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

Average Error: 2.0 → 0.4

Time: 6.2s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := \mathsf{fma}\left(\frac{a}{\left(t - z\right) + 1}, z - y, x\right)\\ t_2 := \left(t + 1\right) - z\\ \mathbf{if}\;a \leq -4.878938494260088 \cdot 10^{+107}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 4.65178715222157 \cdot 10^{-28}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{z}{t_2}, x\right) - \frac{a \cdot y}{t_2}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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
 (let* ((t_1 (fma (/ a (+ (- t z) 1.0)) (- z y) x)) (t_2 (- (+ t 1.0) z)))
   (if (<= a -4.878938494260088e+107)
     t_1
     (if (<= a 4.65178715222157e-28)
       (- (fma a (/ z t_2) x) (/ (* a y) t_2))
       t_1))))

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) {
	double t_1 = fma((a / ((t - z) + 1.0)), (z - y), x);
	double t_2 = (t + 1.0) - z;
	double tmp;
	if (a <= -4.878938494260088e+107) {
		tmp = t_1;
	} else if (a <= 4.65178715222157e-28) {
		tmp = fma(a, (z / t_2), x) - ((a * y) / t_2);
	} else {
		tmp = t_1;
	}
	return tmp;
}

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)
	t_1 = fma(Float64(a / Float64(Float64(t - z) + 1.0)), Float64(z - y), x)
	t_2 = Float64(Float64(t + 1.0) - z)
	tmp = 0.0
	if (a <= -4.878938494260088e+107)
		tmp = t_1;
	elseif (a <= 4.65178715222157e-28)
		tmp = Float64(fma(a, Float64(z / t_2), x) - Float64(Float64(a * y) / t_2));
	else
		tmp = t_1;
	end
	return tmp
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_] := Block[{t$95$1 = N[(N[(a / N[(N[(t - z), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[(z - y), $MachinePrecision] + x), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t + 1.0), $MachinePrecision] - z), $MachinePrecision]}, If[LessEqual[a, -4.878938494260088e+107], t$95$1, If[LessEqual[a, 4.65178715222157e-28], N[(N[(a * N[(z / t$95$2), $MachinePrecision] + x), $MachinePrecision] - N[(N[(a * y), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	2.0
Target	0.2
Herbie	0.4

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

Derivation

1. Split input into 2 regimes

2. if a < -4.8789384942600881e107 or 4.65178715222157008e-28 < a

   1. Initial program 0.2

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

   2. Simplified 0.1

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

   if -4.8789384942600881e107 < a < 4.65178715222157008e-28

   1. Initial program 3.1

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

   2. Simplified 2.9

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

   3. Taylor expanded in a around 0 1.4

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

   4. Applied *-un-lft-identity_binary64 1.4

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

   5. Applied times-frac_binary64 0.6

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

   6. Applied fma-def_binary64 0.6

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{a}{1}, \frac{z}{\left(1 + t\right) - z}, x\right)} - \frac{y \cdot a}{\left(1 + t\right) - z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -4.878938494260088 \cdot 10^{+107}:\\ \;\;\;\;\mathsf{fma}\left(\frac{a}{\left(t - z\right) + 1}, z - y, x\right)\\ \mathbf{elif}\;a \leq 4.65178715222157 \cdot 10^{-28}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{z}{\left(t + 1\right) - z}, x\right) - \frac{a \cdot y}{\left(t + 1\right) - z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{a}{\left(t - z\right) + 1}, z - y, x\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022131 
(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))))