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

Average Error: 2.17% → 1.62%

Time: 31.1s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := \mathsf{fma}\left(y, \frac{z - t}{a - t}, x\right)\\ t_2 := z - t \ne 0\\ \mathbf{if}\;y \leq -5.1 \cdot 10^{+46}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;t_2:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;t_2:\\ \;\;\;\;\frac{y}{a - t} \cdot \left(z - t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{NaN}\\ \end{array} + x\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array}\\ \mathbf{elif}\;y \leq 0.002:\\ \;\;\;\;\frac{\left(z - t\right) \cdot y}{a - t} + x\\ \mathbf{elif}\;t_2:\\ \;\;\;\;\frac{y}{\frac{t - a}{t - z}} + x\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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 (fma y (/ (- z t) (- a t)) x)) (t_2 (!= (- z t) 0.0)))
   (if (<= y -5.1e+46)
     (if t_2 (+ (if t_2 (* (/ y (- a t)) (- z t)) NAN) x) t_1)
     (if (<= y 0.002)
       (+ (/ (* (- z t) y) (- a t)) x)
       (if t_2 (+ (/ y (/ (- t a) (- t z))) x) t_1)))))

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 = fma(y, ((z - t) / (a - t)), x);
	int t_2 = (z - t) != 0.0;
	double tmp_3;
	if (y <= -5.1e+46) {
		double tmp_5;
		if (t_2) {
			double tmp_6;
			if (t_2) {
				tmp_6 = (y / (a - t)) * (z - t);
			} else {
				tmp_6 = (double) NAN;
			}
			tmp_5 = tmp_6 + x;
		} else {
			tmp_5 = t_1;
		}
		tmp_3 = tmp_5;
	} else if (y <= 0.002) {
		tmp_3 = (((z - t) * y) / (a - t)) + x;
	} else if (t_2) {
		tmp_3 = (y / ((t - a) / (t - z))) + x;
	} else {
		tmp_3 = t_1;
	}
	return tmp_3;
}

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 = fma(y, Float64(Float64(z - t) / Float64(a - t)), x)
	t_2 = Float64(z - t) != 0.0
	tmp_3 = 0.0
	if (y <= -5.1e+46)
		tmp_5 = 0.0
		if (t_2)
			tmp_6 = 0.0
			if (t_2)
				tmp_6 = Float64(Float64(y / Float64(a - t)) * Float64(z - t));
			else
				tmp_6 = NaN;
			end
			tmp_5 = Float64(tmp_6 + x);
		else
			tmp_5 = t_1;
		end
		tmp_3 = tmp_5;
	elseif (y <= 0.002)
		tmp_3 = Float64(Float64(Float64(Float64(z - t) * y) / Float64(a - t)) + x);
	elseif (t_2)
		tmp_3 = Float64(Float64(y / Float64(Float64(t - a) / Float64(t - z))) + x);
	else
		tmp_3 = t_1;
	end
	return tmp_3
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[(y * N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, Block[{t$95$2 = Unequal[N[(z - t), $MachinePrecision], 0.0]}, If[LessEqual[y, -5.1e+46], If[t$95$2, N[(If[t$95$2, N[(N[(y / N[(a - t), $MachinePrecision]), $MachinePrecision] * N[(z - t), $MachinePrecision]), $MachinePrecision], Indeterminate] + x), $MachinePrecision], t$95$1], If[LessEqual[y, 0.002], N[(N[(N[(N[(z - t), $MachinePrecision] * y), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], If[t$95$2, N[(N[(y / N[(N[(t - a), $MachinePrecision] / N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]]]

Target

Original	2.17%
Target	0.64%
Herbie	1.62%

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

Derivation

1. Split input into 3 regimes

2. if y < -5.0999999999999997e46

   1. Initial program 0.81

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

   2. Simplified 0.81

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

   3. Applied egg-rr 0.98

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;z - t \ne 0:\\ \;\;\;\;\frac{y}{\frac{t - a}{t - z}} + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{a - t}, x\right)\\ } \end{array}} \]

   4. Applied egg-rr 4.61

      \[\leadsto \begin{array}{l} \mathbf{if}\;z - t \ne 0:\\ \;\;\;\;\color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;z - t \ne 0:\\ \;\;\;\;\frac{y}{a - t} \cdot \left(z - t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{NaN}\\ } \end{array}} + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{a - t}, x\right)\\ \end{array} \]

   if -5.0999999999999997e46 < y < 2e-3

   1. Initial program 3.15

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

   2. Simplified 3.15

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

   3. Taylor expanded in y around -inf 0.85

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

   if 2e-3 < y

   1. Initial program 0.84

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

   2. Simplified 0.84

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

   3. Applied egg-rr 1.03

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;z - t \ne 0:\\ \;\;\;\;\frac{y}{\frac{t - a}{t - z}} + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{a - t}, x\right)\\ } \end{array}} \]
3. Recombined 3 regimes into one program.

Reproduce

herbie shell --seed 2023136 
(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)))))