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

Average Error: 11.74% → 0.2%

Time: 9.5s

Precision: binary64

Cost: 1865

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{if}\;t_0 \leq -4 \cdot 10^{-284} \lor \neg \left(t_0 \leq 0\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \end{array} \]

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 -4e-284) (not (<= t_0 0.0))) 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 <= -4e-284) || !(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
    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 <= (-4d-284)) .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) {
	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 <= -4e-284) || !(t_0 <= 0.0)) {
		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 <= -4e-284) or not (t_0 <= 0.0):
		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 <= -4e-284) || !(t_0 <= 0.0))
		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 <= -4e-284) || ~((t_0 <= 0.0)))
		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, -4e-284], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], t$95$0, N[(z * N[(-1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Target

Original	11.74%
Target	6.14%
Herbie	0.2%

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z))) < -4.00000000000000015e-284 or -0.0 < (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z)))

   1. Initial program 0.13

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

   if -4.00000000000000015e-284 < (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z))) < -0.0

   1. Initial program 91.83

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

   2. Simplified 91.83

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

      [Start] 91.83	\[ \frac{x + y}{1 - \frac{y}{z}} \]
      +-commutative [=>] 91.83	\[ \frac{\color{blue}{y + x}}{1 - \frac{y}{z}} \]

   3. Taylor expanded in y around inf 0.77

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

   4. Simplified 16.28

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

      [Start] 0.77	\[ \left(-1 \cdot z + -1 \cdot \frac{z \cdot x}{y}\right) - \frac{{z}^{2}}{y} \]
      mul-1-neg [=>] 0.77	\[ \left(-1 \cdot z + \color{blue}{\left(-\frac{z \cdot x}{y}\right)}\right) - \frac{{z}^{2}}{y} \]
      unsub-neg [=>] 0.77	\[ \color{blue}{\left(-1 \cdot z - \frac{z \cdot x}{y}\right)} - \frac{{z}^{2}}{y} \]
      mul-1-neg [=>] 0.77	\[ \left(\color{blue}{\left(-z\right)} - \frac{z \cdot x}{y}\right) - \frac{{z}^{2}}{y} \]
      associate-/l* [=>] 0.65	\[ \left(\left(-z\right) - \color{blue}{\frac{z}{\frac{y}{x}}}\right) - \frac{{z}^{2}}{y} \]
      associate-/r/ [=>] 16.28	\[ \left(\left(-z\right) - \color{blue}{\frac{z}{y} \cdot x}\right) - \frac{{z}^{2}}{y} \]
      unpow2 [=>] 16.28	\[ \left(\left(-z\right) - \frac{z}{y} \cdot x\right) - \frac{\color{blue}{z \cdot z}}{y} \]
      associate-/l* [=>] 16.28	\[ \left(\left(-z\right) - \frac{z}{y} \cdot x\right) - \color{blue}{\frac{z}{\frac{y}{z}}} \]

   5. Taylor expanded in z around 0 0.64

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

   6. Simplified 0.64

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

      [Start] 0.64	\[ -1 \cdot \left(\left(1 + \frac{x}{y}\right) \cdot z\right) \]
      mul-1-neg [=>] 0.64	\[ \color{blue}{-\left(1 + \frac{x}{y}\right) \cdot z} \]
      *-commutative [=>] 0.64	\[ -\color{blue}{z \cdot \left(1 + \frac{x}{y}\right)} \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \leq -4 \cdot 10^{-284} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \leq 0\right):\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	40.34%
Cost	1768

\[\begin{array}{l} t_0 := 1 - \frac{y}{z}\\ t_1 := \frac{x}{t_0}\\ t_2 := \frac{y}{t_0}\\ \mathbf{if}\;z \leq -2.4 \cdot 10^{+27}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq -2 \cdot 10^{-65}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -2.35 \cdot 10^{-86}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -2 \cdot 10^{-220}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq -1.25 \cdot 10^{-272}:\\ \;\;\;\;\frac{z \cdot \left(-x\right)}{y}\\ \mathbf{elif}\;z \leq 5.8 \cdot 10^{-241}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 10^{-112}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 5.8 \cdot 10^{-31}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 2 \cdot 10^{-16}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.85 \cdot 10^{+135}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 2
Error	39.96%
Cost	1108

\[\begin{array}{l} \mathbf{if}\;z \leq -2.4 \cdot 10^{-85}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq -2.25 \cdot 10^{-220}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq -1.5 \cdot 10^{-272}:\\ \;\;\;\;\frac{z \cdot \left(-x\right)}{y}\\ \mathbf{elif}\;z \leq 4.6 \cdot 10^{-239}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 7.8 \cdot 10^{-112}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{elif}\;z \leq 7.6 \cdot 10^{+55}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 3
Error	41.37%
Cost	1044

\[\begin{array}{l} t_0 := \frac{z}{y} \cdot \left(-x\right)\\ \mathbf{if}\;z \leq -8.2 \cdot 10^{-87}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq -2 \cdot 10^{-220}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq -9.5 \cdot 10^{-273}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 4.8 \cdot 10^{-240}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 2.05 \cdot 10^{-162}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 7 \cdot 10^{+55}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 4
Error	41.35%
Cost	1044

\[\begin{array}{l} \mathbf{if}\;z \leq -3.9 \cdot 10^{-86}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq -2.3 \cdot 10^{-220}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq -1.6 \cdot 10^{-272}:\\ \;\;\;\;\frac{z}{y} \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 4.9 \cdot 10^{-241}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{-162}:\\ \;\;\;\;\frac{-x}{\frac{y}{z}}\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{+55}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 5
Error	41.11%
Cost	1044

\[\begin{array}{l} \mathbf{if}\;z \leq -2.45 \cdot 10^{-85}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq -2.3 \cdot 10^{-220}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq -1.35 \cdot 10^{-272}:\\ \;\;\;\;\frac{z \cdot \left(-x\right)}{y}\\ \mathbf{elif}\;z \leq 4.8 \cdot 10^{-240}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{-162}:\\ \;\;\;\;\frac{-x}{\frac{y}{z}}\\ \mathbf{elif}\;z \leq 7 \cdot 10^{+55}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 6
Error	26.55%
Cost	777

\[\begin{array}{l} \mathbf{if}\;z \leq -6.7 \cdot 10^{-9} \lor \neg \left(z \leq 9.5 \cdot 10^{+55}\right):\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) - \frac{x \cdot z}{y}\\ \end{array} \]

Alternative 7
Error	34.5%
Cost	721

\[\begin{array}{l} \mathbf{if}\;y \leq -2.6 \cdot 10^{+95}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq -1.95 \cdot 10^{+40} \lor \neg \left(y \leq -3.35 \cdot 10^{-47}\right) \land y \leq 8 \cdot 10^{+139}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 8
Error	27.39%
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -7 \cdot 10^{-9} \lor \neg \left(z \leq 8.2 \cdot 10^{+55}\right):\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \end{array} \]

Alternative 9
Error	43.54%
Cost	656

\[\begin{array}{l} \mathbf{if}\;y \leq -2.6 \cdot 10^{+94}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq -1.95 \cdot 10^{+39}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq -2.75 \cdot 10^{-58}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 4.4 \cdot 10^{-33}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 10
Error	58.22%
Cost	328

\[\begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{-121}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 8.2 \cdot 10^{-153}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 11
Error	64.7%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023088 
(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))))