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

?

Percentage Accurate: 87.9% → 99.6%
Time: 10.4s
Precision: binary64
Cost: 2185

?

\[\frac{x + y}{1 - \frac{y}{z}} \]
↓
\[\begin{array}{l} t_0 := \frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{-263} \lor \neg \left(t_0 \leq 0\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-z\right) - \frac{z}{\frac{y}{x}}\right) - \frac{z}{\frac{y}{z}}\\ \end{array} \]
(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 (or (<= t_0 -5e-263) (not (<= t_0 0.0)))
     t_0
     (- (- (- z) (/ z (/ y x))) (/ z (/ y z))))))
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-263) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = (-z - (z / (y / x))) - (z / (y / z));
	}
	return tmp;
}

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-263)) .or. (.not. (t_0 <= 0.0d0))) then
        tmp = t_0
    else
        tmp = (-z - (z / (y / x))) - (z / (y / z))
    end if
    code = tmp
end function

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-263) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = (-z - (z / (y / x))) - (z / (y / z));
	}
	return tmp;
}

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-263) or not (t_0 <= 0.0):
		tmp = t_0
	else:
		tmp = (-z - (z / (y / x))) - (z / (y / z))
	return tmp

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-263) || !(t_0 <= 0.0))
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(-z) - Float64(z / Float64(y / x))) - Float64(z / Float64(y / z)));
	end
	return tmp
end

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-263) || ~((t_0 <= 0.0)))
		tmp = t_0;
	else
		tmp = (-z - (z / (y / x))) - (z / (y / z));
	end
	tmp_2 = tmp;
end

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[Or[LessEqual[t$95$0, -5e-263], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], t$95$0, N[(N[((-z) - N[(z / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(z / N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

\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^{-263} \lor \neg \left(t_0 \leq 0\right):\\
\;\;\;\;t_0\\

\mathbf{else}:\\
\;\;\;\;\left(\left(-z\right) - \frac{z}{\frac{y}{x}}\right) - \frac{z}{\frac{y}{z}}\\


\end{array}

Local Percentage Accuracy vs ?

The average percentage accuracy by input value. Horizontal axis shows value of an input variable; the variable is choosen in the title. Vertical axis is accuracy; higher is better. Red represent the original program, while blue represents Herbie's suggestion. These can be toggled with buttons below the plot. The line is an average while dots represent individual samples.

Herbie found 12 alternatives:

AlternativeAccuracySpeedup

Accuracy vs Speed

The accuracy (vertical axis) and speed (horizontal axis) of each alternatives. Up and to the right is better. The red square shows the initial program, and each blue circle shows an alternative.The line shows the best available speed-accuracy tradeoffs.

Bogosity?

Bogosity

Try it out?

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original87.9%
Target93.6%
Herbie99.6%
\[\begin{array}{l} \mathbf{if}\;y < -3.7429310762689856 \cdot 10^{+171}:\\ \;\;\;\;\frac{y + x}{-y} \cdot z\\ \mathbf{elif}\;y < 3.5534662456086734 \cdot 10^{+168}:\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y + x}{-y} \cdot z\\ \end{array} \]

Derivation?

  1. Split input into 2 regimes

  2. if (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z))) < -5.00000000000000006e-263 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 -5.00000000000000006e-263 < (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z))) < 0.0

