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

Percentage Accurate: 88.1% → 99.7%

Time: 6.4s

Alternatives: 9

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: 88.1% 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: 99.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{-267} \lor \neg \left(t_0 \leq 0\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (+ x y) (- 1.0 (/ y z)))))
   (if (or (<= t_0 -2e-267) (not (<= t_0 0.0))) t_0 (* z (- -1.0 (/ x y))))))

C

double code(double x, double y, double z) {
	double t_0 = (x + y) / (1.0 - (y / z));
	double tmp;
	if ((t_0 <= -2e-267) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = z * (-1.0 - (x / y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x + y) / (1.0d0 - (y / z))
    if ((t_0 <= (-2d-267)) .or. (.not. (t_0 <= 0.0d0))) then
        tmp = t_0
    else
        tmp = z * ((-1.0d0) - (x / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (x + y) / (1.0 - (y / z));
	double tmp;
	if ((t_0 <= -2e-267) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = z * (-1.0 - (x / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x + y) / (1.0 - (y / z))
	tmp = 0
	if (t_0 <= -2e-267) or not (t_0 <= 0.0):
		tmp = t_0
	else:
		tmp = z * (-1.0 - (x / y))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x + y) / Float64(1.0 - Float64(y / z)))
	tmp = 0.0
	if ((t_0 <= -2e-267) || !(t_0 <= 0.0))
		tmp = t_0;
	else
		tmp = Float64(z * Float64(-1.0 - Float64(x / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x + y) / (1.0 - (y / z));
	tmp = 0.0;
	if ((t_0 <= -2e-267) || ~((t_0 <= 0.0)))
		tmp = t_0;
	else
		tmp = z * (-1.0 - (x / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x + y), $MachinePrecision] / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -2e-267], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], t$95$0, N[(z * N[(-1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z))) < -2e-267 or -0.0 < (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z)))

   1. Initial program 99.8%

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

   if -2e-267 < (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z))) < -0.0

   1. Initial program 15.5%

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

   2. Taylor expanded in y around inf 99.9%

      \[\leadsto \color{blue}{\left(-1 \cdot z + -1 \cdot \frac{x \cdot z}{y}\right) - \frac{{z}^{2}}{y}} \]
   3. Step-by-step derivation

      1. associate--l+ 99.9%

         \[\leadsto \color{blue}{-1 \cdot z + \left(-1 \cdot \frac{x \cdot z}{y} - \frac{{z}^{2}}{y}\right)} \]

      2. +-commutative 99.9%

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{x \cdot z}{y} - \frac{{z}^{2}}{y}\right) + -1 \cdot z} \]

      3. associate-*r/ 99.9%

         \[\leadsto \left(\color{blue}{\frac{-1 \cdot \left(x \cdot z\right)}{y}} - \frac{{z}^{2}}{y}\right) + -1 \cdot z \]

      4. div-sub 99.9%

         \[\leadsto \color{blue}{\frac{-1 \cdot \left(x \cdot z\right) - {z}^{2}}{y}} + -1 \cdot z \]

      5. sub-neg 99.9%

         \[\leadsto \frac{\color{blue}{-1 \cdot \left(x \cdot z\right) + \left(-{z}^{2}\right)}}{y} + -1 \cdot z \]

      6. mul-1-neg 99.9%

         \[\leadsto \frac{\color{blue}{\left(-x \cdot z\right)} + \left(-{z}^{2}\right)}{y} + -1 \cdot z \]

      7. distribute-neg-out 99.9%

         \[\leadsto \frac{\color{blue}{-\left(x \cdot z + {z}^{2}\right)}}{y} + -1 \cdot z \]

      8. *-lft-identity 99.9%

         \[\leadsto \frac{-\left(x \cdot z + \color{blue}{1 \cdot {z}^{2}}\right)}{y} + -1 \cdot z \]

      9. metadata-eval 99.9%

         \[\leadsto \frac{-\left(x \cdot z + \color{blue}{\left(--1\right)} \cdot {z}^{2}\right)}{y} + -1 \cdot z \]

      10. cancel-sign-sub-inv 99.9%

          \[\leadsto \frac{-\color{blue}{\left(x \cdot z - -1 \cdot {z}^{2}\right)}}{y} + -1 \cdot z \]

      11. distribute-neg-frac 99.9%

          \[\leadsto \color{blue}{\left(-\frac{x \cdot z - -1 \cdot {z}^{2}}{y}\right)} + -1 \cdot z \]

      12. mul-1-neg 99.9%

          \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot z - -1 \cdot {z}^{2}}{y}} + -1 \cdot z \]

