Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTick from plot-0.2.3.4, A

Average Error: 10.4 → 0.5

Time: 5.4s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := \frac{y - z}{a - z}\\ \mathbf{if}\;t \leq -1.2875828197469994 \cdot 10^{-69}:\\ \;\;\;\;x + t \cdot t_1\\ \mathbf{elif}\;t \leq 1.0453891552587748 \cdot 10^{-110}:\\ \;\;\;\;x + \frac{t \cdot \left(y - z\right)}{a - z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, t_1, x\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 (+ x (/ (* (- y z) t) (- a z))))

↓

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- y z) (- a z))))
   (if (<= t -1.2875828197469994e-69)
     (+ x (* t t_1))
     (if (<= t 1.0453891552587748e-110)
       (+ x (/ (* t (- y z)) (- a z)))
       (fma t t_1 x)))))

C

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

↓

double code(double x, double y, double z, double t, double a) {
	double t_1 = (y - z) / (a - z);
	double tmp;
	if (t <= -1.2875828197469994e-69) {
		tmp = x + (t * t_1);
	} else if (t <= 1.0453891552587748e-110) {
		tmp = x + ((t * (y - z)) / (a - z));
	} else {
		tmp = fma(t, t_1, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	return Float64(x + Float64(Float64(Float64(y - z) * t) / Float64(a - z)))
end

↓

function code(x, y, z, t, a)
	t_1 = Float64(Float64(y - z) / Float64(a - z))
	tmp = 0.0
	if (t <= -1.2875828197469994e-69)
		tmp = Float64(x + Float64(t * t_1));
	elseif (t <= 1.0453891552587748e-110)
		tmp = Float64(x + Float64(Float64(t * Float64(y - z)) / Float64(a - z)));
	else
		tmp = fma(t, t_1, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(x + N[(N[(N[(y - z), $MachinePrecision] * t), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y - z), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1.2875828197469994e-69], N[(x + N[(t * t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.0453891552587748e-110], N[(x + N[(N[(t * N[(y - z), $MachinePrecision]), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t * t$95$1 + x), $MachinePrecision]]]]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	10.4
Target	0.6
Herbie	0.5

\[\begin{array}{l} \mathbf{if}\;t < -1.0682974490174067 \cdot 10^{-39}:\\ \;\;\;\;x + \frac{y - z}{a - z} \cdot t\\ \mathbf{elif}\;t < 3.9110949887586375 \cdot 10^{-141}:\\ \;\;\;\;x + \frac{\left(y - z\right) \cdot t}{a - z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - z}{a - z} \cdot t\\ \end{array} \]

Derivation

1. Split input into 3 regimes

2. if t < -1.287582819746999e-69

   1. Initial program 17.6

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

   2. Taylor expanded in y around 0 17.6

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

   3. Simplified 0.5

      \[\leadsto x + \color{blue}{t \cdot \frac{y - z}{a - z}} \]

   if -1.287582819746999e-69 < t < 1.0453891552587748e-110

   1. Initial program 0.3

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

   if 1.0453891552587748e-110 < t

   1. Initial program 16.4

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

   2. Taylor expanded in y around 0 16.4

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

   3. Simplified 0.6

      \[\leadsto x + \color{blue}{t \cdot \frac{y - z}{a - z}} \]

   4. Taylor expanded in x around 0 16.4

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

   5. Simplified 0.6

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

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.2875828197469994 \cdot 10^{-69}:\\ \;\;\;\;x + t \cdot \frac{y - z}{a - z}\\ \mathbf{elif}\;t \leq 1.0453891552587748 \cdot 10^{-110}:\\ \;\;\;\;x + \frac{t \cdot \left(y - z\right)}{a - z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{y - z}{a - z}, x\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022129 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTick from plot-0.2.3.4, A"
  :precision binary64

  :herbie-target
  (if (< t -1.0682974490174067e-39) (+ x (* (/ (- y z) (- a z)) t)) (if (< t 3.9110949887586375e-141) (+ x (/ (* (- y z) t) (- a z))) (+ x (* (/ (- y z) (- a z)) t))))

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