Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTicks from plot-0.2.3.4, A

Average Error: 16.92% → 1.4%

Time: 10.5s

Precision: binary64

Cost: 7620

Math

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

↓

\[\begin{array}{l} t_1 := \frac{y \cdot \left(z - t\right)}{z - a}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{z - a}, x\right)\\ \mathbf{elif}\;t_1 \leq 3.2 \cdot 10^{-64}:\\ \;\;\;\;t_1 + x\\ \mathbf{else}:\\ \;\;\;\;x + \left(z - t\right) \cdot \frac{y}{z - a}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (* y (- z t)) (- z a))))
   (if (<= t_1 (- INFINITY))
     (fma y (/ (- z t) (- z a)) x)
     (if (<= t_1 3.2e-64) (+ t_1 x) (+ x (* (- z t) (/ y (- z a))))))))

C

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

↓

double code(double x, double y, double z, double t, double a) {
	double t_1 = (y * (z - t)) / (z - a);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma(y, ((z - t) / (z - a)), x);
	} else if (t_1 <= 3.2e-64) {
		tmp = t_1 + x;
	} else {
		tmp = x + ((z - t) * (y / (z - a)));
	}
	return tmp;
}

Julia

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

↓

function code(x, y, z, t, a)
	t_1 = Float64(Float64(y * Float64(z - t)) / Float64(z - a))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = fma(y, Float64(Float64(z - t) / Float64(z - a)), x);
	elseif (t_1 <= 3.2e-64)
		tmp = Float64(t_1 + x);
	else
		tmp = Float64(x + Float64(Float64(z - t) * Float64(y / Float64(z - a))));
	end
	return tmp
end

Wolfram

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

↓

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(y * N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 3.2e-64], N[(t$95$1 + x), $MachinePrecision], N[(x + N[(N[(z - t), $MachinePrecision] * N[(y / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	16.92%
Target	2.13%
Herbie	1.4%

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

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 y (-.f64 z t)) (-.f64 z a)) < -inf.0

   1. Initial program 100

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

   2. Simplified 0.18

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

      [Start] 100	\[ x + \frac{y \cdot \left(z - t\right)}{z - a} \]
      +-commutative [=>] 100	\[ \color{blue}{\frac{y \cdot \left(z - t\right)}{z - a} + x} \]
      associate-*r/ [<=] 0.18	\[ \color{blue}{y \cdot \frac{z - t}{z - a}} + x \]
      fma-def [=>] 0.18	\[ \color{blue}{\mathsf{fma}\left(y, \frac{z - t}{z - a}, x\right)} \]

   if -inf.0 < (/.f64 (*.f64 y (-.f64 z t)) (-.f64 z a)) < 3.19999999999999975e-64

   1. Initial program 0.5

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

   if 3.19999999999999975e-64 < (/.f64 (*.f64 y (-.f64 z t)) (-.f64 z a))

   1. Initial program 29.53

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

   2. Simplified 3.75

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

      [Start] 29.53	\[ x + \frac{y \cdot \left(z - t\right)}{z - a} \]
      associate-*l/ [<=] 3.75	\[ x + \color{blue}{\frac{y}{z - a} \cdot \left(z - t\right)} \]

3. Recombined 3 regimes into one program.

4. Final simplification 1.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z - t\right)}{z - a} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{z - a}, x\right)\\ \mathbf{elif}\;\frac{y \cdot \left(z - t\right)}{z - a} \leq 3.2 \cdot 10^{-64}:\\ \;\;\;\;\frac{y \cdot \left(z - t\right)}{z - a} + x\\ \mathbf{else}:\\ \;\;\;\;x + \left(z - t\right) \cdot \frac{y}{z - a}\\ \end{array} \]

Alternatives

Alternative 1
Error	1.39%
Cost	1992

\[\begin{array}{l} t_1 := \frac{y \cdot \left(z - t\right)}{z - a}\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{+307}:\\ \;\;\;\;x + \frac{y}{\frac{z - a}{z - t}}\\ \mathbf{elif}\;t_1 \leq 3.2 \cdot 10^{-64}:\\ \;\;\;\;t_1 + x\\ \mathbf{else}:\\ \;\;\;\;x + \left(z - t\right) \cdot \frac{y}{z - a}\\ \end{array} \]

Alternative 2
Error	18.5%
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -1350000 \lor \neg \left(z \leq 2.4 \cdot 10^{-70}\right):\\ \;\;\;\;x + y \cdot \frac{z - t}{z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{t}{\frac{a}{y}}\\ \end{array} \]

Alternative 3
Error	16.01%
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -4.2 \cdot 10^{-99} \lor \neg \left(z \leq 5.8 \cdot 10^{-43}\right):\\ \;\;\;\;x + \frac{y}{\frac{z - a}{z}}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{t}{\frac{a}{y}}\\ \end{array} \]

Alternative 4
Error	13.47%
Cost	841

\[\begin{array}{l} \mathbf{if}\;t \leq -4.9 \cdot 10^{-20} \lor \neg \left(t \leq 9.2 \cdot 10^{+90}\right):\\ \;\;\;\;x - t \cdot \frac{y}{z - a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{z - a}{z}}\\ \end{array} \]

Alternative 5
Error	2.23%
Cost	836

\[\begin{array}{l} \mathbf{if}\;t \leq -1.2 \cdot 10^{-103}:\\ \;\;\;\;x + \left(z - t\right) \cdot \frac{y}{z - a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{z - a}{z - t}}\\ \end{array} \]

Alternative 6
Error	22.59%
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -230000 \lor \neg \left(z \leq 7400000000000\right):\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{t}{a}\\ \end{array} \]

Alternative 7
Error	22.46%
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -4200000 \lor \neg \left(z \leq 30000000000\right):\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;x + t \cdot \frac{y}{a}\\ \end{array} \]

Alternative 8
Error	22.44%
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -390000:\\ \;\;\;\;y + x\\ \mathbf{elif}\;z \leq 29000000000000:\\ \;\;\;\;x + \frac{t}{\frac{a}{y}}\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]

Alternative 9
Error	22.38%
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -125000:\\ \;\;\;\;y + x\\ \mathbf{elif}\;z \leq 6600000000:\\ \;\;\;\;x + \frac{y}{\frac{a}{t}}\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]

Alternative 10
Error	4.38%
Cost	704

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

Alternative 11
Error	31.46%
Cost	456

\[\begin{array}{l} \mathbf{if}\;z \leq -115000:\\ \;\;\;\;y + x\\ \mathbf{elif}\;z \leq 8200000000:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]

Alternative 12
Error	45.46%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023088 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTicks from plot-0.2.3.4, A"
  :precision binary64

  :herbie-target
  (+ x (/ y (/ (- z a) (- z t))))

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