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

Average Accuracy: 96.9% → 99.6%

Time: 20.5s

Precision: binary64

Cost: 832

Math

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

↓

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

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
 (+ x (* a (/ (- z y) (+ (- t z) 1.0)))))

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 x + (a * ((z - y) / ((t - z) + 1.0)));
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = x - ((y - z) / (((t - z) + 1.0d0) / a))
end function

↓

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = x + (a * ((z - y) / ((t - z) + 1.0d0)))
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return x - ((y - z) / (((t - z) + 1.0) / a));
}

↓

public static double code(double x, double y, double z, double t, double a) {
	return x + (a * ((z - y) / ((t - z) + 1.0)));
}

Python

def code(x, y, z, t, a):
	return x - ((y - z) / (((t - z) + 1.0) / a))

↓

def code(x, y, z, t, a):
	return x + (a * ((z - y) / ((t - z) + 1.0)))

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 Float64(x + Float64(a * Float64(Float64(z - y) / Float64(Float64(t - z) + 1.0))))
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = x - ((y - z) / (((t - z) + 1.0) / a));
end

↓

function tmp = code(x, y, z, t, a)
	tmp = x + (a * ((z - y) / ((t - z) + 1.0)));
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[(x + N[(a * N[(N[(z - y), $MachinePrecision] / N[(N[(t - z), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	96.9%
Target	99.6%
Herbie	99.6%

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

Derivation

1. Initial program 96.9%

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

2. Simplified 99.6%

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

   [Start] 96.9	\[ x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   associate-/r/ [=>] 99.6	\[ x - \color{blue}{\frac{y - z}{\left(t - z\right) + 1} \cdot a} \]

3. Final simplification 99.6%

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

Alternatives

Alternative 1
Accuracy	74.2%
Cost	1368

\[\begin{array}{l} t_1 := x + \frac{a}{t} \cdot \left(z - y\right)\\ t_2 := \frac{a}{1 - z}\\ \mathbf{if}\;t \leq -1:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -1.5 \cdot 10^{-220}:\\ \;\;\;\;x + a \cdot \left(z - y\right)\\ \mathbf{elif}\;t \leq -9 \cdot 10^{-279}:\\ \;\;\;\;t_2 \cdot \left(z - y\right)\\ \mathbf{elif}\;t \leq 9.8 \cdot 10^{-285}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;t \leq 6.4 \cdot 10^{-36}:\\ \;\;\;\;x - y \cdot t_2\\ \mathbf{elif}\;t \leq 0.0028:\\ \;\;\;\;x - a\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Accuracy	69.9%
Cost	1240

\[\begin{array}{l} t_1 := x - \frac{a}{\frac{t}{y}}\\ \mathbf{if}\;t \leq -7.5 \cdot 10^{-5}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -6.2 \cdot 10^{-223}:\\ \;\;\;\;x + a \cdot \left(z - y\right)\\ \mathbf{elif}\;t \leq -2.1 \cdot 10^{-278}:\\ \;\;\;\;\frac{a}{1 - z} \cdot \left(z - y\right)\\ \mathbf{elif}\;t \leq 8 \cdot 10^{-285}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;t \leq 9.3 \cdot 10^{-38}:\\ \;\;\;\;x - y \cdot a\\ \mathbf{elif}\;t \leq 0.0026:\\ \;\;\;\;x - a\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Accuracy	71.6%
Cost	1240

\[\begin{array}{l} t_1 := x - \frac{a}{\frac{t}{y}}\\ t_2 := \frac{a}{1 - z}\\ \mathbf{if}\;t \leq -0.018:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -1.4 \cdot 10^{-206}:\\ \;\;\;\;x + a \cdot \left(z - y\right)\\ \mathbf{elif}\;t \leq -3.2 \cdot 10^{-278}:\\ \;\;\;\;t_2 \cdot \left(z - y\right)\\ \mathbf{elif}\;t \leq 8.2 \cdot 10^{-285}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;t \leq 3.2 \cdot 10^{-36}:\\ \;\;\;\;x - y \cdot t_2\\ \mathbf{elif}\;t \leq 0.0028:\\ \;\;\;\;x - a\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Accuracy	89.5%
Cost	969

\[\begin{array}{l} \mathbf{if}\;z \leq -9.4 \cdot 10^{+21} \lor \neg \left(z \leq 5.3 \cdot 10^{-16}\right):\\ \;\;\;\;x + a \cdot \frac{z}{\left(t - z\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;x - a \cdot \frac{y}{t + 1}\\ \end{array} \]

Alternative 5
Accuracy	89.6%
Cost	969

\[\begin{array}{l} \mathbf{if}\;t \leq -1.1 \cdot 10^{+75} \lor \neg \left(t \leq 0.0028\right):\\ \;\;\;\;x + \frac{a}{t} \cdot \left(z - y\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{a}{1 - z} \cdot \left(z - y\right)\\ \end{array} \]

Alternative 6
Accuracy	90.9%
Cost	969

\[\begin{array}{l} \mathbf{if}\;t \leq -1.02 \cdot 10^{+75} \lor \neg \left(t \leq 0.0028\right):\\ \;\;\;\;x + \frac{a}{t} \cdot \left(z - y\right)\\ \mathbf{else}:\\ \;\;\;\;x + a \cdot \frac{z - y}{1 - z}\\ \end{array} \]

Alternative 7
Accuracy	86.6%
Cost	905

\[\begin{array}{l} \mathbf{if}\;z \leq -270000000000 \lor \neg \left(z \leq 1.32 \cdot 10^{+15}\right):\\ \;\;\;\;x + \frac{z - y}{\frac{-z}{a}}\\ \mathbf{else}:\\ \;\;\;\;x - a \cdot \frac{y}{t + 1}\\ \end{array} \]

Alternative 8
Accuracy	74.1%
Cost	844

\[\begin{array}{l} \mathbf{if}\;z \leq -13000000000000:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq -3.6 \cdot 10^{-205}:\\ \;\;\;\;x - \frac{a}{\frac{t}{y}}\\ \mathbf{elif}\;z \leq 0.49:\\ \;\;\;\;x + a \cdot \left(z - y\right)\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \]

Alternative 9
Accuracy	84.6%
Cost	840

\[\begin{array}{l} \mathbf{if}\;z \leq -1.45 \cdot 10^{+55}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{+16}:\\ \;\;\;\;x - a \cdot \frac{y}{t + 1}\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \]

Alternative 10
Accuracy	74.8%
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -0.0066:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 0.185:\\ \;\;\;\;x + a \cdot \left(z - y\right)\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \]

Alternative 11
Accuracy	73.3%
Cost	584

\[\begin{array}{l} \mathbf{if}\;z \leq -8.9 \cdot 10^{+21}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 0.0059:\\ \;\;\;\;x - y \cdot a\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \]

Alternative 12
Accuracy	69.6%
Cost	456

\[\begin{array}{l} \mathbf{if}\;z \leq -3.15 \cdot 10^{+46}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{+20}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \]

Alternative 13
Accuracy	57.7%
Cost	260

\[\begin{array}{l} \mathbf{if}\;a \leq 1.7 \cdot 10^{+158}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;-a\\ \end{array} \]

Alternative 14
Accuracy	56.6%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023135 
(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))))