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

Average Error: 7.4 → 0.1

Time: 8.8s

Precision: binary64

Cost: 1865

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{-281} \lor \neg \left(t_0 \leq 0\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \end{array} \]

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-281) (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-281) || !(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-281)) .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-281) || !(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-281) 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-281) || !(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-281) || ~((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-281], 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	7.4
Target	3.8
Herbie	0.1

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z))) < -1e-281 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 -1e-281 < (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z))) < -0.0

   1. Initial program 58.5

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

   2. Simplified 58.5

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

      [Start] 58.5	\[ \frac{x + y}{1 - \frac{y}{z}} \]
      +-commutative [=>] 58.5	\[ \frac{\color{blue}{y + x}}{1 - \frac{y}{z}} \]

   3. Taylor expanded in z around 0 1.6

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

   4. Simplified 59.0

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

      [Start] 1.6	\[ -1 \cdot \frac{\left(y + x\right) \cdot z}{y} \]
      associate-/l* [=>] 59.0	\[ -1 \cdot \color{blue}{\frac{y + x}{\frac{y}{z}}} \]
      associate-*r/ [=>] 59.0	\[ \color{blue}{\frac{-1 \cdot \left(y + x\right)}{\frac{y}{z}}} \]
      neg-mul-1 [<=] 59.0	\[ \frac{\color{blue}{-\left(y + x\right)}}{\frac{y}{z}} \]
      distribute-neg-in [=>] 59.0	\[ \frac{\color{blue}{\left(-y\right) + \left(-x\right)}}{\frac{y}{z}} \]
      sub-neg [<=] 59.0	\[ \frac{\color{blue}{\left(-y\right) - x}}{\frac{y}{z}} \]

   5. Taylor expanded in y around 0 0.5

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

   6. Simplified 0.5

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

      [Start] 0.5	\[ -1 \cdot z + -1 \cdot \frac{z \cdot x}{y} \]
      mul-1-neg [=>] 0.5	\[ \color{blue}{\left(-z\right)} + -1 \cdot \frac{z \cdot x}{y} \]
      associate-*r/ [<=] 0.5	\[ \left(-z\right) + -1 \cdot \color{blue}{\left(z \cdot \frac{x}{y}\right)} \]
      mul-1-neg [=>] 0.5	\[ \left(-z\right) + \color{blue}{\left(-z \cdot \frac{x}{y}\right)} \]
      unsub-neg [=>] 0.5	\[ \color{blue}{\left(-z\right) - z \cdot \frac{x}{y}} \]
      mul-1-neg [<=] 0.5	\[ \color{blue}{-1 \cdot z} - z \cdot \frac{x}{y} \]
      *-commutative [=>] 0.5	\[ \color{blue}{z \cdot -1} - z \cdot \frac{x}{y} \]
      distribute-lft-out-- [=>] 0.5	\[ \color{blue}{z \cdot \left(-1 - \frac{x}{y}\right)} \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \leq -1 \cdot 10^{-281} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \leq 0\right):\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	16.7
Cost	1241

\[\begin{array}{l} t_0 := 1 - \frac{y}{z}\\ t_1 := z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{if}\;y \leq -3.3 \cdot 10^{+63}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 6.8 \cdot 10^{-138}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{-41}:\\ \;\;\;\;\frac{x}{t_0}\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{-18} \lor \neg \left(y \leq 7 \cdot 10^{+80}\right) \land y \leq 1.95 \cdot 10^{+116}:\\ \;\;\;\;\frac{y}{t_0}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	16.8
Cost	908

\[\begin{array}{l} t_0 := z \cdot \frac{-x}{y} - z\\ \mathbf{if}\;y \leq -4.4 \cdot 10^{+64}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{-138}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{-43}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	16.9
Cost	908

\[\begin{array}{l} t_0 := z \cdot \frac{-x}{y} - z\\ \mathbf{if}\;y \leq -2.85 \cdot 10^{+63}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{-141}:\\ \;\;\;\;\left(1 + \frac{y}{z}\right) \cdot \left(x + y\right)\\ \mathbf{elif}\;y \leq 4.4 \cdot 10^{-48}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	16.8
Cost	844

\[\begin{array}{l} t_0 := z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{if}\;y \leq -5.8 \cdot 10^{+63}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{-137}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{-43}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	17.1
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -2.75 \cdot 10^{+66} \lor \neg \left(y \leq 1.3 \cdot 10^{-45}\right):\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 6
Error	21.1
Cost	456

\[\begin{array}{l} \mathbf{if}\;y \leq -1.34 \cdot 10^{+67}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{-15}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 7
Error	26.6
Cost	392

\[\begin{array}{l} \mathbf{if}\;y \leq -3.5 \cdot 10^{-53}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{-40}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 8
Error	37.8
Cost	328

\[\begin{array}{l} \mathbf{if}\;x \leq -4.1 \cdot 10^{-79}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-141}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 9
Error	41.3
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023016 
(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))))