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

Average Error: 1.7 → 0.3

Time: 17.9s

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	1.7
Target	0.3
Herbie	0.3

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

Derivation

1. Initial program 1.7

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

2. Simplified 0.3

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

   [Start] 1.7	\[ x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   sub-neg [=>] 1.7	\[ \color{blue}{x + \left(-\frac{y - z}{\frac{\left(t - z\right) + 1}{a}}\right)} \]
   +-commutative [=>] 1.7	\[ \color{blue}{\left(-\frac{y - z}{\frac{\left(t - z\right) + 1}{a}}\right) + x} \]
   associate-/r/ [=>] 0.3	\[ \left(-\color{blue}{\frac{y - z}{\left(t - z\right) + 1} \cdot a}\right) + x \]
   distribute-lft-neg-in [=>] 0.3	\[ \color{blue}{\left(-\frac{y - z}{\left(t - z\right) + 1}\right) \cdot a} + x \]
   *-commutative [=>] 0.3	\[ \color{blue}{a \cdot \left(-\frac{y - z}{\left(t - z\right) + 1}\right)} + x \]
   fma-def [=>] 0.3	\[ \color{blue}{\mathsf{fma}\left(a, -\frac{y - z}{\left(t - z\right) + 1}, x\right)} \]
   neg-sub0 [=>] 0.3	\[ \mathsf{fma}\left(a, \color{blue}{0 - \frac{y - z}{\left(t - z\right) + 1}}, x\right) \]
   div-sub [=>] 0.3	\[ \mathsf{fma}\left(a, 0 - \color{blue}{\left(\frac{y}{\left(t - z\right) + 1} - \frac{z}{\left(t - z\right) + 1}\right)}, x\right) \]
   associate--r- [=>] 0.3	\[ \mathsf{fma}\left(a, \color{blue}{\left(0 - \frac{y}{\left(t - z\right) + 1}\right) + \frac{z}{\left(t - z\right) + 1}}, x\right) \]
   neg-sub0 [<=] 0.3	\[ \mathsf{fma}\left(a, \color{blue}{\left(-\frac{y}{\left(t - z\right) + 1}\right)} + \frac{z}{\left(t - z\right) + 1}, x\right) \]
   +-commutative [=>] 0.3	\[ \mathsf{fma}\left(a, \color{blue}{\frac{z}{\left(t - z\right) + 1} + \left(-\frac{y}{\left(t - z\right) + 1}\right)}, x\right) \]
   sub-neg [<=] 0.3	\[ \mathsf{fma}\left(a, \color{blue}{\frac{z}{\left(t - z\right) + 1} - \frac{y}{\left(t - z\right) + 1}}, x\right) \]
   div-sub [<=] 0.3	\[ \mathsf{fma}\left(a, \color{blue}{\frac{z - y}{\left(t - z\right) + 1}}, x\right) \]

3. Final simplification 0.3

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

Alternatives

Alternative 1
Error	15.4
Cost	1108

\[\begin{array}{l} \mathbf{if}\;t \leq -1.2 \cdot 10^{-12}:\\ \;\;\;\;x - \frac{a}{\frac{t}{y}}\\ \mathbf{elif}\;t \leq 8 \cdot 10^{+25}:\\ \;\;\;\;x + a \cdot \frac{z}{1 - z}\\ \mathbf{elif}\;t \leq 7.5 \cdot 10^{+121}:\\ \;\;\;\;x - \frac{a \cdot y}{t}\\ \mathbf{elif}\;t \leq 1.52 \cdot 10^{+156}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;t \leq 7.6 \cdot 10^{+202}:\\ \;\;\;\;x + z \cdot \frac{a}{t}\\ \mathbf{else}:\\ \;\;\;\;x - a \cdot \frac{y}{t}\\ \end{array} \]

Alternative 2
Error	15.4
Cost	1108

\[\begin{array}{l} \mathbf{if}\;t \leq -1.2 \cdot 10^{-12}:\\ \;\;\;\;x - \frac{a}{\frac{t}{y}}\\ \mathbf{elif}\;t \leq 7.5 \cdot 10^{+25}:\\ \;\;\;\;x + \frac{a}{\frac{1 - z}{z}}\\ \mathbf{elif}\;t \leq 9 \cdot 10^{+121}:\\ \;\;\;\;x - \frac{a \cdot y}{t}\\ \mathbf{elif}\;t \leq 6.8 \cdot 10^{+154}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;t \leq 5.4 \cdot 10^{+202}:\\ \;\;\;\;x + z \cdot \frac{a}{t}\\ \mathbf{else}:\\ \;\;\;\;x - a \cdot \frac{y}{t}\\ \end{array} \]

Alternative 3
Error	11.0
Cost	1104

\[\begin{array}{l} t_1 := x - y \cdot \frac{a}{t + 1}\\ \mathbf{if}\;z \leq -2.4 \cdot 10^{+51}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 5400000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.02 \cdot 10^{+51}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{a}{z}\\ \mathbf{elif}\;z \leq 1.65 \cdot 10^{+131}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \]

