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

Average Error: 7.7 → 0.2

Time: 8.6s

Precision: binary64

Cost: 1865

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{-268} \lor \neg \left(t_0 \leq 0\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \end{array} \]

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 -1e-268) (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 <= -1e-268) || !(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 <= (-1d-268)) .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 <= -1e-268) || !(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 <= -1e-268) 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 <= -1e-268) || !(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 <= -1e-268) || ~((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, -1e-268], 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	7.7
Target	3.9
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))) < -9.99999999999999958e-269 or 0.0 < (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z)))

   1. Initial program 0.1

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

   if -9.99999999999999958e-269 < (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z))) < 0.0

   1. Initial program 56.5

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

   2. Simplified 56.5

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

      [Start] 56.5	\[ \frac{x + y}{1 - \frac{y}{z}} \]
      +-commutative [=>] 56.5	\[ \frac{\color{blue}{y + x}}{1 - \frac{y}{z}} \]

   3. Taylor expanded in z around 0 2.7

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

   4. Simplified 57.5

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

      [Start] 2.7	\[ -1 \cdot \frac{\left(y + x\right) \cdot z}{y} \]
      associate-/l* [=>] 57.5	\[ -1 \cdot \color{blue}{\frac{y + x}{\frac{y}{z}}} \]
      associate-*r/ [=>] 57.5	\[ \color{blue}{\frac{-1 \cdot \left(y + x\right)}{\frac{y}{z}}} \]
      neg-mul-1 [<=] 57.5	\[ \frac{\color{blue}{-\left(y + x\right)}}{\frac{y}{z}} \]
      distribute-neg-in [=>] 57.5	\[ \frac{\color{blue}{\left(-y\right) + \left(-x\right)}}{\frac{y}{z}} \]
      sub-neg [<=] 57.5	\[ \frac{\color{blue}{\left(-y\right) - x}}{\frac{y}{z}} \]

   5. Taylor expanded in y around 0 1.3

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

   6. Simplified 1.0

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

      [Start] 1.3	\[ -1 \cdot z + -1 \cdot \frac{z \cdot x}{y} \]
      mul-1-neg [=>] 1.3	\[ \color{blue}{\left(-z\right)} + -1 \cdot \frac{z \cdot x}{y} \]
      associate-*r/ [<=] 1.0	\[ \left(-z\right) + -1 \cdot \color{blue}{\left(z \cdot \frac{x}{y}\right)} \]
      mul-1-neg [=>] 1.0	\[ \left(-z\right) + \color{blue}{\left(-z \cdot \frac{x}{y}\right)} \]
      unsub-neg [=>] 1.0	\[ \color{blue}{\left(-z\right) - z \cdot \frac{x}{y}} \]
      mul-1-neg [<=] 1.0	\[ \color{blue}{-1 \cdot z} - z \cdot \frac{x}{y} \]
      *-commutative [=>] 1.0	\[ -1 \cdot z - \color{blue}{\frac{x}{y} \cdot z} \]
      distribute-rgt-out-- [=>] 1.0	\[ \color{blue}{z \cdot \left(-1 - \frac{x}{y}\right)} \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \leq -1 \cdot 10^{-268} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \leq 0\right):\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	18.2
Cost	1240

\[\begin{array}{l} t_0 := \frac{x}{1 - \frac{y}{z}}\\ t_1 := z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{if}\;z \leq -2.1 \cdot 10^{-36}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq -1.9 \cdot 10^{-121}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1.75 \cdot 10^{-157}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 7.2 \cdot 10^{-107}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 780:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 4.8 \cdot 10^{+41}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 2
Error	18.2
Cost	1240

\[\begin{array}{l} t_0 := 1 - \frac{y}{z}\\ t_1 := \frac{x}{t_0}\\ t_2 := z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{if}\;z \leq -6.5 \cdot 10^{-33}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq -3.7 \cdot 10^{-122}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -2 \cdot 10^{-158}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{-107}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 1.15 \cdot 10^{-5}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{+42}:\\ \;\;\;\;\frac{y}{t_0}\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 3
Error	25.5
Cost	1117

\[\begin{array}{l} \mathbf{if}\;z \leq -7.2 \cdot 10^{-37}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq -1.1 \cdot 10^{-122}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq -3.5 \cdot 10^{-150}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -7.5 \cdot 10^{-160}:\\ \;\;\;\;-x \cdot \frac{z}{y}\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{-103} \lor \neg \left(z \leq 92000\right) \land z \leq 1.2 \cdot 10^{+42}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 4
Error	25.6
Cost	1117

\[\begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{-36}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq -4 \cdot 10^{-122}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq -1.4 \cdot 10^{-149}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -2.3 \cdot 10^{-160}:\\ \;\;\;\;\frac{-x}{\frac{y}{z}}\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{-102} \lor \neg \left(z \leq 6500\right) \land z \leq 1.04 \cdot 10^{+43}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 5
Error	17.6
Cost	978

\[\begin{array}{l} \mathbf{if}\;z \leq -8.5 \cdot 10^{-33} \lor \neg \left(z \leq 7.6 \cdot 10^{-54}\right) \land \left(z \leq 160000 \lor \neg \left(z \leq 2.7 \cdot 10^{+40}\right)\right):\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \end{array} \]

Alternative 6
Error	18.0
Cost	976

\[\begin{array}{l} t_0 := 1 - \frac{y}{z}\\ \mathbf{if}\;z \leq -7.2 \cdot 10^{-32}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq 6.2 \cdot 10^{-108}:\\ \;\;\;\;\frac{z \cdot \left(\left(-x\right) - y\right)}{y}\\ \mathbf{elif}\;z \leq 8 \cdot 10^{-5}:\\ \;\;\;\;\frac{x}{t_0}\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{+42}:\\ \;\;\;\;\frac{y}{t_0}\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 7
Error	20.5
Cost	456

\[\begin{array}{l} \mathbf{if}\;y \leq -2.4 \cdot 10^{+76}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 5.7 \cdot 10^{+78}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 8
Error	27.1
Cost	392

\[\begin{array}{l} \mathbf{if}\;y \leq -6.2 \cdot 10^{+25}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 10^{-37}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 9
Error	38.1
Cost	328

\[\begin{array}{l} \mathbf{if}\;x \leq -6.8 \cdot 10^{-115}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-120}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 10
Error	41.8
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023031 
(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))))