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

Average Error: 2.0 → 0.1

Time: 6.9s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := \mathsf{fma}\left(\frac{a}{t + \left(1 - z\right)}, z - y, x\right)\\ t_2 := \frac{y - z}{\frac{\left(t - z\right) + 1}{a}}\\ t_3 := \left(t + 1\right) - z\\ \mathbf{if}\;t_2 \leq -2.768622796203182 \cdot 10^{-119}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_2 \leq 0:\\ \;\;\;\;\left(x + \frac{z \cdot a}{t_3}\right) - \frac{y \cdot a}{t_3}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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 (- 1.0 z))) (- z y) x))
        (t_2 (/ (- y z) (/ (+ (- t z) 1.0) a)))
        (t_3 (- (+ t 1.0) z)))
   (if (<= t_2 -2.768622796203182e-119)
     t_1
     (if (<= t_2 0.0) (- (+ x (/ (* z a) t_3)) (/ (* y a) t_3)) 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 + (1.0 - z))), (z - y), x);
	double t_2 = (y - z) / (((t - z) + 1.0) / a);
	double t_3 = (t + 1.0) - z;
	double tmp;
	if (t_2 <= -2.768622796203182e-119) {
		tmp = t_1;
	} else if (t_2 <= 0.0) {
		tmp = (x + ((z * a) / t_3)) - ((y * a) / t_3);
	} 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(t + Float64(1.0 - z))), Float64(z - y), x)
	t_2 = Float64(Float64(y - z) / Float64(Float64(Float64(t - z) + 1.0) / a))
	t_3 = Float64(Float64(t + 1.0) - z)
	tmp = 0.0
	if (t_2 <= -2.768622796203182e-119)
		tmp = t_1;
	elseif (t_2 <= 0.0)
		tmp = Float64(Float64(x + Float64(Float64(z * a) / t_3)) - Float64(Float64(y * a) / t_3));
	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[(t + N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(z - y), $MachinePrecision] + x), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y - z), $MachinePrecision] / N[(N[(N[(t - z), $MachinePrecision] + 1.0), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(t + 1.0), $MachinePrecision] - z), $MachinePrecision]}, If[LessEqual[t$95$2, -2.768622796203182e-119], t$95$1, If[LessEqual[t$95$2, 0.0], N[(N[(x + N[(N[(z * a), $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision] - N[(N[(y * a), $MachinePrecision] / t$95$3), $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	0.1

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 y z) (/.f64 (+.f64 (-.f64 t z) 1) a)) < -2.768622796203182e-119 or -0.0 < (/.f64 (-.f64 y z) (/.f64 (+.f64 (-.f64 t z) 1) a))

   1. Initial program 0.1

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

   2. Simplified 0.1

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

   if -2.768622796203182e-119 < (/.f64 (-.f64 y z) (/.f64 (+.f64 (-.f64 t z) 1) a)) < -0.0

   1. Initial program 5.6

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

   2. Simplified 5.2

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

   3. Taylor expanded in a around 0 0.0

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \leq -2.768622796203182 \cdot 10^{-119}:\\ \;\;\;\;\mathsf{fma}\left(\frac{a}{t + \left(1 - z\right)}, z - y, x\right)\\ \mathbf{elif}\;\frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \leq 0:\\ \;\;\;\;\left(x + \frac{z \cdot a}{\left(t + 1\right) - z}\right) - \frac{y \cdot a}{\left(t + 1\right) - z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{a}{t + \left(1 - z\right)}, z - y, x\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022153 
(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))))