    1. Initial program 12.7%

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

    2. Taylor expanded in y around inf 99.8%

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

    3. Simplified100.0%

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

      [Start]99.8%

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

      mul-1-neg [=>]99.8%

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

      unsub-neg [=>]99.8%

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

      mul-1-neg [=>]99.8%

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

      associate-/l* [=>]100.0%

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

      unpow2 [=>]100.0%

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

      associate-/l* [=>]100.0%

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

  4. Final simplification99.8%

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \leq -5 \cdot 10^{-263} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \leq 0\right):\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-z\right) - \frac{z}{\frac{y}{x}}\right) - \frac{z}{\frac{y}{z}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy99.6%
Cost2185
\[\begin{array}{l} t_0 := \frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{-263} \lor \neg \left(t_0 \leq 0\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-z\right) - \frac{z}{\frac{y}{x}}\right) - \frac{z}{\frac{y}{z}}\\ \end{array} \]
Alternative 2
Accuracy99.6%
Cost1865
\[\begin{array}{l} t_0 := \frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{-263} \lor \neg \left(t_0 \leq 0\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \end{array} \]
Alternative 3
Accuracy73.1%
Cost1104
\[\begin{array}{l} t_0 := 1 - \frac{y}{z}\\ t_1 := \left(x + y\right) \cdot \left(1 + \frac{y}{z}\right)\\ \mathbf{if}\;z \leq -3.25 \cdot 10^{+63}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -5.8 \cdot 10^{-47}:\\ \;\;\;\;\frac{y}{t_0}\\ \mathbf{elif}\;z \leq -3.5 \cdot 10^{-97}:\\ \;\;\;\;\frac{x}{t_0}\\ \mathbf{elif}\;z \leq 6.8 \cdot 10^{-6}:\\ \;\;\;\;\left(-z\right) - \frac{x \cdot z}{y}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 4
Accuracy58.6%
Cost1044
\[\begin{array}{l} \mathbf{if}\;z \leq -3.9 \cdot 10^{-97}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq -2.2 \cdot 10^{-144}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq -8.5 \cdot 10^{-185}:\\ \;\;\;\;z \cdot \frac{-x}{y}\\ \mathbf{elif}\;z \leq -6 \cdot 10^{-275}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{-204}:\\ \;\;\;\;\frac{x}{-\frac{y}{z}}\\ \mathbf{elif}\;z \leq 7.8 \cdot 10^{-7}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]
Alternative 5
Accuracy58.3%
Cost1044
\[\begin{array}{l} \mathbf{if}\;z \leq -1.06 \cdot 10^{-97}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq -1.6 \cdot 10^{-140}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq -3.4 \cdot 10^{-185}:\\ \;\;\;\;\frac{-z}{\frac{y}{x}}\\ \mathbf{elif}\;z \leq -4 \cdot 10^{-275}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{-197}:\\ \;\;\;\;\frac{x}{-\frac{y}{z}}\\ \mathbf{elif}\;z \leq 2.95 \cdot 10^{-6}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]
Alternative 6
Accuracy73.3%
Cost1040
\[\begin{array}{l} t_0 := 1 - \frac{y}{z}\\ \mathbf{if}\;z \leq -5.4 \cdot 10^{+63}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq -9 \cdot 10^{-48}:\\ \;\;\;\;\frac{y}{t_0}\\ \mathbf{elif}\;z \leq -1.16 \cdot 10^{-97}:\\ \;\;\;\;\frac{x}{t_0}\\ \mathbf{elif}\;z \leq 1.32 \cdot 10^{-6}:\\ \;\;\;\;\left(-z\right) - \frac{x \cdot z}{y}\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]
Alternative 7
Accuracy72.2%
Cost976
\[\begin{array}{l} t_0 := 1 - \frac{y}{z}\\ \mathbf{if}\;z \leq -4.1 \cdot 10^{+63}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq -3 \cdot 10^{-47}:\\ \;\;\;\;\frac{y}{t_0}\\ \mathbf{elif}\;z \leq -4.3 \cdot 10^{-97}:\\ \;\;\;\;\frac{x}{t_0}\\ \mathbf{elif}\;z \leq 1.82 \cdot 10^{-6}:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]
Alternative 8
Accuracy72.7%
Cost712
\[\begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{+63}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{-6}:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]
Alternative 9
Accuracy66.8%
Cost456
\[\begin{array}{l} \mathbf{if}\;y \leq -1.4 \cdot 10^{+98}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 3.25 \cdot 10^{-9}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]
Alternative 10
Accuracy57.7%
Cost392
\[\begin{array}{l} \mathbf{if}\;y \leq -35000000000000:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{-58}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]
Alternative 11
Accuracy40.4%
Cost328
\[\begin{array}{l} \mathbf{if}\;x \leq -1.8 \cdot 10^{-75}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 3.7 \cdot 10^{-87}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
Alternative 12
Accuracy35.0%
Cost64
\[x \]

Reproduce?

herbie shell --seed 2023271 
(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))))