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

Average Error: 11.3 → 0.4

Time: 11.5s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := \mathsf{fma}\left(y, \frac{z - t}{z - a}, x\right)\\ \mathbf{if}\;y \leq -5.3186352085026664 \cdot 10^{-11}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 3.850809288581042 \cdot 10^{-38}:\\ \;\;\;\;\left(x + \frac{y \cdot z}{z - a}\right) - \frac{y \cdot t}{z - a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma y (/ (- z t) (- z a)) x)))
   (if (<= y -5.3186352085026664e-11)
     t_1
     (if (<= y 3.850809288581042e-38)
       (- (+ x (/ (* y z) (- z a))) (/ (* y t) (- z a)))
       t_1))))

C

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

↓

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(y, ((z - t) / (z - a)), x);
	double tmp;
	if (y <= -5.3186352085026664e-11) {
		tmp = t_1;
	} else if (y <= 3.850809288581042e-38) {
		tmp = (x + ((y * z) / (z - a))) - ((y * t) / (z - a));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

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

↓

function code(x, y, z, t, a)
	t_1 = fma(y, Float64(Float64(z - t) / Float64(z - a)), x)
	tmp = 0.0
	if (y <= -5.3186352085026664e-11)
		tmp = t_1;
	elseif (y <= 3.850809288581042e-38)
		tmp = Float64(Float64(x + Float64(Float64(y * z) / Float64(z - a))) - Float64(Float64(y * t) / Float64(z - a)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

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

↓

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(y * N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[y, -5.3186352085026664e-11], t$95$1, If[LessEqual[y, 3.850809288581042e-38], N[(N[(x + N[(N[(y * z), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(y * t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $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	11.3
Target	1.4
Herbie	0.4

\[x + \frac{y}{\frac{z - a}{z - t}} \]

Derivation

1. Split input into 2 regimes

2. if y < -5.31863520850266644e-11 or 3.8508092885810421e-38 < y

   1. Initial program 22.1

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

   2. Simplified 0.6

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

   3. Applied add-cube-cbrt_binary64 1.8

      \[\leadsto \color{blue}{\left(\sqrt[3]{\mathsf{fma}\left(y, \frac{z - t}{z - a}, x\right)} \cdot \sqrt[3]{\mathsf{fma}\left(y, \frac{z - t}{z - a}, x\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(y, \frac{z - t}{z - a}, x\right)}} \]

   4. Applied pow1/3_binary64 34.4

      \[\leadsto \left(\sqrt[3]{\mathsf{fma}\left(y, \frac{z - t}{z - a}, x\right)} \cdot \sqrt[3]{\mathsf{fma}\left(y, \frac{z - t}{z - a}, x\right)}\right) \cdot \color{blue}{{\left(\mathsf{fma}\left(y, \frac{z - t}{z - a}, x\right)\right)}^{0.3333333333333333}} \]

   5. Applied pow1/3_binary64 34.9

      \[\leadsto \left(\sqrt[3]{\mathsf{fma}\left(y, \frac{z - t}{z - a}, x\right)} \cdot \color{blue}{{\left(\mathsf{fma}\left(y, \frac{z - t}{z - a}, x\right)\right)}^{0.3333333333333333}}\right) \cdot {\left(\mathsf{fma}\left(y, \frac{z - t}{z - a}, x\right)\right)}^{0.3333333333333333} \]

   6. Applied pow1/3_binary64 35.2

      \[\leadsto \left(\color{blue}{{\left(\mathsf{fma}\left(y, \frac{z - t}{z - a}, x\right)\right)}^{0.3333333333333333}} \cdot {\left(\mathsf{fma}\left(y, \frac{z - t}{z - a}, x\right)\right)}^{0.3333333333333333}\right) \cdot {\left(\mathsf{fma}\left(y, \frac{z - t}{z - a}, x\right)\right)}^{0.3333333333333333} \]

   7. Applied pow-sqr_binary64 35.2

      \[\leadsto \color{blue}{{\left(\mathsf{fma}\left(y, \frac{z - t}{z - a}, x\right)\right)}^{\left(2 \cdot 0.3333333333333333\right)}} \cdot {\left(\mathsf{fma}\left(y, \frac{z - t}{z - a}, x\right)\right)}^{0.3333333333333333} \]

   8. Applied pow-prod-up_binary64 0.6

      \[\leadsto \color{blue}{{\left(\mathsf{fma}\left(y, \frac{z - t}{z - a}, x\right)\right)}^{\left(2 \cdot 0.3333333333333333 + 0.3333333333333333\right)}} \]

   if -5.31863520850266644e-11 < y < 3.8508092885810421e-38

   1. Initial program 0.3

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

   2. Simplified 2.6

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

   3. Taylor expanded in y around 0 0.3

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.3186352085026664 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{z - a}, x\right)\\ \mathbf{elif}\;y \leq 3.850809288581042 \cdot 10^{-38}:\\ \;\;\;\;\left(x + \frac{y \cdot z}{z - a}\right) - \frac{y \cdot t}{z - a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{z - a}, x\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022131 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTicks from plot-0.2.3.4, A"
  :precision binary64

  :herbie-target
  (+ x (/ y (/ (- z a) (- z t))))

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