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

Average Error: 10.7 → 0.2

Time: 5.3s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := \mathsf{fma}\left(t, \frac{y - z}{a - z}, x\right)\\ t_2 := \frac{\left(y - z\right) \cdot t}{a - z}\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_2 \leq 1.1129374755727955 \cdot 10^{+303}:\\ \;\;\;\;t_2 + x\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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 (fma t (/ (- y z) (- a z)) x)) (t_2 (/ (* (- y z) t) (- a z))))
   (if (<= t_2 (- INFINITY))
     t_1
     (if (<= t_2 1.1129374755727955e+303) (+ t_2 x) t_1))))

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 = fma(t, ((y - z) / (a - z)), x);
	double t_2 = ((y - z) * t) / (a - z);
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_2 <= 1.1129374755727955e+303) {
		tmp = t_2 + x;
	} else {
		tmp = t_1;
	}
	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 = fma(t, Float64(Float64(y - z) / Float64(a - z)), x)
	t_2 = Float64(Float64(Float64(y - z) * t) / Float64(a - z))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_2 <= 1.1129374755727955e+303)
		tmp = Float64(t_2 + x);
	else
		tmp = t_1;
	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[(t * N[(N[(y - z), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(y - z), $MachinePrecision] * t), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$1, If[LessEqual[t$95$2, 1.1129374755727955e+303], N[(t$95$2 + x), $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	10.7
Target	0.5
Herbie	0.2

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (-.f64 y z) t) (-.f64 a z)) < -inf.0 or 1.11293747557279552e303 < (/.f64 (*.f64 (-.f64 y z) t) (-.f64 a z))

   1. Initial program 63.5

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

   2. Applied egg-rr 0.1

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

   if -inf.0 < (/.f64 (*.f64 (-.f64 y z) t) (-.f64 a z)) < 1.11293747557279552e303

   1. Initial program 0.2

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

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(y - z\right) \cdot t}{a - z} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{y - z}{a - z}, x\right)\\ \mathbf{elif}\;\frac{\left(y - z\right) \cdot t}{a - z} \leq 1.1129374755727955 \cdot 10^{+303}:\\ \;\;\;\;\frac{\left(y - z\right) \cdot t}{a - z} + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{y - z}{a - z}, x\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022133 
(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))))