Data.Colour.RGB:hslsv from colour-2.3.3, B

Average Error: 0.66% → 0.23%

Time: 16.6s

Precision: binary64

Cost: 7104

Math

\[\frac{60 \cdot \left(x - y\right)}{z - t} + a \cdot 120 \]

↓

\[\mathsf{fma}\left(60, \frac{x - y}{z - t}, a \cdot 120\right) \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (+ (/ (* 60.0 (- x y)) (- z t)) (* a 120.0)))

↓

(FPCore (x y z t a)
 :precision binary64
 (fma 60.0 (/ (- x y) (- z t)) (* a 120.0)))

C

double code(double x, double y, double z, double t, double a) {
	return ((60.0 * (x - y)) / (z - t)) + (a * 120.0);
}

↓

double code(double x, double y, double z, double t, double a) {
	return fma(60.0, ((x - y) / (z - t)), (a * 120.0));
}

Julia

function code(x, y, z, t, a)
	return Float64(Float64(Float64(60.0 * Float64(x - y)) / Float64(z - t)) + Float64(a * 120.0))
end

↓

function code(x, y, z, t, a)
	return fma(60.0, Float64(Float64(x - y) / Float64(z - t)), Float64(a * 120.0))
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(N[(N[(60.0 * N[(x - y), $MachinePrecision]), $MachinePrecision] / N[(z - t), $MachinePrecision]), $MachinePrecision] + N[(a * 120.0), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_, t_, a_] := N[(60.0 * N[(N[(x - y), $MachinePrecision] / N[(z - t), $MachinePrecision]), $MachinePrecision] + N[(a * 120.0), $MachinePrecision]), $MachinePrecision]

Target

Original	0.66%
Target	0.24%
Herbie	0.23%

\[\frac{60}{\frac{z - t}{x - y}} + a \cdot 120 \]

Derivation

1. Initial program 0.66

   \[\frac{60 \cdot \left(x - y\right)}{z - t} + a \cdot 120 \]

2. Simplified 0.23

   \[\leadsto \color{blue}{\mathsf{fma}\left(60, \frac{x - y}{z - t}, a \cdot 120\right)} \]
   Proof

   [Start] 0.66	\[ \frac{60 \cdot \left(x - y\right)}{z - t} + a \cdot 120 \]
   associate-*r/ [<=] 0.23	\[ \color{blue}{60 \cdot \frac{x - y}{z - t}} + a \cdot 120 \]
   fma-def [=>] 0.23	\[ \color{blue}{\mathsf{fma}\left(60, \frac{x - y}{z - t}, a \cdot 120\right)} \]

3. Final simplification 0.23

   \[\leadsto \mathsf{fma}\left(60, \frac{x - y}{z - t}, a \cdot 120\right) \]

Alternatives

Alternative 1
Error	30.49%
Cost	1500

\[\begin{array}{l} t_1 := 60 \cdot \frac{x - y}{z - t}\\ t_2 := a \cdot 120 + \frac{y}{\frac{t}{60}}\\ t_3 := a \cdot 120 + \frac{-60}{\frac{z}{y}}\\ \mathbf{if}\;t \leq -2.5 \cdot 10^{+31}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -0.32:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -5 \cdot 10^{-190}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq -1.22 \cdot 10^{-219}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 8.1 \cdot 10^{-100}:\\ \;\;\;\;a \cdot 120 + 60 \cdot \frac{x}{z}\\ \mathbf{elif}\;t \leq 2.4 \cdot 10^{-56}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 0.00013:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 2
Error	30.44%
Cost	1500

\[\begin{array}{l} t_1 := \frac{60}{\frac{z - t}{x - y}}\\ t_2 := a \cdot 120 + \frac{y}{\frac{t}{60}}\\ t_3 := a \cdot 120 + \frac{-60}{\frac{z}{y}}\\ \mathbf{if}\;t \leq -1.2 \cdot 10^{+31}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -0.0035:\\ \;\;\;\;60 \cdot \frac{x - y}{z - t}\\ \mathbf{elif}\;t \leq -1.2 \cdot 10^{-189}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq -1.4 \cdot 10^{-219}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.09 \cdot 10^{-100}:\\ \;\;\;\;a \cdot 120 + 60 \cdot \frac{x}{z}\\ \mathbf{elif}\;t \leq 10^{-56}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 9.2 \cdot 10^{-5}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Error	16.54%
Cost	1225

\[\begin{array}{l} \mathbf{if}\;a \cdot 120 \leq -1 \cdot 10^{-89} \lor \neg \left(a \cdot 120 \leq 2 \cdot 10^{-102}\right):\\ \;\;\;\;a \cdot 120 + \frac{y \cdot -60}{z - t}\\ \mathbf{else}:\\ \;\;\;\;60 \cdot \frac{x - y}{z - t}\\ \end{array} \]