      13. mul-1-neg 99.9%

          \[\leadsto -1 \cdot \frac{x \cdot z - -1 \cdot {z}^{2}}{y} + \color{blue}{\left(-z\right)} \]

      14. unsub-neg 99.9%

          \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot z - -1 \cdot {z}^{2}}{y} - z} \]

   4. Simplified 99.9%

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

   5. Taylor expanded in z around 0 99.9%

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

   6. Taylor expanded in x around 0 99.9%

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

      1. mul-1-neg 99.9%

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

      2. +-commutative 99.9%

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

      3. associate-*l/ 99.9%

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

      4. associate-*r* 99.9%

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

      5. mul-1-neg 99.9%

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

      6. distribute-rgt-in 99.9%

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

      7. +-commutative 99.9%

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

      8. mul-1-neg 99.9%

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

      9. unsub-neg 99.9%

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

   8. Simplified 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \leq -2 \cdot 10^{-267} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \leq 0\right):\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \end{array} \]

Alternative 2: 71.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-z\right) - \frac{x \cdot z}{y}\\ t_1 := \left(x + y\right) \cdot \left(1 + \frac{y}{z}\right)\\ \mathbf{if}\;z \leq -5 \cdot 10^{+21}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.55 \cdot 10^{-158}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 5.4 \cdot 10^{-99}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{elif}\;z \leq 2.8 \cdot 10^{+101}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (- z) (/ (* x z) y))) (t_1 (* (+ x y) (+ 1.0 (/ y z)))))
   (if (<= z -5e+21)
     t_1
     (if (<= z 1.55e-158)
       t_0
       (if (<= z 5.4e-99)
         (/ x (- 1.0 (/ y z)))
         (if (<= z 2.8e+101) t_0 t_1))))))

C

double code(double x, double y, double z) {
	double t_0 = -z - ((x * z) / y);
	double t_1 = (x + y) * (1.0 + (y / z));
	double tmp;
	if (z <= -5e+21) {
		tmp = t_1;
	} else if (z <= 1.55e-158) {
		tmp = t_0;
	} else if (z <= 5.4e-99) {
		tmp = x / (1.0 - (y / z));
	} else if (z <= 2.8e+101) {
		tmp = t_0;
	} 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) :: tmp
    t_0 = -z - ((x * z) / y)
    t_1 = (x + y) * (1.0d0 + (y / z))
    if (z <= (-5d+21)) then
        tmp = t_1
    else if (z <= 1.55d-158) then
        tmp = t_0
    else if (z <= 5.4d-99) then
        tmp = x / (1.0d0 - (y / z))
    else if (z <= 2.8d+101) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = -z - ((x * z) / y);
	double t_1 = (x + y) * (1.0 + (y / z));
	double tmp;
	if (z <= -5e+21) {
		tmp = t_1;
	} else if (z <= 1.55e-158) {
		tmp = t_0;
	} else if (z <= 5.4e-99) {
		tmp = x / (1.0 - (y / z));
	} else if (z <= 2.8e+101) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = -z - ((x * z) / y)
	t_1 = (x + y) * (1.0 + (y / z))
	tmp = 0
	if z <= -5e+21:
		tmp = t_1
	elif z <= 1.55e-158:
		tmp = t_0
	elif z <= 5.4e-99:
		tmp = x / (1.0 - (y / z))
	elif z <= 2.8e+101:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(-z) - Float64(Float64(x * z) / y))
	t_1 = Float64(Float64(x + y) * Float64(1.0 + Float64(y / z)))
	tmp = 0.0
	if (z <= -5e+21)
		tmp = t_1;
	elseif (z <= 1.55e-158)
		tmp = t_0;
	elseif (z <= 5.4e-99)
		tmp = Float64(x / Float64(1.0 - Float64(y / z)));
	elseif (z <= 2.8e+101)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = -z - ((x * z) / y);
	t_1 = (x + y) * (1.0 + (y / z));
	tmp = 0.0;
	if (z <= -5e+21)
		tmp = t_1;
	elseif (z <= 1.55e-158)
		tmp = t_0;
	elseif (z <= 5.4e-99)
		tmp = x / (1.0 - (y / z));
	elseif (z <= 2.8e+101)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[((-z) - N[(N[(x * z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x + y), $MachinePrecision] * N[(1.0 + N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -5e+21], t$95$1, If[LessEqual[z, 1.55e-158], t$95$0, If[LessEqual[z, 5.4e-99], N[(x / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.8e+101], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -5e21 or 2.79999999999999981e101 < z

