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

Average Error: 2.3 → 0.2

Time: 22.3s

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	2.3
Target	0.2
Herbie	0.2

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

Derivation

1. Initial program 2.3

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

2. Simplified 0.2

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

   [Start] 2.3	\[ x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   sub-neg [=>] 2.3	\[ \color{blue}{x + \left(-\frac{y - z}{\frac{\left(t - z\right) + 1}{a}}\right)} \]
   +-commutative [=>] 2.3	\[ \color{blue}{\left(-\frac{y - z}{\frac{\left(t - z\right) + 1}{a}}\right) + x} \]
   associate-/r/ [=>] 0.2	\[ \left(-\color{blue}{\frac{y - z}{\left(t - z\right) + 1} \cdot a}\right) + x \]
   distribute-lft-neg-in [=>] 0.2	\[ \color{blue}{\left(-\frac{y - z}{\left(t - z\right) + 1}\right) \cdot a} + x \]
   *-commutative [=>] 0.2	\[ \color{blue}{a \cdot \left(-\frac{y - z}{\left(t - z\right) + 1}\right)} + x \]
   fma-def [=>] 0.2	\[ \color{blue}{\mathsf{fma}\left(a, -\frac{y - z}{\left(t - z\right) + 1}, x\right)} \]
   neg-sub0 [=>] 0.2	\[ \mathsf{fma}\left(a, \color{blue}{0 - \frac{y - z}{\left(t - z\right) + 1}}, x\right) \]
   div-sub [=>] 0.2	\[ \mathsf{fma}\left(a, 0 - \color{blue}{\left(\frac{y}{\left(t - z\right) + 1} - \frac{z}{\left(t - z\right) + 1}\right)}, x\right) \]
   associate--r- [=>] 0.2	\[ \mathsf{fma}\left(a, \color{blue}{\left(0 - \frac{y}{\left(t - z\right) + 1}\right) + \frac{z}{\left(t - z\right) + 1}}, x\right) \]
   neg-sub0 [<=] 0.2	\[ \mathsf{fma}\left(a, \color{blue}{\left(-\frac{y}{\left(t - z\right) + 1}\right)} + \frac{z}{\left(t - z\right) + 1}, x\right) \]
   +-commutative [=>] 0.2	\[ \mathsf{fma}\left(a, \color{blue}{\frac{z}{\left(t - z\right) + 1} + \left(-\frac{y}{\left(t - z\right) + 1}\right)}, x\right) \]
   sub-neg [<=] 0.2	\[ \mathsf{fma}\left(a, \color{blue}{\frac{z}{\left(t - z\right) + 1} - \frac{y}{\left(t - z\right) + 1}}, x\right) \]
   div-sub [<=] 0.2	\[ \mathsf{fma}\left(a, \color{blue}{\frac{z - y}{\left(t - z\right) + 1}}, x\right) \]

3. Final simplification 0.2

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

Alternatives

Alternative 1
Error	19.7
Cost	1372

\[\begin{array}{l} t_1 := x - a \cdot \frac{y}{t}\\ t_2 := x - a \cdot y\\ \mathbf{if}\;t \leq -2.1 \cdot 10^{+26}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -4.6 \cdot 10^{-153}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;t \leq -1.35 \cdot 10^{-206}:\\ \;\;\;\;a \cdot \frac{z - y}{1 - z}\\ \mathbf{elif}\;t \leq 5 \cdot 10^{-246}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 2.2 \cdot 10^{-99}:\\ \;\;\;\;\left(x - a\right) - \frac{a}{z}\\ \mathbf{elif}\;t \leq 3.8 \cdot 10^{-55}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 4.4 \cdot 10^{+135}:\\ \;\;\;\;x - a\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	16.8
Cost	1368

\[\begin{array}{l} t_1 := x + z \cdot \frac{a}{1 - z}\\ t_2 := x - a \cdot \frac{y}{t}\\ \mathbf{if}\;t \leq -5 \cdot 10^{+26}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -1.72 \cdot 10^{-152}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -1.5 \cdot 10^{-231}:\\ \;\;\;\;a \cdot \frac{z - y}{1 - z}\\ \mathbf{elif}\;t \leq -2.8 \cdot 10^{-303}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2.6 \cdot 10^{-248}:\\ \;\;\;\;x - a \cdot y\\ \mathbf{elif}\;t \leq 1.4 \cdot 10^{+60}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Error	15.5
Cost	1368

