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

Average Accuracy: 96.9% → 99.7%

Time: 13.8s

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	96.9%
Target	99.7%
Herbie	99.7%

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

Derivation

1. Initial program 96.9%

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

2. Simplified 99.7%

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

   [Start] 96.9	\[ x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   sub-neg [=>] 96.9	\[ \color{blue}{x + \left(-\frac{y - z}{\frac{\left(t - z\right) + 1}{a}}\right)} \]
   +-commutative [=>] 96.9	\[ \color{blue}{\left(-\frac{y - z}{\frac{\left(t - z\right) + 1}{a}}\right) + x} \]
   associate-/r/ [=>] 99.7	\[ \left(-\color{blue}{\frac{y - z}{\left(t - z\right) + 1} \cdot a}\right) + x \]
   *-commutative [=>] 99.7	\[ \left(-\color{blue}{a \cdot \frac{y - z}{\left(t - z\right) + 1}}\right) + x \]
   distribute-rgt-neg-in [=>] 99.7	\[ \color{blue}{a \cdot \left(-\frac{y - z}{\left(t - z\right) + 1}\right)} + x \]
   fma-def [=>] 99.7	\[ \color{blue}{\mathsf{fma}\left(a, -\frac{y - z}{\left(t - z\right) + 1}, x\right)} \]
   div-sub [=>] 99.7	\[ \mathsf{fma}\left(a, -\color{blue}{\left(\frac{y}{\left(t - z\right) + 1} - \frac{z}{\left(t - z\right) + 1}\right)}, x\right) \]
   sub-neg [=>] 99.7	\[ \mathsf{fma}\left(a, -\color{blue}{\left(\frac{y}{\left(t - z\right) + 1} + \left(-\frac{z}{\left(t - z\right) + 1}\right)\right)}, x\right) \]
   +-commutative [=>] 99.7	\[ \mathsf{fma}\left(a, -\color{blue}{\left(\left(-\frac{z}{\left(t - z\right) + 1}\right) + \frac{y}{\left(t - z\right) + 1}\right)}, x\right) \]
   distribute-neg-in [=>] 99.7	\[ \mathsf{fma}\left(a, \color{blue}{\left(-\left(-\frac{z}{\left(t - z\right) + 1}\right)\right) + \left(-\frac{y}{\left(t - z\right) + 1}\right)}, x\right) \]
   remove-double-neg [=>] 99.7	\[ \mathsf{fma}\left(a, \color{blue}{\frac{z}{\left(t - z\right) + 1}} + \left(-\frac{y}{\left(t - z\right) + 1}\right), x\right) \]
   sub-neg [<=] 99.7	\[ \mathsf{fma}\left(a, \color{blue}{\frac{z}{\left(t - z\right) + 1} - \frac{y}{\left(t - z\right) + 1}}, x\right) \]
   div-sub [<=] 99.7	\[ \mathsf{fma}\left(a, \color{blue}{\frac{z - y}{\left(t - z\right) + 1}}, x\right) \]

3. Final simplification 99.7%

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

Alternatives

Alternative 1
Accuracy	71.4%
Cost	1240

\[\begin{array}{l} t_1 := x - \frac{a \cdot y}{t}\\ t_2 := x - a \cdot y\\ \mathbf{if}\;z \leq -38000000:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq -9.8 \cdot 10^{-230}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 3.9 \cdot 10^{-256}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.06 \cdot 10^{-172}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 4.8 \cdot 10^{-150}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 6 \cdot 10^{+37}:\\ \;\;\;\;x + \frac{a \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \]

Alternative 2
Accuracy	83.7%
Cost	1100

\[\begin{array}{l} t_1 := x + \frac{z - y}{\frac{-z}{a}}\\ \mathbf{if}\;z \leq -2.2 \cdot 10^{+69}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{-30}:\\ \;\;\;\;x - \frac{a \cdot y}{t + 1}\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{+105}:\\ \;\;\;\;x + \frac{a \cdot z}{\left(t + 1\right) - z}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Accuracy	70.8%
Cost	972

\[\begin{array}{l} t_1 := x - \left(y - z\right) \cdot \frac{a}{t}\\ \mathbf{if}\;t \leq -3.2 \cdot 10^{-38}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -1.65 \cdot 10^{-192}:\\ \;\;\;\;\frac{a}{1 - z} \cdot \left(z - y\right)\\ \mathbf{elif}\;t \leq 8.5 \cdot 10^{+100}:\\ \;\;\;\;x - a\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Accuracy	71.5%
Cost	972

\[\begin{array}{l} \mathbf{if}\;t \leq -1.4 \cdot 10^{-42}:\\ \;\;\;\;x - \left(y - z\right) \cdot \frac{a}{t}\\ \mathbf{elif}\;t \leq -4.2 \cdot 10^{-193}:\\ \;\;\;\;\frac{a}{1 - z} \cdot \left(z - y\right)\\ \mathbf{elif}\;t \leq 5.4 \cdot 10^{+40}:\\ \;\;\;\;x - a\\ \mathbf{else}:\\ \;\;\;\;x - a \cdot \frac{y - z}{t}\\ \end{array} \]

Alternative 5
Accuracy	83.7%
Cost	905

\[\begin{array}{l} \mathbf{if}\;z \leq -2.2 \cdot 10^{+69} \lor \neg \left(z \leq 24500000\right):\\ \;\;\;\;x + \frac{z - y}{\frac{-z}{a}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{a \cdot y}{t + 1}\\ \end{array} \]

Alternative 6
Accuracy	72.5%
Cost	844

\[\begin{array}{l} \mathbf{if}\;z \leq -84000000000:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 4.3 \cdot 10^{-153}:\\ \;\;\;\;x - a \cdot y\\ \mathbf{elif}\;z \leq 1.85 \cdot 10^{+39}:\\ \;\;\;\;x + \frac{a \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \]

Alternative 7
Accuracy	82.2%
Cost	840

\[\begin{array}{l} \mathbf{if}\;z \leq -7.6 \cdot 10^{+104}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 18000000000:\\ \;\;\;\;x - \frac{a \cdot y}{t + 1}\\ \mathbf{else}:\\ \;\;\;\;\left(x - a\right) - \frac{a}{z}\\ \end{array} \]

Alternative 8
Accuracy	99.7%
Cost	832

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

Alternative 9
Accuracy	73.6%
Cost	584

\[\begin{array}{l} \mathbf{if}\;z \leq -118000:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 0.00165:\\ \;\;\;\;x - a \cdot y\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \]

Alternative 10
Accuracy	69.2%
Cost	456

\[\begin{array}{l} \mathbf{if}\;z \leq -2.1 \cdot 10^{+39}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 145000000:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \]

Alternative 11
Accuracy	56.5%
Cost	392

\[\begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{-115}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-284}:\\ \;\;\;\;-a\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 12
Accuracy	55.8%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023151 
(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))))