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

Average Accuracy: 96.2% → 96.7%

Time: 8.3s

Precision: binary64

Cost: 7108

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{z}{t} \leq -5 \cdot 10^{-78}:\\ \;\;\;\;\mathsf{fma}\left(y - x, \frac{z}{t}, x\right)\\ \mathbf{elif}\;\frac{z}{t} \leq 5 \cdot 10^{-218}:\\ \;\;\;\;x + z \cdot \frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{z}{t} \cdot \left(y - x\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ z t) -5e-78)
   (fma (- y x) (/ z t) x)
   (if (<= (/ z t) 5e-218) (+ x (* z (/ y t))) (+ x (* (/ z t) (- y 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 ((z / t) <= -5e-78) {
		tmp = fma((y - x), (z / t), x);
	} else if ((z / t) <= 5e-218) {
		tmp = x + (z * (y / t));
	} else {
		tmp = x + ((z / t) * (y - 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 (Float64(z / t) <= -5e-78)
		tmp = fma(Float64(y - x), Float64(z / t), x);
	elseif (Float64(z / t) <= 5e-218)
		tmp = Float64(x + Float64(z * Float64(y / t)));
	else
		tmp = Float64(x + Float64(Float64(z / t) * Float64(y - 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[N[(z / t), $MachinePrecision], -5e-78], N[(N[(y - x), $MachinePrecision] * N[(z / t), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[N[(z / t), $MachinePrecision], 5e-218], N[(x + N[(z * N[(y / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(z / t), $MachinePrecision] * N[(y - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Target

Original	96.2%
Target	96.0%
Herbie	96.7%

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

Derivation

1. Split input into 3 regimes

2. if (/.f64 z t) < -4.9999999999999996e-78

   1. Initial program 94.0%

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

   2. Simplified 94.0%

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

      [Start] 94.0	\[ x + \left(y - x\right) \cdot \frac{z}{t} \]
      +-commutative [=>] 94.0	\[ \color{blue}{\left(y - x\right) \cdot \frac{z}{t} + x} \]
      fma-def [=>] 94.0	\[ \color{blue}{\mathsf{fma}\left(y - x, \frac{z}{t}, x\right)} \]

   if -4.9999999999999996e-78 < (/.f64 z t) < 5.00000000000000041e-218

   1. Initial program 97.4%

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

   2. Taylor expanded in y around inf 98.2%

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

   3. Simplified 98.5%

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

      [Start] 98.2	\[ x + \frac{y \cdot z}{t} \]
      associate-*l/ [<=] 98.5	\[ x + \color{blue}{\frac{y}{t} \cdot z} \]

   if 5.00000000000000041e-218 < (/.f64 z t)

   1. Initial program 96.5%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z}{t} \leq -5 \cdot 10^{-78}:\\ \;\;\;\;\mathsf{fma}\left(y - x, \frac{z}{t}, x\right)\\ \mathbf{elif}\;\frac{z}{t} \leq 5 \cdot 10^{-218}:\\ \;\;\;\;x + z \cdot \frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{z}{t} \cdot \left(y - x\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	91.4%
Cost	1488

\[\begin{array}{l} t_1 := x + \frac{y}{\frac{t}{z}}\\ t_2 := z \cdot \frac{y - x}{t}\\ \mathbf{if}\;\frac{z}{t} \leq -5 \cdot 10^{+75}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\frac{z}{t} \leq -5 \cdot 10^{-78}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{z}{t} \leq 0:\\ \;\;\;\;x + z \cdot \frac{y}{t}\\ \mathbf{elif}\;\frac{z}{t} \leq 0.2:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 2
Accuracy	91.5%
Cost	1488

\[\begin{array}{l} t_1 := x + \frac{y}{\frac{t}{z}}\\ \mathbf{if}\;\frac{z}{t} \leq -5 \cdot 10^{+75}:\\ \;\;\;\;z \cdot \frac{y - x}{t}\\ \mathbf{elif}\;\frac{z}{t} \leq -5 \cdot 10^{-78}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{z}{t} \leq 0:\\ \;\;\;\;x + z \cdot \frac{y}{t}\\ \mathbf{elif}\;\frac{z}{t} \leq 0.2:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{\frac{t}{y - x}}\\ \end{array} \]

