Average Error: 0.4 → 0.1
Time: 18.6s
Precision: binary64
Cost: 7104
\[\frac{60 \cdot \left(x - y\right)}{z - t} + a \cdot 120 \]
↓
\[\mathsf{fma}\left(a, 120, \frac{60}{z - t} \cdot \left(x - y\right)\right) \]
(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 a 120.0 (* (/ 60.0 (- z t)) (- x y))))
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(a, 120.0, ((60.0 / (z - t)) * (x - y)));
}

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(a, 120.0, Float64(Float64(60.0 / Float64(z - t)) * Float64(x - y)))
end

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[(a * 120.0 + N[(N[(60.0 / N[(z - t), $MachinePrecision]), $MachinePrecision] * N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

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

Error

Target

Original0.4
Target0.2
Herbie0.1
\[\frac{60}{\frac{z - t}{x - y}} + a \cdot 120 \]

Derivation

  1. Initial program 0.4

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

  2. Simplified0.1

    \[\leadsto \color{blue}{\mathsf{fma}\left(a, 120, \frac{60}{z - t} \cdot \left(x - y\right)\right)} \]
    Proof
    (fma.f64 a 120 (*.f64 (/.f64 60 (-.f64 z t)) (-.f64 x y))): 0 points increase in error, 0 points decrease in error
    (fma.f64 a 120 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 60 (-.f64 x y)) (-.f64 z t)))): 24 points increase in error, 18 points decrease in error
    (Rewrite<= fma-def_binary64 (+.f64 (*.f64 a 120) (/.f64 (*.f64 60 (-.f64 x y)) (-.f64 z t)))): 4 points increase in error, 0 points decrease in error
    (Rewrite<= +-commutative_binary64 (+.f64 (/.f64 (*.f64 60 (-.f64 x y)) (-.f64 z t)) (*.f64 a 120))): 0 points increase in error, 0 points decrease in error

