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

Percentage Accurate: 88.0% → 99.1%

Time: 10.0s

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: 88.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{-226} \lor \neg \left(t_0 \leq 0\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-z}{\frac{y}{x + y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (+ x y) (- 1.0 (/ y z)))))
   (if (or (<= t_0 -5e-226) (not (<= t_0 0.0))) t_0 (/ (- z) (/ y (+ x y))))))

C

double code(double x, double y, double z) {
	double t_0 = (x + y) / (1.0 - (y / z));
	double tmp;
	if ((t_0 <= -5e-226) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = -z / (y / (x + y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x + y) / (1.0d0 - (y / z))
    if ((t_0 <= (-5d-226)) .or. (.not. (t_0 <= 0.0d0))) then
        tmp = t_0
    else
        tmp = -z / (y / (x + y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (x + y) / (1.0 - (y / z));
	double tmp;
	if ((t_0 <= -5e-226) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = -z / (y / (x + y));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x + y) / (1.0 - (y / z))
	tmp = 0
	if (t_0 <= -5e-226) or not (t_0 <= 0.0):
		tmp = t_0
	else:
		tmp = -z / (y / (x + y))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x + y) / Float64(1.0 - Float64(y / z)))
	tmp = 0.0
	if ((t_0 <= -5e-226) || !(t_0 <= 0.0))
		tmp = t_0;
	else
		tmp = Float64(Float64(-z) / Float64(y / Float64(x + y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x + y) / (1.0 - (y / z));
	tmp = 0.0;
	if ((t_0 <= -5e-226) || ~((t_0 <= 0.0)))
		tmp = t_0;
	else
		tmp = -z / (y / (x + y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x + y), $MachinePrecision] / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -5e-226], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], t$95$0, N[((-z) / N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z))) < -4.9999999999999998e-226 or 0.0 < (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z)))

   1. Initial program 99.9%

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

   if -4.9999999999999998e-226 < (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z))) < 0.0

   1. Initial program 5.7%

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

   2. Taylor expanded in z around 0 99.9%

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

      1. mul-1-neg 99.9%

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

      2. associate-/l* 99.9%

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

      3. distribute-neg-frac 99.9%

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

      4. +-commutative 99.9%

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

   4. Simplified 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \leq -5 \cdot 10^{-226} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \leq 0\right):\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-z}{\frac{y}{x + y}}\\ \end{array} \]

Alternative 2: 75.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{y}{z}\\ t_1 := \frac{x}{t_0}\\ t_2 := \left(x + y\right) \cdot \left(1 + \frac{y}{z}\right)\\ \mathbf{if}\;y \leq -9.8 \cdot 10^{+112}:\\ \;\;\;\;z \cdot \frac{-\left(x + y\right)}{y}\\ \mathbf{elif}\;y \leq -860000000:\\ \;\;\;\;\frac{y}{t_0}\\ \mathbf{elif}\;y \leq -4.7 \cdot 10^{-49}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -1.35 \cdot 10^{-194}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{-133}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 0.027:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{-z}{\frac{y}{x + y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ y z)))
        (t_1 (/ x t_0))
        (t_2 (* (+ x y) (+ 1.0 (/ y z)))))
   (if (<= y -9.8e+112)
     (* z (/ (- (+ x y)) y))
     (if (<= y -860000000.0)
       (/ y t_0)
       (if (<= y -4.7e-49)
         t_1
         (if (<= y -1.35e-194)
           t_2
           (if (<= y 1.25e-133)
             t_1
             (if (<= y 0.027) t_2 (/ (- z) (/ y (+ x y)))))))))))