Alternative 3
Accuracy	91.2%
Cost	1488

\[\begin{array}{l} t_1 := x + \frac{y}{\frac{t}{z}}\\ \mathbf{if}\;\frac{z}{t} \leq -5 \cdot 10^{+75}:\\ \;\;\;\;z \cdot \frac{y - x}{t}\\ \mathbf{elif}\;\frac{z}{t} \leq -5 \cdot 10^{-78}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{z}{t} \leq 0:\\ \;\;\;\;x + z \cdot \frac{y}{t}\\ \mathbf{elif}\;\frac{z}{t} \leq 0.02:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot \left(y - x\right)}{t}\\ \end{array} \]

Alternative 4
Accuracy	64.6%
Cost	1424

\[\begin{array}{l} t_1 := \frac{y}{\frac{t}{z}}\\ t_2 := \frac{-x}{\frac{t}{z}}\\ \mathbf{if}\;\frac{z}{t} \leq -5 \cdot 10^{+75}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\frac{z}{t} \leq -1 \cdot 10^{-82}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{z}{t} \leq 10^{-38}:\\ \;\;\;\;x\\ \mathbf{elif}\;\frac{z}{t} \leq 2 \cdot 10^{+84}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 5
Accuracy	64.5%
Cost	1424

\[\begin{array}{l} t_1 := \frac{y}{\frac{t}{z}}\\ \mathbf{if}\;\frac{z}{t} \leq -5 \cdot 10^{+75}:\\ \;\;\;\;\frac{z}{t} \cdot \left(-x\right)\\ \mathbf{elif}\;\frac{z}{t} \leq -1 \cdot 10^{-82}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{z}{t} \leq 10^{-38}:\\ \;\;\;\;x\\ \mathbf{elif}\;\frac{z}{t} \leq 2 \cdot 10^{+84}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{\frac{t}{z}}\\ \end{array} \]

Alternative 6
Accuracy	96.7%
Cost	1097

\[\begin{array}{l} \mathbf{if}\;\frac{z}{t} \leq -5 \cdot 10^{-78} \lor \neg \left(\frac{z}{t} \leq 5 \cdot 10^{-218}\right):\\ \;\;\;\;x + \frac{z}{t} \cdot \left(y - x\right)\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \frac{y}{t}\\ \end{array} \]

Alternative 7
Accuracy	78.5%
Cost	969

\[\begin{array}{l} \mathbf{if}\;\frac{z}{t} \leq -5 \cdot 10^{-19} \lor \neg \left(\frac{z}{t} \leq 10^{-12}\right):\\ \;\;\;\;z \cdot \frac{y - x}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 8
Accuracy	90.8%
Cost	969

\[\begin{array}{l} \mathbf{if}\;\frac{z}{t} \leq -40000 \lor \neg \left(\frac{z}{t} \leq 0.02\right):\\ \;\;\;\;z \cdot \frac{y - x}{t}\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \frac{y}{t}\\ \end{array} \]

Alternative 9
Accuracy	64.4%
Cost	841

\[\begin{array}{l} \mathbf{if}\;\frac{z}{t} \leq -1 \cdot 10^{-82} \lor \neg \left(\frac{z}{t} \leq 10^{-38}\right):\\ \;\;\;\;\frac{z}{t} \cdot y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 10
Accuracy	64.5%
Cost	841

\[\begin{array}{l} \mathbf{if}\;\frac{z}{t} \leq -1 \cdot 10^{-82} \lor \neg \left(\frac{z}{t} \leq 10^{-38}\right):\\ \;\;\;\;\frac{y}{\frac{t}{z}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 11
Accuracy	49.8%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023141 
(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))))