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

Average Accuracy: 83.4% → 99.5%

Time: 10.3s

Precision: binary64

Cost: 7620

Math

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

↓

\[\begin{array}{l} t_1 := \frac{y \cdot \left(z - t\right)}{a - t}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{a - t}, x\right)\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+260}:\\ \;\;\;\;t_1 + x\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a - t}{z - t}}\\ \end{array} \]

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 (/ (* y (- z t)) (- a t))))
   (if (<= t_1 (- INFINITY))
     (fma y (/ (- z t) (- a t)) x)
     (if (<= t_1 2e+260) (+ t_1 x) (+ x (/ y (/ (- a t) (- z t))))))))

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 = (y * (z - t)) / (a - t);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma(y, ((z - t) / (a - t)), x);
	} else if (t_1 <= 2e+260) {
		tmp = t_1 + x;
	} else {
		tmp = x + (y / ((a - t) / (z - t)));
	}
	return tmp;
}

Julia

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

↓

function code(x, y, z, t, a)
	t_1 = Float64(Float64(y * Float64(z - t)) / Float64(a - t))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = fma(y, Float64(Float64(z - t) / Float64(a - t)), x);
	elseif (t_1 <= 2e+260)
		tmp = Float64(t_1 + x);
	else
		tmp = Float64(x + Float64(y / Float64(Float64(a - t) / Float64(z - t))));
	end
	return tmp
end

Wolfram

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

↓

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(y * N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 2e+260], N[(t$95$1 + x), $MachinePrecision], N[(x + N[(y / N[(N[(a - t), $MachinePrecision] / N[(z - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	83.4%
Target	98.1%
Herbie	99.5%

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 0.0%

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

   2. Simplified 99.8%

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

      [Start] 0.0	\[ x + \frac{y \cdot \left(z - t\right)}{a - t} \]
      +-commutative [=>] 0.0	\[ \color{blue}{\frac{y \cdot \left(z - t\right)}{a - t} + x} \]
      associate-*r/ [<=] 99.8	\[ \color{blue}{y \cdot \frac{z - t}{a - t}} + x \]
      fma-def [=>] 99.8	\[ \color{blue}{\mathsf{fma}\left(y, \frac{z - t}{a - t}, x\right)} \]

   if -inf.0 < (/.f64 (*.f64 y (-.f64 z t)) (-.f64 a t)) < 2.00000000000000013e260

   1. Initial program 99.6%

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

   if 2.00000000000000013e260 < (/.f64 (*.f64 y (-.f64 z t)) (-.f64 a t))

   1. Initial program 11.9%

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

   2. Simplified 97.7%

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

      [Start] 11.9	\[ x + \frac{y \cdot \left(z - t\right)}{a - t} \]
      associate-/l* [=>] 97.7	\[ x + \color{blue}{\frac{y}{\frac{a - t}{z - t}}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z - t\right)}{a - t} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{a - t}, x\right)\\ \mathbf{elif}\;\frac{y \cdot \left(z - t\right)}{a - t} \leq 2 \cdot 10^{+260}:\\ \;\;\;\;\frac{y \cdot \left(z - t\right)}{a - t} + x\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a - t}{z - t}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.5%
Cost	1993

\[\begin{array}{l} t_1 := \frac{y \cdot \left(z - t\right)}{a - t}\\ \mathbf{if}\;t_1 \leq -\infty \lor \neg \left(t_1 \leq 2 \cdot 10^{+260}\right):\\ \;\;\;\;x + \frac{y}{\frac{a - t}{z - t}}\\ \mathbf{else}:\\ \;\;\;\;t_1 + x\\ \end{array} \]

Alternative 2
Accuracy	80.5%
Cost	972

\[\begin{array}{l} t_1 := x - t \cdot \frac{y}{a - t}\\ \mathbf{if}\;a \leq -7.2 \cdot 10^{-130}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 2.25 \cdot 10^{-90}:\\ \;\;\;\;x + y \cdot \frac{t - z}{t}\\ \mathbf{elif}\;a \leq 5.5 \cdot 10^{+121}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z}}\\ \end{array} \]

Alternative 3
Accuracy	81.7%
Cost	841

\[\begin{array}{l} \mathbf{if}\;t \leq -9.5 \cdot 10^{-126} \lor \neg \left(t \leq 1.5 \cdot 10^{-106}\right):\\ \;\;\;\;x - t \cdot \frac{y}{a - t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot z}{a}\\ \end{array} \]

Alternative 4
Accuracy	77.5%
Cost	713

\[\begin{array}{l} \mathbf{if}\;t \leq -2.9 \cdot 10^{-69} \lor \neg \left(t \leq 5 \cdot 10^{-20}\right):\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z}}\\ \end{array} \]

Alternative 5
Accuracy	76.7%
Cost	713

\[\begin{array}{l} \mathbf{if}\;t \leq -3.8 \cdot 10^{-75} \lor \neg \left(t \leq 1.6 \cdot 10^{-23}\right):\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot z}{a}\\ \end{array} \]

Alternative 6
Accuracy	95.1%
Cost	704

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

Alternative 7
Accuracy	98.1%
Cost	704

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

Alternative 8
Accuracy	69.0%
Cost	456

\[\begin{array}{l} \mathbf{if}\;t \leq -3 \cdot 10^{-57}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;t \leq 1.5 \cdot 10^{-96}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]

Alternative 9
Accuracy	58.0%
Cost	328

\[\begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{-223}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 8.8 \cdot 10^{-129}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 10
Accuracy	54.7%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023151 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTicks from plot-0.2.3.4, B"
  :precision binary64

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

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