Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisLine from plot-0.2.3.4, B

Average Error: 1.5 → 0.6

Time: 6.7s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq -163.86311167990965:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{a - t}, x\right)\\ \mathbf{elif}\;y \leq 1.72501828309298 \cdot 10^{+46}:\\ \;\;\;\;\left(x + \frac{y \cdot z}{a - t}\right) - \frac{y \cdot t}{a - t}\\ \mathbf{else}:\\ \;\;\;\;x + \sqrt{y} \cdot \left(\left(z - t\right) \cdot \frac{\sqrt{y}}{a - t}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t a)
 :precision binary64
 (if (<= y -163.86311167990965)
   (fma y (/ (- z t) (- a t)) x)
   (if (<= y 1.72501828309298e+46)
     (- (+ x (/ (* y z) (- a t))) (/ (* y t) (- a t)))
     (+ x (* (sqrt y) (* (- z t) (/ (sqrt y) (- a t))))))))

C

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

↓

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (y <= -163.86311167990965) {
		tmp = fma(y, ((z - t) / (a - t)), x);
	} else if (y <= 1.72501828309298e+46) {
		tmp = (x + ((y * z) / (a - t))) - ((y * t) / (a - t));
	} else {
		tmp = x + (sqrt(y) * ((z - t) * (sqrt(y) / (a - t))));
	}
	return tmp;
}

Julia

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

↓

function code(x, y, z, t, a)
	tmp = 0.0
	if (y <= -163.86311167990965)
		tmp = fma(y, Float64(Float64(z - t) / Float64(a - t)), x);
	elseif (y <= 1.72501828309298e+46)
		tmp = Float64(Float64(x + Float64(Float64(y * z) / Float64(a - t))) - Float64(Float64(y * t) / Float64(a - t)));
	else
		tmp = Float64(x + Float64(sqrt(y) * Float64(Float64(z - t) * Float64(sqrt(y) / Float64(a - t)))));
	end
	return tmp
end

Wolfram

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

↓

code[x_, y_, z_, t_, a_] := If[LessEqual[y, -163.86311167990965], N[(y * N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[y, 1.72501828309298e+46], N[(N[(x + N[(N[(y * z), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(y * t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[Sqrt[y], $MachinePrecision] * N[(N[(z - t), $MachinePrecision] * N[(N[Sqrt[y], $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	1.5
Target	0.5
Herbie	0.6

\[\begin{array}{l} \mathbf{if}\;y < -8.508084860551241 \cdot 10^{-17}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;y < 2.894426862792089 \cdot 10^{-49}:\\ \;\;\;\;x + \left(y \cdot \left(z - t\right)\right) \cdot \frac{1}{a - t}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a - t}\\ \end{array} \]

Derivation

1. Split input into 3 regimes

2. if y < -163.86311167990965

   1. Initial program 0.7

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

   2. Simplified 0.7

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

   3. Applied *-un-lft-identity_binary64 0.7

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

   4. Applied add-cube-cbrt_binary64 1.5

      \[\leadsto \mathsf{fma}\left(y, \frac{\color{blue}{\left(\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}\right) \cdot \sqrt[3]{z - t}}}{1 \cdot \left(a - t\right)}, x\right) \]

   5. Applied times-frac_binary64 1.5

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}}{1} \cdot \frac{\sqrt[3]{z - t}}{a - t}}, x\right) \]

   6. Simplified 1.5

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

   7. Applied pow1_binary64 1.5

      \[\leadsto \mathsf{fma}\left(y, \left(\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}\right) \cdot \color{blue}{{\left(\frac{\sqrt[3]{z - t}}{a - t}\right)}^{1}}, x\right) \]

   8. Applied pow1_binary64 1.5

      \[\leadsto \mathsf{fma}\left(y, \left(\sqrt[3]{z - t} \cdot \color{blue}{{\left(\sqrt[3]{z - t}\right)}^{1}}\right) \cdot {\left(\frac{\sqrt[3]{z - t}}{a - t}\right)}^{1}, x\right) \]

   9. Applied pow1_binary64 1.5

      \[\leadsto \mathsf{fma}\left(y, \left(\color{blue}{{\left(\sqrt[3]{z - t}\right)}^{1}} \cdot {\left(\sqrt[3]{z - t}\right)}^{1}\right) \cdot {\left(\frac{\sqrt[3]{z - t}}{a - t}\right)}^{1}, x\right) \]

   10. Applied pow-prod-down_binary64 1.5

       \[\leadsto \mathsf{fma}\left(y, \color{blue}{{\left(\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}\right)}^{1}} \cdot {\left(\frac{\sqrt[3]{z - t}}{a - t}\right)}^{1}, x\right) \]

   11. Applied pow-prod-down_binary64 1.5

       \[\leadsto \mathsf{fma}\left(y, \color{blue}{{\left(\left(\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}\right) \cdot \frac{\sqrt[3]{z - t}}{a - t}\right)}^{1}}, x\right) \]

   12. Simplified 0.7

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

   if -163.86311167990965 < y < 1.72501828309297999e46

   1. Initial program 2.1

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

   2. Simplified 2.1

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

   3. Taylor expanded in y around 0 0.5

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

   if 1.72501828309297999e46 < y

   1. Initial program 0.4

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

   2. Simplified 0.4

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

   3. Taylor expanded in y around 0 26.5

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

   4. Simplified 3.1

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

   5. Applied *-un-lft-identity_binary64 3.1

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

   6. Applied add-sqr-sqrt_binary64 3.3

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

   7. Applied times-frac_binary64 3.4

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

   8. Applied associate-*l*_binary64 0.8

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

   9. Simplified 0.8

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

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -163.86311167990965:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{a - t}, x\right)\\ \mathbf{elif}\;y \leq 1.72501828309298 \cdot 10^{+46}:\\ \;\;\;\;\left(x + \frac{y \cdot z}{a - t}\right) - \frac{y \cdot t}{a - t}\\ \mathbf{else}:\\ \;\;\;\;x + \sqrt{y} \cdot \left(\left(z - t\right) \cdot \frac{\sqrt{y}}{a - t}\right)\\ \end{array} \]

Reproduce

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

  :herbie-target
  (if (< y -8.508084860551241e-17) (+ x (* y (/ (- z t) (- a t)))) (if (< y 2.894426862792089e-49) (+ x (* (* y (- z t)) (/ 1.0 (- a t)))) (+ x (* y (/ (- z t) (- a t))))))

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