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

Average Error: 11.85% → 1.58%

Time: 7.4s

Precision: binary64

Cost: 1865

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{-217} \lor \neg \left(t_0 \leq 2 \cdot 10^{-240}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \end{array} \]

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 -1e-217) (not (<= t_0 2e-240)))
     t_0
     (* z (- -1.0 (/ x 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 <= -1e-217) || !(t_0 <= 2e-240)) {
		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
    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 <= (-1d-217)) .or. (.not. (t_0 <= 2d-240))) 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) {
	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 <= -1e-217) || !(t_0 <= 2e-240)) {
		tmp = t_0;
	} else {
		tmp = z * (-1.0 - (x / 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 <= -1e-217) or not (t_0 <= 2e-240):
		tmp = t_0
	else:
		tmp = z * (-1.0 - (x / 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 <= -1e-217) || !(t_0 <= 2e-240))
		tmp = t_0;
	else
		tmp = Float64(z * Float64(-1.0 - Float64(x / 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 <= -1e-217) || ~((t_0 <= 2e-240)))
		tmp = t_0;
	else
		tmp = z * (-1.0 - (x / 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, -1e-217], N[Not[LessEqual[t$95$0, 2e-240]], $MachinePrecision]], t$95$0, N[(z * N[(-1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Target

Original	11.85%
Target	6.55%
Herbie	1.58%

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z))) < -1.00000000000000008e-217 or 1.9999999999999999e-240 < (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z)))

   1. Initial program 0.13

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

   if -1.00000000000000008e-217 < (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z))) < 1.9999999999999999e-240

   1. Initial program 70.99

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

   2. Simplified 70.99

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

      [Start] 70.99	\[ \frac{x + y}{1 - \frac{y}{z}} \]
      +-commutative [=>] 70.99	\[ \frac{\color{blue}{y + x}}{1 - \frac{y}{z}} \]

   3. Taylor expanded in z around 0 16.52

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

   4. Simplified 79.9

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

      [Start] 16.52	\[ -1 \cdot \frac{\left(y + x\right) \cdot z}{y} \]
      associate-/l* [=>] 79.9	\[ -1 \cdot \color{blue}{\frac{y + x}{\frac{y}{z}}} \]
      associate-*r/ [=>] 79.9	\[ \color{blue}{\frac{-1 \cdot \left(y + x\right)}{\frac{y}{z}}} \]
      neg-mul-1 [<=] 79.9	\[ \frac{\color{blue}{-\left(y + x\right)}}{\frac{y}{z}} \]
      distribute-neg-in [=>] 79.9	\[ \frac{\color{blue}{\left(-y\right) + \left(-x\right)}}{\frac{y}{z}} \]
      sub-neg [<=] 79.9	\[ \frac{\color{blue}{\left(-y\right) - x}}{\frac{y}{z}} \]

   5. Taylor expanded in y around 0 10.64

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

   6. Simplified 8.9

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

      [Start] 10.64	\[ -1 \cdot z + -1 \cdot \frac{z \cdot x}{y} \]
      mul-1-neg [=>] 10.64	\[ \color{blue}{\left(-z\right)} + -1 \cdot \frac{z \cdot x}{y} \]
      associate-*r/ [<=] 8.89	\[ \left(-z\right) + -1 \cdot \color{blue}{\left(z \cdot \frac{x}{y}\right)} \]
      mul-1-neg [=>] 8.89	\[ \left(-z\right) + \color{blue}{\left(-z \cdot \frac{x}{y}\right)} \]
      unsub-neg [=>] 8.89	\[ \color{blue}{\left(-z\right) - z \cdot \frac{x}{y}} \]
      mul-1-neg [<=] 8.89	\[ \color{blue}{-1 \cdot z} - z \cdot \frac{x}{y} \]
      *-commutative [=>] 8.89	\[ \color{blue}{z \cdot -1} - z \cdot \frac{x}{y} \]
      distribute-lft-out-- [=>] 8.9	\[ \color{blue}{z \cdot \left(-1 - \frac{x}{y}\right)} \]

3. Recombined 2 regimes into one program.

4. Final simplification 1.58

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \leq -1 \cdot 10^{-217} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \leq 2 \cdot 10^{-240}\right):\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	28.61%
Cost	1373

\[\begin{array}{l} t_0 := z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{if}\;y \leq -1.05 \cdot 10^{+34}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -2.3 \cdot 10^{-11}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq -1.35 \cdot 10^{-53}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 10^{-269}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{elif}\;y \leq 9.6 \cdot 10^{-78} \lor \neg \left(y \leq 5.2 \cdot 10^{+48}\right) \land y \leq 1.45 \cdot 10^{+136}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	29.3%
Cost	1243

\[\begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{+34} \lor \neg \left(y \leq -1.6 \cdot 10^{-11} \lor \neg \left(y \leq -1.2 \cdot 10^{-61}\right) \land \left(y \leq 4.2 \cdot 10^{-76} \lor \neg \left(y \leq 4 \cdot 10^{+48}\right) \land y \leq 1.1 \cdot 10^{+136}\right)\right):\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 3
Error	34.11%
Cost	780

\[\begin{array}{l} \mathbf{if}\;y \leq -2.8 \cdot 10^{+47}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq -5 \cdot 10^{-12}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq -6.4 \cdot 10^{-58}:\\ \;\;\;\;x \cdot \frac{-z}{y}\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{+136}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 4
Error	34.19%
Cost	780

\[\begin{array}{l} \mathbf{if}\;y \leq -1.65 \cdot 10^{+47}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq -5.1 \cdot 10^{-12}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq -4 \cdot 10^{-58}:\\ \;\;\;\;\frac{-z}{\frac{y}{x}}\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{+137}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 5
Error	33.27%
Cost	456

\[\begin{array}{l} \mathbf{if}\;y \leq -3.8 \cdot 10^{+47}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{+136}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 6
Error	42.57%
Cost	392

\[\begin{array}{l} \mathbf{if}\;y \leq -5.5 \cdot 10^{+18}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{-64}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 7
Error	59.56%
Cost	328

\[\begin{array}{l} \mathbf{if}\;x \leq -4.3 \cdot 10^{-194}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-127}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 8
Error	65.68%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023121 
(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))))