C

double code(double x, double y, double z) {
	double t_0 = 1.0 - (y / z);
	double t_1 = x / t_0;
	double t_2 = (x + y) * (1.0 + (y / z));
	double tmp;
	if (y <= -9.8e+112) {
		tmp = z * (-(x + y) / y);
	} else if (y <= -860000000.0) {
		tmp = y / t_0;
	} else if (y <= -4.7e-49) {
		tmp = t_1;
	} else if (y <= -1.35e-194) {
		tmp = t_2;
	} else if (y <= 1.25e-133) {
		tmp = t_1;
	} else if (y <= 0.027) {
		tmp = t_2;
	} else {
		tmp = -z / (y / (x + y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = 1.0d0 - (y / z)
    t_1 = x / t_0
    t_2 = (x + y) * (1.0d0 + (y / z))
    if (y <= (-9.8d+112)) then
        tmp = z * (-(x + y) / y)
    else if (y <= (-860000000.0d0)) then
        tmp = y / t_0
    else if (y <= (-4.7d-49)) then
        tmp = t_1
    else if (y <= (-1.35d-194)) then
        tmp = t_2
    else if (y <= 1.25d-133) then
        tmp = t_1
    else if (y <= 0.027d0) then
        tmp = t_2
    else
        tmp = -z / (y / (x + y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = 1.0 - (y / z);
	double t_1 = x / t_0;
	double t_2 = (x + y) * (1.0 + (y / z));
	double tmp;
	if (y <= -9.8e+112) {
		tmp = z * (-(x + y) / y);
	} else if (y <= -860000000.0) {
		tmp = y / t_0;
	} else if (y <= -4.7e-49) {
		tmp = t_1;
	} else if (y <= -1.35e-194) {
		tmp = t_2;
	} else if (y <= 1.25e-133) {
		tmp = t_1;
	} else if (y <= 0.027) {
		tmp = t_2;
	} else {
		tmp = -z / (y / (x + y));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 1.0 - (y / z)
	t_1 = x / t_0
	t_2 = (x + y) * (1.0 + (y / z))
	tmp = 0
	if y <= -9.8e+112:
		tmp = z * (-(x + y) / y)
	elif y <= -860000000.0:
		tmp = y / t_0
	elif y <= -4.7e-49:
		tmp = t_1
	elif y <= -1.35e-194:
		tmp = t_2
	elif y <= 1.25e-133:
		tmp = t_1
	elif y <= 0.027:
		tmp = t_2
	else:
		tmp = -z / (y / (x + y))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(1.0 - Float64(y / z))
	t_1 = Float64(x / t_0)
	t_2 = Float64(Float64(x + y) * Float64(1.0 + Float64(y / z)))
	tmp = 0.0
	if (y <= -9.8e+112)
		tmp = Float64(z * Float64(Float64(-Float64(x + y)) / y));
	elseif (y <= -860000000.0)
		tmp = Float64(y / t_0);
	elseif (y <= -4.7e-49)
		tmp = t_1;
	elseif (y <= -1.35e-194)
		tmp = t_2;
	elseif (y <= 1.25e-133)
		tmp = t_1;
	elseif (y <= 0.027)
		tmp = t_2;
	else
		tmp = Float64(Float64(-z) / Float64(y / Float64(x + y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 1.0 - (y / z);
	t_1 = x / t_0;
	t_2 = (x + y) * (1.0 + (y / z));
	tmp = 0.0;
	if (y <= -9.8e+112)
		tmp = z * (-(x + y) / y);
	elseif (y <= -860000000.0)
		tmp = y / t_0;
	elseif (y <= -4.7e-49)
		tmp = t_1;
	elseif (y <= -1.35e-194)
		tmp = t_2;
	elseif (y <= 1.25e-133)
		tmp = t_1;
	elseif (y <= 0.027)
		tmp = t_2;
	else
		tmp = -z / (y / (x + y));
	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[(x / t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x + y), $MachinePrecision] * N[(1.0 + N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -9.8e+112], N[(z * N[((-N[(x + y), $MachinePrecision]) / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -860000000.0], N[(y / t$95$0), $MachinePrecision], If[LessEqual[y, -4.7e-49], t$95$1, If[LessEqual[y, -1.35e-194], t$95$2, If[LessEqual[y, 1.25e-133], t$95$1, If[LessEqual[y, 0.027], t$95$2, N[((-z) / N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if y < -9.80000000000000008e112

   1. Initial program 67.2%

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

      1. sub-neg 67.2%

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

      2. +-commutative 67.2%

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

      3. div-inv 67.0%

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

      4. distribute-lft-neg-in 67.0%

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

      5. fma-def 67.0%

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

   3. Applied egg-rr 67.0%

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

   4. Taylor expanded in z around 0 62.6%

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

      1. mul-1-neg 62.6%

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

      2. associate-*r/ 86.7%

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

      3. distribute-rgt-neg-in 86.7%

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

      4. +-commutative 86.7%

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

   6. Simplified 86.7%

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

   if -9.80000000000000008e112 < y < -8.6e8

   1. Initial program 95.4%

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

   2. Taylor expanded in x around 0 79.7%

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

   if -8.6e8 < y < -4.70000000000000021e-49 or -1.35e-194 < y < 1.25e-133

   1. Initial program 99.9%

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

   2. Taylor expanded in x around inf 88.2%

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

   if -4.70000000000000021e-49 < y < -1.35e-194 or 1.25e-133 < y < 0.0269999999999999997

