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

Average Error: 0.4 → 0.1

Time: 23.6s

Precision: binary64

Cost: 7104

Math

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

↓

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

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 (/ (- y x) (- t z)) (* 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, ((y - x) / (t - z)), (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(y - x) / Float64(t - z)), 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[(y - x), $MachinePrecision] / N[(t - z), $MachinePrecision]), $MachinePrecision] + N[(a * 120.0), $MachinePrecision]), $MachinePrecision]

Target

Original	0.4
Target	0.1
Herbie	0.1

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

Derivation

1. Initial program 0.4

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

2. Simplified 0.1

   \[\leadsto \color{blue}{\mathsf{fma}\left(60, \frac{y - x}{t - z}, a \cdot 120\right)} \]
   Proof
   (fma.f64 60 (/.f64 (-.f64 y x) (-.f64 t z)) (*.f64 a 120)): 0 points increase in error, 0 points decrease in error
   (fma.f64 60 (Rewrite<= *-lft-identity_binary64 (*.f64 1 (/.f64 (-.f64 y x) (-.f64 t z)))) (*.f64 a 120)): 0 points increase in error, 0 points decrease in error
   (fma.f64 60 (*.f64 (Rewrite<= metadata-eval (/.f64 -1 -1)) (/.f64 (-.f64 y x) (-.f64 t z))) (*.f64 a 120)): 0 points increase in error, 0 points decrease in error
   (fma.f64 60 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 -1 (-.f64 y x)) (*.f64 -1 (-.f64 t z)))) (*.f64 a 120)): 0 points increase in error, 0 points decrease in error
   (fma.f64 60 (/.f64 (Rewrite<= neg-mul-1_binary64 (neg.f64 (-.f64 y x))) (*.f64 -1 (-.f64 t z))) (*.f64 a 120)): 0 points increase in error, 0 points decrease in error
   (fma.f64 60 (/.f64 (Rewrite<= sub0-neg_binary64 (-.f64 0 (-.f64 y x))) (*.f64 -1 (-.f64 t z))) (*.f64 a 120)): 0 points increase in error, 0 points decrease in error
   (fma.f64 60 (/.f64 (Rewrite<= associate-+l-_binary64 (+.f64 (-.f64 0 y) x)) (*.f64 -1 (-.f64 t z))) (*.f64 a 120)): 0 points increase in error, 0 points decrease in error
   (fma.f64 60 (/.f64 (+.f64 (Rewrite<= neg-sub0_binary64 (neg.f64 y)) x) (*.f64 -1 (-.f64 t z))) (*.f64 a 120)): 0 points increase in error, 0 points decrease in error
   (fma.f64 60 (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 x (neg.f64 y))) (*.f64 -1 (-.f64 t z))) (*.f64 a 120)): 0 points increase in error, 0 points decrease in error
   (fma.f64 60 (/.f64 (Rewrite<= sub-neg_binary64 (-.f64 x y)) (*.f64 -1 (-.f64 t z))) (*.f64 a 120)): 0 points increase in error, 0 points decrease in error
   (fma.f64 60 (/.f64 (-.f64 x y) (Rewrite<= neg-mul-1_binary64 (neg.f64 (-.f64 t z)))) (*.f64 a 120)): 0 points increase in error, 0 points decrease in error
   (fma.f64 60 (/.f64 (-.f64 x y) (Rewrite<= sub0-neg_binary64 (-.f64 0 (-.f64 t z)))) (*.f64 a 120)): 0 points increase in error, 0 points decrease in error
   (fma.f64 60 (/.f64 (-.f64 x y) (Rewrite<= associate-+l-_binary64 (+.f64 (-.f64 0 t) z))) (*.f64 a 120)): 0 points increase in error, 0 points decrease in error
   (fma.f64 60 (/.f64 (-.f64 x y) (+.f64 (Rewrite<= neg-sub0_binary64 (neg.f64 t)) z)) (*.f64 a 120)): 0 points increase in error, 0 points decrease in error
   (fma.f64 60 (/.f64 (-.f64 x y) (Rewrite<= +-commutative_binary64 (+.f64 z (neg.f64 t)))) (*.f64 a 120)): 0 points increase in error, 0 points decrease in error
   (fma.f64 60 (/.f64 (-.f64 x y) (Rewrite<= sub-neg_binary64 (-.f64 z t))) (*.f64 a 120)): 0 points increase in error, 0 points decrease in error
   (Rewrite<= fma-def_binary64 (+.f64 (*.f64 60 (/.f64 (-.f64 x y) (-.f64 z t))) (*.f64 a 120))): 8 points increase in error, 2 points decrease in error
   (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 (-.f64 x y) (-.f64 z t)) 60)) (*.f64 a 120)): 0 points increase in error, 0 points decrease in error
   (+.f64 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 (-.f64 x y) 60) (-.f64 z t))) (*.f64 a 120)): 19 points increase in error, 24 points decrease in error
   (+.f64 (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 60 (-.f64 x y))) (-.f64 z t)) (*.f64 a 120)): 0 points increase in error, 0 points decrease in error

