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

Average Accuracy: 83.3% → 98.1%

Time: 14.7s

Precision: binary64

Cost: 6976

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

Julia

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

↓

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

Wolfram

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

↓

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

Target

Original	83.3%
Target	99.3%
Herbie	98.1%

\[\begin{array}{l} \mathbf{if}\;t < -1.0682974490174067 \cdot 10^{-39}:\\ \;\;\;\;x + \frac{y - z}{a - z} \cdot t\\ \mathbf{elif}\;t < 3.9110949887586375 \cdot 10^{-141}:\\ \;\;\;\;x + \frac{\left(y - z\right) \cdot t}{a - z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - z}{a - z} \cdot t\\ \end{array} \]

Derivation

1. Initial program 83.3%

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

2. Simplified 98.1%

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

   [Start] 83.3	\[ x + \frac{\left(y - z\right) \cdot t}{a - z} \]
   +-commutative [=>] 83.3	\[ \color{blue}{\frac{\left(y - z\right) \cdot t}{a - z} + x} \]
   associate-*l/ [<=] 98.1	\[ \color{blue}{\frac{y - z}{a - z} \cdot t} + x \]
   fma-def [=>] 98.1	\[ \color{blue}{\mathsf{fma}\left(\frac{y - z}{a - z}, t, x\right)} \]

3. Final simplification 98.1%

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

Alternatives

Alternative 1
Accuracy	81.3%
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -1550 \lor \neg \left(z \leq 6.6 \cdot 10^{-105}\right):\\ \;\;\;\;x + t \cdot \left(1 - \frac{y}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{t}{\frac{a}{y}}\\ \end{array} \]

Alternative 2
Accuracy	82.3%
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -2.25 \cdot 10^{+18} \lor \neg \left(z \leq 4.3 \cdot 10^{-105}\right):\\ \;\;\;\;x + t \cdot \left(1 - \frac{y}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{t}{\frac{a}{y - z}}\\ \end{array} \]

Alternative 3
Accuracy	84.5%
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -3.2 \cdot 10^{-69} \lor \neg \left(z \leq 3.1 \cdot 10^{-109}\right):\\ \;\;\;\;x - \frac{t}{\frac{a}{z} + -1}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{t}{\frac{a}{y - z}}\\ \end{array} \]

Alternative 4
Accuracy	76.1%
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -2.65 \cdot 10^{-70}:\\ \;\;\;\;t + x\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{-105}:\\ \;\;\;\;x + t \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;t + x\\ \end{array} \]

Alternative 5
Accuracy	76.5%
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -5.8 \cdot 10^{-37}:\\ \;\;\;\;t + x\\ \mathbf{elif}\;z \leq 8 \cdot 10^{-105}:\\ \;\;\;\;x + \frac{t}{\frac{a}{y}}\\ \mathbf{else}:\\ \;\;\;\;t + x\\ \end{array} \]

Alternative 6
Accuracy	98.1%
Cost	704

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

Alternative 7
Accuracy	68.2%
Cost	456

\[\begin{array}{l} \mathbf{if}\;z \leq -1.25 \cdot 10^{-47}:\\ \;\;\;\;t + x\\ \mathbf{elif}\;z \leq 2.25 \cdot 10^{-105}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t + x\\ \end{array} \]

Alternative 8
Accuracy	56.0%
Cost	196

\[\begin{array}{l} \mathbf{if}\;t \leq 8.5 \cdot 10^{+172}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \]

Alternative 9
Accuracy	20.0%
Cost	64

\[t \]

Reproduce

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

  :herbie-target
  (if (< t -1.0682974490174067e-39) (+ x (* (/ (- y z) (- a z)) t)) (if (< t 3.9110949887586375e-141) (+ x (/ (* (- y z) t) (- a z))) (+ x (* (/ (- y z) (- a z)) t))))

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