Average Error: 0.4 → 0.1
Time: 7.2s
Precision: binary64
\[\frac{60 \cdot \left(x - y\right)}{z - t} + a \cdot 120 \]
↓
\[\mathsf{fma}\left(120, a, 60 \cdot \left(\frac{x}{z - t} - \frac{y}{z - t}\right)\right) \]
(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 120.0 a (* 60.0 (- (/ x (- z t)) (/ y (- z t))))))
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(120.0, a, (60.0 * ((x / (z - t)) - (y / (z - t)))));
}

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(120.0, a, Float64(60.0 * Float64(Float64(x / Float64(z - t)) - Float64(y / Float64(z - t)))))
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[(120.0 * a + N[(60.0 * N[(N[(x / N[(z - t), $MachinePrecision]), $MachinePrecision] - N[(y / N[(z - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

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

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original0.4
Target0.1
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. Simplified0.4

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

  3. Taylor expanded in x around 0 0.1

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

  4. Taylor expanded in a around 0 0.1

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

  5. Simplified0.1

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

  6. Final simplification0.1

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

Reproduce

herbie shell --seed 2022131 
(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)))