   1. Initial program 100.0%

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

   2. Taylor expanded in z around inf 73.0%

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

      1. associate-+r+ 73.0%

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

      2. *-lft-identity 73.0%

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

      3. associate-/l* 83.9%

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

      4. associate-/r/ 83.9%

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

      5. distribute-rgt-in 83.8%

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

      6. +-commutative 83.8%

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

   4. Simplified 83.8%

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

   if -5e21 < z < 1.55000000000000009e-158 or 5.4e-99 < z < 2.79999999999999981e101

   1. Initial program 76.1%

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

   2. Taylor expanded in y around inf 77.6%

      \[\leadsto \color{blue}{\left(-1 \cdot z + -1 \cdot \frac{x \cdot z}{y}\right) - \frac{{z}^{2}}{y}} \]
   3. Step-by-step derivation

      1. associate--l+ 77.6%

         \[\leadsto \color{blue}{-1 \cdot z + \left(-1 \cdot \frac{x \cdot z}{y} - \frac{{z}^{2}}{y}\right)} \]

      2. +-commutative 77.6%

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{x \cdot z}{y} - \frac{{z}^{2}}{y}\right) + -1 \cdot z} \]

      3. associate-*r/ 77.6%

         \[\leadsto \left(\color{blue}{\frac{-1 \cdot \left(x \cdot z\right)}{y}} - \frac{{z}^{2}}{y}\right) + -1 \cdot z \]

      4. div-sub 77.6%

         \[\leadsto \color{blue}{\frac{-1 \cdot \left(x \cdot z\right) - {z}^{2}}{y}} + -1 \cdot z \]

      5. sub-neg 77.6%

         \[\leadsto \frac{\color{blue}{-1 \cdot \left(x \cdot z\right) + \left(-{z}^{2}\right)}}{y} + -1 \cdot z \]

      6. mul-1-neg 77.6%

         \[\leadsto \frac{\color{blue}{\left(-x \cdot z\right)} + \left(-{z}^{2}\right)}{y} + -1 \cdot z \]

      7. distribute-neg-out 77.6%

         \[\leadsto \frac{\color{blue}{-\left(x \cdot z + {z}^{2}\right)}}{y} + -1 \cdot z \]

      8. *-lft-identity 77.6%

         \[\leadsto \frac{-\left(x \cdot z + \color{blue}{1 \cdot {z}^{2}}\right)}{y} + -1 \cdot z \]

      9. metadata-eval 77.6%

         \[\leadsto \frac{-\left(x \cdot z + \color{blue}{\left(--1\right)} \cdot {z}^{2}\right)}{y} + -1 \cdot z \]

      10. cancel-sign-sub-inv 77.6%

          \[\leadsto \frac{-\color{blue}{\left(x \cdot z - -1 \cdot {z}^{2}\right)}}{y} + -1 \cdot z \]

      11. distribute-neg-frac 77.6%

          \[\leadsto \color{blue}{\left(-\frac{x \cdot z - -1 \cdot {z}^{2}}{y}\right)} + -1 \cdot z \]

      12. mul-1-neg 77.6%

          \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot z - -1 \cdot {z}^{2}}{y}} + -1 \cdot z \]

      13. mul-1-neg 77.6%

          \[\leadsto -1 \cdot \frac{x \cdot z - -1 \cdot {z}^{2}}{y} + \color{blue}{\left(-z\right)} \]

      14. unsub-neg 77.6%

          \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot z - -1 \cdot {z}^{2}}{y} - z} \]

   4. Simplified 77.6%

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

   5. Taylor expanded in z around 0 77.8%

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

      1. associate-*r/ 77.8%

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

      2. mul-1-neg 77.8%

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

      3. distribute-rgt-neg-in 77.8%

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

   7. Simplified 77.8%

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

   if 1.55000000000000009e-158 < z < 5.4e-99

   1. Initial program 99.9%

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

   2. Taylor expanded in x around inf 94.1%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{+21}:\\ \;\;\;\;\left(x + y\right) \cdot \left(1 + \frac{y}{z}\right)\\ \mathbf{elif}\;z \leq 1.55 \cdot 10^{-158}:\\ \;\;\;\;\left(-z\right) - \frac{x \cdot z}{y}\\ \mathbf{elif}\;z \leq 5.4 \cdot 10^{-99}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{elif}\;z \leq 2.8 \cdot 10^{+101}:\\ \;\;\;\;\left(-z\right) - \frac{x \cdot z}{y}\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) \cdot \left(1 + \frac{y}{z}\right)\\ \end{array} \]

Alternative 3: 71.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-z\right) - \frac{x \cdot z}{y}\\ \mathbf{if}\;z \leq -7.8 \cdot 10^{+21}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq 1.55 \cdot 10^{-158}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 5.4 \cdot 10^{-99}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{elif}\;z \leq 1.15 \cdot 10^{+90}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (- z) (/ (* x z) y))))
   (if (<= z -7.8e+21)
     (+ x y)
     (if (<= z 1.55e-158)
       t_0
       (if (<= z 5.4e-99)
         (/ x (- 1.0 (/ y z)))
         (if (<= z 1.15e+90) t_0 (+ x y)))))))