3. Final simplification 0.1

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

Alternatives

Alternative 1
Error	21.9
Cost	2424

\[\begin{array}{l} t_1 := \left(y - x\right) \cdot \frac{60}{t - z}\\ t_2 := a \cdot 120 + y \cdot \frac{-60}{z}\\ t_3 := a \cdot 120 + 60 \cdot \frac{y}{t}\\ t_4 := a \cdot 120 + x \cdot \frac{-60}{t}\\ \mathbf{if}\;t \leq -4.5913389687455985 \cdot 10^{+145}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq -1.004347875769466 \cdot 10^{+86}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -2028989355.305274:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq -9.058827771087351 \cdot 10^{-55}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -3.9298432965934975 \cdot 10^{-72}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq -4.2 \cdot 10^{-184}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -6 \cdot 10^{-209}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 5.2 \cdot 10^{-204}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 5 \cdot 10^{-178}:\\ \;\;\;\;\frac{-60}{\frac{t - z}{x}}\\ \mathbf{elif}\;t \leq 1.58 \cdot 10^{-102}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 1.1684188798241903 \cdot 10^{-45}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 5.037881870787659 \cdot 10^{+55}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;t \leq 1.4269620906786866 \cdot 10^{+73}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.8683560630599017 \cdot 10^{+149}:\\ \;\;\;\;t_4\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 2
Error	15.8
Cost	1744

\[\begin{array}{l} t_1 := \left(y - x\right) \cdot \frac{60}{t - z}\\ t_2 := \frac{-60}{\frac{z - t}{y}} + a \cdot 120\\ \mathbf{if}\;z - t \leq -2 \cdot 10^{-20}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z - t \leq 2 \cdot 10^{-195}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z - t \leq 2 \cdot 10^{-67}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z - t \leq 5 \cdot 10^{+133}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Error	20.3
Cost	1632

\[\begin{array}{l} t_1 := a \cdot 120 + x \cdot \frac{-60}{t}\\ t_2 := a \cdot 120 + \frac{60 \cdot x}{z}\\ t_3 := a \cdot 120 + y \cdot \frac{-60}{z}\\ t_4 := a \cdot 120 + 60 \cdot \frac{y}{t}\\ \mathbf{if}\;z \leq -5.310034397914631 \cdot 10^{+35}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq -1.95 \cdot 10^{-86}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -2.8 \cdot 10^{-94}:\\ \;\;\;\;\frac{60 \cdot y}{t - z}\\ \mathbf{elif}\;z \leq -1.1 \cdot 10^{-220}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 9.2 \cdot 10^{-290}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{-253}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.9665924776624043 \cdot 10^{-27}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;z \leq 2.631589801836488 \cdot 10^{+204}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 4
Error	20.7
Cost	1500

\[\begin{array}{l} t_1 := a \cdot 120 + x \cdot \frac{-60}{t}\\ t_2 := a \cdot 120 + \frac{60 \cdot x}{z}\\ t_3 := a \cdot 120 + y \cdot \frac{-60}{z}\\ \mathbf{if}\;z \leq -5.310034397914631 \cdot 10^{+35}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq -1.95 \cdot 10^{-86}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -2.8 \cdot 10^{-94}:\\ \;\;\;\;\frac{60 \cdot y}{t - z}\\ \mathbf{elif}\;z \leq -8.6 \cdot 10^{-291}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -9.4 \cdot 10^{-303}:\\ \;\;\;\;\frac{60}{\frac{t - z}{y}}\\ \mathbf{elif}\;z \leq 6.611938104638012 \cdot 10^{-40}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.631589801836488 \cdot 10^{+204}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 5
Error	25.5
Cost	1488

\[\begin{array}{l} \mathbf{if}\;a \cdot 120 \leq -5 \cdot 10^{-63}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \cdot 120 \leq -5 \cdot 10^{-232}:\\ \;\;\;\;a \cdot 120 + x \cdot \frac{-60}{t}\\ \mathbf{elif}\;a \cdot 120 \leq 2 \cdot 10^{-97}:\\ \;\;\;\;\frac{60 \cdot y}{t - z}\\ \mathbf{elif}\;a \cdot 120 \leq 5 \cdot 10^{-30}:\\ \;\;\;\;\frac{-60}{\frac{t - z}{x}}\\ \mathbf{else}:\\ \;\;\;\;a \cdot 120\\ \end{array} \]

