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

Average Error: 7.4 → 0.1

Time: 8.6s

Precision: binary64

Cost: 22985

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x + y}{1 - \frac{y}{z}}\\ t_1 := \mathsf{fma}\left(x, z, z \cdot z\right)\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{-298} \lor \neg \left(t_0 \leq 0\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-z}{\frac{y}{x}} - \left(\frac{t_1}{\frac{{y}^{3}}{z \cdot z}} + \left(z + \frac{t_1}{y} \cdot \frac{z}{y}\right)\right)\right) - \frac{z}{\frac{y}{z}}\\ \end{array} \]

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)))) (t_1 (fma x z (* z z))))
   (if (or (<= t_0 -2e-298) (not (<= t_0 0.0)))
     t_0
     (-
      (-
       (/ (- z) (/ y x))
       (+ (/ t_1 (/ (pow y 3.0) (* z z))) (+ z (* (/ t_1 y) (/ z y)))))
      (/ z (/ y z))))))

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 t_1 = fma(x, z, (z * z));
	double tmp;
	if ((t_0 <= -2e-298) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = ((-z / (y / x)) - ((t_1 / (pow(y, 3.0) / (z * z))) + (z + ((t_1 / y) * (z / y))))) - (z / (y / z));
	}
	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)))
	t_1 = fma(x, z, Float64(z * z))
	tmp = 0.0
	if ((t_0 <= -2e-298) || !(t_0 <= 0.0))
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(Float64(-z) / Float64(y / x)) - Float64(Float64(t_1 / Float64((y ^ 3.0) / Float64(z * z))) + Float64(z + Float64(Float64(t_1 / y) * Float64(z / y))))) - Float64(z / Float64(y / z)));
	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]}, Block[{t$95$1 = N[(x * z + N[(z * z), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -2e-298], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], t$95$0, N[(N[(N[((-z) / N[(y / x), $MachinePrecision]), $MachinePrecision] - N[(N[(t$95$1 / N[(N[Power[y, 3.0], $MachinePrecision] / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(z + N[(N[(t$95$1 / y), $MachinePrecision] * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(z / N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	7.4
Target	3.7
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))) < -1.99999999999999982e-298 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 -1.99999999999999982e-298 < (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z))) < -0.0

   1. Initial program 59.5

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

   2. Simplified 59.5

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

      [Start] 59.5	\[ \frac{x + y}{1 - \frac{y}{z}} \]
      +-commutative [=>] 59.5	\[ \frac{\color{blue}{y + x}}{1 - \frac{y}{z}} \]

   3. Taylor expanded in y around inf 0.2

      \[\leadsto \color{blue}{\left(-1 \cdot z + \left(-1 \cdot \frac{z \cdot x}{y} + \left(\frac{\left(-1 \cdot \left(z \cdot x\right) - {z}^{2}\right) \cdot {z}^{2}}{{y}^{3}} + \frac{\left(-1 \cdot \left(z \cdot x\right) - {z}^{2}\right) \cdot z}{{y}^{2}}\right)\right)\right) - \frac{{z}^{2}}{y}} \]

   4. Simplified 0.2

      \[\leadsto \color{blue}{\left(\frac{-z}{\frac{y}{x}} + \left(\left(\left(-z\right) - \frac{\mathsf{fma}\left(x, z, z \cdot z\right)}{y} \cdot \frac{z}{y}\right) - \frac{\mathsf{fma}\left(x, z, z \cdot z\right)}{\frac{{y}^{3}}{z \cdot z}}\right)\right) - \frac{z}{\frac{y}{z}}} \]
      Proof

      [Start] 0.2	\[ \left(-1 \cdot z + \left(-1 \cdot \frac{z \cdot x}{y} + \left(\frac{\left(-1 \cdot \left(z \cdot x\right) - {z}^{2}\right) \cdot {z}^{2}}{{y}^{3}} + \frac{\left(-1 \cdot \left(z \cdot x\right) - {z}^{2}\right) \cdot z}{{y}^{2}}\right)\right)\right) - \frac{{z}^{2}}{y} \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \leq -2 \cdot 10^{-298} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \leq 0\right):\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-z}{\frac{y}{x}} - \left(\frac{\mathsf{fma}\left(x, z, z \cdot z\right)}{\frac{{y}^{3}}{z \cdot z}} + \left(z + \frac{\mathsf{fma}\left(x, z, z \cdot z\right)}{y} \cdot \frac{z}{y}\right)\right)\right) - \frac{z}{\frac{y}{z}}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.1
Cost	1865

\[\begin{array}{l} t_0 := \frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{if}\;t_0 \leq -4 \cdot 10^{-282} \lor \neg \left(t_0 \leq 0\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \end{array} \]

Alternative 2
Error	16.3
Cost	1240

\[\begin{array}{l} t_0 := 1 - \frac{y}{z}\\ t_1 := \frac{y}{t_0}\\ t_2 := z \cdot \left(-1 - \frac{x}{y}\right)\\ t_3 := \frac{x}{t_0}\\ \mathbf{if}\;y \leq -1.14 \cdot 10^{+117}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -1.28 \cdot 10^{-11}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{-176}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{-68}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 6.6 \cdot 10^{-22}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y \leq 5.3 \cdot 10^{+17}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Error	16.5
Cost	1108

\[\begin{array}{l} t_0 := \frac{x}{1 - \frac{y}{z}}\\ t_1 := z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{if}\;y \leq -1.7 \cdot 10^{+70}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -4 \cdot 10^{-30}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 1.06 \cdot 10^{-179}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{-68}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 2 \cdot 10^{-12}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	16.7
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -1.35 \cdot 10^{+70} \lor \neg \left(y \leq 40000000000000\right):\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 5
Error	27.9
Cost	656

\[\begin{array}{l} \mathbf{if}\;y \leq -5.5 \cdot 10^{+104}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq -1.22 \cdot 10^{-56}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{-87}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{+46}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 6
Error	20.4
Cost	456

\[\begin{array}{l} \mathbf{if}\;y \leq -1.05 \cdot 10^{+71}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{+84}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 7
Error	40.2
Cost	328

\[\begin{array}{l} \mathbf{if}\;y \leq -9.8 \cdot 10^{-58}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{-87}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 8
Error	41.9
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023073 
(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))))