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

Average Error: 15.8 → 4.6

Time: 16.3s

Precision: binary64

Cost: 9804

Math

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

↓

\[\begin{array}{l} t_1 := x - \frac{y}{\frac{t}{a - z}}\\ t_2 := \left(x + y\right) + \frac{y \cdot \left(t - z\right)}{a - t}\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_2 \leq -2 \cdot 10^{-199}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_2 \leq 0:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x + \mathsf{fma}\left(\frac{t - z}{a - t}, y, y\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- x (/ y (/ t (- a z)))))
        (t_2 (+ (+ x y) (/ (* y (- t z)) (- a t)))))
   (if (<= t_2 (- INFINITY))
     t_1
     (if (<= t_2 -2e-199)
       t_2
       (if (<= t_2 0.0) t_1 (+ x (fma (/ (- t z) (- a t)) y y)))))))

C

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

↓

double code(double x, double y, double z, double t, double a) {
	double t_1 = x - (y / (t / (a - z)));
	double t_2 = (x + y) + ((y * (t - z)) / (a - t));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_2 <= -2e-199) {
		tmp = t_2;
	} else if (t_2 <= 0.0) {
		tmp = t_1;
	} else {
		tmp = x + fma(((t - z) / (a - t)), y, y);
	}
	return tmp;
}

Julia

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

↓

function code(x, y, z, t, a)
	t_1 = Float64(x - Float64(y / Float64(t / Float64(a - z))))
	t_2 = Float64(Float64(x + y) + Float64(Float64(y * Float64(t - z)) / Float64(a - t)))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_2 <= -2e-199)
		tmp = t_2;
	elseif (t_2 <= 0.0)
		tmp = t_1;
	else
		tmp = Float64(x + fma(Float64(Float64(t - z) / Float64(a - t)), y, y));
	end
	return tmp
end

Wolfram

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

↓

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x - N[(y / N[(t / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x + y), $MachinePrecision] + N[(N[(y * N[(t - z), $MachinePrecision]), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$1, If[LessEqual[t$95$2, -2e-199], t$95$2, If[LessEqual[t$95$2, 0.0], t$95$1, N[(x + N[(N[(N[(t - z), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision] * y + y), $MachinePrecision]), $MachinePrecision]]]]]]

Target

Original	15.8
Target	8.3
Herbie	4.6

\[\begin{array}{l} \mathbf{if}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} < -1.3664970889390727 \cdot 10^{-7}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \mathbf{elif}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} < 1.4754293444577233 \cdot 10^{-239}:\\ \;\;\;\;\frac{y \cdot \left(a - z\right) - x \cdot t}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \end{array} \]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (+.f64 x y) (/.f64 (*.f64 (-.f64 z t) y) (-.f64 a t))) < -inf.0 or -1.99999999999999996e-199 < (-.f64 (+.f64 x y) (/.f64 (*.f64 (-.f64 z t) y) (-.f64 a t))) < 0.0

   1. Initial program 58.5

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

   2. Simplified 42.1

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

      [Start] 58.5	\[ \left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \]
      associate-/l* [=>] 42.1	\[ \left(x + y\right) - \color{blue}{\frac{z - t}{\frac{a - t}{y}}} \]

   3. Taylor expanded in t around -inf 22.0

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

   4. Simplified 12.0

      \[\leadsto \color{blue}{x - \frac{y}{\frac{t}{a - z}}} \]
      Proof

      [Start] 22.0	\[ -1 \cdot \frac{y \cdot a - y \cdot z}{t} + x \]
      +-commutative [=>] 22.0	\[ \color{blue}{x + -1 \cdot \frac{y \cdot a - y \cdot z}{t}} \]
      mul-1-neg [=>] 22.0	\[ x + \color{blue}{\left(-\frac{y \cdot a - y \cdot z}{t}\right)} \]
      unsub-neg [=>] 22.0	\[ \color{blue}{x - \frac{y \cdot a - y \cdot z}{t}} \]
      distribute-lft-out-- [=>] 22.0	\[ x - \frac{\color{blue}{y \cdot \left(a - z\right)}}{t} \]
      associate-/l* [=>] 12.0	\[ x - \color{blue}{\frac{y}{\frac{t}{a - z}}} \]

   if -inf.0 < (-.f64 (+.f64 x y) (/.f64 (*.f64 (-.f64 z t) y) (-.f64 a t))) < -1.99999999999999996e-199

   1. Initial program 1.1

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

   if 0.0 < (-.f64 (+.f64 x y) (/.f64 (*.f64 (-.f64 z t) y) (-.f64 a t)))

