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

Average Error: 7.4 → 0.7

Time: 8.8s

Precision: binary64

Cost: 1865

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{-217} \lor \neg \left(t_0 \leq 0\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) - z \cdot \frac{x}{y}\\ \end{array} \]

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 -1e-217) (not (<= t_0 0.0))) t_0 (- (- z) (* z (/ 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 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = -z - (z * (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 <= 0.0d0))) then
        tmp = t_0
    else
        tmp = -z - (z * (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 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = -z - (z * (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 <= 0.0):
		tmp = t_0
	else:
		tmp = -z - (z * (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 <= 0.0))
		tmp = t_0;
	else
		tmp = Float64(Float64(-z) - Float64(z * 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 <= 0.0)))
		tmp = t_0;
	else
		tmp = -z - (z * (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, 0.0]], $MachinePrecision]], t$95$0, N[((-z) - N[(z * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Target

Original	7.4
Target	4.0
Herbie	0.7

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z))) < -1.00000000000000008e-217 or -0.0 < (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z)))

   1. Initial program 0.1

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

   if -1.00000000000000008e-217 < (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z))) < -0.0

   1. Initial program 49.7

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

   2. Simplified 49.7

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

      [Start] 49.7	\[ \frac{x + y}{1 - \frac{y}{z}} \]
      +-commutative [=>] 49.7	\[ \frac{\color{blue}{y + x}}{1 - \frac{y}{z}} \]

   3. Taylor expanded in z around 0 8.0

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

   4. Simplified 54.0

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

      [Start] 8.0	\[ -1 \cdot \frac{\left(y + x\right) \cdot z}{y} \]
      associate-/l* [=>] 54.0	\[ -1 \cdot \color{blue}{\frac{y + x}{\frac{y}{z}}} \]
      associate-*r/ [=>] 54.0	\[ \color{blue}{\frac{-1 \cdot \left(y + x\right)}{\frac{y}{z}}} \]
      neg-mul-1 [<=] 54.0	\[ \frac{\color{blue}{-\left(y + x\right)}}{\frac{y}{z}} \]
      distribute-neg-in [=>] 54.0	\[ \frac{\color{blue}{\left(-y\right) + \left(-x\right)}}{\frac{y}{z}} \]
      sub-neg [<=] 54.0	\[ \frac{\color{blue}{\left(-y\right) - x}}{\frac{y}{z}} \]

   5. Taylor expanded in y around 0 5.0

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

   6. Simplified 4.3

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

      [Start] 5.0	\[ -1 \cdot z + -1 \cdot \frac{z \cdot x}{y} \]
      mul-1-neg [=>] 5.0	\[ \color{blue}{\left(-z\right)} + -1 \cdot \frac{z \cdot x}{y} \]
      associate-*r/ [<=] 4.3	\[ \left(-z\right) + -1 \cdot \color{blue}{\left(z \cdot \frac{x}{y}\right)} \]
      mul-1-neg [=>] 4.3	\[ \left(-z\right) + \color{blue}{\left(-z \cdot \frac{x}{y}\right)} \]
      +-commutative [<=] 4.3	\[ \color{blue}{\left(-z \cdot \frac{x}{y}\right) + \left(-z\right)} \]
      unsub-neg [=>] 4.3	\[ \color{blue}{\left(-z \cdot \frac{x}{y}\right) - z} \]
      distribute-rgt-neg-in [=>] 4.3	\[ \color{blue}{z \cdot \left(-\frac{x}{y}\right)} - z \]
      distribute-neg-frac [=>] 4.3	\[ z \cdot \color{blue}{\frac{-x}{y}} - z \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \leq -1 \cdot 10^{-217} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \leq 0\right):\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) - z \cdot \frac{x}{y}\\ \end{array} \]

Alternatives

Alternative 1
Error	18.0
Cost	1172

\[\begin{array}{l} t_0 := 1 - \frac{y}{z}\\ t_1 := \frac{y}{t_0}\\ \mathbf{if}\;y \leq -7 \cdot 10^{+190}:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{elif}\;y \leq -1.95 \cdot 10^{-22}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{-110}:\\ \;\;\;\;\frac{x}{t_0}\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{-64}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 4.4 \cdot 10^{+88}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) - z \cdot \frac{x}{y}\\ \end{array} \]

Alternative 2
Error	18.0
Cost	1108

\[\begin{array}{l} t_0 := 1 - \frac{y}{z}\\ t_1 := \frac{y}{t_0}\\ t_2 := z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{if}\;y \leq -2.2 \cdot 10^{+187}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -4.8 \cdot 10^{-25}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{-110}:\\ \;\;\;\;\frac{x}{t_0}\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{-64}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{+88}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Error	16.8
Cost	978

\[\begin{array}{l} \mathbf{if}\;y \leq -4000000000 \lor \neg \left(y \leq 5.8 \cdot 10^{-64}\right) \land \left(y \leq 4.2 \cdot 10^{-26} \lor \neg \left(y \leq 4 \cdot 10^{+82}\right)\right):\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 4
Error	27.7
Cost	524

\[\begin{array}{l} \mathbf{if}\;y \leq -6.5 \cdot 10^{-18}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 9.2 \cdot 10^{-112}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.12 \cdot 10^{+77}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 5
Error	20.8
Cost	456

\[\begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+73}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{+81}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 6
Error	40.6
Cost	328

\[\begin{array}{l} \mathbf{if}\;y \leq -2.45 \cdot 10^{+59}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 1.42 \cdot 10^{-111}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 7
Error	41.9
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023060 
(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))))