C

double code(double x, double y, double z) {
	double t_0 = -z - ((x * z) / y);
	double tmp;
	if (z <= -7.8e+21) {
		tmp = x + y;
	} else if (z <= 1.55e-158) {
		tmp = t_0;
	} else if (z <= 5.4e-99) {
		tmp = x / (1.0 - (y / z));
	} else if (z <= 1.15e+90) {
		tmp = t_0;
	} 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 = -z - ((x * z) / y)
    if (z <= (-7.8d+21)) then
        tmp = x + y
    else if (z <= 1.55d-158) then
        tmp = t_0
    else if (z <= 5.4d-99) then
        tmp = x / (1.0d0 - (y / z))
    else if (z <= 1.15d+90) then
        tmp = t_0
    else
        tmp = x + y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = -z - ((x * z) / y);
	double tmp;
	if (z <= -7.8e+21) {
		tmp = x + y;
	} else if (z <= 1.55e-158) {
		tmp = t_0;
	} else if (z <= 5.4e-99) {
		tmp = x / (1.0 - (y / z));
	} else if (z <= 1.15e+90) {
		tmp = t_0;
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = -z - ((x * z) / y)
	tmp = 0
	if z <= -7.8e+21:
		tmp = x + y
	elif z <= 1.55e-158:
		tmp = t_0
	elif z <= 5.4e-99:
		tmp = x / (1.0 - (y / z))
	elif z <= 1.15e+90:
		tmp = t_0
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(-z) - Float64(Float64(x * z) / y))
	tmp = 0.0
	if (z <= -7.8e+21)
		tmp = Float64(x + y);
	elseif (z <= 1.55e-158)
		tmp = t_0;
	elseif (z <= 5.4e-99)
		tmp = Float64(x / Float64(1.0 - Float64(y / z)));
	elseif (z <= 1.15e+90)
		tmp = t_0;
	else
		tmp = Float64(x + y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = -z - ((x * z) / y);
	tmp = 0.0;
	if (z <= -7.8e+21)
		tmp = x + y;
	elseif (z <= 1.55e-158)
		tmp = t_0;
	elseif (z <= 5.4e-99)
		tmp = x / (1.0 - (y / z));
	elseif (z <= 1.15e+90)
		tmp = t_0;
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[((-z) - N[(N[(x * z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -7.8e+21], N[(x + y), $MachinePrecision], If[LessEqual[z, 1.55e-158], t$95$0, If[LessEqual[z, 5.4e-99], N[(x / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.15e+90], t$95$0, N[(x + y), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -7.8e21 or 1.15e90 < z

   1. Initial program 99.9%

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

   2. Taylor expanded in z around inf 82.5%

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

      1. +-commutative 82.5%

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

   4. Simplified 82.5%

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

   if -7.8e21 < z < 1.55000000000000009e-158 or 5.4e-99 < z < 1.15e90

   1. Initial program 75.8%

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

   2. Taylor expanded in y around inf 78.0%

      \[\leadsto \color{blue}{\left(-1 \cdot z + -1 \cdot \frac{x \cdot z}{y}\right) - \frac{{z}^{2}}{y}} \]
   3. Step-by-step derivation

      1. associate--l+ 78.0%

         \[\leadsto \color{blue}{-1 \cdot z + \left(-1 \cdot \frac{x \cdot z}{y} - \frac{{z}^{2}}{y}\right)} \]

      2. +-commutative 78.0%

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{x \cdot z}{y} - \frac{{z}^{2}}{y}\right) + -1 \cdot z} \]

      3. associate-*r/ 78.0%

         \[\leadsto \left(\color{blue}{\frac{-1 \cdot \left(x \cdot z\right)}{y}} - \frac{{z}^{2}}{y}\right) + -1 \cdot z \]

      4. div-sub 78.0%

         \[\leadsto \color{blue}{\frac{-1 \cdot \left(x \cdot z\right) - {z}^{2}}{y}} + -1 \cdot z \]

      5. sub-neg 78.0%

         \[\leadsto \frac{\color{blue}{-1 \cdot \left(x \cdot z\right) + \left(-{z}^{2}\right)}}{y} + -1 \cdot z \]

      6. mul-1-neg 78.0%

         \[\leadsto \frac{\color{blue}{\left(-x \cdot z\right)} + \left(-{z}^{2}\right)}{y} + -1 \cdot z \]

      7. distribute-neg-out 78.0%

         \[\leadsto \frac{\color{blue}{-\left(x \cdot z + {z}^{2}\right)}}{y} + -1 \cdot z \]

