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

Average Error: 10.8 → 0.5

Time: 5.1s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := \frac{y \cdot \left(z - t\right)}{z - a}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{z - a}, x\right)\\ \mathbf{elif}\;t_1 \leq 7.177294426869409 \cdot 10^{+236}:\\ \;\;\;\;t_1 + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z - a}, z - t, x\right)\\ \end{array} \]

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 (/ (* y (- z t)) (- z a))))
   (if (<= t_1 (- INFINITY))
     (fma y (/ (- z t) (- z a)) x)
     (if (<= t_1 7.177294426869409e+236)
       (+ t_1 x)
       (fma (/ y (- z a)) (- z t) x)))))

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 = (y * (z - t)) / (z - a);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma(y, ((z - t) / (z - a)), x);
	} else if (t_1 <= 7.177294426869409e+236) {
		tmp = t_1 + x;
	} else {
		tmp = fma((y / (z - a)), (z - t), x);
	}
	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 = Float64(Float64(y * Float64(z - t)) / Float64(z - a))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = fma(y, Float64(Float64(z - t) / Float64(z - a)), x);
	elseif (t_1 <= 7.177294426869409e+236)
		tmp = Float64(t_1 + x);
	else
		tmp = fma(Float64(y / Float64(z - a)), Float64(z - t), x);
	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[(N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(y * N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 7.177294426869409e+236], N[(t$95$1 + x), $MachinePrecision], N[(N[(y / N[(z - a), $MachinePrecision]), $MachinePrecision] * N[(z - t), $MachinePrecision] + 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	10.8
Target	1.2
Herbie	0.5

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 64.0

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

   2. Simplified 0.1

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

   if -inf.0 < (/.f64 (*.f64 y (-.f64 z t)) (-.f64 z a)) < 7.17729442686940866e236

   1. Initial program 0.3

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

   if 7.17729442686940866e236 < (/.f64 (*.f64 y (-.f64 z t)) (-.f64 z a))

   1. Initial program 53.9

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

   2. Simplified 1.5

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

   3. Applied clear-num_binary64 1.5

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

   4. Applied div-inv_binary64 1.6

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

   5. Applied *-un-lft-identity_binary64 1.6

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

   6. Applied times-frac_binary64 1.6

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

   7. Taylor expanded in y around 0 53.9

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

   8. Simplified 3.2

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

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z - t\right)}{z - a} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{z - a}, x\right)\\ \mathbf{elif}\;\frac{y \cdot \left(z - t\right)}{z - a} \leq 7.177294426869409 \cdot 10^{+236}:\\ \;\;\;\;\frac{y \cdot \left(z - t\right)}{z - a} + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z - a}, z - t, x\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022129 
(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))))