   1. Initial program 99.9%

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

   2. Taylor expanded in z around inf 82.0%

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

      1. associate-+r+ 82.0%

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

      2. *-lft-identity 82.0%

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

      3. associate-/l* 80.7%

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

      4. associate-/r/ 82.0%

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

      5. distribute-rgt-in 82.0%

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

      6. +-commutative 82.0%

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

   4. Simplified 82.0%

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

   if 0.0269999999999999997 < y

   1. Initial program 72.4%

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

   2. Taylor expanded in z around 0 66.5%

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

      1. mul-1-neg 66.5%

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

      2. associate-/l* 82.5%

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

      3. distribute-neg-frac 82.5%

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

      4. +-commutative 82.5%

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

   4. Simplified 82.5%

      \[\leadsto \color{blue}{\frac{-z}{\frac{y}{y + x}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9.8 \cdot 10^{+112}:\\ \;\;\;\;z \cdot \frac{-\left(x + y\right)}{y}\\ \mathbf{elif}\;y \leq -860000000:\\ \;\;\;\;\frac{y}{1 - \frac{y}{z}}\\ \mathbf{elif}\;y \leq -4.7 \cdot 10^{-49}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{elif}\;y \leq -1.35 \cdot 10^{-194}:\\ \;\;\;\;\left(x + y\right) \cdot \left(1 + \frac{y}{z}\right)\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{-133}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{elif}\;y \leq 0.027:\\ \;\;\;\;\left(x + y\right) \cdot \left(1 + \frac{y}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-z}{\frac{y}{x + y}}\\ \end{array} \]

Alternative 3: 75.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{y}{z}\\ t_1 := z \cdot \frac{-\left(x + y\right)}{y}\\ t_2 := \frac{x}{t_0}\\ \mathbf{if}\;y \leq -5 \cdot 10^{+113}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -140000000000:\\ \;\;\;\;\frac{y}{t_0}\\ \mathbf{elif}\;y \leq -2.25 \cdot 10^{-48}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -1.4 \cdot 10^{-194}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{-133}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 0.001:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ y z))) (t_1 (* z (/ (- (+ x y)) y))) (t_2 (/ x t_0)))
   (if (<= y -5e+113)
     t_1
     (if (<= y -140000000000.0)
       (/ y t_0)
       (if (<= y -2.25e-48)
         t_2
         (if (<= y -1.4e-194)
           (+ x y)
           (if (<= y 1.25e-133) t_2 (if (<= y 0.001) (+ x y) t_1))))))))

C

double code(double x, double y, double z) {
	double t_0 = 1.0 - (y / z);
	double t_1 = z * (-(x + y) / y);
	double t_2 = x / t_0;
	double tmp;
	if (y <= -5e+113) {
		tmp = t_1;
	} else if (y <= -140000000000.0) {
		tmp = y / t_0;
	} else if (y <= -2.25e-48) {
		tmp = t_2;
	} else if (y <= -1.4e-194) {
		tmp = x + y;
	} else if (y <= 1.25e-133) {
		tmp = t_2;
	} else if (y <= 0.001) {
		tmp = x + y;
	} else {
		tmp = t_1;
	}
	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) :: t_2
    real(8) :: tmp
    t_0 = 1.0d0 - (y / z)
    t_1 = z * (-(x + y) / y)
    t_2 = x / t_0
    if (y <= (-5d+113)) then
        tmp = t_1
    else if (y <= (-140000000000.0d0)) then
        tmp = y / t_0
    else if (y <= (-2.25d-48)) then
        tmp = t_2
    else if (y <= (-1.4d-194)) then
        tmp = x + y
    else if (y <= 1.25d-133) then
        tmp = t_2
    else if (y <= 0.001d0) then
        tmp = x + y
    else
        tmp = t_1
    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 = z * (-(x + y) / y);
	double t_2 = x / t_0;
	double tmp;
	if (y <= -5e+113) {
		tmp = t_1;
	} else if (y <= -140000000000.0) {
		tmp = y / t_0;
	} else if (y <= -2.25e-48) {
		tmp = t_2;
	} else if (y <= -1.4e-194) {
		tmp = x + y;
	} else if (y <= 1.25e-133) {
		tmp = t_2;
	} else if (y <= 0.001) {
		tmp = x + y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 1.0 - (y / z)
	t_1 = z * (-(x + y) / y)
	t_2 = x / t_0
	tmp = 0
	if y <= -5e+113:
		tmp = t_1
	elif y <= -140000000000.0:
		tmp = y / t_0
	elif y <= -2.25e-48:
		tmp = t_2
	elif y <= -1.4e-194:
		tmp = x + y
	elif y <= 1.25e-133:
		tmp = t_2
	elif y <= 0.001:
		tmp = x + y
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(1.0 - Float64(y / z))
	t_1 = Float64(z * Float64(Float64(-Float64(x + y)) / y))
	t_2 = Float64(x / t_0)
	tmp = 0.0
	if (y <= -5e+113)
		tmp = t_1;
	elseif (y <= -140000000000.0)
		tmp = Float64(y / t_0);
	elseif (y <= -2.25e-48)
		tmp = t_2;
	elseif (y <= -1.4e-194)
		tmp = Float64(x + y);
	elseif (y <= 1.25e-133)
		tmp = t_2;
	elseif (y <= 0.001)
		tmp = Float64(x + y);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 1.0 - (y / z);
	t_1 = z * (-(x + y) / y);
	t_2 = x / t_0;
	tmp = 0.0;
	if (y <= -5e+113)
		tmp = t_1;
	elseif (y <= -140000000000.0)
		tmp = y / t_0;
	elseif (y <= -2.25e-48)
		tmp = t_2;
	elseif (y <= -1.4e-194)
		tmp = x + y;
	elseif (y <= 1.25e-133)
		tmp = t_2;
	elseif (y <= 0.001)
		tmp = x + y;
	else
		tmp = t_1;
	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[(z * N[((-N[(x + y), $MachinePrecision]) / y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x / t$95$0), $MachinePrecision]}, If[LessEqual[y, -5e+113], t$95$1, If[LessEqual[y, -140000000000.0], N[(y / t$95$0), $MachinePrecision], If[LessEqual[y, -2.25e-48], t$95$2, If[LessEqual[y, -1.4e-194], N[(x + y), $MachinePrecision], If[LessEqual[y, 1.25e-133], t$95$2, If[LessEqual[y, 0.001], N[(x + y), $MachinePrecision], t$95$1]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -5e113 or 1e-3 < y

