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

Average Error: 1.3 → 1.5

Time: 14.4s

Precision: binary64

Cost: 14600

Math

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

↓

\[\begin{array}{l} t_1 := \frac{z - t}{a - t}\\ t_2 := \mathsf{fma}\left(z - t, \frac{y}{a - t}, x\right)\\ \mathbf{if}\;t_1 \leq 2 \cdot 10^{-15}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 1:\\ \;\;\;\;x + y \cdot {\left(\sqrt[3]{t_1}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- a t))) (t_2 (fma (- z t) (/ y (- a t)) x)))
   (if (<= t_1 2e-15)
     t_2
     (if (<= t_1 1.0) (+ x (* y (pow (cbrt t_1) 3.0))) t_2))))

C

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

↓

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (a - t);
	double t_2 = fma((z - t), (y / (a - t)), x);
	double tmp;
	if (t_1 <= 2e-15) {
		tmp = t_2;
	} else if (t_1 <= 1.0) {
		tmp = x + (y * pow(cbrt(t_1), 3.0));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

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

↓

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(a - t))
	t_2 = fma(Float64(z - t), Float64(y / Float64(a - t)), x)
	tmp = 0.0
	if (t_1 <= 2e-15)
		tmp = t_2;
	elseif (t_1 <= 1.0)
		tmp = Float64(x + Float64(y * (cbrt(t_1) ^ 3.0)));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

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

↓

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(z - t), $MachinePrecision] * N[(y / N[(a - t), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t$95$1, 2e-15], t$95$2, If[LessEqual[t$95$1, 1.0], N[(x + N[(y * N[Power[N[Power[t$95$1, 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Target

Original	1.3
Target	0.4
Herbie	1.5

\[\begin{array}{l} \mathbf{if}\;y < -8.508084860551241 \cdot 10^{-17}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;y < 2.894426862792089 \cdot 10^{-49}:\\ \;\;\;\;x + \left(y \cdot \left(z - t\right)\right) \cdot \frac{1}{a - t}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a - t}\\ \end{array} \]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 z t) (-.f64 a t)) < 2.0000000000000002e-15 or 1 < (/.f64 (-.f64 z t) (-.f64 a t))

   1. Initial program 2.0

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

   2. Simplified 2.4

      \[\leadsto \color{blue}{\mathsf{fma}\left(z - t, \frac{y}{a - t}, x\right)} \]
      Proof
      (fma.f64 (-.f64 z t) (/.f64 y (-.f64 a t)) x): 0 points increase in error, 0 points decrease in error
      (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (-.f64 z t) (/.f64 y (-.f64 a t))) x)): 2 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 y (-.f64 a t)) (-.f64 z t))) x): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 y (-.f64 z t)) (-.f64 a t))) x): 50 points increase in error, 15 points decrease in error
      (+.f64 (Rewrite<= associate-*r/_binary64 (*.f64 y (/.f64 (-.f64 z t) (-.f64 a t)))) x): 20 points increase in error, 51 points decrease in error
      (Rewrite<= +-commutative_binary64 (+.f64 x (*.f64 y (/.f64 (-.f64 z t) (-.f64 a t))))): 0 points increase in error, 0 points decrease in error

   if 2.0000000000000002e-15 < (/.f64 (-.f64 z t) (-.f64 a t)) < 1

   1. Initial program 0.0

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

   2. Applied egg-rr 0.0

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

4. Final simplification 1.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{a - t} \leq 2 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(z - t, \frac{y}{a - t}, x\right)\\ \mathbf{elif}\;\frac{z - t}{a - t} \leq 1:\\ \;\;\;\;x + y \cdot {\left(\sqrt[3]{\frac{z - t}{a - t}}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z - t, \frac{y}{a - t}, x\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	1.4
Cost	8008