  3. Final simplification0.1

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

Alternatives

Alternative 1
Error15.7
Cost1356
\[\begin{array}{l} \mathbf{if}\;a \cdot 120 \leq -1 \cdot 10^{-47}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \cdot 120 \leq 10^{-122}:\\ \;\;\;\;60 \cdot \frac{x - y}{z - t}\\ \mathbf{elif}\;a \cdot 120 \leq 10^{+52}:\\ \;\;\;\;x \cdot \frac{60}{z} + a \cdot 120\\ \mathbf{else}:\\ \;\;\;\;a \cdot 120\\ \end{array} \]
Alternative 2
Error15.7
Cost1356
\[\begin{array}{l} \mathbf{if}\;a \cdot 120 \leq -1 \cdot 10^{-47}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \cdot 120 \leq 10^{-122}:\\ \;\;\;\;\frac{60}{\frac{z - t}{x - y}}\\ \mathbf{elif}\;a \cdot 120 \leq 10^{+52}:\\ \;\;\;\;x \cdot \frac{60}{z} + a \cdot 120\\ \mathbf{else}:\\ \;\;\;\;a \cdot 120\\ \end{array} \]
Alternative 3
Error16.3
Cost1232
\[\begin{array}{l} t_1 := \left(x - y\right) \cdot \frac{-60}{t} + a \cdot 120\\ \mathbf{if}\;t \leq -120000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -2.6 \cdot 10^{-84}:\\ \;\;\;\;\frac{60}{\frac{z - t}{x - y}}\\ \mathbf{elif}\;t \leq -1.1 \cdot 10^{-210}:\\ \;\;\;\;y \cdot \frac{-60}{z} + a \cdot 120\\ \mathbf{elif}\;t \leq 6.4 \cdot 10^{+65}:\\ \;\;\;\;x \cdot \frac{60}{z} + a \cdot 120\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 4
Error8.5
Cost1232
\[\begin{array}{l} t_1 := \frac{60 \cdot x}{z - t} + a \cdot 120\\ t_2 := \frac{60}{\frac{t - z}{y}} + a \cdot 120\\ \mathbf{if}\;y \leq -6 \cdot 10^{+187}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-52}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 7.6 \cdot 10^{+49}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{+89}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]
Alternative 5
Error10.5
Cost1224
\[\begin{array}{l} t_1 := \frac{60 \cdot x}{z - t} + a \cdot 120\\ \mathbf{if}\;a \cdot 120 \leq -5 \cdot 10^{-188}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \cdot 120 \leq 10^{-122}:\\ \;\;\;\;60 \cdot \frac{x - y}{z - t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 6
Error28.4
Cost1112
\[\begin{array}{l} t_1 := -60 \cdot \frac{x}{t}\\ \mathbf{if}\;a \leq -3.2 \cdot 10^{-133}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \leq 2.3 \cdot 10^{-284}:\\ \;\;\;\;60 \cdot \frac{x}{z}\\ \mathbf{elif}\;a \leq 1.55 \cdot 10^{-214}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \leq 4.1 \cdot 10^{-191}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 4 \cdot 10^{-140}:\\ \;\;\;\;60 \cdot \frac{y}{t}\\ \mathbf{elif}\;a \leq 9.2 \cdot 10^{-118}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;a \cdot 120\\ \end{array} \]
Alternative 7
Error15.5
Cost1096
\[\begin{array}{l} \mathbf{if}\;a \cdot 120 \leq -1 \cdot 10^{-47}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \cdot 120 \leq 2 \cdot 10^{+68}:\\ \;\;\;\;60 \cdot \frac{x - y}{z - t}\\ \mathbf{else}:\\ \;\;\;\;a \cdot 120\\ \end{array} \]
Alternative 8
Error28.2
Cost980
\[\begin{array}{l} \mathbf{if}\;a \leq -8.2 \cdot 10^{-133}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \leq 3.5 \cdot 10^{-284}:\\ \;\;\;\;60 \cdot \frac{x}{z}\\ \mathbf{elif}\;a \leq 1.8 \cdot 10^{-215}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \leq 3.25 \cdot 10^{-189}:\\ \;\;\;\;-60 \cdot \frac{x}{t}\\ \mathbf{elif}\;a \leq 3.7 \cdot 10^{-137}:\\ \;\;\;\;\frac{-60}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;a \cdot 120\\ \end{array} \]
Alternative 9
Error25.3
Cost976
\[\begin{array}{l} t_1 := -60 \cdot \frac{x - y}{t}\\ \mathbf{if}\;a \leq -1.54 \cdot 10^{-189}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \leq 4.3 \cdot 10^{-187}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 6 \cdot 10^{-170}:\\ \;\;\;\;\frac{-60}{\frac{z}{y}}\\ \mathbf{elif}\;a \leq 7 \cdot 10^{-118}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;a \cdot 120\\ \end{array} \]
Alternative 10
Error10.6
Cost968
\[\begin{array}{l} t_1 := \left(x - y\right) \cdot \frac{-60}{t} + a \cdot 120\\ \mathbf{if}\;t \leq -1.7 \cdot 10^{-11}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.75 \cdot 10^{+64}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{60}{z} + a \cdot 120\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 11
Error24.6
Cost844
\[\begin{array}{l} \mathbf{if}\;a \leq -2.5 \cdot 10^{-52}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \leq 1.95 \cdot 10^{-290}:\\ \;\;\;\;60 \cdot \frac{x}{z - t}\\ \mathbf{elif}\;a \leq 1.15 \cdot 10^{-117}:\\ \;\;\;\;-60 \cdot \frac{x - y}{t}\\ \mathbf{else}:\\ \;\;\;\;a \cdot 120\\ \end{array} \]
Alternative 12
Error0.2
Cost832
\[\frac{60}{\frac{z - t}{x - y}} + a \cdot 120 \]
Alternative 13
Error27.8
Cost716
\[\begin{array}{l} \mathbf{if}\;a \leq -4.6 \cdot 10^{-133}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \leq 3.4 \cdot 10^{-279}:\\ \;\;\;\;60 \cdot \frac{x}{z}\\ \mathbf{elif}\;a \leq 7.8 \cdot 10^{-118}:\\ \;\;\;\;-60 \cdot \frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;a \cdot 120\\ \end{array} \]
Alternative 14
Error28.0
Cost584
\[\begin{array}{l} \mathbf{if}\;a \leq -8.2 \cdot 10^{-190}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \leq 1.1 \cdot 10^{-117}:\\ \;\;\;\;-60 \cdot \frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;a \cdot 120\\ \end{array} \]
Alternative 15
Error28.9
Cost192
\[a \cdot 120 \]

Error

Reproduce

herbie shell --seed 2022329 
(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)))