   1. Initial program 69.8%

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

      1. sub-neg 69.8%

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

      2. +-commutative 69.8%

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

      3. div-inv 69.7%

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

      4. distribute-lft-neg-in 69.7%

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

      5. fma-def 69.7%

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

   3. Applied egg-rr 69.7%

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

   4. Taylor expanded in z around 0 64.6%

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

      1. mul-1-neg 64.6%

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

      2. associate-*r/ 84.5%

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

      3. distribute-rgt-neg-in 84.5%

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

      4. +-commutative 84.5%

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

   6. Simplified 84.5%

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

   if -5e113 < y < -1.4e11

   1. Initial program 95.4%

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

   2. Taylor expanded in x around 0 79.7%

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

   if -1.4e11 < y < -2.24999999999999994e-48 or -1.40000000000000006e-194 < y < 1.25e-133

   1. Initial program 99.9%

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

   2. Taylor expanded in x around inf 88.2%

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

   if -2.24999999999999994e-48 < y < -1.40000000000000006e-194 or 1.25e-133 < y < 1e-3

   1. Initial program 99.9%

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

   2. Taylor expanded in z around inf 81.5%

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

      1. +-commutative 81.5%

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

   4. Simplified 81.5%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+113}:\\ \;\;\;\;z \cdot \frac{-\left(x + y\right)}{y}\\ \mathbf{elif}\;y \leq -140000000000:\\ \;\;\;\;\frac{y}{1 - \frac{y}{z}}\\ \mathbf{elif}\;y \leq -2.25 \cdot 10^{-48}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{elif}\;y \leq -1.4 \cdot 10^{-194}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{-133}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{elif}\;y \leq 0.001:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{-\left(x + y\right)}{y}\\ \end{array} \]

Alternative 4: 75.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{y}{z}\\ t_1 := \frac{x}{t_0}\\ \mathbf{if}\;y \leq -3.5 \cdot 10^{+113}:\\ \;\;\;\;z \cdot \frac{-\left(x + y\right)}{y}\\ \mathbf{elif}\;y \leq -80000000000000:\\ \;\;\;\;\frac{y}{t_0}\\ \mathbf{elif}\;y \leq -6.4 \cdot 10^{-51}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -1.35 \cdot 10^{-194}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{-132}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 2100000000:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;\frac{-z}{\frac{y}{x + y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ y z))) (t_1 (/ x t_0)))
   (if (<= y -3.5e+113)
     (* z (/ (- (+ x y)) y))
     (if (<= y -80000000000000.0)
       (/ y t_0)
       (if (<= y -6.4e-51)
         t_1
         (if (<= y -1.35e-194)
           (+ x y)
           (if (<= y 1.15e-132)
             t_1
             (if (<= y 2100000000.0) (+ x y) (/ (- z) (/ y (+ x y)))))))))))