\[\begin{array}{l} t_1 := \frac{z - t}{a - t}\\ t_2 := \mathsf{fma}\left(z - t, \frac{y}{a - t}, x\right)\\ \mathbf{if}\;t_1 \leq 4 \cdot 10^{-66}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 1:\\ \;\;\;\;x + t_1 \cdot y\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 2
Error	0.2
Cost	1736

\[\begin{array}{l} t_1 := \frac{z - t}{a - t}\\ t_2 := x + \frac{z}{\left(a - t\right) \cdot \frac{1}{y}}\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{+213}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+266}:\\ \;\;\;\;x + t_1 \cdot y\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Error	14.5
Cost	1108

\[\begin{array}{l} t_1 := x + \frac{z}{\frac{a}{y}}\\ \mathbf{if}\;t \leq -1.2648237186566223 \cdot 10^{-54}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;t \leq 5.582685282189044 \cdot 10^{-90}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2.4950669394191283 \cdot 10^{-53}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;t \leq 1.3665214784922365 \cdot 10^{-40}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.6127033775985273 \cdot 10^{+39}:\\ \;\;\;\;x - y \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]

Alternative 4
Error	14.5
Cost	1108

\[\begin{array}{l} t_1 := x + z \cdot \frac{y}{a}\\ \mathbf{if}\;t \leq -1.2648237186566223 \cdot 10^{-54}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;t \leq 5.582685282189044 \cdot 10^{-90}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2.4950669394191283 \cdot 10^{-53}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;t \leq 1.3665214784922365 \cdot 10^{-40}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.6127033775985273 \cdot 10^{+39}:\\ \;\;\;\;x - y \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]

Alternative 5
Error	7.7
Cost	968

\[\begin{array}{l} \mathbf{if}\;z \leq -84813552095543460:\\ \;\;\;\;x + \frac{z}{\frac{a - t}{y}}\\ \mathbf{elif}\;z \leq 2.1044262832077266 \cdot 10^{+86}:\\ \;\;\;\;x - \frac{y}{\frac{a}{t} + -1}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{z}{\left(a - t\right) \cdot \frac{1}{y}}\\ \end{array} \]

Alternative 6
Error	10.9
Cost	840

\[\begin{array}{l} \mathbf{if}\;t \leq -8.169849564537166 \cdot 10^{-11}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;t \leq 1.4603977639294056 \cdot 10^{+40}:\\ \;\;\;\;x + \frac{z}{\frac{a - t}{y}}\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]

Alternative 7
Error	8.4
Cost	840

\[\begin{array}{l} t_1 := x + \left(y - y \cdot \frac{z}{t}\right)\\ \mathbf{if}\;t \leq -8.169849564537166 \cdot 10^{-11}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 7.379609680963355 \cdot 10^{-12}:\\ \;\;\;\;x + \frac{z}{\frac{a - t}{y}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	7.7
Cost	840

\[\begin{array}{l} t_1 := x + \frac{z}{\frac{a - t}{y}}\\ \mathbf{if}\;z \leq -84813552095543460:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.1044262832077266 \cdot 10^{+86}:\\ \;\;\;\;x - \frac{y}{\frac{a}{t} + -1}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	14.4
Cost	712

\[\begin{array}{l} \mathbf{if}\;t \leq -1.2648237186566223 \cdot 10^{-54}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;t \leq 5.582685282189044 \cdot 10^{-90}:\\ \;\;\;\;x + \frac{z}{\frac{a}{y}}\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]

Alternative 10
Error	19.4
Cost	456

\[\begin{array}{l} \mathbf{if}\;t \leq -5.131901334223244 \cdot 10^{-97}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;t \leq 5.582685282189044 \cdot 10^{-90}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]

Alternative 11
Error	28.9
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022316 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisLine from plot-0.2.3.4, B"
  :precision binary64

  :herbie-target
  (if (< y -8.508084860551241e-17) (+ x (* y (/ (- z t) (- a t)))) (if (< y 2.894426862792089e-49) (+ x (* (* y (- z t)) (/ 1.0 (- a t)))) (+ x (* y (/ (- z t) (- a t))))))

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