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

Percentage Accurate: 88.3% → 99.6%

Time: 7.1s

Precision: binary64

Cost: 1929

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{-270} \lor \neg \left(t_0 \leq 0\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) - \frac{z \cdot \left(x + z\right)}{y}\\ \end{array} \]

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 (or (<= t_0 -5e-270) (not (<= t_0 0.0)))
     t_0
     (- (- z) (/ (* z (+ x z)) y)))))

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-270) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = -z - ((z * (x + z)) / 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
    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-270)) .or. (.not. (t_0 <= 0.0d0))) then
        tmp = t_0
    else
        tmp = -z - ((z * (x + z)) / y)
    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-270) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = -z - ((z * (x + z)) / y);
	}
	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-270) or not (t_0 <= 0.0):
		tmp = t_0
	else:
		tmp = -z - ((z * (x + z)) / y)
	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-270) || !(t_0 <= 0.0))
		tmp = t_0;
	else
		tmp = Float64(Float64(-z) - Float64(Float64(z * Float64(x + z)) / y));
	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-270) || ~((t_0 <= 0.0)))
		tmp = t_0;
	else
		tmp = -z - ((z * (x + z)) / y);
	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[Or[LessEqual[t$95$0, -5e-270], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], t$95$0, N[((-z) - N[(N[(z * N[(x + z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]

Target

Original	88.3%
Target	93.7%
Herbie	99.6%

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z))) < -4.9999999999999998e-270 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 -4.9999999999999998e-270 < (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z))) < -0.0

   1. Initial program 15.4%

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

   2. Taylor expanded in y around inf 99.9%

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

   3. Simplified 99.9%

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

      [Start] 99.9%	\[ \left(-1 \cdot z + -1 \cdot \frac{z \cdot x}{y}\right) - \frac{{z}^{2}}{y} \]
      sub-neg [=>] 99.9%	\[ \color{blue}{\left(-1 \cdot z + -1 \cdot \frac{z \cdot x}{y}\right) + \left(-\frac{{z}^{2}}{y}\right)} \]
      mul-1-neg [=>] 99.9%	\[ \left(-1 \cdot z + \color{blue}{\left(-\frac{z \cdot x}{y}\right)}\right) + \left(-\frac{{z}^{2}}{y}\right) \]
      unsub-neg [=>] 99.9%	\[ \color{blue}{\left(-1 \cdot z - \frac{z \cdot x}{y}\right)} + \left(-\frac{{z}^{2}}{y}\right) \]
      associate-+l- [=>] 99.9%	\[ \color{blue}{-1 \cdot z - \left(\frac{z \cdot x}{y} - \left(-\frac{{z}^{2}}{y}\right)\right)} \]
      mul-1-neg [=>] 99.9%	\[ \color{blue}{\left(-z\right)} - \left(\frac{z \cdot x}{y} - \left(-\frac{{z}^{2}}{y}\right)\right) \]
      distribute-frac-neg [<=] 99.9%	\[ \left(-z\right) - \left(\frac{z \cdot x}{y} - \color{blue}{\frac{-{z}^{2}}{y}}\right) \]
      mul-1-neg [<=] 99.9%	\[ \left(-z\right) - \left(\frac{z \cdot x}{y} - \frac{\color{blue}{-1 \cdot {z}^{2}}}{y}\right) \]
      div-sub [<=] 99.9%	\[ \left(-z\right) - \color{blue}{\frac{z \cdot x - -1 \cdot {z}^{2}}{y}} \]
      sub-neg [=>] 99.9%	\[ \left(-z\right) - \frac{\color{blue}{z \cdot x + \left(--1 \cdot {z}^{2}\right)}}{y} \]
      mul-1-neg [=>] 99.9%	\[ \left(-z\right) - \frac{z \cdot x + \left(-\color{blue}{\left(-{z}^{2}\right)}\right)}{y} \]
      remove-double-neg [=>] 99.9%	\[ \left(-z\right) - \frac{z \cdot x + \color{blue}{{z}^{2}}}{y} \]
      unpow2 [=>] 99.9%	\[ \left(-z\right) - \frac{z \cdot x + \color{blue}{z \cdot z}}{y} \]
      distribute-lft-out [=>] 99.9%	\[ \left(-z\right) - \frac{\color{blue}{z \cdot \left(x + z\right)}}{y} \]

3. Recombined 2 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \leq -5 \cdot 10^{-270} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \leq 0\right):\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) - \frac{z \cdot \left(x + z\right)}{y}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.6%
Cost	1929

\[\begin{array}{l} t_0 := \frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{-270} \lor \neg \left(t_0 \leq 0\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) - \frac{z \cdot \left(x + z\right)}{y}\\ \end{array} \]

Alternative 2
Accuracy	99.6%
Cost	1865

\[\begin{array}{l} t_0 := \frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{-270} \lor \neg \left(t_0 \leq 0\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \end{array} \]

Alternative 3
Accuracy	59.3%
Cost	1108

\[\begin{array}{l} t_0 := \frac{x}{1 - \frac{y}{z}}\\ \mathbf{if}\;x \leq -4.2 \cdot 10^{-69}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-221}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;x \leq 2.15 \cdot 10^{-200}:\\ \;\;\;\;-z\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-174}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;x \leq 3.25 \cdot 10^{-105}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Accuracy	67.1%
Cost	713

\[\begin{array}{l} t_0 := 1 - \frac{y}{z}\\ \mathbf{if}\;x \leq -8.5 \cdot 10^{-54} \lor \neg \left(x \leq 1.1 \cdot 10^{-40}\right):\\ \;\;\;\;\frac{x}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t_0}\\ \end{array} \]

Alternative 5
Accuracy	54.4%
Cost	524

\[\begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{+19}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{-90}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{+180}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 6
Accuracy	65.9%
Cost	456

\[\begin{array}{l} \mathbf{if}\;y \leq -3.6 \cdot 10^{+29}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{+180}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 7
Accuracy	40.5%
Cost	328

\[\begin{array}{l} \mathbf{if}\;x \leq -3.8 \cdot 10^{-90}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{-110}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 8
Accuracy	34.5%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023171 
(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))))