C

double code(double x, double y, double z) {
	double t_0 = 1.0 - (y / z);
	double t_1 = x / t_0;
	double tmp;
	if (y <= -3.5e+113) {
		tmp = z * (-(x + y) / y);
	} else if (y <= -80000000000000.0) {
		tmp = y / t_0;
	} else if (y <= -6.4e-51) {
		tmp = t_1;
	} else if (y <= -1.35e-194) {
		tmp = x + y;
	} else if (y <= 1.15e-132) {
		tmp = t_1;
	} else if (y <= 2100000000.0) {
		tmp = x + y;
	} else {
		tmp = -z / (y / (x + y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 - (y / z)
    t_1 = x / t_0
    if (y <= (-3.5d+113)) then
        tmp = z * (-(x + y) / y)
    else if (y <= (-80000000000000.0d0)) then
        tmp = y / t_0
    else if (y <= (-6.4d-51)) then
        tmp = t_1
    else if (y <= (-1.35d-194)) then
        tmp = x + y
    else if (y <= 1.15d-132) then
        tmp = t_1
    else if (y <= 2100000000.0d0) then
        tmp = x + y
    else
        tmp = -z / (y / (x + y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = 1.0 - (y / z);
	double t_1 = x / t_0;
	double tmp;
	if (y <= -3.5e+113) {
		tmp = z * (-(x + y) / y);
	} else if (y <= -80000000000000.0) {
		tmp = y / t_0;
	} else if (y <= -6.4e-51) {
		tmp = t_1;
	} else if (y <= -1.35e-194) {
		tmp = x + y;
	} else if (y <= 1.15e-132) {
		tmp = t_1;
	} else if (y <= 2100000000.0) {
		tmp = x + y;
	} else {
		tmp = -z / (y / (x + y));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 1.0 - (y / z)
	t_1 = x / t_0
	tmp = 0
	if y <= -3.5e+113:
		tmp = z * (-(x + y) / y)
	elif y <= -80000000000000.0:
		tmp = y / t_0
	elif y <= -6.4e-51:
		tmp = t_1
	elif y <= -1.35e-194:
		tmp = x + y
	elif y <= 1.15e-132:
		tmp = t_1
	elif y <= 2100000000.0:
		tmp = x + y
	else:
		tmp = -z / (y / (x + y))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(1.0 - Float64(y / z))
	t_1 = Float64(x / t_0)
	tmp = 0.0
	if (y <= -3.5e+113)
		tmp = Float64(z * Float64(Float64(-Float64(x + y)) / y));
	elseif (y <= -80000000000000.0)
		tmp = Float64(y / t_0);
	elseif (y <= -6.4e-51)
		tmp = t_1;
	elseif (y <= -1.35e-194)
		tmp = Float64(x + y);
	elseif (y <= 1.15e-132)
		tmp = t_1;
	elseif (y <= 2100000000.0)
		tmp = Float64(x + y);
	else
		tmp = Float64(Float64(-z) / Float64(y / Float64(x + y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 1.0 - (y / z);
	t_1 = x / t_0;
	tmp = 0.0;
	if (y <= -3.5e+113)
		tmp = z * (-(x + y) / y);
	elseif (y <= -80000000000000.0)
		tmp = y / t_0;
	elseif (y <= -6.4e-51)
		tmp = t_1;
	elseif (y <= -1.35e-194)
		tmp = x + y;
	elseif (y <= 1.15e-132)
		tmp = t_1;
	elseif (y <= 2100000000.0)
		tmp = x + y;
	else
		tmp = -z / (y / (x + y));
	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[(x / t$95$0), $MachinePrecision]}, If[LessEqual[y, -3.5e+113], N[(z * N[((-N[(x + y), $MachinePrecision]) / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -80000000000000.0], N[(y / t$95$0), $MachinePrecision], If[LessEqual[y, -6.4e-51], t$95$1, If[LessEqual[y, -1.35e-194], N[(x + y), $MachinePrecision], If[LessEqual[y, 1.15e-132], t$95$1, If[LessEqual[y, 2100000000.0], N[(x + y), $MachinePrecision], N[((-z) / N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if y < -3.5000000000000001e113

