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

Average Error: 10.5 → 0.8

Time: 10.3s

Precision: binary64

Cost: 8264

Math

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

↓

\[\begin{array}{l} t_1 := \mathsf{fma}\left(\frac{z - t}{z - a}, y, x\right)\\ t_2 := \frac{y \cdot \left(z - t\right)}{z - a}\\ \mathbf{if}\;t_2 \leq -5 \cdot 10^{+276}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_2 \leq 5 \cdot 10^{+97}:\\ \;\;\;\;x - \frac{y \cdot \left(t - z\right)}{z - a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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 (/ (- z t) (- z a)) y x)) (t_2 (/ (* y (- z t)) (- z a))))
   (if (<= t_2 -5e+276)
     t_1
     (if (<= t_2 5e+97) (- x (/ (* y (- t z)) (- 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(((z - t) / (z - a)), y, x);
	double t_2 = (y * (z - t)) / (z - a);
	double tmp;
	if (t_2 <= -5e+276) {
		tmp = t_1;
	} else if (t_2 <= 5e+97) {
		tmp = x - ((y * (t - z)) / (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(Float64(Float64(z - t) / Float64(z - a)), y, x)
	t_2 = Float64(Float64(y * Float64(z - t)) / Float64(z - a))
	tmp = 0.0
	if (t_2 <= -5e+276)
		tmp = t_1;
	elseif (t_2 <= 5e+97)
		tmp = Float64(x - Float64(Float64(y * Float64(t - z)) / 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[(N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+276], t$95$1, If[LessEqual[t$95$2, 5e+97], N[(x - N[(N[(y * N[(t - z), $MachinePrecision]), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Target

Original	10.5
Target	1.4
Herbie	0.8

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 y (-.f64 z t)) (-.f64 z a)) < -5.00000000000000001e276 or 4.99999999999999999e97 < (/.f64 (*.f64 y (-.f64 z t)) (-.f64 z a))

   1. Initial program 42.1

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

   2. Simplified 2.6

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z - t}{z - a}, y, x\right)} \]
      Proof
      (fma.f64 (/.f64 (-.f64 z t) (-.f64 z a)) y x): 0 points increase in error, 0 points decrease in error
      (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (/.f64 (-.f64 z t) (-.f64 z a)) y) x)): 0 points increase in error, 2 points decrease in error
      (+.f64 (Rewrite=> *-commutative_binary64 (*.f64 y (/.f64 (-.f64 z t) (-.f64 z a)))) x): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 y (-.f64 z t)) (-.f64 z a))) x): 59 points increase in error, 18 points decrease in error
      (Rewrite<= +-commutative_binary64 (+.f64 x (/.f64 (*.f64 y (-.f64 z t)) (-.f64 z a)))): 0 points increase in error, 0 points decrease in error

   if -5.00000000000000001e276 < (/.f64 (*.f64 y (-.f64 z t)) (-.f64 z a)) < 4.99999999999999999e97

   1. Initial program 0.2

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

4. Final simplification 0.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z - t\right)}{z - a} \leq -5 \cdot 10^{+276}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{z - a}, y, x\right)\\ \mathbf{elif}\;\frac{y \cdot \left(z - t\right)}{z - a} \leq 5 \cdot 10^{+97}:\\ \;\;\;\;x - \frac{y \cdot \left(t - z\right)}{z - a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{z - a}, y, x\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.6
Cost	1992

\[\begin{array}{l} t_1 := x + \frac{y}{\frac{z - a}{z - t}}\\ t_2 := \frac{y \cdot \left(z - t\right)}{z - a}\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_2 \leq 5 \cdot 10^{+97}:\\ \;\;\;\;x - \frac{y \cdot \left(t - z\right)}{z - a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	14.8
Cost	1368