      8. *-lft-identity 78.0%

         \[\leadsto \frac{-\left(x \cdot z + \color{blue}{1 \cdot {z}^{2}}\right)}{y} + -1 \cdot z \]

      9. metadata-eval 78.0%

         \[\leadsto \frac{-\left(x \cdot z + \color{blue}{\left(--1\right)} \cdot {z}^{2}\right)}{y} + -1 \cdot z \]

      10. cancel-sign-sub-inv 78.0%

          \[\leadsto \frac{-\color{blue}{\left(x \cdot z - -1 \cdot {z}^{2}\right)}}{y} + -1 \cdot z \]

      11. distribute-neg-frac 78.0%

          \[\leadsto \color{blue}{\left(-\frac{x \cdot z - -1 \cdot {z}^{2}}{y}\right)} + -1 \cdot z \]

      12. mul-1-neg 78.0%

          \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot z - -1 \cdot {z}^{2}}{y}} + -1 \cdot z \]

      13. mul-1-neg 78.0%

          \[\leadsto -1 \cdot \frac{x \cdot z - -1 \cdot {z}^{2}}{y} + \color{blue}{\left(-z\right)} \]

      14. unsub-neg 78.0%

          \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot z - -1 \cdot {z}^{2}}{y} - z} \]

   4. Simplified 78.0%

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

   5. Taylor expanded in z around 0 78.2%

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

      1. associate-*r/ 78.2%

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

      2. mul-1-neg 78.2%

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

      3. distribute-rgt-neg-in 78.2%

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

   7. Simplified 78.2%

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

   if 1.55000000000000009e-158 < z < 5.4e-99

   1. Initial program 99.9%

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

   2. Taylor expanded in x around inf 94.1%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.8 \cdot 10^{+21}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq 1.55 \cdot 10^{-158}:\\ \;\;\;\;\left(-z\right) - \frac{x \cdot z}{y}\\ \mathbf{elif}\;z \leq 5.4 \cdot 10^{-99}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{elif}\;z \leq 1.15 \cdot 10^{+90}:\\ \;\;\;\;\left(-z\right) - \frac{x \cdot z}{y}\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 4: 74.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{if}\;y \leq -7.5 \cdot 10^{+62}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -4.2 \cdot 10^{-166}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 59000000000000:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (- -1.0 (/ x y)))))
   (if (<= y -7.5e+62)
     t_0
     (if (<= y -4.2e-166)
       (+ x y)
       (if (<= y 59000000000000.0) (/ x (- 1.0 (/ y z))) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = z * (-1.0 - (x / y));
	double tmp;
	if (y <= -7.5e+62) {
		tmp = t_0;
	} else if (y <= -4.2e-166) {
		tmp = x + y;
	} else if (y <= 59000000000000.0) {
		tmp = x / (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 = z * ((-1.0d0) - (x / y))
    if (y <= (-7.5d+62)) then
        tmp = t_0
    else if (y <= (-4.2d-166)) then
        tmp = x + y
    else if (y <= 59000000000000.0d0) then
        tmp = x / (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 = z * (-1.0 - (x / y));
	double tmp;
	if (y <= -7.5e+62) {
		tmp = t_0;
	} else if (y <= -4.2e-166) {
		tmp = x + y;
	} else if (y <= 59000000000000.0) {
		tmp = x / (1.0 - (y / z));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * (-1.0 - (x / y))
	tmp = 0
	if y <= -7.5e+62:
		tmp = t_0
	elif y <= -4.2e-166:
		tmp = x + y
	elif y <= 59000000000000.0:
		tmp = x / (1.0 - (y / z))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(-1.0 - Float64(x / y)))
	tmp = 0.0
	if (y <= -7.5e+62)
		tmp = t_0;
	elseif (y <= -4.2e-166)
		tmp = Float64(x + y);
	elseif (y <= 59000000000000.0)
		tmp = Float64(x / Float64(1.0 - Float64(y / z)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * (-1.0 - (x / y));
	tmp = 0.0;
	if (y <= -7.5e+62)
		tmp = t_0;
	elseif (y <= -4.2e-166)
		tmp = x + y;
	elseif (y <= 59000000000000.0)
		tmp = x / (1.0 - (y / z));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[(-1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -7.5e+62], t$95$0, If[LessEqual[y, -4.2e-166], N[(x + y), $MachinePrecision], If[LessEqual[y, 59000000000000.0], N[(x / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -7.49999999999999998e62 or 5.9e13 < y