   1. Initial program 67.2%

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

      1. sub-neg 67.2%

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

      2. +-commutative 67.2%

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

      3. div-inv 67.0%

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

      4. distribute-lft-neg-in 67.0%

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

      5. fma-def 67.0%

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

   3. Applied egg-rr 67.0%

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

   4. Taylor expanded in z around 0 62.6%

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

      1. mul-1-neg 62.6%

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

      2. associate-*r/ 86.7%

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

      3. distribute-rgt-neg-in 86.7%

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

      4. +-commutative 86.7%

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

   6. Simplified 86.7%

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

   if -3.5000000000000001e113 < y < -8e13

   1. Initial program 95.4%

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

   2. Taylor expanded in x around 0 79.7%

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

   if -8e13 < y < -6.4e-51 or -1.35e-194 < y < 1.15000000000000002e-132

   1. Initial program 99.9%

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

   2. Taylor expanded in x around inf 88.2%

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

   if -6.4e-51 < y < -1.35e-194 or 1.15000000000000002e-132 < y < 2.1e9

   1. Initial program 99.9%

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

   2. Taylor expanded in z around inf 81.5%

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

      1. +-commutative 81.5%

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

   4. Simplified 81.5%

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

   if 2.1e9 < y

   1. Initial program 72.4%

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

   2. Taylor expanded in z around 0 66.5%

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

      1. mul-1-neg 66.5%

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

      2. associate-/l* 82.5%

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

      3. distribute-neg-frac 82.5%

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

      4. +-commutative 82.5%

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

   4. Simplified 82.5%

      \[\leadsto \color{blue}{\frac{-z}{\frac{y}{y + x}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.5 \cdot 10^{+113}:\\ \;\;\;\;z \cdot \frac{-\left(x + y\right)}{y}\\ \mathbf{elif}\;y \leq -80000000000000:\\ \;\;\;\;\frac{y}{1 - \frac{y}{z}}\\ \mathbf{elif}\;y \leq -6.4 \cdot 10^{-51}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{elif}\;y \leq -1.35 \cdot 10^{-194}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{-132}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{elif}\;y \leq 2100000000:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;\frac{-z}{\frac{y}{x + y}}\\ \end{array} \]

Alternative 5: 69.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{y}{z}\\ \mathbf{if}\;y \leq -9.4 \cdot 10^{+144}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq -440000000000:\\ \;\;\;\;\frac{y}{t_0}\\ \mathbf{elif}\;y \leq -1.35 \cdot 10^{-194}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 1.85 \cdot 10^{-132}:\\ \;\;\;\;\frac{x}{t_0}\\ \mathbf{elif}\;y \leq 6.4 \cdot 10^{+53}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ y z))))
   (if (<= y -9.4e+144)
     (- z)
     (if (<= y -440000000000.0)
       (/ y t_0)
       (if (<= y -1.35e-194)
         (+ x y)
         (if (<= y 1.85e-132) (/ x t_0) (if (<= y 6.4e+53) (+ x y) (- z))))))))

C

double code(double x, double y, double z) {
	double t_0 = 1.0 - (y / z);
	double tmp;
	if (y <= -9.4e+144) {
		tmp = -z;
	} else if (y <= -440000000000.0) {
		tmp = y / t_0;
	} else if (y <= -1.35e-194) {
		tmp = x + y;
	} else if (y <= 1.85e-132) {
		tmp = x / t_0;
	} else if (y <= 6.4e+53) {
		tmp = x + y;
	} else {
		tmp = -z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - (y / z)
    if (y <= (-9.4d+144)) then
        tmp = -z
    else if (y <= (-440000000000.0d0)) then
        tmp = y / t_0
    else if (y <= (-1.35d-194)) then
        tmp = x + y
    else if (y <= 1.85d-132) then
        tmp = x / t_0
    else if (y <= 6.4d+53) then
        tmp = x + y
    else
        tmp = -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = 1.0 - (y / z);
	double tmp;
	if (y <= -9.4e+144) {
		tmp = -z;
	} else if (y <= -440000000000.0) {
		tmp = y / t_0;
	} else if (y <= -1.35e-194) {
		tmp = x + y;
	} else if (y <= 1.85e-132) {
		tmp = x / t_0;
	} else if (y <= 6.4e+53) {
		tmp = x + y;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 1.0 - (y / z)
	tmp = 0
	if y <= -9.4e+144:
		tmp = -z
	elif y <= -440000000000.0:
		tmp = y / t_0
	elif y <= -1.35e-194:
		tmp = x + y
	elif y <= 1.85e-132:
		tmp = x / t_0
	elif y <= 6.4e+53:
		tmp = x + y
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(1.0 - Float64(y / z))
	tmp = 0.0
	if (y <= -9.4e+144)
		tmp = Float64(-z);
	elseif (y <= -440000000000.0)
		tmp = Float64(y / t_0);
	elseif (y <= -1.35e-194)
		tmp = Float64(x + y);
	elseif (y <= 1.85e-132)
		tmp = Float64(x / t_0);
	elseif (y <= 6.4e+53)
		tmp = Float64(x + y);
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 1.0 - (y / z);
	tmp = 0.0;
	if (y <= -9.4e+144)
		tmp = -z;
	elseif (y <= -440000000000.0)
		tmp = y / t_0;
	elseif (y <= -1.35e-194)
		tmp = x + y;
	elseif (y <= 1.85e-132)
		tmp = x / t_0;
	elseif (y <= 6.4e+53)
		tmp = x + y;
	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]}, If[LessEqual[y, -9.4e+144], (-z), If[LessEqual[y, -440000000000.0], N[(y / t$95$0), $MachinePrecision], If[LessEqual[y, -1.35e-194], N[(x + y), $MachinePrecision], If[LessEqual[y, 1.85e-132], N[(x / t$95$0), $MachinePrecision], If[LessEqual[y, 6.4e+53], N[(x + y), $MachinePrecision], (-z)]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -9.4000000000000004e144 or 6.4e53 < y

