%

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

   3. Taylor expanded in z around inf 73.8%

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

      1. +-commutative 73.8%

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

   5. Simplified 73.8%

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

   if -4.7999999999999999e-21 < z < 2.29999999999999979e-59

   1. Initial program 72.4%

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

      1. clear-num 72.2%

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

      2. inv-pow 72.2%

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

   4. Applied egg-rr 72.2%

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

   5. Taylor expanded in z around 0 77.7%

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

      1. associate-*r/ 77.7%

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

      2. +-commutative 77.7%

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

      3. associate-*r* 77.7%

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

      4. associate-*r/ 80.5%

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

      5. mul-1-neg 80.5%

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

      6. +-commutative 80.5%

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

   7. Simplified 80.5%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.8 \cdot 10^{-21} \lor \neg \left(z \leq 2.3 \cdot 10^{-59}\right):\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-\frac{x + y}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.4 \cdot 10^{-15} \lor \neg \left(z \leq 3.8 \cdot 10^{-59}\right):\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(-z\right)}{y} - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -5.4e-15) (not (<= z 3.8e-59)))
   (+ x y)
   (- (/ (* x (- z)) y) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -5.4e-15) || !(z <= 3.8e-59)) {
		tmp = x + y;
	} else {
		tmp = ((x * -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) :: tmp
    if ((z <= (-5.4d-15)) .or. (.not. (z <= 3.8d-59))) then
        tmp = x + y
    else
        tmp = ((x * -z) / y) - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -5.4e-15) || !(z <= 3.8e-59)) {
		tmp = x + y;
	} else {
		tmp = ((x * -z) / y) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -5.4e-15) or not (z <= 3.8e-59):
		tmp = x + y
	else:
		tmp = ((x * -z) / y) - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -5.4e-15) || !(z <= 3.8e-59))
		tmp = Float64(x + y);
	else
		tmp = Float64(Float64(Float64(x * Float64(-z)) / y) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -5.4e-15) || ~((z <= 3.8e-59)))
		tmp = x + y;
	else
		tmp = ((x * -z) / y) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -5.4e-15], N[Not[LessEqual[z, 3.8e-59]], $MachinePrecision]], N[(x + y), $MachinePrecision], N[(N[(N[(x * (-z)), $MachinePrecision] / y), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -5.40000000000000018e-15 or 3.79999999999999983e-59 < z

   1. Initial program 99.2%

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

   3. Taylor expanded in z around inf 73.8%

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

      1. +-commutative 73.8%

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

   5. Simplified 73.8%

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

   if -5.40000000000000018e-15 < z < 3.79999999999999983e-59

   1. Initial program 72.4%

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

   3. Taylor expanded in z around 0 77.7%

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

      1. mul-1-neg 77.7%

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

      2. associate-/l* 80.6%

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

      3. distribute-neg-frac 80.6%

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

      4. +-commutative 80.6%

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

   5. Simplified 80.6%

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

   6. Taylor expanded in y around 0 80.8%

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

      1. mul-1-neg 80.8%

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

      2. associate-*r/ 76.9%

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

      3. unsub-neg 76.9%

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

      4. mul-1-neg 76.9%

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

      5. associate-*r/ 80.8%

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

   8. Simplified 80.8%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.4 \cdot 10^{-15} \lor \neg \left(z \leq 3.8 \cdot 10^{-59}\right):\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(-z\right)}{y} - z\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 58.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.7 \cdot 10^{-40} \lor \neg \left(y \leq 7 \cdot 10^{+22}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -3.7e-40) (not (<= y 7e+22))) (- z) x))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -3.7e-40) || !(y <= 7e+22)) {
		tmp = -z;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((y <= (-3.7d-40)) .or. (.not. (y <= 7d+22))) then
        tmp = -z
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -3.7e-40) || !(y <= 7e+22)) {
		tmp = -z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -3.7e-40) or not (y <= 7e+22):
		tmp = -z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -3.7e-40) || !(y <= 7e+22))
		tmp = Float64(-z);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -3.7e-40) || ~((y <= 7e+22)))
		tmp = -z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -3.7e-40], N[Not[LessEqual[y, 7e+22]], $MachinePrecision]], (-z), x]

Derivation

1. Split input into 2 regimes

2. if y < -3.69999999999999998e-40 or 7e22 < y

   1. Initial program 73.0%

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

   3. Taylor expanded in y around inf 56.8%

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

      1. mul-1-neg 56.8%

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

   5. Simplified 56.8%

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

   if -3.69999999999999998e-40 < y < 7e22

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0 54.1%

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

4. Final simplification 55.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.7 \cdot 10^{-40} \lor \neg \left(y \leq 7 \cdot 10^{+22}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 41.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.5 \cdot 10^{-132}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 3.25 \cdot 10^{-111}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -7.5e-132) x (if (<= x 3.25e-111) y x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -7.5e-132) {
		tmp = x;
	} else if (x <= 3.25e-111) {
		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 <= (-7.5d-132)) then
        tmp = x
    else if (x <= 3.25d-111) 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 <= -7.5e-132) {
		tmp = x;
	} else if (x <= 3.25e-111) {
		tmp = y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -7.5e-132:
		tmp = x
	elif x <= 3.25e-111:
		tmp = y
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -7.5e-132)
		tmp = x;
	elseif (x <= 3.25e-111)
		tmp = y;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -7.5e-132)
		tmp = x;
	elseif (x <= 3.25e-111)
		tmp = y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -7.5e-132], x, If[LessEqual[x, 3.25e-111], y, x]]

Derivation

1. Split input into 2 regimes

2. if x < -7.49999999999999989e-132 or 3.24999999999999987e-111 < x

   1. Initial program 86.1%

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

   3. Taylor expanded in y around 0 37.1%

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

   if -7.49999999999999989e-132 < x < 3.24999999999999987e-111

   1. Initial program 87.8%

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

   3. Taylor expanded in x around 0 75.8%

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

   4. Taylor expanded in y around 0 43.4%

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

4. Final simplification 39.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.5 \cdot 10^{-132}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 3.25 \cdot 10^{-111}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 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 86.6%

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

3. Taylor expanded in y around 0 30.2%

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

4. Final simplification 30.2%

   \[\leadsto x \]
5. Add Preprocessing

Developer target: 93.4% 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 2024039 
(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))))