Average Error: 0.4 → 0.1
Time: 17.4s
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)))): 19 points increase in error, 22 points decrease in error
    (Rewrite<= fma-def_binary64 (+.f64 (*.f64 a 120) (/.f64 (*.f64 60 (-.f64 x y)) (-.f64 z t)))): 8 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
Error0.1
Cost7104
\[\mathsf{fma}\left(60, \frac{y - x}{t - z}, a \cdot 120\right) \]
Alternative 2
Error16.6
Cost2136
\[\begin{array}{l} t_1 := a \cdot 120 + 60 \cdot \frac{x}{z}\\ \mathbf{if}\;a \cdot 120 \leq -2 \cdot 10^{+106}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \cdot 120 \leq -5 \cdot 10^{+33}:\\ \;\;\;\;a \cdot 120 + \frac{x \cdot -60}{t}\\ \mathbf{elif}\;a \cdot 120 \leq -5 \cdot 10^{-45}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \cdot 120 \leq 5 \cdot 10^{-22}:\\ \;\;\;\;\left(y - x\right) \cdot \frac{60}{t - z}\\ \mathbf{elif}\;a \cdot 120 \leq 5 \cdot 10^{+63}:\\ \;\;\;\;a \cdot 120 + -60 \cdot \frac{x}{t}\\ \mathbf{elif}\;a \cdot 120 \leq 2 \cdot 10^{+125}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;a \cdot 120\\ \end{array} \]
Alternative 3
Error16.6
Cost1368
\[\begin{array}{l} t_1 := a \cdot 120 + -60 \cdot \frac{x}{t}\\ t_2 := a \cdot 120 + 60 \cdot \frac{x}{z}\\ \mathbf{if}\;a \leq -1.5 \cdot 10^{+104}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq -2.25 \cdot 10^{+30}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -3.6 \cdot 10^{-50}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 1.26 \cdot 10^{-18}:\\ \;\;\;\;\left(y - x\right) \cdot \frac{60}{t - z}\\ \mathbf{elif}\;a \leq 7.5 \cdot 10^{+62}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 1.12 \cdot 10^{+123}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;a \cdot 120\\ \end{array} \]
Alternative 4
Error10.3
Cost1224
\[\begin{array}{l} t_1 := \frac{y \cdot -60}{z - t} + a \cdot 120\\ \mathbf{if}\;a \cdot 120 \leq -1 \cdot 10^{-96}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \cdot 120 \leq 5 \cdot 10^{-103}:\\ \;\;\;\;\left(y - x\right) \cdot \frac{60}{t - z}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 5
Error15.0
Cost1104
\[\begin{array}{l} \mathbf{if}\;a \leq -1.42 \cdot 10^{+104}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \leq -4 \cdot 10^{+52}:\\ \;\;\;\;a \cdot 120 + -60 \cdot \frac{x}{t}\\ \mathbf{elif}\;a \leq -1.1 \cdot 10^{+28}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \leq 9 \cdot 10^{-19}:\\ \;\;\;\;\left(y - x\right) \cdot \frac{60}{t - z}\\ \mathbf{else}:\\ \;\;\;\;a \cdot 120\\ \end{array} \]
Alternative 6
Error6.7
Cost968
\[\begin{array}{l} t_1 := \frac{60 \cdot x}{z - t} + a \cdot 120\\ \mathbf{if}\;x \leq -20500:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 2.25 \cdot 10^{+26}:\\ \;\;\;\;\frac{y \cdot -60}{z - t} + a \cdot 120\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 7
Error14.9
Cost840
\[\begin{array}{l} \mathbf{if}\;a \leq -6.4 \cdot 10^{+27}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \leq 2.6 \cdot 10^{-23}:\\ \;\;\;\;\left(y - x\right) \cdot \frac{60}{t - z}\\ \mathbf{else}:\\ \;\;\;\;a \cdot 120\\ \end{array} \]
Alternative 8
Error0.2
Cost832
\[\frac{60}{\frac{z - t}{x - y}} + a \cdot 120 \]
Alternative 9
Error0.1
Cost832
\[\frac{x - y}{\frac{z - t}{60}} + a \cdot 120 \]
Alternative 10
Error24.7
Cost712
\[\begin{array}{l} \mathbf{if}\;a \leq -1.3 \cdot 10^{-99}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \leq 5.8 \cdot 10^{-105}:\\ \;\;\;\;-60 \cdot \frac{x}{t - z}\\ \mathbf{else}:\\ \;\;\;\;a \cdot 120\\ \end{array} \]
Alternative 11
Error24.8
Cost712
\[\begin{array}{l} \mathbf{if}\;a \leq -9.8 \cdot 10^{-99}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \leq 5.4 \cdot 10^{-105}:\\ \;\;\;\;x \cdot \frac{-60}{t - z}\\ \mathbf{else}:\\ \;\;\;\;a \cdot 120\\ \end{array} \]
Alternative 12
Error24.8
Cost712
\[\begin{array}{l} \mathbf{if}\;a \leq -1.22 \cdot 10^{-98}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \leq 5.7 \cdot 10^{-105}:\\ \;\;\;\;\frac{x \cdot -60}{t - z}\\ \mathbf{else}:\\ \;\;\;\;a \cdot 120\\ \end{array} \]
Alternative 13
Error28.7
Cost584
\[\begin{array}{l} \mathbf{if}\;a \leq -1.02 \cdot 10^{-116}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \leq 2.05 \cdot 10^{-222}:\\ \;\;\;\;-60 \cdot \frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;a \cdot 120\\ \end{array} \]
Alternative 14
Error29.4
Cost192
\[a \cdot 120 \]

Error

Reproduce

herbie shell --seed 2022331 
(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)))