   1. Initial program 66.1%

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

   2. Taylor expanded in y around inf 76.5%

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

      1. mul-1-neg 76.5%

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

   4. Simplified 76.5%

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

   if -9.4000000000000004e144 < y < -4.4e11

   1. Initial program 90.1%

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

   2. Taylor expanded in x around 0 75.4%

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

   if -4.4e11 < y < -1.35e-194 or 1.8500000000000001e-132 < y < 6.4e53

   1. Initial program 99.9%

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

   2. Taylor expanded in z around inf 74.7%

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

      1. +-commutative 74.7%

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

   4. Simplified 74.7%

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

   if -1.35e-194 < y < 1.8500000000000001e-132

   1. Initial program 99.9%

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

   2. Taylor expanded in x around inf 92.3%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9.4 \cdot 10^{+144}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq -440000000000:\\ \;\;\;\;\frac{y}{1 - \frac{y}{z}}\\ \mathbf{elif}\;y \leq -1.35 \cdot 10^{-194}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 1.85 \cdot 10^{-132}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{elif}\;y \leq 6.4 \cdot 10^{+53}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 6: 68.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.04 \cdot 10^{+26}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq -1.35 \cdot 10^{-194}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 3.4 \cdot 10^{-133}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{+53}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.04e+26)
   (- z)
   (if (<= y -1.35e-194)
     (+ x y)
     (if (<= y 3.4e-133)
       (/ x (- 1.0 (/ y z)))
       (if (<= y 1.9e+53) (+ x y) (- z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.04e+26) {
		tmp = -z;
	} else if (y <= -1.35e-194) {
		tmp = x + y;
	} else if (y <= 3.4e-133) {
		tmp = x / (1.0 - (y / z));
	} else if (y <= 1.9e+53) {
		tmp = x + y;
	} else {
		tmp = -z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-1.04d+26)) then
        tmp = -z
    else if (y <= (-1.35d-194)) then
        tmp = x + y
    else if (y <= 3.4d-133) then
        tmp = x / (1.0d0 - (y / z))
    else if (y <= 1.9d+53) then
        tmp = x + y
    else
        tmp = -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.04e+26) {
		tmp = -z;
	} else if (y <= -1.35e-194) {
		tmp = x + y;
	} else if (y <= 3.4e-133) {
		tmp = x / (1.0 - (y / z));
	} else if (y <= 1.9e+53) {
		tmp = x + y;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1.04e+26:
		tmp = -z
	elif y <= -1.35e-194:
		tmp = x + y
	elif y <= 3.4e-133:
		tmp = x / (1.0 - (y / z))
	elif y <= 1.9e+53:
		tmp = x + y
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.04e+26)
		tmp = Float64(-z);
	elseif (y <= -1.35e-194)
		tmp = Float64(x + y);
	elseif (y <= 3.4e-133)
		tmp = Float64(x / Float64(1.0 - Float64(y / z)));
	elseif (y <= 1.9e+53)
		tmp = Float64(x + y);
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -1.04e+26)
		tmp = -z;
	elseif (y <= -1.35e-194)
		tmp = x + y;
	elseif (y <= 3.4e-133)
		tmp = x / (1.0 - (y / z));
	elseif (y <= 1.9e+53)
		tmp = x + y;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.04e+26], (-z), If[LessEqual[y, -1.35e-194], N[(x + y), $MachinePrecision], If[LessEqual[y, 3.4e-133], N[(x / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.9e+53], N[(x + y), $MachinePrecision], (-z)]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.0399999999999999e26 or 1.89999999999999999e53 < y

   1. Initial program 71.8%

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

   2. Taylor expanded in y around inf 70.5%

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

      1. mul-1-neg 70.5%

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

   4. Simplified 70.5%

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

   if -1.0399999999999999e26 < y < -1.35e-194 or 3.40000000000000006e-133 < y < 1.89999999999999999e53

