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

Average Error: 3.28% → 0.31%

Time: 18.1s

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	3.28%
Target	0.31%
Herbie	0.31%

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

Derivation

1. Initial program 3.28

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

2. Simplified 0.31

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

   [Start] 3.28	\[ x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   sub-neg [=>] 3.28	\[ \color{blue}{x + \left(-\frac{y - z}{\frac{\left(t - z\right) + 1}{a}}\right)} \]
   +-commutative [=>] 3.28	\[ \color{blue}{\left(-\frac{y - z}{\frac{\left(t - z\right) + 1}{a}}\right) + x} \]
   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	27.59%
Cost	1240

\[\begin{array}{l} t_1 := x - a \cdot y\\ t_2 := x - a \cdot \frac{y}{t}\\ \mathbf{if}\;z \leq -6 \cdot 10^{+16}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq -7.2 \cdot 10^{-172}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -4.5 \cdot 10^{-261}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.05 \cdot 10^{-286}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 4 \cdot 10^{-214}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 9.2 \cdot 10^{+20}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \]

Alternative 2
Error	27.6%
Cost	1240

\[\begin{array}{l} t_1 := x - a \cdot \frac{y}{t}\\ \mathbf{if}\;z \leq -4.4 \cdot 10^{+16}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq -3.1 \cdot 10^{-173}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -2.8 \cdot 10^{-266}:\\ \;\;\;\;x - \frac{a}{\frac{1}{y}}\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{-290}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 8.2 \cdot 10^{-216}:\\ \;\;\;\;x - a \cdot y\\ \mathbf{elif}\;z \leq 1.15 \cdot 10^{+21}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \]

Alternative 3
Error	27.52%
Cost	1240

\[\begin{array}{l} t_1 := x - \frac{a}{\frac{t}{y}}\\ \mathbf{if}\;z \leq -4.2 \cdot 10^{+16}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq -3.9 \cdot 10^{-170}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -7.4 \cdot 10^{-266}:\\ \;\;\;\;x - \frac{a}{\frac{1}{y}}\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{-288}:\\ \;\;\;\;x - a \cdot \frac{y}{t}\\ \mathbf{elif}\;z \leq 1.14 \cdot 10^{-215}:\\ \;\;\;\;x - a \cdot y\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{+22}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \]

Alternative 4
Error	27.55%
Cost	1240

\[\begin{array}{l} t_1 := x - \frac{a}{\frac{t}{y}}\\ \mathbf{if}\;z \leq -1.45 \cdot 10^{+17}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq -2.35 \cdot 10^{-170}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1.35 \cdot 10^{-266}:\\ \;\;\;\;x - \frac{a}{\frac{1}{y}}\\ \mathbf{elif}\;z \leq 2 \cdot 10^{-286}:\\ \;\;\;\;x - a \cdot \frac{y}{t}\\ \mathbf{elif}\;z \leq 7.6 \cdot 10^{-216}:\\ \;\;\;\;x - a \cdot y\\ \mathbf{elif}\;z \leq 25000000000000:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x - \left(a + \frac{a}{z}\right)\\ \end{array} \]

Alternative 5
Error	27.44%
Cost	1240

\[\begin{array}{l} t_1 := x - \left(a + \frac{a}{z}\right)\\ \mathbf{if}\;z \leq -42:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1.9 \cdot 10^{-174}:\\ \;\;\;\;x + \frac{a}{t} \cdot \left(z - y\right)\\ \mathbf{elif}\;z \leq -8.2 \cdot 10^{-265}:\\ \;\;\;\;x - \frac{a}{\frac{1}{y}}\\ \mathbf{elif}\;z \leq 9.8 \cdot 10^{-287}:\\ \;\;\;\;x - a \cdot \frac{y}{t}\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{-215}:\\ \;\;\;\;x - a \cdot y\\ \mathbf{elif}\;z \leq 58000000000000:\\ \;\;\;\;x - \frac{a}{\frac{t}{y}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	12.24%
Cost	969

\[\begin{array}{l} \mathbf{if}\;z \leq -9.6 \cdot 10^{-37} \lor \neg \left(z \leq 1050000000\right):\\ \;\;\;\;x + z \cdot \frac{a}{\left(t - z\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;x - a \cdot \frac{y}{t + 1}\\ \end{array} \]

Alternative 7
Error	8.67%
Cost	968

\[\begin{array}{l} \mathbf{if}\;t \leq -1.08 \cdot 10^{+49}:\\ \;\;\;\;x - a \cdot \frac{y - z}{t}\\ \mathbf{elif}\;t \leq 6.2 \cdot 10^{+75}:\\ \;\;\;\;x - \frac{a}{\frac{1 - z}{y - z}}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{a}{t} \cdot \left(z - y\right)\\ \end{array} \]

Alternative 8
Error	13.56%
Cost	905

\[\begin{array}{l} \mathbf{if}\;z \leq -1.35 \cdot 10^{+16} \lor \neg \left(z \leq 7.5 \cdot 10^{+21}\right):\\ \;\;\;\;x - \frac{y - z}{\frac{-z}{a}}\\ \mathbf{else}:\\ \;\;\;\;x - a \cdot \frac{y}{t + 1}\\ \end{array} \]

Alternative 9
Error	14.98%
Cost	840

\[\begin{array}{l} \mathbf{if}\;z \leq -1.95 \cdot 10^{+21}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{+14}:\\ \;\;\;\;x - a \cdot \frac{y}{t + 1}\\ \mathbf{else}:\\ \;\;\;\;x - \left(a + \frac{a}{z}\right)\\ \end{array} \]

Alternative 10
Error	0.31%
Cost	832

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

Alternative 11
Error	27.24%
Cost	584

\[\begin{array}{l} \mathbf{if}\;z \leq -3.1 \cdot 10^{-32}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 10^{+21}:\\ \;\;\;\;x - a \cdot y\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \]

Alternative 12
Error	31.33%
Cost	456

\[\begin{array}{l} \mathbf{if}\;z \leq -4.3 \cdot 10^{+18}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 6.4 \cdot 10^{+68}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \]

Alternative 13
Error	42.16%
Cost	392

\[\begin{array}{l} \mathbf{if}\;a \leq -2.8 \cdot 10^{+273}:\\ \;\;\;\;-a\\ \mathbf{elif}\;a \leq 4.5 \cdot 10^{+213}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;-a\\ \end{array} \]

Alternative 14
Error	43.38%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023121 
(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))))