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

Average Accuracy: 97.9% → 97.9%

Time: 14.1s

Precision: binary64

Cost: 6976

Math

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

↓

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

FPCore

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

↓

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

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) {
	return fma(y, ((z - t) / (a - t)), x);
}

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)
	return fma(y, Float64(Float64(z - t) / Float64(a - t)), x)
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_] := N[(y * N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]

Target

Original	97.9%
Target	99.4%
Herbie	97.9%

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

Derivation

1. Initial program 97.9%

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

2. Simplified 97.9%

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

   [Start] 97.9	\[ x + y \cdot \frac{z - t}{a - t} \]
   +-commutative [=>] 97.9	\[ \color{blue}{y \cdot \frac{z - t}{a - t} + x} \]
   fma-def [=>] 97.9	\[ \color{blue}{\mathsf{fma}\left(y, \frac{z - t}{a - t}, x\right)} \]

3. Final simplification 97.9%

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

Alternatives

Alternative 1
Accuracy	67.4%
Cost	1372

\[\begin{array}{l} t_1 := \left(z - t\right) \cdot \frac{y}{a - t}\\ t_2 := x + z \cdot \frac{y}{a}\\ \mathbf{if}\;a \leq -8.5 \cdot 10^{+17}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq -9 \cdot 10^{-140}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 6.2 \cdot 10^{-224}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;a \leq 4.3 \cdot 10^{-17}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 2.95 \cdot 10^{+16}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;a \leq 8.2 \cdot 10^{+71}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 1.7 \cdot 10^{+95}:\\ \;\;\;\;x - t \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 2
Accuracy	70.4%
Cost	1368

\[\begin{array}{l} t_1 := \left(z - t\right) \cdot \frac{y}{a - t}\\ \mathbf{if}\;a \leq -3.2 \cdot 10^{+18}:\\ \;\;\;\;x + z \cdot \frac{y}{a}\\ \mathbf{elif}\;a \leq -1.75 \cdot 10^{-140}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 7.4 \cdot 10^{-224}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;a \leq 3.6 \cdot 10^{-19}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 4.5 \cdot 10^{+16}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;a \leq 1.95 \cdot 10^{+18}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x + \frac{z - t}{\frac{a}{y}}\\ \end{array} \]

Alternative 3
Accuracy	84.9%
Cost	1169

\[\begin{array}{l} t_1 := x - y \cdot \frac{t}{a - t}\\ \mathbf{if}\;t \leq -1.9 \cdot 10^{-29}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 7.4 \cdot 10^{-29}:\\ \;\;\;\;x + \frac{z - t}{\frac{a}{y}}\\ \mathbf{elif}\;t \leq 7.6 \cdot 10^{+70} \lor \neg \left(t \leq 4.1 \cdot 10^{+188}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x + \frac{z - t}{\frac{-t}{y}}\\ \end{array} \]

Alternative 4
Accuracy	60.9%
Cost	976

\[\begin{array}{l} t_1 := z \cdot \frac{y}{a - t}\\ \mathbf{if}\;a \leq -5.3 \cdot 10^{-67}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;a \leq -3 \cdot 10^{-145}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 9.5 \cdot 10^{-223}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;a \leq 1.3 \cdot 10^{-119}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 6.6 \cdot 10^{+71}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5
Accuracy	84.9%
Cost	841

\[\begin{array}{l} \mathbf{if}\;t \leq -1.75 \cdot 10^{-29} \lor \neg \left(t \leq 1.1 \cdot 10^{-28}\right):\\ \;\;\;\;x - y \cdot \frac{t}{a - t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{z - t}{\frac{a}{y}}\\ \end{array} \]

Alternative 6
Accuracy	77.7%
Cost	712

\[\begin{array}{l} \mathbf{if}\;t \leq -4.5 \cdot 10^{+35}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;t \leq 6.9 \cdot 10^{+16}:\\ \;\;\;\;x + z \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]

Alternative 7
Accuracy	97.9%
Cost	704

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

Alternative 8
Accuracy	69.3%
Cost	456

\[\begin{array}{l} \mathbf{if}\;t \leq -2.05 \cdot 10^{-29}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;t \leq 1400000000000:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]

Alternative 9
Accuracy	57.0%
Cost	328

\[\begin{array}{l} \mathbf{if}\;x \leq -6 \cdot 10^{-134}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{-79}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 10
Accuracy	55.6%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023138 
(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)))))