   1. Initial program 99.9%

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

   2. Taylor expanded in z around inf 74.5%

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

      1. +-commutative 74.5%

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

   4. Simplified 74.5%

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

   if -1.35e-194 < y < 3.40000000000000006e-133

   1. Initial program 99.9%

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

   2. Taylor expanded in x around inf 92.3%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.04 \cdot 10^{+26}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq -1.35 \cdot 10^{-194}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 3.4 \cdot 10^{-133}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{+53}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 7: 68.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.35 \cdot 10^{+26}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 1.12 \cdot 10^{+55}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2.35e+26) (- z) (if (<= y 1.12e+55) (+ x y) (- z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.35e+26) {
		tmp = -z;
	} else if (y <= 1.12e+55) {
		tmp = x + y;
	} else {
		tmp = -z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-2.35d+26)) then
        tmp = -z
    else if (y <= 1.12d+55) then
        tmp = x + y
    else
        tmp = -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.35e+26) {
		tmp = -z;
	} else if (y <= 1.12e+55) {
		tmp = x + y;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -2.35e+26:
		tmp = -z
	elif y <= 1.12e+55:
		tmp = x + y
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -2.35e+26)
		tmp = Float64(-z);
	elseif (y <= 1.12e+55)
		tmp = Float64(x + y);
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -2.35e+26)
		tmp = -z;
	elseif (y <= 1.12e+55)
		tmp = x + y;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -2.35e+26], (-z), If[LessEqual[y, 1.12e+55], N[(x + y), $MachinePrecision], (-z)]]

Derivation

1. Split input into 2 regimes

2. if y < -2.3499999999999999e26 or 1.12000000000000006e55 < y

   1. Initial program 71.8%

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

   2. Taylor expanded in y around inf 70.5%

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

      1. mul-1-neg 70.5%

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

   4. Simplified 70.5%

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

   if -2.3499999999999999e26 < y < 1.12000000000000006e55

   1. Initial program 99.9%

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

   2. Taylor expanded in z around inf 76.8%

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

      1. +-commutative 76.8%

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

   4. Simplified 76.8%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.35 \cdot 10^{+26}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 1.12 \cdot 10^{+55}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 8: 58.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -370000000000:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{+52}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -370000000000.0) (- z) (if (<= y 2.6e+52) x (- z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -370000000000.0) {
		tmp = -z;
	} else if (y <= 2.6e+52) {
		tmp = x;
	} 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 <= (-370000000000.0d0)) then
        tmp = -z
    else if (y <= 2.6d+52) then
        tmp = x
    else
        tmp = -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -370000000000.0) {
		tmp = -z;
	} else if (y <= 2.6e+52) {
		tmp = x;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -370000000000.0:
		tmp = -z
	elif y <= 2.6e+52:
		tmp = x
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -370000000000.0)
		tmp = Float64(-z);
	elseif (y <= 2.6e+52)
		tmp = x;
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -370000000000.0)
		tmp = -z;
	elseif (y <= 2.6e+52)
		tmp = x;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -370000000000.0], (-z), If[LessEqual[y, 2.6e+52], x, (-z)]]

Derivation

1. Split input into 2 regimes

2. if y < -3.7e11 or 2.6e52 < y

   1. Initial program 72.6%

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

   2. Taylor expanded in y around inf 69.5%

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

      1. mul-1-neg 69.5%

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

   4. Simplified 69.5%

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

   if -3.7e11 < y < 2.6e52

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 57.8%

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

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -370000000000:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{+52}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 9: 40.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.3 \cdot 10^{-181}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-116}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -2.3e-181) x (if (<= x 7.5e-116) y x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -2.3e-181) {
		tmp = x;
	} else if (x <= 7.5e-116) {
		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 <= (-2.3d-181)) then
        tmp = x
    else if (x <= 7.5d-116) 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 <= -2.3e-181) {
		tmp = x;
	} else if (x <= 7.5e-116) {
		tmp = y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -2.3e-181:
		tmp = x
	elif x <= 7.5e-116:
		tmp = y
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -2.3e-181)
		tmp = x;
	elseif (x <= 7.5e-116)
		tmp = y;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -2.3e-181)
		tmp = x;
	elseif (x <= 7.5e-116)
		tmp = y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -2.3e-181], x, If[LessEqual[x, 7.5e-116], y, x]]

Derivation

1. Split input into 2 regimes

2. if x < -2.29999999999999991e-181 or 7.5000000000000004e-116 < x

   1. Initial program 89.8%

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

   2. Taylor expanded in y around 0 47.4%

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

   if -2.29999999999999991e-181 < x < 7.5000000000000004e-116

   1. Initial program 85.4%

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

   2. Taylor expanded in x around 0 74.1%

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

   3. Taylor expanded in y around 0 44.3%

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

4. Final simplification 46.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.3 \cdot 10^{-181}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-116}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 10: 35.2% 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 88.5%

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

2. Taylor expanded in y around 0 36.2%

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

3. Final simplification 36.2%

   \[\leadsto x \]

Developer target: 93.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023275 
(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))))