   1. Initial program 71.7%

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

   2. Taylor expanded in y around inf 71.3%

      \[\leadsto \color{blue}{\left(-1 \cdot z + -1 \cdot \frac{x \cdot z}{y}\right) - \frac{{z}^{2}}{y}} \]
   3. Step-by-step derivation

      1. associate--l+ 71.3%

         \[\leadsto \color{blue}{-1 \cdot z + \left(-1 \cdot \frac{x \cdot z}{y} - \frac{{z}^{2}}{y}\right)} \]

      2. +-commutative 71.3%

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{x \cdot z}{y} - \frac{{z}^{2}}{y}\right) + -1 \cdot z} \]

      3. associate-*r/ 71.3%

         \[\leadsto \left(\color{blue}{\frac{-1 \cdot \left(x \cdot z\right)}{y}} - \frac{{z}^{2}}{y}\right) + -1 \cdot z \]

      4. div-sub 71.3%

         \[\leadsto \color{blue}{\frac{-1 \cdot \left(x \cdot z\right) - {z}^{2}}{y}} + -1 \cdot z \]

      5. sub-neg 71.3%

         \[\leadsto \frac{\color{blue}{-1 \cdot \left(x \cdot z\right) + \left(-{z}^{2}\right)}}{y} + -1 \cdot z \]

      6. mul-1-neg 71.3%

         \[\leadsto \frac{\color{blue}{\left(-x \cdot z\right)} + \left(-{z}^{2}\right)}{y} + -1 \cdot z \]

      7. distribute-neg-out 71.3%

         \[\leadsto \frac{\color{blue}{-\left(x \cdot z + {z}^{2}\right)}}{y} + -1 \cdot z \]

      8. *-lft-identity 71.3%

         \[\leadsto \frac{-\left(x \cdot z + \color{blue}{1 \cdot {z}^{2}}\right)}{y} + -1 \cdot z \]

      9. metadata-eval 71.3%

         \[\leadsto \frac{-\left(x \cdot z + \color{blue}{\left(--1\right)} \cdot {z}^{2}\right)}{y} + -1 \cdot z \]

      10. cancel-sign-sub-inv 71.3%

          \[\leadsto \frac{-\color{blue}{\left(x \cdot z - -1 \cdot {z}^{2}\right)}}{y} + -1 \cdot z \]

      11. distribute-neg-frac 71.3%

          \[\leadsto \color{blue}{\left(-\frac{x \cdot z - -1 \cdot {z}^{2}}{y}\right)} + -1 \cdot z \]

      12. mul-1-neg 71.3%

          \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot z - -1 \cdot {z}^{2}}{y}} + -1 \cdot z \]

      13. mul-1-neg 71.3%

          \[\leadsto -1 \cdot \frac{x \cdot z - -1 \cdot {z}^{2}}{y} + \color{blue}{\left(-z\right)} \]

      14. unsub-neg 71.3%

          \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot z - -1 \cdot {z}^{2}}{y} - z} \]

   4. Simplified 71.4%

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

   5. Taylor expanded in z around 0 80.3%

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

   6. Taylor expanded in x around 0 75.6%

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

      1. mul-1-neg 75.6%

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

      2. +-commutative 75.6%

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

      3. associate-*l/ 80.3%

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

      4. associate-*r* 80.3%

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

      5. mul-1-neg 80.3%

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

      6. distribute-rgt-in 80.3%

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

      7. +-commutative 80.3%

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

      8. mul-1-neg 80.3%

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

      9. unsub-neg 80.3%

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

   8. Simplified 80.3%

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

   if -7.49999999999999998e62 < y < -4.1999999999999999e-166

   1. Initial program 99.9%

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

   2. Taylor expanded in z around inf 68.4%

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

      1. +-commutative 68.4%

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

   4. Simplified 68.4%

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

   if -4.1999999999999999e-166 < y < 5.9e13

   1. Initial program 99.8%

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

   2. Taylor expanded in x around inf 80.6%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.5 \cdot 10^{+62}:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{elif}\;y \leq -4.2 \cdot 10^{-166}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 59000000000000:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \end{array} \]

Alternative 5: 74.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.2 \cdot 10^{+67} \lor \neg \left(y \leq 3000000000000\right):\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -4.2e+67) (not (<= y 3000000000000.0)))
   (* z (- -1.0 (/ x y)))
   (+ x y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -4.2e+67) || !(y <= 3000000000000.0)) {
		tmp = z * (-1.0 - (x / y));
	} else {
		tmp = x + y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((y <= (-4.2d+67)) .or. (.not. (y <= 3000000000000.0d0))) then
        tmp = z * ((-1.0d0) - (x / y))
    else
        tmp = x + y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -4.2e+67) || !(y <= 3000000000000.0)) {
		tmp = z * (-1.0 - (x / y));
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -4.2e+67) or not (y <= 3000000000000.0):
		tmp = z * (-1.0 - (x / y))
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -4.2e+67) || !(y <= 3000000000000.0))
		tmp = Float64(z * Float64(-1.0 - Float64(x / y)));
	else
		tmp = Float64(x + y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -4.2e+67) || ~((y <= 3000000000000.0)))
		tmp = z * (-1.0 - (x / y));
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -4.2e+67], N[Not[LessEqual[y, 3000000000000.0]], $MachinePrecision]], N[(z * N[(-1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.2000000000000003e67 or 3e12 < y

