Average Error: 0.5 → 0.1
Time: 35.7s
Precision: binary64
Cost: 7104
\[\frac{60 \cdot \left(x - y\right)}{z - t} + a \cdot 120 \]
↓
\[\mathsf{fma}\left(60, \frac{y - x}{t - z}, a \cdot 120\right) \]
(FPCore (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)))
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));
}

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

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]

\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)

Error

Target

Original0.5
Target0.2
Herbie0.1
\[\frac{60}{\frac{z - t}{x - y}} + a \cdot 120 \]

Derivation

  1. Initial program 0.5

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

  2. Simplified0.1

    \[\leadsto \color{blue}{\mathsf{fma}\left(60, \frac{y - x}{t - z}, a \cdot 120\right)} \]
    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))): 6 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)): 20 points increase in error, 18 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 simplification0.1

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

Alternatives

Alternative 1
Error19.7
Cost5728
\[\begin{array}{l} t_1 := a \cdot 120 + \frac{60 \cdot x}{z}\\ t_2 := \frac{60 \cdot \left(x - y\right)}{z - t}\\ \mathbf{if}\;t_2 \leq -1 \cdot 10^{+162}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_2 \leq -4 \cdot 10^{+84}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_2 \leq -2 \cdot 10^{+60}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_2 \leq -1 \cdot 10^{+50}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;t_2 \leq -2 \cdot 10^{-41}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_2 \leq 5 \cdot 10^{-52}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;t_2 \leq 2 \cdot 10^{-14}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_2 \leq 10^{+57}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]
Alternative 2
Error19.5
Cost5728
\[\begin{array}{l} t_1 := \left(y - x\right) \cdot \frac{60}{t - z}\\ t_2 := \frac{60 \cdot \left(x - y\right)}{z - t}\\ t_3 := a \cdot 120 + \frac{60 \cdot x}{z}\\ \mathbf{if}\;t_2 \leq -1 \cdot 10^{+162}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_2 \leq -4 \cdot 10^{+84}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t_2 \leq -2 \cdot 10^{+60}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_2 \leq -1 \cdot 10^{+50}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;t_2 \leq -2 \cdot 10^{-41}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t_2 \leq 5 \cdot 10^{-52}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;t_2 \leq 2 \cdot 10^{-14}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_2 \leq 10^{+57}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 3
Error19.5
Cost5728
\[\begin{array}{l} t_1 := \frac{60 \cdot \left(x - y\right)}{z - t}\\ t_2 := a \cdot 120 + \frac{60 \cdot x}{z}\\ t_3 := \left(y - x\right) \cdot \frac{60}{t - z}\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{+162}:\\ \;\;\;\;60 \cdot \frac{y - x}{t - z}\\ \mathbf{elif}\;t_1 \leq -4 \cdot 10^{+84}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq -2 \cdot 10^{+60}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq -1 \cdot 10^{+50}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;t_1 \leq -2 \cdot 10^{-41}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{-52}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{-14}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t_1 \leq 10^{+57}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]
Alternative 4
Error18.7
Cost1628
\[\begin{array}{l} t_1 := \frac{60}{\frac{z}{x - y}} + a \cdot 120\\ \mathbf{if}\;z \leq -2.8557740785384106 \cdot 10^{-59}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -7 \cdot 10^{-230}:\\ \;\;\;\;\frac{60 \cdot \left(x - y\right)}{z - t}\\ \mathbf{elif}\;z \leq -1.45 \cdot 10^{-295}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;z \leq 3.7 \cdot 10^{-258}:\\ \;\;\;\;60 \cdot \frac{y - x}{t - z}\\ \mathbf{elif}\;z \leq 5.8 \cdot 10^{-126}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{-112}:\\ \;\;\;\;\left(y - x\right) \cdot \frac{60}{t - z}\\ \mathbf{elif}\;z \leq 6.952335156555223 \cdot 10^{-41}:\\ \;\;\;\;a \cdot 120\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 5
