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

Average Accuracy: 96.8% → 99.6%

Time: 14.2s

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	96.8%
Target	99.6%
Herbie	99.6%

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

Derivation

1. Initial program 96.8%

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

2. Simplified 99.6%

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

   [Start] 96.8	\[ x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   sub-neg [=>] 96.8	\[ \color{blue}{x + \left(-\frac{y - z}{\frac{\left(t - z\right) + 1}{a}}\right)} \]
   +-commutative [=>] 96.8	\[ \color{blue}{\left(-\frac{y - z}{\frac{\left(t - z\right) + 1}{a}}\right) + x} \]
   associate-/r/ [=>] 99.6	\[ \left(-\color{blue}{\frac{y - z}{\left(t - z\right) + 1} \cdot a}\right) + x \]
   distribute-lft-neg-in [=>] 99.6	\[ \color{blue}{\left(-\frac{y - z}{\left(t - z\right) + 1}\right) \cdot a} + x \]
   *-commutative [=>] 99.6	\[ \color{blue}{a \cdot \left(-\frac{y - z}{\left(t - z\right) + 1}\right)} + x \]
   fma-def [=>] 99.6	\[ \color{blue}{\mathsf{fma}\left(a, -\frac{y - z}{\left(t - z\right) + 1}, x\right)} \]
   neg-sub0 [=>] 99.6	\[ \mathsf{fma}\left(a, \color{blue}{0 - \frac{y - z}{\left(t - z\right) + 1}}, x\right) \]
   div-sub [=>] 99.6	\[ \mathsf{fma}\left(a, 0 - \color{blue}{\left(\frac{y}{\left(t - z\right) + 1} - \frac{z}{\left(t - z\right) + 1}\right)}, x\right) \]
   associate--r- [=>] 99.6	\[ \mathsf{fma}\left(a, \color{blue}{\left(0 - \frac{y}{\left(t - z\right) + 1}\right) + \frac{z}{\left(t - z\right) + 1}}, x\right) \]
   neg-sub0 [<=] 99.6	\[ \mathsf{fma}\left(a, \color{blue}{\left(-\frac{y}{\left(t - z\right) + 1}\right)} + \frac{z}{\left(t - z\right) + 1}, x\right) \]
   +-commutative [=>] 99.6	\[ \mathsf{fma}\left(a, \color{blue}{\frac{z}{\left(t - z\right) + 1} + \left(-\frac{y}{\left(t - z\right) + 1}\right)}, x\right) \]
   sub-neg [<=] 99.6	\[ \mathsf{fma}\left(a, \color{blue}{\frac{z}{\left(t - z\right) + 1} - \frac{y}{\left(t - z\right) + 1}}, x\right) \]
   div-sub [<=] 99.6	\[ \mathsf{fma}\left(a, \color{blue}{\frac{z - y}{\left(t - z\right) + 1}}, x\right) \]

3. Final simplification 99.6%

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

Alternatives

Alternative 1
Accuracy	72.4%
Cost	1112

\[\begin{array}{l} t_1 := x - \frac{y}{\frac{t}{a}}\\ t_2 := x - a \cdot y\\ \mathbf{if}\;z \leq -5.5 \cdot 10^{+67}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq -0.053:\\ \;\;\;\;x + \frac{a}{\frac{z}{y}}\\ \mathbf{elif}\;z \leq -3.9 \cdot 10^{-292}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{-165}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{-131}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 0.002:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \]

Alternative 2
Accuracy	86.2%
Cost	1104

\[\begin{array}{l} t_1 := x + \left(y \cdot \frac{a}{z} - a\right)\\ \mathbf{if}\;z \leq -1.6 \cdot 10^{+16}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -5 \cdot 10^{-14}:\\ \;\;\;\;a \cdot \frac{-y}{\left(t - z\right) + 1}\\ \mathbf{elif}\;z \leq -8 \cdot 10^{-63}:\\ \;\;\;\;x + \frac{a \cdot z}{1 - z}\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{+15}:\\ \;\;\;\;x - \frac{a}{\frac{t + 1}{y}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Accuracy	72.4%
Cost	980

\[\begin{array}{l} t_1 := x - a \cdot y\\ t_2 := x - \frac{y}{\frac{t}{a}}\\ \mathbf{if}\;z \leq -0.052:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq -4.8 \cdot 10^{-293}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 2.35 \cdot 10^{-166}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.42 \cdot 10^{-132}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 0.0018:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \]

Alternative 4
Accuracy	84.4%
Cost	972

\[\begin{array}{l} \mathbf{if}\;z \leq -1.15 \cdot 10^{+68}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq -1.35 \cdot 10^{+14}:\\ \;\;\;\;x + \frac{a}{\frac{z}{y}}\\ \mathbf{elif}\;z \leq 6 \cdot 10^{+14}:\\ \;\;\;\;x - y \cdot \frac{a}{t + 1}\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \]

Alternative 5
Accuracy	87.2%
Cost	972

\[\begin{array}{l} t_1 := x + \left(y \cdot \frac{a}{z} - a\right)\\ \mathbf{if}\;z \leq -1.6 \cdot 10^{+16}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1.1 \cdot 10^{-26}:\\ \;\;\;\;a \cdot \frac{z - y}{t + \left(1 - z\right)}\\ \mathbf{elif}\;z \leq 1.45 \cdot 10^{+14}:\\ \;\;\;\;x - \frac{a}{\frac{t + 1}{y}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Accuracy	88.0%
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -27500000 \lor \neg \left(z \leq 2.4 \cdot 10^{+18}\right):\\ \;\;\;\;x + \left(y \cdot \frac{a}{z} - a\right)\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \frac{a}{t + 1}\\ \end{array} \]

Alternative 7
Accuracy	88.2%
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -3800000 \lor \neg \left(z \leq 2.9 \cdot 10^{+16}\right):\\ \;\;\;\;x + \left(y \cdot \frac{a}{z} - a\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{a}{\frac{t + 1}{y}}\\ \end{array} \]

Alternative 8
Accuracy	99.6%
Cost	832

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

Alternative 9
Accuracy	72.9%
Cost	585

\[\begin{array}{l} \mathbf{if}\;z \leq -4.6 \cdot 10^{+27} \lor \neg \left(z \leq 0.0019\right):\\ \;\;\;\;x - a\\ \mathbf{else}:\\ \;\;\;\;x - a \cdot y\\ \end{array} \]

Alternative 10
Accuracy	69.2%
Cost	456

\[\begin{array}{l} \mathbf{if}\;z \leq -4.2 \cdot 10^{-63}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{+23}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \]

Alternative 11
Accuracy	56.6%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023133 
(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))))