   1. Initial program 71.7%

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

   2. Taylor expanded in y around inf 71.3%

      \[\leadsto \color{blue}{\left(-1 \cdot z + -1 \cdot \frac{x \cdot z}{y}\right) - \frac{{z}^{2}}{y}} \]
   3. Step-by-step derivation

      1. associate--l+ 71.3%

         \[\leadsto \color{blue}{-1 \cdot z + \left(-1 \cdot \frac{x \cdot z}{y} - \frac{{z}^{2}}{y}\right)} \]

      2. +-commutative 71.3%

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{x \cdot z}{y} - \frac{{z}^{2}}{y}\right) + -1 \cdot z} \]

      3. associate-*r/ 71.3%

         \[\leadsto \left(\color{blue}{\frac{-1 \cdot \left(x \cdot z\right)}{y}} - \frac{{z}^{2}}{y}\right) + -1 \cdot z \]

      4. div-sub 71.3%

         \[\leadsto \color{blue}{\frac{-1 \cdot \left(x \cdot z\right) - {z}^{2}}{y}} + -1 \cdot z \]

      5. sub-neg 71.3%

         \[\leadsto \frac{\color{blue}{-1 \cdot \left(x \cdot z\right) + \left(-{z}^{2}\right)}}{y} + -1 \cdot z \]

      6. mul-1-neg 71.3%

         \[\leadsto \frac{\color{blue}{\left(-x \cdot z\right)} + \left(-{z}^{2}\right)}{y} + -1 \cdot z \]

      7. distribute-neg-out 71.3%

         \[\leadsto \frac{\color{blue}{-\left(x \cdot z + {z}^{2}\right)}}{y} + -1 \cdot z \]

      8. *-lft-identity 71.3%

         \[\leadsto \frac{-\left(x \cdot z + \color{blue}{1 \cdot {z}^{2}}\right)}{y} + -1 \cdot z \]

      9. metadata-eval 71.3%

         \[\leadsto \frac{-\left(x \cdot z + \color{blue}{\left(--1\right)} \cdot {z}^{2}\right)}{y} + -1 \cdot z \]

      10. cancel-sign-sub-inv 71.3%

          \[\leadsto \frac{-\color{blue}{\left(x \cdot z - -1 \cdot {z}^{2}\right)}}{y} + -1 \cdot z \]

      11. distribute-neg-frac 71.3%

          \[\leadsto \color{blue}{\left(-\frac{x \cdot z - -1 \cdot {z}^{2}}{y}\right)} + -1 \cdot z \]

      12. mul-1-neg 71.3%

          \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot z - -1 \cdot {z}^{2}}{y}} + -1 \cdot z \]

      13. mul-1-neg 71.3%

          \[\leadsto -1 \cdot \frac{x \cdot z - -1 \cdot {z}^{2}}{y} + \color{blue}{\left(-z\right)} \]

      14. unsub-neg 71.3%

          \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot z - -1 \cdot {z}^{2}}{y} - z} \]