Error9.7
Cost1100
\[\begin{array}{l} \mathbf{if}\;z \leq -3.727572365816789 \cdot 10^{+76}:\\ \;\;\;\;\frac{60}{\frac{z}{x - y}} + a \cdot 120\\ \mathbf{elif}\;z \leq -6.19036267756672 \cdot 10^{-56}:\\ \;\;\;\;\frac{-60}{\frac{z - t}{y}} + a \cdot 120\\ \mathbf{elif}\;z \leq 137812.85364904528:\\ \;\;\;\;a \cdot 120 + -60 \cdot \frac{x - y}{t}\\ \mathbf{else}:\\ \;\;\;\;a \cdot 120 + \frac{x - y}{z \cdot 0.016666666666666666}\\ \end{array} \]
Alternative 6
Error9.7
Cost1100
\[\begin{array}{l} \mathbf{if}\;z \leq -3.727572365816789 \cdot 10^{+76}:\\ \;\;\;\;\frac{60}{\frac{z}{x - y}} + a \cdot 120\\ \mathbf{elif}\;z \leq -6.19036267756672 \cdot 10^{-56}:\\ \;\;\;\;\frac{-60}{\frac{z - t}{y}} + a \cdot 120\\ \mathbf{elif}\;z \leq 137812.85364904528:\\ \;\;\;\;a \cdot 120 + -60 \cdot \frac{x - y}{t}\\ \mathbf{else}:\\ \;\;\;\;a \cdot 120 + \left(x - y\right) \cdot \frac{60}{z}\\ \end{array} \]
Alternative 7
Error26.5
Cost976
\[\begin{array}{l} t_1 := \frac{60 \cdot x}{z - t}\\ \mathbf{if}\;x \leq -5.69367494401534 \cdot 10^{+159}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.7443792890551808 \cdot 10^{+145}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+204}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{+244}:\\ \;\;\;\;a \cdot 120\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 8
Error26.5
Cost976
\[\begin{array}{l} t_1 := \frac{60 \cdot x}{z - t}\\ \mathbf{if}\;x \leq -5.69367494401534 \cdot 10^{+159}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.7443792890551808 \cdot 10^{+145}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+204}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{+244}:\\ \;\;\;\;a \cdot 120\\ \mathbf{else}:\\ \;\;\;\;\frac{-60}{\frac{t - z}{x}}\\ \end{array} \]
Alternative 9
Error26.2
Cost972
\[\begin{array}{l} \mathbf{if}\;x \leq -5.69367494401534 \cdot 10^{+159}:\\ \;\;\;\;\frac{60 \cdot x}{z - t}\\ \mathbf{elif}\;x \leq 3.6560464854083027 \cdot 10^{-34}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;x \leq 10^{+242}:\\ \;\;\;\;a \cdot 120 + \frac{60 \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{-60}{\frac{t - z}{x}}\\ \end{array} \]
Alternative 10
Error10.2
Cost968
\[\begin{array}{l} t_1 := \frac{60}{\frac{z}{x - y}} + a \cdot 120\\ \mathbf{if}\;z \leq -55188552221769270:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 137812.85364904528:\\ \;\;\;\;a \cdot 120 + -60 \cdot \frac{x - y}{t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 11
Error10.2
Cost968
\[\begin{array}{l} \mathbf{if}\;z \leq -55188552221769270:\\ \;\;\;\;\frac{60}{\frac{z}{x - y}} + a \cdot 120\\ \mathbf{elif}\;z \leq 137812.85364904528:\\ \;\;\;\;a \cdot 120 + -60 \cdot \frac{x - y}{t}\\ \mathbf{else}:\\ \;\;\;\;a \cdot 120 + \frac{x - y}{z \cdot 0.016666666666666666}\\ \end{array} \]
Alternative 12
Error0.5
Cost832
\[\frac{60 \cdot \left(x - y\right)}{z - t} + a \cdot 120 \]
Alternative 13
Error0.1
Cost832
\[\frac{x - y}{\frac{z - t}{60}} + a \cdot 120 \]
Alternative 14
Error25.8
Cost712
\[\begin{array}{l} \mathbf{if}\;a \leq -9.477138800784602 \cdot 10^{-136}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \leq 1.538980465400714 \cdot 10^{-247}:\\ \;\;\;\;\frac{y}{\left(t - z\right) \cdot 0.016666666666666666}\\ \mathbf{else}:\\ \;\;\;\;a \cdot 120\\ \end{array} \]
Alternative 15
Error28.3
Cost584
\[\begin{array}{l} \mathbf{if}\;a \leq -3.339572478831351 \cdot 10^{-158}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \leq 1.2210759271188218 \cdot 10^{-256}:\\ \;\;\;\;y \cdot \frac{-60}{z}\\ \mathbf{else}:\\ \;\;\;\;a \cdot 120\\ \end{array} \]
Alternative 16
Error28.9
Cost192
\[a \cdot 120 \]

Error

Reproduce

herbie shell --seed 2022297 
(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)))