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

Average Error: 1.9 → 1.4

Time: 14.3s

Precision: binary64

Cost: 8780

Math

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

↓

\[\begin{array}{l} t_1 := x + \frac{\left(y - x\right) \cdot z}{t}\\ t_2 := x + \left(y - x\right) \cdot \frac{z}{t}\\ \mathbf{if}\;t_2 \leq -2 \cdot 10^{-199}:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \mathbf{elif}\;t_2 \leq 10^{-140}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_2 \leq 2 \cdot 10^{+287}:\\ \;\;\;\;\mathsf{fma}\left(y - x, \frac{z}{t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ x (/ (* (- y x) z) t))) (t_2 (+ x (* (- y x) (/ z t)))))
   (if (<= t_2 -2e-199)
     (+ x (/ (- y x) (/ t z)))
     (if (<= t_2 1e-140)
       t_1
       (if (<= t_2 2e+287) (fma (- y x) (/ z t) x) t_1)))))

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 t_1 = x + (((y - x) * z) / t);
	double t_2 = x + ((y - x) * (z / t));
	double tmp;
	if (t_2 <= -2e-199) {
		tmp = x + ((y - x) / (t / z));
	} else if (t_2 <= 1e-140) {
		tmp = t_1;
	} else if (t_2 <= 2e+287) {
		tmp = fma((y - x), (z / t), x);
	} else {
		tmp = t_1;
	}
	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)
	t_1 = Float64(x + Float64(Float64(Float64(y - x) * z) / t))
	t_2 = Float64(x + Float64(Float64(y - x) * Float64(z / t)))
	tmp = 0.0
	if (t_2 <= -2e-199)
		tmp = Float64(x + Float64(Float64(y - x) / Float64(t / z)));
	elseif (t_2 <= 1e-140)
		tmp = t_1;
	elseif (t_2 <= 2e+287)
		tmp = fma(Float64(y - x), Float64(z / t), x);
	else
		tmp = t_1;
	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_] := Block[{t$95$1 = N[(x + N[(N[(N[(y - x), $MachinePrecision] * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x + N[(N[(y - x), $MachinePrecision] * N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e-199], N[(x + N[(N[(y - x), $MachinePrecision] / N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e-140], t$95$1, If[LessEqual[t$95$2, 2e+287], N[(N[(y - x), $MachinePrecision] * N[(z / t), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]]]

Target

Original	1.9
Target	2.2
Herbie	1.4

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 1.5

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

   2. Applied egg-rr 1.5

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

   if -1.99999999999999996e-199 < (+.f64 x (*.f64 (-.f64 y x) (/.f64 z t))) < 9.9999999999999998e-141 or 2.0000000000000002e287 < (+.f64 x (*.f64 (-.f64 y x) (/.f64 z t)))

   1. Initial program 9.0

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

   2. Applied egg-rr 8.3

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

   3. Taylor expanded in t around 0 5.3

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

   if 9.9999999999999998e-141 < (+.f64 x (*.f64 (-.f64 y x) (/.f64 z t))) < 2.0000000000000002e287

   1. Initial program 0.2

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

   2. Simplified 0.2

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z}{t}, x\right)} \]
      Proof
      (fma.f64 (-.f64 y x) (/.f64 z t) x): 0 points increase in error, 0 points decrease in error
      (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (-.f64 y x) (/.f64 z t)) x)): 4 points increase in error, 1 points decrease in error
      (Rewrite<= +-commutative_binary64 (+.f64 x (*.f64 (-.f64 y x) (/.f64 z t)))): 0 points increase in error, 0 points decrease in error
3. Recombined 3 regimes into one program.