Alternative 4
Error	10.02%
Cost	1097

\[\begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-12} \lor \neg \left(y \leq 2.4 \cdot 10^{+21}\right):\\ \;\;\;\;a \cdot 120 + \frac{60}{\left(z - t\right) \cdot \frac{-1}{y}}\\ \mathbf{else}:\\ \;\;\;\;a \cdot 120 + \frac{60 \cdot x}{z - t}\\ \end{array} \]

Alternative 5
Error	23.97%
Cost	1096

\[\begin{array}{l} \mathbf{if}\;a \cdot 120 \leq -2 \cdot 10^{+42}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \cdot 120 \leq 5 \cdot 10^{-26}:\\ \;\;\;\;60 \cdot \frac{x - y}{z - t}\\ \mathbf{else}:\\ \;\;\;\;a \cdot 120\\ \end{array} \]

Alternative 6
Error	10.11%
Cost	969

\[\begin{array}{l} \mathbf{if}\;y \leq -5.8 \cdot 10^{-10} \lor \neg \left(y \leq 5.5 \cdot 10^{+25}\right):\\ \;\;\;\;a \cdot 120 + \frac{y \cdot -60}{z - t}\\ \mathbf{else}:\\ \;\;\;\;a \cdot 120 + \frac{60 \cdot x}{z - t}\\ \end{array} \]

Alternative 7
Error	0.24%
Cost	832

\[a \cdot 120 + \frac{60}{\frac{z - t}{x - y}} \]

Alternative 8
Error	43.6%
Cost	716

\[\begin{array}{l} \mathbf{if}\;a \leq -4.6 \cdot 10^{-179}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \leq 4.2 \cdot 10^{-306}:\\ \;\;\;\;-60 \cdot \frac{y}{z}\\ \mathbf{elif}\;a \leq 1.14 \cdot 10^{-197}:\\ \;\;\;\;-60 \cdot \frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;a \cdot 120\\ \end{array} \]

Alternative 9
Error	43.36%
Cost	716

\[\begin{array}{l} \mathbf{if}\;a \leq -4.9 \cdot 10^{-179}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \leq -2.9 \cdot 10^{-229}:\\ \;\;\;\;-60 \cdot \frac{y}{z}\\ \mathbf{elif}\;a \leq 7.5 \cdot 10^{-193}:\\ \;\;\;\;60 \cdot \frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;a \cdot 120\\ \end{array} \]

Alternative 10
Error	43.35%
Cost	716

\[\begin{array}{l} \mathbf{if}\;a \leq -2.6 \cdot 10^{-179}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \leq -3.1 \cdot 10^{-230}:\\ \;\;\;\;y \cdot \frac{-60}{z}\\ \mathbf{elif}\;a \leq 1.68 \cdot 10^{-192}:\\ \;\;\;\;60 \cdot \frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;a \cdot 120\\ \end{array} \]

Alternative 11
Error	38.37%
Cost	712

\[\begin{array}{l} \mathbf{if}\;a \leq -7.2 \cdot 10^{-89}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \leq 1.05 \cdot 10^{-100}:\\ \;\;\;\;-60 \cdot \frac{y}{z - t}\\ \mathbf{else}:\\ \;\;\;\;a \cdot 120\\ \end{array} \]

Alternative 12
Error	43.51%
Cost	584

\[\begin{array}{l} \mathbf{if}\;a \leq -2.8 \cdot 10^{-187}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \leq 1.06 \cdot 10^{-197}:\\ \;\;\;\;-60 \cdot \frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;a \cdot 120\\ \end{array} \]

Alternative 13
Error	44.85%
Cost	192

\[a \cdot 120 \]

Reproduce

herbie shell --seed 2023090 
(FPCore (x y z t a)
  :name "Data.Colour.RGB:hslsv from colour-2.3.3, B"
  :precision binary64

  :herbie-target
  (+ (/ 60.0 (/ (- z t) (- x y))) (* a 120.0))

  (+ (/ (* 60.0 (- x y)) (- z t)) (* a 120.0)))