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

Average Accuracy: 88.2% → 98.2%

Time: 9.0s

Precision: binary64

Cost: 8969

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{-201} \lor \neg \left(t_0 \leq 0\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(z \cdot z, \frac{1}{y} + \frac{x}{y \cdot y}, z \cdot \frac{x + y}{y}\right)\\ \end{array} \]

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-201) (not (<= t_0 0.0)))
     t_0
     (- (fma (* z z) (+ (/ 1.0 y) (/ x (* y y))) (* z (/ (+ x y) 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-201) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = -fma((z * z), ((1.0 / y) + (x / (y * y))), (z * ((x + y) / 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-201) || !(t_0 <= 0.0))
		tmp = t_0;
	else
		tmp = Float64(-fma(Float64(z * z), Float64(Float64(1.0 / y) + Float64(x / Float64(y * y))), Float64(z * Float64(Float64(x + y) / y))));
	end
	return 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-201], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], t$95$0, (-N[(N[(z * z), $MachinePrecision] * N[(N[(1.0 / y), $MachinePrecision] + N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(z * N[(N[(x + y), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])]]

Target

Original	88.2%
Target	94.1%
Herbie	98.2%

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z))) < -9.99999999999999946e-202 or 0.0 < (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z)))

   1. Initial program 99.9%

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

   if -9.99999999999999946e-202 < (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z))) < 0.0

   1. Initial program 25.5%

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

   2. Simplified 25.5%

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

      [Start] 25.5	\[ \frac{x + y}{1 - \frac{y}{z}} \]
      +-commutative [=>] 25.5	\[ \frac{\color{blue}{y + x}}{1 - \frac{y}{z}} \]

   3. Taylor expanded in z around 0 86.1%

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

   4. Simplified 89.1%

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

      [Start] 86.1	\[ -1 \cdot \left(\left(\frac{1}{y} + \frac{x}{{y}^{2}}\right) \cdot {z}^{2}\right) + -1 \cdot \frac{\left(y + x\right) \cdot z}{y} \]
      distribute-lft-out [=>] 86.1	\[ \color{blue}{-1 \cdot \left(\left(\frac{1}{y} + \frac{x}{{y}^{2}}\right) \cdot {z}^{2} + \frac{\left(y + x\right) \cdot z}{y}\right)} \]
      mul-1-neg [=>] 86.1	\[ \color{blue}{-\left(\left(\frac{1}{y} + \frac{x}{{y}^{2}}\right) \cdot {z}^{2} + \frac{\left(y + x\right) \cdot z}{y}\right)} \]
      *-commutative [=>] 86.1	\[ -\left(\color{blue}{{z}^{2} \cdot \left(\frac{1}{y} + \frac{x}{{y}^{2}}\right)} + \frac{\left(y + x\right) \cdot z}{y}\right) \]
      fma-def [=>] 86.1	\[ -\color{blue}{\mathsf{fma}\left({z}^{2}, \frac{1}{y} + \frac{x}{{y}^{2}}, \frac{\left(y + x\right) \cdot z}{y}\right)} \]
      unpow2 [=>] 86.1	\[ -\mathsf{fma}\left(\color{blue}{z \cdot z}, \frac{1}{y} + \frac{x}{{y}^{2}}, \frac{\left(y + x\right) \cdot z}{y}\right) \]
      unpow2 [=>] 86.1	\[ -\mathsf{fma}\left(z \cdot z, \frac{1}{y} + \frac{x}{\color{blue}{y \cdot y}}, \frac{\left(y + x\right) \cdot z}{y}\right) \]
      associate-/l* [=>] 14.7	\[ -\mathsf{fma}\left(z \cdot z, \frac{1}{y} + \frac{x}{y \cdot y}, \color{blue}{\frac{y + x}{\frac{y}{z}}}\right) \]
      associate-/r/ [=>] 89.1	\[ -\mathsf{fma}\left(z \cdot z, \frac{1}{y} + \frac{x}{y \cdot y}, \color{blue}{\frac{y + x}{y} \cdot z}\right) \]
      +-commutative [=>] 89.1	\[ -\mathsf{fma}\left(z \cdot z, \frac{1}{y} + \frac{x}{y \cdot y}, \frac{\color{blue}{x + y}}{y} \cdot z\right) \]

3. Recombined 2 regimes into one program.

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \leq -1 \cdot 10^{-201} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \leq 0\right):\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(z \cdot z, \frac{1}{y} + \frac{x}{y \cdot y}, z \cdot \frac{x + y}{y}\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.8%
Cost	1865

\[\begin{array}{l} t_0 := \frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{-285} \lor \neg \left(t_0 \leq 0\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \left(-z\right) - z\\ \end{array} \]

Alternative 2
Accuracy	75.8%
Cost	972

\[\begin{array}{l} t_0 := \frac{x}{y} \cdot \left(-z\right) - z\\ \mathbf{if}\;y \leq -1.1 \cdot 10^{+27}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -1.05 \cdot 10^{-126}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{elif}\;y \leq 1.52 \cdot 10^{-10}:\\ \;\;\;\;\left(1 + \frac{y}{z}\right) \cdot \left(x + y\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Accuracy	75.8%
Cost	908

\[\begin{array}{l} t_0 := \frac{x}{y} \cdot \left(-z\right) - z\\ \mathbf{if}\;y \leq -1.05 \cdot 10^{+25}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -2 \cdot 10^{-126}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{-10}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Accuracy	75.8%
Cost	844

\[\begin{array}{l} t_0 := z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{if}\;y \leq -8.5 \cdot 10^{+23}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -4.4 \cdot 10^{-126}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{elif}\;y \leq 1.85 \cdot 10^{-10}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Accuracy	68.3%
Cost	780

\[\begin{array}{l} \mathbf{if}\;y \leq -3.3 \cdot 10^{+52}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{+45}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{+84}:\\ \;\;\;\;\frac{x}{y} \cdot \left(-z\right)\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{+102}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 6
Accuracy	75.7%
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -5.8 \cdot 10^{+25} \lor \neg \left(y \leq 2.5 \cdot 10^{-10}\right):\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 7
Accuracy	69.0%
Cost	456

\[\begin{array}{l} \mathbf{if}\;y \leq -1.6 \cdot 10^{+50}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 6.8 \cdot 10^{+102}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 8
Accuracy	59.6%
Cost	392

\[\begin{array}{l} \mathbf{if}\;y \leq -2.3 \cdot 10^{+24}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 12000000000000:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 9
Accuracy	41.2%
Cost	328

\[\begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{-42}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{-121}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 10
Accuracy	35.9%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023133 
(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))))