4. Final simplification 1.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \left(y - x\right) \cdot \frac{z}{t} \leq -2 \cdot 10^{-199}:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \mathbf{elif}\;x + \left(y - x\right) \cdot \frac{z}{t} \leq 10^{-140}:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot z}{t}\\ \mathbf{elif}\;x + \left(y - x\right) \cdot \frac{z}{t} \leq 2 \cdot 10^{+287}:\\ \;\;\;\;\mathsf{fma}\left(y - x, \frac{z}{t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot z}{t}\\ \end{array} \]

Alternatives

Alternative 1
Error	21.9
Cost	2984

\[\begin{array}{l} t_1 := \frac{y}{\frac{t}{z}}\\ t_2 := x \cdot \frac{-z}{t}\\ \mathbf{if}\;\frac{z}{t} \leq -4 \cdot 10^{+275}:\\ \;\;\;\;\frac{z}{\frac{-t}{x}}\\ \mathbf{elif}\;\frac{z}{t} \leq -2 \cdot 10^{+62}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{z}{t} \leq -5 \cdot 10^{+17}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\frac{z}{t} \leq -4 \cdot 10^{-11}:\\ \;\;\;\;y \cdot \frac{z}{t}\\ \mathbf{elif}\;\frac{z}{t} \leq -1 \cdot 10^{-96}:\\ \;\;\;\;x\\ \mathbf{elif}\;\frac{z}{t} \leq -5 \cdot 10^{-135}:\\ \;\;\;\;\frac{z}{\frac{t}{y}}\\ \mathbf{elif}\;\frac{z}{t} \leq 3.9 \cdot 10^{-40}:\\ \;\;\;\;x\\ \mathbf{elif}\;\frac{z}{t} \leq 4 \cdot 10^{+41}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{z}{t} \leq 2 \cdot 10^{+109}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\frac{z}{t} \leq 10^{+141}:\\ \;\;\;\;\frac{y \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(-z\right)}{t}\\ \end{array} \]

Alternative 2
Error	1.4
Cost	2508

\[\begin{array}{l} t_1 := x + \frac{\left(y - x\right) \cdot z}{t}\\ t_2 := x + \left(y - x\right) \cdot \frac{z}{t}\\ \mathbf{if}\;t_2 \leq -2 \cdot 10^{-199}:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \mathbf{elif}\;t_2 \leq 10^{-140}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_2 \leq 2 \cdot 10^{+287}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	22.2
Cost	2464

\[\begin{array}{l} t_1 := \frac{y}{\frac{t}{z}}\\ t_2 := \frac{x \cdot \left(-z\right)}{t}\\ \mathbf{if}\;\frac{z}{t} \leq -4 \cdot 10^{+275}:\\ \;\;\;\;\frac{z}{\frac{-t}{x}}\\ \mathbf{elif}\;\frac{z}{t} \leq -4 \cdot 10^{-11}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{z}{t} \leq -1 \cdot 10^{-96}:\\ \;\;\;\;x\\ \mathbf{elif}\;\frac{z}{t} \leq -5 \cdot 10^{-135}:\\ \;\;\;\;\frac{z}{\frac{t}{y}}\\ \mathbf{elif}\;\frac{z}{t} \leq 3.9 \cdot 10^{-40}:\\ \;\;\;\;x\\ \mathbf{elif}\;\frac{z}{t} \leq 4 \cdot 10^{+41}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{z}{t} \leq 2 \cdot 10^{+109}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\frac{z}{t} \leq 10^{+141}:\\ \;\;\;\;\frac{y \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 4
Error	22.4
Cost	2204

\[\begin{array}{l} t_1 := \frac{y}{\frac{t}{z}}\\ t_2 := \frac{z}{\frac{-t}{x}}\\ \mathbf{if}\;\frac{z}{t} \leq -4 \cdot 10^{+275}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\frac{z}{t} \leq -4 \cdot 10^{-11}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{z}{t} \leq -1 \cdot 10^{-96}:\\ \;\;\;\;x\\ \mathbf{elif}\;\frac{z}{t} \leq -5 \cdot 10^{-135}:\\ \;\;\;\;\frac{z}{\frac{t}{y}}\\ \mathbf{elif}\;\frac{z}{t} \leq 3.9 \cdot 10^{-40}:\\ \;\;\;\;x\\ \mathbf{elif}\;\frac{z}{t} \leq 4 \cdot 10^{+41}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{z}{t} \leq 2 \cdot 10^{+109}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot z}{t}\\ \end{array} \]