\[\begin{array}{l} t_1 := y \cdot \frac{z - t}{z - a}\\ t_2 := x + \frac{z}{\frac{z - a}{y}}\\ \mathbf{if}\;a \leq -1.95 \cdot 10^{+74}:\\ \;\;\;\;x + t \cdot \frac{y}{a}\\ \mathbf{elif}\;a \leq -1.3 \cdot 10^{+30}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -8 \cdot 10^{-32}:\\ \;\;\;\;x + \frac{t}{\frac{a}{y}}\\ \mathbf{elif}\;a \leq -1.32 \cdot 10^{-61}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -1.5 \cdot 10^{-158}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 2.5 \cdot 10^{+44}:\\ \;\;\;\;x + y \cdot \left(1 - \frac{t}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Error	17.4
Cost	1240

\[\begin{array}{l} t_1 := x + \frac{t}{\frac{a}{y}}\\ \mathbf{if}\;a \leq -1.95 \cdot 10^{+74}:\\ \;\;\;\;x + t \cdot \frac{y}{a}\\ \mathbf{elif}\;a \leq -1.6 \cdot 10^{+29}:\\ \;\;\;\;y \cdot \frac{z - t}{z - a}\\ \mathbf{elif}\;a \leq -1.05 \cdot 10^{-30}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -1.02 \cdot 10^{-258}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;a \leq 1.06 \cdot 10^{-270}:\\ \;\;\;\;x - \frac{y}{\frac{z}{t}}\\ \mathbf{elif}\;a \leq 3 \cdot 10^{+43}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	14.5
Cost	1236

\[\begin{array}{l} t_1 := x + \frac{t}{\frac{a}{y}}\\ \mathbf{if}\;a \leq -2.6 \cdot 10^{+74}:\\ \;\;\;\;x + t \cdot \frac{y}{a}\\ \mathbf{elif}\;a \leq -2.8 \cdot 10^{+30}:\\ \;\;\;\;y \cdot \frac{z - t}{z - a}\\ \mathbf{elif}\;a \leq -5.4 \cdot 10^{-31}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -2.9 \cdot 10^{-159}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;a \leq 10^{+33}:\\ \;\;\;\;x + y \cdot \left(1 - \frac{t}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	16.8
Cost	976

\[\begin{array}{l} \mathbf{if}\;a \leq -2.4 \cdot 10^{-31}:\\ \;\;\;\;x + t \cdot \frac{y}{a}\\ \mathbf{elif}\;a \leq -1.96 \cdot 10^{-258}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;a \leq 6 \cdot 10^{-268}:\\ \;\;\;\;x - \frac{y}{\frac{z}{t}}\\ \mathbf{elif}\;a \leq 9 \cdot 10^{+32}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;x + \frac{t}{\frac{a}{y}}\\ \end{array} \]

Alternative 6
Error	1.7
Cost	968

\[\begin{array}{l} t_1 := x + \frac{y}{\frac{z - a}{z - t}}\\ \mathbf{if}\;z \leq -1.16 \cdot 10^{-172}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2 \cdot 10^{-285}:\\ \;\;\;\;x + \frac{y}{a} \cdot \left(t - z\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	11.0
Cost	840

\[\begin{array}{l} \mathbf{if}\;z \leq -2.1 \cdot 10^{+17}:\\ \;\;\;\;x + y \cdot \left(1 - \frac{t}{z}\right)\\ \mathbf{elif}\;z \leq 3.9 \cdot 10^{-127}:\\ \;\;\;\;x + \frac{y}{a} \cdot \left(t - z\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{z}{\frac{z - a}{y}}\\ \end{array} \]

Alternative 8
Error	14.4
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -1.76 \cdot 10^{+16}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;z \leq 410:\\ \;\;\;\;x + \frac{t}{\frac{a}{y}}\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]

Alternative 9
Error	14.5
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -1.9 \cdot 10^{+16}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;z \leq 13.2:\\ \;\;\;\;x + t \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]

Alternative 10
Error	20.2
Cost	456

\[\begin{array}{l} \mathbf{if}\;a \leq -1.02 \cdot 10^{+142}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 9.2 \cdot 10^{+172}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 11
Error	28.6
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022329 
(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))))