   1. Initial program 12.0

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

   2. Simplified 4.7

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

      [Start] 12.0	\[ \left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \]
      associate--l+ [=>] 11.8	\[ \color{blue}{x + \left(y - \frac{\left(z - t\right) \cdot y}{a - t}\right)} \]
      sub-neg [=>] 11.8	\[ x + \color{blue}{\left(y + \left(-\frac{\left(z - t\right) \cdot y}{a - t}\right)\right)} \]
      +-commutative [=>] 11.8	\[ x + \color{blue}{\left(\left(-\frac{\left(z - t\right) \cdot y}{a - t}\right) + y\right)} \]
      neg-mul-1 [=>] 11.8	\[ x + \left(\color{blue}{-1 \cdot \frac{\left(z - t\right) \cdot y}{a - t}} + y\right) \]
      associate-*l/ [<=] 4.7	\[ x + \left(-1 \cdot \color{blue}{\left(\frac{z - t}{a - t} \cdot y\right)} + y\right) \]
      associate-*r* [=>] 4.7	\[ x + \left(\color{blue}{\left(-1 \cdot \frac{z - t}{a - t}\right) \cdot y} + y\right) \]
      fma-def [=>] 4.7	\[ x + \color{blue}{\mathsf{fma}\left(-1 \cdot \frac{z - t}{a - t}, y, y\right)} \]
      mul-1-neg [=>] 4.7	\[ x + \mathsf{fma}\left(\color{blue}{-\frac{z - t}{a - t}}, y, y\right) \]
      neg-sub0 [=>] 4.7	\[ x + \mathsf{fma}\left(\color{blue}{0 - \frac{z - t}{a - t}}, y, y\right) \]
      div-sub [=>] 4.7	\[ x + \mathsf{fma}\left(0 - \color{blue}{\left(\frac{z}{a - t} - \frac{t}{a - t}\right)}, y, y\right) \]
      associate--r- [=>] 4.7	\[ x + \mathsf{fma}\left(\color{blue}{\left(0 - \frac{z}{a - t}\right) + \frac{t}{a - t}}, y, y\right) \]
      neg-sub0 [<=] 4.7	\[ x + \mathsf{fma}\left(\color{blue}{\left(-\frac{z}{a - t}\right)} + \frac{t}{a - t}, y, y\right) \]
      +-commutative [=>] 4.7	\[ x + \mathsf{fma}\left(\color{blue}{\frac{t}{a - t} + \left(-\frac{z}{a - t}\right)}, y, y\right) \]
      sub-neg [<=] 4.7	\[ x + \mathsf{fma}\left(\color{blue}{\frac{t}{a - t} - \frac{z}{a - t}}, y, y\right) \]
      div-sub [<=] 4.7	\[ x + \mathsf{fma}\left(\color{blue}{\frac{t - z}{a - t}}, y, y\right) \]

3. Recombined 3 regimes into one program.

4. Final simplification 4.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x + y\right) + \frac{y \cdot \left(t - z\right)}{a - t} \leq -\infty:\\ \;\;\;\;x - \frac{y}{\frac{t}{a - z}}\\ \mathbf{elif}\;\left(x + y\right) + \frac{y \cdot \left(t - z\right)}{a - t} \leq -2 \cdot 10^{-199}:\\ \;\;\;\;\left(x + y\right) + \frac{y \cdot \left(t - z\right)}{a - t}\\ \mathbf{elif}\;\left(x + y\right) + \frac{y \cdot \left(t - z\right)}{a - t} \leq 0:\\ \;\;\;\;x - \frac{y}{\frac{t}{a - z}}\\ \mathbf{else}:\\ \;\;\;\;x + \mathsf{fma}\left(\frac{t - z}{a - t}, y, y\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	4.5
Cost	3532

\[\begin{array}{l} t_1 := x - \frac{y}{\frac{t}{a - z}}\\ t_2 := \left(x + y\right) + \frac{y \cdot \left(t - z\right)}{a - t}\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_2 \leq -2 \cdot 10^{-199}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_2 \leq 0:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x + \left(y - \frac{y}{\frac{a - t}{z - t}}\right)\\ \end{array} \]

