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

Average Error: 12.18% → 0.39%

Time: 11.3s

Precision: binary64

Cost: 2184

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{-268}:\\ \;\;\;\;\frac{x + y}{\left(2 - \frac{y}{z}\right) + -1}\\ \mathbf{elif}\;t_0 \leq 10^{-295}:\\ \;\;\;\;\left(\left(-z\right) - \frac{z}{\frac{y}{x}}\right) - \frac{z \cdot z}{y}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (+ x y) (- 1.0 (/ y z)))))
   (if (<= t_0 -5e-268)
     (/ (+ x y) (+ (- 2.0 (/ y z)) -1.0))
     (if (<= t_0 1e-295) (- (- (- z) (/ z (/ y x))) (/ (* z z) y)) t_0))))

C

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

↓

double code(double x, double y, double z) {
	double t_0 = (x + y) / (1.0 - (y / z));
	double tmp;
	if (t_0 <= -5e-268) {
		tmp = (x + y) / ((2.0 - (y / z)) + -1.0);
	} else if (t_0 <= 1e-295) {
		tmp = (-z - (z / (y / x))) - ((z * z) / y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

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

↓

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-268)) then
        tmp = (x + y) / ((2.0d0 - (y / z)) + (-1.0d0))
    else if (t_0 <= 1d-295) then
        tmp = (-z - (z / (y / x))) - ((z * z) / y)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

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

↓

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-268) {
		tmp = (x + y) / ((2.0 - (y / z)) + -1.0);
	} else if (t_0 <= 1e-295) {
		tmp = (-z - (z / (y / x))) - ((z * z) / y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

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

↓

def code(x, y, z):
	t_0 = (x + y) / (1.0 - (y / z))
	tmp = 0
	if t_0 <= -5e-268:
		tmp = (x + y) / ((2.0 - (y / z)) + -1.0)
	elif t_0 <= 1e-295:
		tmp = (-z - (z / (y / x))) - ((z * z) / y)
	else:
		tmp = t_0
	return tmp

Julia

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

↓

function code(x, y, z)
	t_0 = Float64(Float64(x + y) / Float64(1.0 - Float64(y / z)))
	tmp = 0.0
	if (t_0 <= -5e-268)
		tmp = Float64(Float64(x + y) / Float64(Float64(2.0 - Float64(y / z)) + -1.0));
	elseif (t_0 <= 1e-295)
		tmp = Float64(Float64(Float64(-z) - Float64(z / Float64(y / x))) - Float64(Float64(z * z) / y));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x, y, z)
	t_0 = (x + y) / (1.0 - (y / z));
	tmp = 0.0;
	if (t_0 <= -5e-268)
		tmp = (x + y) / ((2.0 - (y / z)) + -1.0);
	elseif (t_0 <= 1e-295)
		tmp = (-z - (z / (y / x))) - ((z * z) / y);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x + y), $MachinePrecision] / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-268], N[(N[(x + y), $MachinePrecision] / N[(N[(2.0 - N[(y / z), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e-295], N[(N[((-z) - N[(z / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(z * z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Target

Original	12.18%
Target	6.34%
Herbie	0.39%

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

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z))) < -4.9999999999999999e-268

   1. Initial program 0.13

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

   2. Simplified 0.13

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

      [Start] 0.13	\[ \frac{x + y}{1 - \frac{y}{z}} \]
      +-commutative [=>] 0.13	\[ \frac{\color{blue}{y + x}}{1 - \frac{y}{z}} \]

   3. Applied egg-rr 0.14

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

   4. Applied egg-rr 0.14

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

   if -4.9999999999999999e-268 < (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z))) < 1.00000000000000006e-295

   1. Initial program 87.46

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

   2. Simplified 87.46

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

      [Start] 87.46	\[ \frac{x + y}{1 - \frac{y}{z}} \]
      +-commutative [=>] 87.46	\[ \frac{\color{blue}{y + x}}{1 - \frac{y}{z}} \]

   3. Taylor expanded in y around inf 2.39

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

   4. Simplified 1.97

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

      [Start] 2.39	\[ \left(-1 \cdot z + -1 \cdot \frac{z \cdot x}{y}\right) - \frac{{z}^{2}}{y} \]
      distribute-lft-out [=>] 2.39	\[ \color{blue}{-1 \cdot \left(z + \frac{z \cdot x}{y}\right)} - \frac{{z}^{2}}{y} \]
      associate-/l* [=>] 1.97	\[ -1 \cdot \left(z + \color{blue}{\frac{z}{\frac{y}{x}}}\right) - \frac{{z}^{2}}{y} \]
      unpow2 [=>] 1.97	\[ -1 \cdot \left(z + \frac{z}{\frac{y}{x}}\right) - \frac{\color{blue}{z \cdot z}}{y} \]

   if 1.00000000000000006e-295 < (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z)))

   1. Initial program 0.14

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

4. Final simplification 0.39

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \leq -5 \cdot 10^{-268}:\\ \;\;\;\;\frac{x + y}{\left(2 - \frac{y}{z}\right) + -1}\\ \mathbf{elif}\;\frac{x + y}{1 - \frac{y}{z}} \leq 10^{-295}:\\ \;\;\;\;\left(\left(-z\right) - \frac{z}{\frac{y}{x}}\right) - \frac{z \cdot z}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.31%
Cost	1865

\[\begin{array}{l} t_0 := \frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{-276} \lor \neg \left(t_0 \leq 10^{-295}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{\frac{-y}{x + y}}\\ \end{array} \]

Alternative 2
Error	0.31%
Cost	1864

\[\begin{array}{l} t_0 := \frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{-276}:\\ \;\;\;\;\frac{x + y}{\left(2 - \frac{y}{z}\right) + -1}\\ \mathbf{elif}\;t_0 \leq 10^{-295}:\\ \;\;\;\;\frac{z}{\frac{-y}{x + y}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	26.36%
Cost	1040

\[\begin{array}{l} t_0 := 1 - \frac{y}{z}\\ t_1 := \frac{z}{\frac{-y}{x + y}}\\ \mathbf{if}\;y \leq -7200000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-157}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{-29}:\\ \;\;\;\;\frac{x}{t_0}\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{+144}:\\ \;\;\;\;\frac{y}{t_0}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	27.92%
Cost	976

\[\begin{array}{l} t_0 := z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{if}\;y \leq -110000000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 7 \cdot 10^{-157}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 2 \cdot 10^{-50}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{+144}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	26.41%
Cost	976

\[\begin{array}{l} t_0 := 1 - \frac{y}{z}\\ t_1 := z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{if}\;y \leq -24500000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{-157}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{-29}:\\ \;\;\;\;\frac{x}{t_0}\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{+144}:\\ \;\;\;\;\frac{y}{t_0}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	26.46%
Cost	841

\[\begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{-13} \lor \neg \left(x \leq 1.6 \cdot 10^{+15}\right):\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{y} + \frac{-1}{z}}\\ \end{array} \]

Alternative 7
Error	27.24%
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -2.4 \cdot 10^{-84}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq 5.1 \cdot 10^{+38}:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 8
Error	43.17%
Cost	524

\[\begin{array}{l} \mathbf{if}\;y \leq -5.4 \cdot 10^{+26}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{+15}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{+144}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 9
Error	32.83%
Cost	456

\[\begin{array}{l} \mathbf{if}\;y \leq -9.5 \cdot 10^{+133}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{+144}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 10
Error	59.77%
Cost	328

\[\begin{array}{l} \mathbf{if}\;x \leq -3.7 \cdot 10^{-42}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{-94}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 11
Error	65.31%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023115 
(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))))