Alternative 6
Error	21.4
Cost	1104

\[\begin{array}{l} t_1 := a \cdot 120 + y \cdot \frac{-60}{z}\\ t_2 := a \cdot 120 + x \cdot \frac{-60}{t}\\ \mathbf{if}\;t \leq -3.9298432965934975 \cdot 10^{-72}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 5.2 \cdot 10^{-204}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 5 \cdot 10^{-178}:\\ \;\;\;\;\frac{-60}{\frac{t - z}{x}}\\ \mathbf{elif}\;t \leq 2.7 \cdot 10^{-110}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 7
Error	6.9
Cost	968

\[\begin{array}{l} t_1 := \frac{-60}{\frac{z - t}{y}} + a \cdot 120\\ \mathbf{if}\;y \leq -1.0398980316801028 \cdot 10^{+35}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 3.4160917836170812 \cdot 10^{-34}:\\ \;\;\;\;\frac{60 \cdot x}{z - t} + a \cdot 120\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	6.9
Cost	968

\[\begin{array}{l} \mathbf{if}\;y \leq -1.0398980316801028 \cdot 10^{+35}:\\ \;\;\;\;\frac{y \cdot -60}{z - t} + a \cdot 120\\ \mathbf{elif}\;y \leq 3.4160917836170812 \cdot 10^{-34}:\\ \;\;\;\;\frac{60 \cdot x}{z - t} + a \cdot 120\\ \mathbf{else}:\\ \;\;\;\;\frac{-60}{\frac{z - t}{y}} + a \cdot 120\\ \end{array} \]

Alternative 9
Error	26.3
Cost	844

\[\begin{array}{l} \mathbf{if}\;a \leq -6.289745688542691 \cdot 10^{-234}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \leq 8.34135025301754 \cdot 10^{-100}:\\ \;\;\;\;\frac{60 \cdot y}{t - z}\\ \mathbf{elif}\;a \leq 2.180646645156897 \cdot 10^{-32}:\\ \;\;\;\;x \cdot \frac{-60}{t - z}\\ \mathbf{else}:\\ \;\;\;\;a \cdot 120\\ \end{array} \]

Alternative 10
Error	26.3
Cost	844

\[\begin{array}{l} \mathbf{if}\;a \leq -6.289745688542691 \cdot 10^{-234}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \leq 8.34135025301754 \cdot 10^{-100}:\\ \;\;\;\;\frac{60 \cdot y}{t - z}\\ \mathbf{elif}\;a \leq 2.180646645156897 \cdot 10^{-32}:\\ \;\;\;\;\frac{-60}{\frac{t - z}{x}}\\ \mathbf{else}:\\ \;\;\;\;a \cdot 120\\ \end{array} \]

Alternative 11
Error	0.4
Cost	832

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

Alternative 12
Error	25.7
Cost	712

\[\begin{array}{l} \mathbf{if}\;a \leq -6.289745688542691 \cdot 10^{-234}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \leq 1.9273520504971495 \cdot 10^{-99}:\\ \;\;\;\;\frac{60 \cdot y}{t - z}\\ \mathbf{else}:\\ \;\;\;\;a \cdot 120\\ \end{array} \]

Alternative 13
Error	28.6
Cost	584

\[\begin{array}{l} \mathbf{if}\;a \leq -6.289745688542691 \cdot 10^{-234}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \leq 1.9273520504971495 \cdot 10^{-99}:\\ \;\;\;\;-60 \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;a \cdot 120\\ \end{array} \]

Alternative 14
Error	28.9
Cost	584

\[\begin{array}{l} \mathbf{if}\;a \leq -7.263835656568478 \cdot 10^{-285}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \leq 1.9273520504971495 \cdot 10^{-99}:\\ \;\;\;\;y \cdot \frac{60}{t}\\ \mathbf{else}:\\ \;\;\;\;a \cdot 120\\ \end{array} \]

Alternative 15
Error	29.0
Cost	192

\[a \cdot 120 \]

Reproduce

herbie shell --seed 2022308 
(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)))