Alternative 4
Error	10.9
Cost	1104

\[\begin{array}{l} \mathbf{if}\;z \leq -1.9 \cdot 10^{+52}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 5400000000:\\ \;\;\;\;x - a \cdot \frac{y}{t + 1}\\ \mathbf{elif}\;z \leq 1.02 \cdot 10^{+51}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{a}{z}\\ \mathbf{elif}\;z \leq 3.9 \cdot 10^{+130}:\\ \;\;\;\;x - y \cdot \frac{a}{t + 1}\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \]

Alternative 5
Error	6.8
Cost	1100

\[\begin{array}{l} t_1 := x + a \cdot \frac{z}{t + \left(1 - z\right)}\\ \mathbf{if}\;z \leq -6.3 \cdot 10^{+35}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 4800000000:\\ \;\;\;\;x - a \cdot \frac{y}{t + 1}\\ \mathbf{elif}\;z \leq 1.02 \cdot 10^{+51}:\\ \;\;\;\;x - \frac{y - z}{\frac{-z}{a}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	18.6
Cost	976

\[\begin{array}{l} \mathbf{if}\;z \leq -3.7 \cdot 10^{-47}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 460000000:\\ \;\;\;\;x - a \cdot y\\ \mathbf{elif}\;z \leq 1.02 \cdot 10^{+51}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{a}{z}\\ \mathbf{elif}\;z \leq 3.9 \cdot 10^{+130}:\\ \;\;\;\;x + z \cdot \frac{a}{t}\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \]

Alternative 7
Error	18.9
Cost	976

\[\begin{array}{l} t_1 := x - a \cdot y\\ t_2 := x - a \cdot \frac{y}{t}\\ \mathbf{if}\;t \leq -1.95 \cdot 10^{+31}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -2.5 \cdot 10^{-83}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -4.2 \cdot 10^{-243}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;t \leq 5 \cdot 10^{+25}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 8
Error	18.9
Cost	976

\[\begin{array}{l} t_1 := x - a \cdot y\\ \mathbf{if}\;t \leq -1.95 \cdot 10^{+31}:\\ \;\;\;\;x - \frac{a}{\frac{t}{y}}\\ \mathbf{elif}\;t \leq -3.1 \cdot 10^{-83}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -6.2 \cdot 10^{-243}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;t \leq 5 \cdot 10^{+25}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x - a \cdot \frac{y}{t}\\ \end{array} \]

Alternative 9
Error	5.9
Cost	969

\[\begin{array}{l} \mathbf{if}\;t \leq -2.8 \cdot 10^{+23} \lor \neg \left(t \leq 1.52 \cdot 10^{+26}\right):\\ \;\;\;\;x - \frac{a}{\frac{t}{y - z}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{a}{\frac{1 - z}{y - z}}\\ \end{array} \]

Alternative 10
Error	8.6
Cost	904

\[\begin{array}{l} \mathbf{if}\;z \leq -1.2 \cdot 10^{+53}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 4600000000:\\ \;\;\;\;x - a \cdot \frac{y}{t + 1}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y - z}{\frac{-z}{a}}\\ \end{array} \]

Alternative 11
Error	18.3
Cost	844

\[\begin{array}{l} \mathbf{if}\;z \leq -5.4 \cdot 10^{-47}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 460000000:\\ \;\;\;\;x - a \cdot y\\ \mathbf{elif}\;z \leq 1.02 \cdot 10^{+51}:\\ \;\;\;\;a \cdot \frac{y - z}{z}\\ \mathbf{elif}\;z \leq 3.9 \cdot 10^{+130}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \]

Alternative 12
Error	18.3
Cost	844

\[\begin{array}{l} \mathbf{if}\;z \leq -4.9 \cdot 10^{-47}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 300000000:\\ \;\;\;\;x - a \cdot y\\ \mathbf{elif}\;z \leq 1.02 \cdot 10^{+51}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{a}{z}\\ \mathbf{elif}\;z \leq 3.9 \cdot 10^{+130}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \]

Alternative 13
Error	13.2
Cost	841

\[\begin{array}{l} \mathbf{if}\;t \leq -1.15 \cdot 10^{+16} \lor \neg \left(t \leq 1.1 \cdot 10^{+26}\right):\\ \;\;\;\;x - \left(y - z\right) \cdot \frac{a}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{a}{\frac{1 - z}{z}}\\ \end{array} \]

Alternative 14
Error	0.3
Cost	832

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

Alternative 15
Error	16.7
Cost	584

\[\begin{array}{l} \mathbf{if}\;z \leq -1.32 \cdot 10^{-47}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{-6}:\\ \;\;\;\;x - a \cdot y\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \]

Alternative 16
Error	18.9
Cost	456

\[\begin{array}{l} \mathbf{if}\;z \leq -5.2 \cdot 10^{-47}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 3.35 \cdot 10^{-6}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \]

Alternative 17
Error	26.2
Cost	392

\[\begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{-151}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 4.4 \cdot 10^{-183}:\\ \;\;\;\;-a\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 18
Error	27.4
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023031 
(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))))