   4. Simplified 71.4%

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

   5. Taylor expanded in z around 0 80.3%

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

   6. Taylor expanded in x around 0 75.6%

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

      1. mul-1-neg 75.6%

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

      2. +-commutative 75.6%

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

      3. associate-*l/ 80.3%

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

      4. associate-*r* 80.3%

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

      5. mul-1-neg 80.3%

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

      6. distribute-rgt-in 80.3%

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

      7. +-commutative 80.3%

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

      8. mul-1-neg 80.3%

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

      9. unsub-neg 80.3%

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

   8. Simplified 80.3%

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

   if -4.2000000000000003e67 < y < 3e12

   1. Initial program 99.8%

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

   2. Taylor expanded in z around inf 73.0%

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

      1. +-commutative 73.0%

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

   4. Simplified 73.0%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.2 \cdot 10^{+67} \lor \neg \left(y \leq 3000000000000\right):\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 6: 58.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.6 \cdot 10^{+55}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq -2.85 \cdot 10^{-64}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 480000000:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.6e+55)
   (- z)
   (if (<= y -2.85e-64) y (if (<= y 480000000.0) x (- z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.6e+55) {
		tmp = -z;
	} else if (y <= -2.85e-64) {
		tmp = y;
	} else if (y <= 480000000.0) {
		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 <= (-1.6d+55)) then
        tmp = -z
    else if (y <= (-2.85d-64)) then
        tmp = y
    else if (y <= 480000000.0d0) 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 <= -1.6e+55) {
		tmp = -z;
	} else if (y <= -2.85e-64) {
		tmp = y;
	} else if (y <= 480000000.0) {
		tmp = x;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1.6e+55:
		tmp = -z
	elif y <= -2.85e-64:
		tmp = y
	elif y <= 480000000.0:
		tmp = x
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.6e+55)
		tmp = Float64(-z);
	elseif (y <= -2.85e-64)
		tmp = y;
	elseif (y <= 480000000.0)
		tmp = x;
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -1.6e+55)
		tmp = -z;
	elseif (y <= -2.85e-64)
		tmp = y;
	elseif (y <= 480000000.0)
		tmp = x;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.6e+55], (-z), If[LessEqual[y, -2.85e-64], y, If[LessEqual[y, 480000000.0], x, (-z)]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.6000000000000001e55 or 4.8e8 < y

   1. Initial program 72.5%

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

   2. Taylor expanded in y around inf 63.7%

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

      1. mul-1-neg 63.7%

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

   4. Simplified 63.7%

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

   if -1.6000000000000001e55 < y < -2.8500000000000001e-64

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 55.9%

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

   3. Taylor expanded in y around 0 48.0%

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

   if -2.8500000000000001e-64 < y < 4.8e8

   1. Initial program 99.8%

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

   2. Taylor expanded in y around 0 60.4%

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

4. Final simplification 60.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.6 \cdot 10^{+55}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq -2.85 \cdot 10^{-64}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 480000000:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 7: 67.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.6 \cdot 10^{+59} \lor \neg \left(y \leq 4 \cdot 10^{+14}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -2.6e+59) (not (<= y 4e+14))) (- z) (+ x y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -2.6e+59) || !(y <= 4e+14)) {
		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 <= (-2.6d+59)) .or. (.not. (y <= 4d+14))) 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 <= -2.6e+59) || !(y <= 4e+14)) {
		tmp = -z;
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -2.6e+59) or not (y <= 4e+14):
		tmp = -z
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -2.6e+59) || !(y <= 4e+14))
		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 <= -2.6e+59) || ~((y <= 4e+14)))
		tmp = -z;
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -2.6e+59], N[Not[LessEqual[y, 4e+14]], $MachinePrecision]], (-z), N[(x + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.59999999999999999e59 or 4e14 < y

   1. Initial program 71.7%

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

   2. Taylor expanded in y around inf 65.2%

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

      1. mul-1-neg 65.2%

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

   4. Simplified 65.2%

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

   if -2.59999999999999999e59 < y < 4e14

   1. Initial program 99.8%

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

   2. Taylor expanded in z around inf 73.0%

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

      1. +-commutative 73.0%

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

   4. Simplified 73.0%

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

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.6 \cdot 10^{+59} \lor \neg \left(y \leq 4 \cdot 10^{+14}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 8: 41.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{-84}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-45}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -4.4e-84) x (if (<= x 2.3e-45) y x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -4.4e-84) {
		tmp = x;
	} else if (x <= 2.3e-45) {
		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 <= (-4.4d-84)) then
        tmp = x
    else if (x <= 2.3d-45) 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 <= -4.4e-84) {
		tmp = x;
	} else if (x <= 2.3e-45) {
		tmp = y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -4.4e-84:
		tmp = x
	elif x <= 2.3e-45:
		tmp = y
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -4.4e-84)
		tmp = x;
	elseif (x <= 2.3e-45)
		tmp = y;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -4.4e-84)
		tmp = x;
	elseif (x <= 2.3e-45)
		tmp = y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -4.4e-84], x, If[LessEqual[x, 2.3e-45], y, x]]

Derivation

1. Split input into 2 regimes

2. if x < -4.3999999999999998e-84 or 2.29999999999999992e-45 < x

   1. Initial program 85.2%

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

   2. Taylor expanded in y around 0 43.0%

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

   if -4.3999999999999998e-84 < x < 2.29999999999999992e-45

   1. Initial program 90.4%

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

   2. Taylor expanded in x around 0 79.8%

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

   3. Taylor expanded in y around 0 44.0%

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

4. Final simplification 43.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{-84}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-45}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 9: 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 87.0%

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

2. Taylor expanded in y around 0 33.3%

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

3. Final simplification 33.3%

   \[\leadsto x \]

Developer target: 94.0% 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 2023322 
(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))))