Alternative 2
Error	23.3
Cost	1240

\[\begin{array}{l} t_1 := \frac{y}{t} \cdot \left(z - a\right)\\ \mathbf{if}\;x \leq -1.4 \cdot 10^{-174}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;x \leq -7.1 \cdot 10^{-242}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -1.5 \cdot 10^{-288}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{-263}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 7.6 \cdot 10^{-185}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{-174}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 3 \cdot 10^{-103}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 3
Error	22.2
Cost	1240

\[\begin{array}{l} t_1 := y \cdot \frac{z}{t - a}\\ \mathbf{if}\;a \leq -1.8 \cdot 10^{-148}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;a \leq -1.3 \cdot 10^{-189}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -6 \cdot 10^{-224}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;a \leq -1 \cdot 10^{-259}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 1.1 \cdot 10^{-134}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 4.6 \cdot 10^{-74}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 4
Error	11.3
Cost	1104

\[\begin{array}{l} t_1 := x - \frac{y}{\frac{t}{a - z}}\\ \mathbf{if}\;t \leq -85000000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.65 \cdot 10^{-44}:\\ \;\;\;\;\left(x + y\right) - \frac{z}{\frac{a}{y}}\\ \mathbf{elif}\;t \leq 6.7 \cdot 10^{+19}:\\ \;\;\;\;x + z \cdot \frac{y}{t}\\ \mathbf{elif}\;t \leq 8.8 \cdot 10^{+59}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	11.4
Cost	1104

\[\begin{array}{l} t_1 := x - \frac{y}{\frac{t}{a - z}}\\ \mathbf{if}\;t \leq -19500000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 4.1 \cdot 10^{-48}:\\ \;\;\;\;\left(x + y\right) - \frac{z}{\frac{a}{y}}\\ \mathbf{elif}\;t \leq 7.2 \cdot 10^{-27}:\\ \;\;\;\;x + \frac{y \cdot \left(z - a\right)}{t}\\ \mathbf{elif}\;t \leq 1.35 \cdot 10^{+60}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	5.6
Cost	1097

\[\begin{array}{l} \mathbf{if}\;t \leq -1.5 \cdot 10^{+174} \lor \neg \left(t \leq 1.5 \cdot 10^{+144}\right):\\ \;\;\;\;x - \frac{y}{\frac{t}{a - z}}\\ \mathbf{else}:\\ \;\;\;\;x + \left(y + \frac{t - z}{\frac{a - t}{y}}\right)\\ \end{array} \]

Alternative 7
Error	5.9
Cost	964

\[\begin{array}{l} \mathbf{if}\;t \leq -3.15 \cdot 10^{+174}:\\ \;\;\;\;x - \frac{y}{\frac{t}{a - z}}\\ \mathbf{else}:\\ \;\;\;\;x + \left(y - \frac{y}{\frac{a - t}{z - t}}\right)\\ \end{array} \]

Alternative 8
Error	10.3
Cost	841

\[\begin{array}{l} \mathbf{if}\;a \leq -5.5 \cdot 10^{-30} \lor \neg \left(a \leq 2.9 \cdot 10^{-14}\right):\\ \;\;\;\;\left(x + y\right) - \frac{y}{\frac{a}{z}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{t}{a - z}}\\ \end{array} \]

Alternative 9
Error	13.1
Cost	840

\[\begin{array}{l} \mathbf{if}\;a \leq -1.65 \cdot 10^{-31}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;a \leq 7.5 \cdot 10^{-15}:\\ \;\;\;\;x - \frac{y}{\frac{t}{a - z}}\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 10
Error	14.1
Cost	712

\[\begin{array}{l} \mathbf{if}\;a \leq -3.4 \cdot 10^{-35}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;a \leq 3 \cdot 10^{-15}:\\ \;\;\;\;x + z \cdot \frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 11
Error	14.1
Cost	712

\[\begin{array}{l} \mathbf{if}\;a \leq -5.2 \cdot 10^{-34}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;a \leq 4 \cdot 10^{-14}:\\ \;\;\;\;x + y \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 12
Error	19.6
Cost	456

\[\begin{array}{l} \mathbf{if}\;a \leq -2.2 \cdot 10^{+38}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;a \leq 3.6 \cdot 10^{-18}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 13
Error	28.6
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022364 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTick from plot-0.2.3.4, B"
  :precision binary64

  :herbie-target
  (if (< (- (+ x y) (/ (* (- z t) y) (- a t))) -1.3664970889390727e-7) (- (+ y x) (* (* (- z t) (/ 1.0 (- a t))) y)) (if (< (- (+ x y) (/ (* (- z t) y) (- a t))) 1.4754293444577233e-239) (/ (- (* y (- a z)) (* x t)) (- a t)) (- (+ y x) (* (* (- z t) (/ 1.0 (- a t))) y))))

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