Graphics.Rendering.Plot.Render.Plot.Axis:tickPosition from plot-0.2.3.4

Average Error: 2.0 → 1.7

Time: 10.5s

Precision: binary64

Cost: 7112

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \leq -1.3 \cdot 10^{-44}:\\ \;\;\;\;x + \frac{z}{\frac{t}{y - x}}\\ \mathbf{elif}\;t \leq 5 \cdot 10^{-262}:\\ \;\;\;\;x + \frac{1}{t} \cdot \frac{y - x}{\frac{1}{z}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y - x, \frac{z}{t}, x\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (if (<= t -1.3e-44)
   (+ x (/ z (/ t (- y x))))
   (if (<= t 5e-262)
     (+ x (* (/ 1.0 t) (/ (- y x) (/ 1.0 z))))
     (fma (- y x) (/ z t) x))))

C

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

↓

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -1.3e-44) {
		tmp = x + (z / (t / (y - x)));
	} else if (t <= 5e-262) {
		tmp = x + ((1.0 / t) * ((y - x) / (1.0 / z)));
	} else {
		tmp = fma((y - x), (z / t), x);
	}
	return tmp;
}

Julia

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

↓

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -1.3e-44)
		tmp = Float64(x + Float64(z / Float64(t / Float64(y - x))));
	elseif (t <= 5e-262)
		tmp = Float64(x + Float64(Float64(1.0 / t) * Float64(Float64(y - x) / Float64(1.0 / z))));
	else
		tmp = fma(Float64(y - x), Float64(z / t), x);
	end
	return tmp
end

Wolfram

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

↓

code[x_, y_, z_, t_] := If[LessEqual[t, -1.3e-44], N[(x + N[(z / N[(t / N[(y - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 5e-262], N[(x + N[(N[(1.0 / t), $MachinePrecision] * N[(N[(y - x), $MachinePrecision] / N[(1.0 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y - x), $MachinePrecision] * N[(z / t), $MachinePrecision] + x), $MachinePrecision]]]

Target

Original	2.0
Target	2.2
Herbie	1.7

\[\begin{array}{l} \mathbf{if}\;\left(y - x\right) \cdot \frac{z}{t} < -1013646692435.8867:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \mathbf{elif}\;\left(y - x\right) \cdot \frac{z}{t} < 0:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \end{array} \]

Derivation

1. Split input into 3 regimes

2. if t < -1.2999999999999999e-44

   1. Initial program 1.2

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

   2. Applied egg-rr 1.2

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

   if -1.2999999999999999e-44 < t < 4.99999999999999992e-262

   1. Initial program 4.5

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

   2. Applied egg-rr 4.8

      \[\leadsto x + \color{blue}{\frac{\frac{y - x}{\sqrt[3]{\frac{t}{z}} \cdot \sqrt[3]{\frac{t}{z}}}}{\sqrt[3]{\frac{t}{z}}}} \]

   3. Simplified 4.8

      \[\leadsto x + \color{blue}{\frac{\frac{y - x}{\sqrt[3]{\frac{t}{z}}}}{\sqrt[3]{\frac{t}{z}} \cdot \sqrt[3]{\frac{t}{z}}}} \]
      Proof

      [Start] 4.8	\[ x + \frac{\frac{y - x}{\sqrt[3]{\frac{t}{z}} \cdot \sqrt[3]{\frac{t}{z}}}}{\sqrt[3]{\frac{t}{z}}} \]
      associate-/l/ [=>] 4.8	\[ x + \color{blue}{\frac{y - x}{\sqrt[3]{\frac{t}{z}} \cdot \left(\sqrt[3]{\frac{t}{z}} \cdot \sqrt[3]{\frac{t}{z}}\right)}} \]
      associate-/r* [=>] 4.8	\[ x + \color{blue}{\frac{\frac{y - x}{\sqrt[3]{\frac{t}{z}}}}{\sqrt[3]{\frac{t}{z}} \cdot \sqrt[3]{\frac{t}{z}}}} \]