Alternative 5
Error	14.1
Cost	2008

\[\begin{array}{l} t_1 := x \cdot \left(1 - \frac{z}{t}\right)\\ t_2 := \frac{z}{\frac{t}{y - x}}\\ \mathbf{if}\;\frac{z}{t} \leq -5000:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\frac{z}{t} \leq -1 \cdot 10^{-96}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{z}{t} \leq -5 \cdot 10^{-135}:\\ \;\;\;\;\frac{z}{\frac{t}{y}}\\ \mathbf{elif}\;\frac{z}{t} \leq 3.9 \cdot 10^{-40}:\\ \;\;\;\;x\\ \mathbf{elif}\;\frac{z}{t} \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\frac{y}{\frac{t}{z}}\\ \mathbf{elif}\;\frac{z}{t} \leq 20000000:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 6
Error	22.4
Cost	1360

\[\begin{array}{l} t_1 := \frac{y}{\frac{t}{z}}\\ \mathbf{if}\;\frac{z}{t} \leq -4 \cdot 10^{-11}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{z}{t} \leq -1 \cdot 10^{-96}:\\ \;\;\;\;x\\ \mathbf{elif}\;\frac{z}{t} \leq -5 \cdot 10^{-135}:\\ \;\;\;\;\frac{z}{\frac{t}{y}}\\ \mathbf{elif}\;\frac{z}{t} \leq 3.9 \cdot 10^{-40}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	1.5
Cost	1356

\[\begin{array}{l} t_1 := x + \frac{y - x}{\frac{t}{z}}\\ \mathbf{if}\;\frac{z}{t} \leq -2 \cdot 10^{+238}:\\ \;\;\;\;\frac{z}{\frac{t}{y - x}}\\ \mathbf{elif}\;\frac{z}{t} \leq -5 \cdot 10^{-27}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{z}{t} \leq 0:\\ \;\;\;\;x + z \cdot \frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	18.7
Cost	976

\[\begin{array}{l} t_1 := x \cdot \left(1 - \frac{z}{t}\right)\\ \mathbf{if}\;x \leq -7.79245494227672 \cdot 10^{-106}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 4.743296979692854 \cdot 10^{-161}:\\ \;\;\;\;\frac{z}{\frac{t}{y}}\\ \mathbf{elif}\;x \leq 1.7271485835444998 \cdot 10^{-92}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.1130836454440602 \cdot 10^{-56}:\\ \;\;\;\;\frac{y}{\frac{t}{z}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	5.9
Cost	968

\[\begin{array}{l} t_1 := \frac{z}{\frac{t}{y - x}}\\ \mathbf{if}\;\frac{z}{t} \leq -5000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{z}{t} \leq 5 \cdot 10^{-17}:\\ \;\;\;\;x + z \cdot \frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	6.4
Cost	968

\[\begin{array}{l} t_1 := \frac{\left(y - x\right) \cdot z}{t}\\ \mathbf{if}\;\frac{z}{t} \leq -500:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{z}{t} \leq 5 \cdot 10^{-18}:\\ \;\;\;\;x + z \cdot \frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Error	6.4
Cost	968

\[\begin{array}{l} \mathbf{if}\;\frac{z}{t} \leq -5000:\\ \;\;\;\;\frac{z}{\frac{t}{y - x}}\\ \mathbf{elif}\;\frac{z}{t} \leq 5 \cdot 10^{-17}:\\ \;\;\;\;x + \frac{y \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(y - x\right) \cdot z}{t}\\ \end{array} \]

Alternative 12
Error	5.4
Cost	708

\[\begin{array}{l} \mathbf{if}\;t \leq 1.1858974925383363 \cdot 10^{+64}:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \frac{y}{t}\\ \end{array} \]

Alternative 13
Error	26.7
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -2.1800384038544185 \cdot 10^{-42}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 4.743296979692854 \cdot 10^{-161}:\\ \;\;\;\;\frac{z}{\frac{t}{y}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 14
Error	31.7
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022316 
(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))))