\[\begin{array}{l} t_1 := x + z \cdot \frac{a}{1 - z}\\ t_2 := x - \left(y - z\right) \cdot \frac{a}{t}\\ \mathbf{if}\;t \leq -2.7 \cdot 10^{+99}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -1.3 \cdot 10^{-153}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -8 \cdot 10^{-226}:\\ \;\;\;\;a \cdot \frac{z - y}{1 - z}\\ \mathbf{elif}\;t \leq -7.5 \cdot 10^{-305}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.8 \cdot 10^{-248}:\\ \;\;\;\;x - a \cdot y\\ \mathbf{elif}\;t \leq 1.6 \cdot 10^{+101}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 4
Error	17.4
Cost	1108

\[\begin{array}{l} t_1 := x - a \cdot \frac{y}{t}\\ t_2 := \left(x - a\right) - \frac{a}{z}\\ t_3 := x - a \cdot y\\ \mathbf{if}\;z \leq -13600000000:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -1.7 \cdot 10^{-157}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -2.5 \cdot 10^{-275}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{-204}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{-14}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 5
Error	17.4
Cost	980

\[\begin{array}{l} t_1 := x - a \cdot y\\ t_2 := x - a \cdot \frac{y}{t}\\ \mathbf{if}\;z \leq -75000000:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq -3 \cdot 10^{-155}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -1.3 \cdot 10^{-278}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{-204}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 7.5 \cdot 10^{-16}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \]

Alternative 6
Error	8.3
Cost	969

\[\begin{array}{l} \mathbf{if}\;z \leq -1.8 \cdot 10^{-49} \lor \neg \left(z \leq 3.6 \cdot 10^{-22}\right):\\ \;\;\;\;x + z \cdot \frac{a}{\left(t - z\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{t + 1}{a}}\\ \end{array} \]

Alternative 7
Error	6.4
Cost	968

\[\begin{array}{l} \mathbf{if}\;t \leq -6 \cdot 10^{+98}:\\ \;\;\;\;x + a \cdot \frac{z - y}{t}\\ \mathbf{elif}\;t \leq 1.1 \cdot 10^{+136}:\\ \;\;\;\;x - \frac{a}{\frac{1 - z}{y - z}}\\ \mathbf{else}:\\ \;\;\;\;x - \left(y - z\right) \cdot \frac{a}{t}\\ \end{array} \]

Alternative 8
Error	8.9
Cost	905

\[\begin{array}{l} \mathbf{if}\;z \leq -920000000 \lor \neg \left(z \leq 7.6 \cdot 10^{+23}\right):\\ \;\;\;\;x - \frac{y - z}{\frac{-z}{a}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{t + 1}{a}}\\ \end{array} \]

Alternative 9
Error	10.8
Cost	840

\[\begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{-49}:\\ \;\;\;\;x + z \cdot \frac{a}{1 - z}\\ \mathbf{elif}\;z \leq 4.1 \cdot 10^{-14}:\\ \;\;\;\;x - a \cdot \frac{y}{t + 1}\\ \mathbf{else}:\\ \;\;\;\;\left(x - a\right) - \frac{a}{z}\\ \end{array} \]

Alternative 10
Error	11.0
Cost	840

\[\begin{array}{l} \mathbf{if}\;z \leq -1.8 \cdot 10^{-49}:\\ \;\;\;\;x + z \cdot \frac{a}{1 - z}\\ \mathbf{elif}\;z \leq 4.1 \cdot 10^{-14}:\\ \;\;\;\;x - \frac{y}{\frac{t + 1}{a}}\\ \mathbf{else}:\\ \;\;\;\;\left(x - a\right) - \frac{a}{z}\\ \end{array} \]

Alternative 11
Error	0.2
Cost	832

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

Alternative 12
Error	19.4
Cost	584

\[\begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 7.6 \cdot 10^{-56}:\\ \;\;\;\;x + a \cdot z\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \]

Alternative 13
Error	17.8
Cost	584

\[\begin{array}{l} \mathbf{if}\;z \leq -1.05 \cdot 10^{-80}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{-23}:\\ \;\;\;\;x - a \cdot y\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \]

Alternative 14
Error	19.6
Cost	456

\[\begin{array}{l} \mathbf{if}\;z \leq -23000000000:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 7.6 \cdot 10^{-56}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \]

Alternative 15
Error	27.5
Cost	392

\[\begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{-105}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{-110}:\\ \;\;\;\;-a\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 16
Error	28.1
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023018 
(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))))