   4. Applied egg-rr 2.7

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

   if 4.99999999999999992e-262 < t

   1. Initial program 1.7

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

   2. Simplified 1.7

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

      [Start] 1.7	\[ x + \left(y - x\right) \cdot \frac{z}{t} \]
      +-commutative [=>] 1.7	\[ \color{blue}{\left(y - x\right) \cdot \frac{z}{t} + x} \]
      fma-def [=>] 1.7	\[ \color{blue}{\mathsf{fma}\left(y - x, \frac{z}{t}, x\right)} \]

3. Recombined 3 regimes into one program.

4. Final simplification 1.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.3 \cdot 10^{-44}:\\ \;\;\;\;x + \frac{z}{\frac{t}{y - x}}\\ \mathbf{elif}\;t \leq 5 \cdot 10^{-262}:\\ \;\;\;\;x + \frac{1}{t} \cdot \frac{y - x}{\frac{1}{z}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y - x, \frac{z}{t}, x\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	25.9
Cost	1944

\[\begin{array}{l} t_1 := \frac{z \cdot y}{t}\\ t_2 := \frac{-x}{\frac{t}{z}}\\ \mathbf{if}\;\frac{z}{t} \leq -2 \cdot 10^{+17}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\frac{z}{t} \leq -2 \cdot 10^{-166}:\\ \;\;\;\;\frac{y}{\frac{t}{z}}\\ \mathbf{elif}\;\frac{z}{t} \leq 5 \cdot 10^{-107}:\\ \;\;\;\;x\\ \mathbf{elif}\;\frac{z}{t} \leq 10^{-26}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{z}{t} \leq 0.0004:\\ \;\;\;\;x\\ \mathbf{elif}\;\frac{z}{t} \leq 10^{+91}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	26.3
Cost	1944

\[\begin{array}{l} t_1 := \frac{z \cdot y}{t}\\ \mathbf{if}\;\frac{z}{t} \leq -1 \cdot 10^{+52}:\\ \;\;\;\;\frac{-z}{\frac{t}{x}}\\ \mathbf{elif}\;\frac{z}{t} \leq -2 \cdot 10^{-166}:\\ \;\;\;\;\frac{y}{\frac{t}{z}}\\ \mathbf{elif}\;\frac{z}{t} \leq 5 \cdot 10^{-107}:\\ \;\;\;\;x\\ \mathbf{elif}\;\frac{z}{t} \leq 10^{-26}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{z}{t} \leq 0.0004:\\ \;\;\;\;x\\ \mathbf{elif}\;\frac{z}{t} \leq 10^{+91}:\\ \;\;\;\;\frac{-x}{\frac{t}{z}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	18.4
Cost	1748

\[\begin{array}{l} t_1 := z \cdot \frac{y - x}{t}\\ \mathbf{if}\;\frac{z}{t} \leq -200000000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{z}{t} \leq -2 \cdot 10^{-166}:\\ \;\;\;\;\frac{y}{\frac{t}{z}}\\ \mathbf{elif}\;\frac{z}{t} \leq 5 \cdot 10^{-107}:\\ \;\;\;\;x\\ \mathbf{elif}\;\frac{z}{t} \leq 10^{-26}:\\ \;\;\;\;\frac{z \cdot y}{t}\\ \mathbf{elif}\;\frac{z}{t} \leq 2 \cdot 10^{-14}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	16.1
Cost	1748

\[\begin{array}{l} t_1 := \left(y - x\right) \cdot \frac{z}{t}\\ \mathbf{if}\;\frac{z}{t} \leq -2 \cdot 10^{+124}:\\ \;\;\;\;z \cdot \frac{y - x}{t}\\ \mathbf{elif}\;\frac{z}{t} \leq -2 \cdot 10^{-176}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{z}{t} \leq 5 \cdot 10^{-107}:\\ \;\;\;\;x\\ \mathbf{elif}\;\frac{z}{t} \leq 10^{-26}:\\ \;\;\;\;\frac{z \cdot y}{t}\\ \mathbf{elif}\;\frac{z}{t} \leq 2 \cdot 10^{-14}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	2.8
Cost	1229

\[\begin{array}{l} \mathbf{if}\;\frac{z}{t} \leq -2 \cdot 10^{+124}:\\ \;\;\;\;z \cdot \frac{y - x}{t}\\ \mathbf{elif}\;\frac{z}{t} \leq -100000000000 \lor \neg \left(\frac{z}{t} \leq 0.0004\right):\\ \;\;\;\;\left(y - x\right) \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z}{t}\\ \end{array} \]

