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

Average Error: 11.4 → 0.9

Time: 19.2s

Precision: binary64

Cost: 7240

Math

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

↓

\[\begin{array}{l} t_1 := \mathsf{fma}\left(y, \frac{z - t}{z - a}, x\right)\\ \mathbf{if}\;y \leq -2.6 \cdot 10^{+45}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 10^{-161}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{z - a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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 (fma y (/ (- z t) (- z a)) x)))
   (if (<= y -2.6e+45)
     t_1
     (if (<= y 1e-161) (+ x (/ (* y (- z t)) (- z a))) t_1))))

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 = fma(y, ((z - t) / (z - a)), x);
	double tmp;
	if (y <= -2.6e+45) {
		tmp = t_1;
	} else if (y <= 1e-161) {
		tmp = x + ((y * (z - t)) / (z - a));
	} else {
		tmp = t_1;
	}
	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 = fma(y, Float64(Float64(z - t) / Float64(z - a)), x)
	tmp = 0.0
	if (y <= -2.6e+45)
		tmp = t_1;
	elseif (y <= 1e-161)
		tmp = Float64(x + Float64(Float64(y * Float64(z - t)) / Float64(z - a)));
	else
		tmp = t_1;
	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[(y * N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[y, -2.6e+45], t$95$1, If[LessEqual[y, 1e-161], N[(x + N[(N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Target

Original	11.4
Target	1.2
Herbie	0.9

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

Derivation

1. Split input into 2 regimes

2. if y < -2.60000000000000007e45 or 1.00000000000000003e-161 < y

   1. Initial program 19.3

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

   2. Simplified 0.9

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

   if -2.60000000000000007e45 < y < 1.00000000000000003e-161

   1. Initial program 0.8

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

Alternatives

Alternative 1
Error	9.9
Cost	1168

\[\begin{array}{l} t_1 := x + \frac{y}{z - a} \cdot z\\ \mathbf{if}\;z \leq -2.8 \cdot 10^{-51}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.18 \cdot 10^{-147}:\\ \;\;\;\;x + t \cdot \frac{y}{a - z}\\ \mathbf{elif}\;z \leq 7 \cdot 10^{-11}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.65 \cdot 10^{+44}:\\ \;\;\;\;x + \frac{-y \cdot t}{z - a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{z - t}{z} \cdot y\\ \end{array} \]

Alternative 2
Error	17.0
Cost	976

\[\begin{array}{l} t_1 := x + \frac{t}{a} \cdot y\\ \mathbf{if}\;a \leq -2 \cdot 10^{+109}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -3.6 \cdot 10^{-272}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;a \leq 5.1 \cdot 10^{-279}:\\ \;\;\;\;x + \left(-\frac{t}{z} \cdot y\right)\\ \mathbf{elif}\;a \leq 1.9 \cdot 10^{-57}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	1.2
Cost	968

\[\begin{array}{l} t_1 := x + \frac{z - t}{z - a} \cdot y\\ \mathbf{if}\;y \leq -5 \cdot 10^{+81}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 2 \cdot 10^{-164}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{z - a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	10.6
Cost	840

\[\begin{array}{l} \mathbf{if}\;z \leq -9.2 \cdot 10^{+17}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{+39}:\\ \;\;\;\;x + t \cdot \frac{y}{a - z}\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]

Alternative 5
Error	9.6
Cost	840

\[\begin{array}{l} t_1 := x + \frac{y}{z} \cdot \left(z - t\right)\\ \mathbf{if}\;z \leq -3.7 \cdot 10^{+19}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.8 \cdot 10^{-30}:\\ \;\;\;\;x + t \cdot \frac{y}{a - z}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	9.7
Cost	840

\[\begin{array}{l} \mathbf{if}\;z \leq -1.16 \cdot 10^{-46}:\\ \;\;\;\;x + \frac{y}{z - a} \cdot z\\ \mathbf{elif}\;z \leq 1.45 \cdot 10^{-36}:\\ \;\;\;\;x + t \cdot \frac{y}{a - z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{z} \cdot \left(z - t\right)\\ \end{array} \]

Alternative 7
Error	9.0
Cost	840

\[\begin{array}{l} \mathbf{if}\;z \leq -3.15 \cdot 10^{-46}:\\ \;\;\;\;x + \frac{y}{z - a} \cdot z\\ \mathbf{elif}\;z \leq 4 \cdot 10^{-35}:\\ \;\;\;\;x + t \cdot \frac{y}{a - z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{z - t}{z} \cdot y\\ \end{array} \]

Alternative 8
Error	14.5
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -3.5 \cdot 10^{+18}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{-88}:\\ \;\;\;\;x + \frac{t}{a} \cdot y\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]

Alternative 9
Error	14.5
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -660000000000:\\ \;\;\;\;y + x\\ \mathbf{elif}\;z \leq 6.2 \cdot 10^{-84}:\\ \;\;\;\;x + \frac{y}{a} \cdot t\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]

Alternative 10
Error	3.0
Cost	704

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

Alternative 11
Error	1.3
Cost	704

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

Alternative 12
Error	20.0
Cost	456

\[\begin{array}{l} \mathbf{if}\;z \leq -2.9 \cdot 10^{-59}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;z \leq 3.65 \cdot 10^{-127}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]

Alternative 13
Error	28.6
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023010 
(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))))