Graphics.Rendering.Chart.SparkLine:renderSparkLine from Chart-1.5.3

Average Error: 3.2% → 0.31%

Time: 15.3s

Precision: binary64

Cost: 7104

Math

\[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]

↓

\[\mathsf{fma}\left(a, \frac{z - y}{\left(t - z\right) + 1}, x\right) \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (- x (/ (- y z) (/ (+ (- t z) 1.0) a))))

↓

(FPCore (x y z t a) :precision binary64 (fma a (/ (- z y) (+ (- t z) 1.0)) x))

C

double code(double x, double y, double z, double t, double a) {
	return x - ((y - z) / (((t - z) + 1.0) / a));
}

↓

double code(double x, double y, double z, double t, double a) {
	return fma(a, ((z - y) / ((t - z) + 1.0)), x);
}

Julia

function code(x, y, z, t, a)
	return Float64(x - Float64(Float64(y - z) / Float64(Float64(Float64(t - z) + 1.0) / a)))
end

↓

function code(x, y, z, t, a)
	return fma(a, Float64(Float64(z - y) / Float64(Float64(t - z) + 1.0)), x)
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(x - N[(N[(y - z), $MachinePrecision] / N[(N[(N[(t - z), $MachinePrecision] + 1.0), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_, t_, a_] := N[(a * N[(N[(z - y), $MachinePrecision] / N[(N[(t - z), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]

Target

Original	3.2%
Target	0.31%
Herbie	0.31%

\[x - \frac{y - z}{\left(t - z\right) + 1} \cdot a \]

Derivation

1. Initial program 3.2

   \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]

2. Simplified 0.31

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

   [Start] 3.2	\[ x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   sub-neg [=>] 3.2	\[ \color{blue}{x + \left(-\frac{y - z}{\frac{\left(t - z\right) + 1}{a}}\right)} \]
   +-commutative [=>] 3.2	\[ \color{blue}{\left(-\frac{y - z}{\frac{\left(t - z\right) + 1}{a}}\right) + x} \]
   associate-/r/ [=>] 0.31	\[ \left(-\color{blue}{\frac{y - z}{\left(t - z\right) + 1} \cdot a}\right) + x \]
   distribute-lft-neg-in [=>] 0.31	\[ \color{blue}{\left(-\frac{y - z}{\left(t - z\right) + 1}\right) \cdot a} + x \]
   *-commutative [=>] 0.31	\[ \color{blue}{a \cdot \left(-\frac{y - z}{\left(t - z\right) + 1}\right)} + x \]
   fma-def [=>] 0.31	\[ \color{blue}{\mathsf{fma}\left(a, -\frac{y - z}{\left(t - z\right) + 1}, x\right)} \]
   neg-sub0 [=>] 0.31	\[ \mathsf{fma}\left(a, \color{blue}{0 - \frac{y - z}{\left(t - z\right) + 1}}, x\right) \]
   div-sub [=>] 0.31	\[ \mathsf{fma}\left(a, 0 - \color{blue}{\left(\frac{y}{\left(t - z\right) + 1} - \frac{z}{\left(t - z\right) + 1}\right)}, x\right) \]
   associate--r- [=>] 0.31	\[ \mathsf{fma}\left(a, \color{blue}{\left(0 - \frac{y}{\left(t - z\right) + 1}\right) + \frac{z}{\left(t - z\right) + 1}}, x\right) \]
   neg-sub0 [<=] 0.31	\[ \mathsf{fma}\left(a, \color{blue}{\left(-\frac{y}{\left(t - z\right) + 1}\right)} + \frac{z}{\left(t - z\right) + 1}, x\right) \]
   +-commutative [=>] 0.31	\[ \mathsf{fma}\left(a, \color{blue}{\frac{z}{\left(t - z\right) + 1} + \left(-\frac{y}{\left(t - z\right) + 1}\right)}, x\right) \]
   sub-neg [<=] 0.31	\[ \mathsf{fma}\left(a, \color{blue}{\frac{z}{\left(t - z\right) + 1} - \frac{y}{\left(t - z\right) + 1}}, x\right) \]
   div-sub [<=] 0.31	\[ \mathsf{fma}\left(a, \color{blue}{\frac{z - y}{\left(t - z\right) + 1}}, x\right) \]

3. Final simplification 0.31

   \[\leadsto \mathsf{fma}\left(a, \frac{z - y}{\left(t - z\right) + 1}, x\right) \]

Alternatives

Alternative 1
Error	28.98%
Cost	1108

\[\begin{array}{l} t_1 := x - a \cdot y\\ \mathbf{if}\;z \leq -3.5 \cdot 10^{+21}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 6.4 \cdot 10^{-258}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{-58}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 7.5 \cdot 10^{+16}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.65 \cdot 10^{+39}:\\ \;\;\;\;a \cdot \frac{z - y}{t}\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{+133}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \]

Alternative 2
Error	14.7%
Cost	1096

\[\begin{array}{l} \mathbf{if}\;z \leq -2.95 \cdot 10^{+29}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 6 \cdot 10^{+90}:\\ \;\;\;\;x - y \cdot \frac{a}{t + 1}\\ \mathbf{else}:\\ \;\;\;\;\left(x - a\right) + \frac{a}{z} \cdot \left(y + \left(-1 - t\right)\right)\\ \end{array} \]

Alternative 3
Error	30.95%
Cost	976

\[\begin{array}{l} t_1 := x - \frac{a \cdot y}{t}\\ \mathbf{if}\;t \leq -86000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -3 \cdot 10^{-225}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;t \leq 5 \cdot 10^{-151}:\\ \;\;\;\;x - a \cdot y\\ \mathbf{elif}\;t \leq 30000000:\\ \;\;\;\;x - a\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	15.02%
Cost	904

\[\begin{array}{l} \mathbf{if}\;z \leq -8.8 \cdot 10^{+29}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 6.6 \cdot 10^{+72}:\\ \;\;\;\;x - y \cdot \frac{a}{t + 1}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{z - y}{\frac{-z}{a}}\\ \end{array} \]

Alternative 5
Error	15.15%
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -9.8 \cdot 10^{+29} \lor \neg \left(z \leq 8 \cdot 10^{+74}\right):\\ \;\;\;\;x - a\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \frac{a}{t + 1}\\ \end{array} \]

Alternative 6
Error	0.31%
Cost	832

\[x + a \cdot \frac{z - y}{\left(t - z\right) + 1} \]

Alternative 7
Error	28.55%
Cost	588

\[\begin{array}{l} \mathbf{if}\;z \leq -1.8 \cdot 10^{+23}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 8 \cdot 10^{-258}:\\ \;\;\;\;x - a \cdot y\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{+133}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \]

Alternative 8
Error	30.84%
Cost	456

\[\begin{array}{l} \mathbf{if}\;z \leq -2.8 \cdot 10^{+26}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{+133}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \]

Alternative 9
Error	42.65%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023088 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Chart.SparkLine:renderSparkLine from Chart-1.5.3"
  :precision binary64

  :herbie-target
  (- x (* (/ (- y z) (+ (- t z) 1.0)) a))

  (- x (/ (- y z) (/ (+ (- t z) 1.0) a))))