Alternative 6
Error	2.9
Cost	1229

\[\begin{array}{l} \mathbf{if}\;\frac{z}{t} \leq -2 \cdot 10^{+124}:\\ \;\;\;\;z \cdot \frac{y - x}{t}\\ \mathbf{elif}\;\frac{z}{t} \leq -100000000000 \lor \neg \left(\frac{z}{t} \leq 0.0004\right):\\ \;\;\;\;\left(y - x\right) \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{t}{z}}\\ \end{array} \]

Alternative 7
Error	3.8
Cost	1229

\[\begin{array}{l} \mathbf{if}\;\frac{z}{t} \leq -2 \cdot 10^{+124}:\\ \;\;\;\;z \cdot \frac{y - x}{t}\\ \mathbf{elif}\;\frac{z}{t} \leq -40000 \lor \neg \left(\frac{z}{t} \leq 0.0004\right):\\ \;\;\;\;\left(y - x\right) \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{z \cdot y}{t}\\ \end{array} \]

Alternative 8
Error	1.7
Cost	1096

\[\begin{array}{l} \mathbf{if}\;t \leq -1.4 \cdot 10^{-44}:\\ \;\;\;\;x + \frac{z}{\frac{t}{y - x}}\\ \mathbf{elif}\;t \leq 6.2 \cdot 10^{-262}:\\ \;\;\;\;x + \frac{1}{t} \cdot \frac{y - x}{\frac{1}{z}}\\ \mathbf{else}:\\ \;\;\;\;x + \left(y - x\right) \cdot \frac{z}{t}\\ \end{array} \]

Alternative 9
Error	2.1
Cost	968

\[\begin{array}{l} \mathbf{if}\;x \leq 2.7 \cdot 10^{-267}:\\ \;\;\;\;x + \left(y - x\right) \cdot \frac{z}{t}\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-55}:\\ \;\;\;\;x + \frac{z}{\frac{t}{y - x}}\\ \mathbf{else}:\\ \;\;\;\;x + \left(y - x\right) \cdot \left(z \cdot \frac{1}{t}\right)\\ \end{array} \]

Alternative 10
Error	25.0
Cost	841

\[\begin{array}{l} \mathbf{if}\;\frac{z}{t} \leq -2 \cdot 10^{-166} \lor \neg \left(\frac{z}{t} \leq 5 \cdot 10^{-107}\right):\\ \;\;\;\;y \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 11
Error	25.0
Cost	841

\[\begin{array}{l} \mathbf{if}\;\frac{z}{t} \leq -2 \cdot 10^{-166} \lor \neg \left(\frac{z}{t} \leq 5 \cdot 10^{-107}\right):\\ \;\;\;\;\frac{y}{\frac{t}{z}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 12
Error	2.1
Cost	841

\[\begin{array}{l} \mathbf{if}\;x \leq 9.5 \cdot 10^{-269} \lor \neg \left(x \leq 3 \cdot 10^{-55}\right):\\ \;\;\;\;x + \left(y - x\right) \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{z}{\frac{t}{y - x}}\\ \end{array} \]

Alternative 13
Error	1.7
Cost	836

\[\begin{array}{l} \mathbf{if}\;\frac{z}{t} \leq -2 \cdot 10^{+124}:\\ \;\;\;\;z \cdot \frac{y - x}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \left(y - x\right) \cdot \frac{z}{t}\\ \end{array} \]

Alternative 14
Error	31.6
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023018 
(FPCore (x y z t)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:tickPosition from plot-0.2.3.4"
  :precision binary64

  :herbie-target
  (if (< (* (- y x) (/ z t)) -1013646692435.8867) (+ x (/ (- y x) (/ t z))) (if (< (* (- y x) (/ z t)) 0.0) (+ x (/ (* (- y x) z) t)) (+ x (/ (- y x) (/ t z)))))

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