Average Error: 7.4 → 0.1
Time: 6.9s
Precision: binary64
\[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
↓
\[\frac{y}{\left(x + 1\right) \cdot \left(t - \frac{x}{z}\right)} + \frac{x - \frac{x}{\mathsf{fma}\left(z, t, -x\right)}}{x + 1} \]
(FPCore (x y z t)
 :precision binary64
 (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0)))
↓
(FPCore (x y z t)
 :precision binary64
 (+
  (/ y (* (+ x 1.0) (- t (/ x z))))
  (/ (- x (/ x (fma z t (- x)))) (+ x 1.0))))
double code(double x, double y, double z, double t) {
	return (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0);
}
↓
double code(double x, double y, double z, double t) {
	return (y / ((x + 1.0) * (t - (x / z)))) + ((x - (x / fma(z, t, -x))) / (x + 1.0));
}

function code(x, y, z, t)
	return Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / Float64(Float64(t * z) - x))) / Float64(x + 1.0))
end
↓
function code(x, y, z, t)
	return Float64(Float64(y / Float64(Float64(x + 1.0) * Float64(t - Float64(x / z)))) + Float64(Float64(x - Float64(x / fma(z, t, Float64(-x)))) / Float64(x + 1.0)))
end

code[x_, y_, z_, t_] := N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]
↓
code[x_, y_, z_, t_] := N[(N[(y / N[(N[(x + 1.0), $MachinePrecision] * N[(t - N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(x - N[(x / N[(z * t + (-x)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}
↓
\frac{y}{\left(x + 1\right) \cdot \left(t - \frac{x}{z}\right)} + \frac{x - \frac{x}{\mathsf{fma}\left(z, t, -x\right)}}{x + 1}

Error

Target

Original7.4
Target0.4
Herbie0.1
\[\frac{x + \left(\frac{y}{t - \frac{x}{z}} - \frac{x}{t \cdot z - x}\right)}{x + 1} \]

Derivation

  1. Initial program 7.4

    \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]

  2. Taylor expanded in y around 0 7.4

    \[\leadsto \frac{x + \color{blue}{\left(\frac{y \cdot z}{t \cdot z - x} + -1 \cdot \frac{x}{t \cdot z - x}\right)}}{x + 1} \]

  3. Simplified2.3

    \[\leadsto \frac{x + \color{blue}{\left(\frac{y}{\frac{t \cdot z - x}{z}} - \frac{x}{t \cdot z - x}\right)}}{x + 1} \]

  4. Taylor expanded in t around 0 0.4

    \[\leadsto \frac{x + \left(\frac{y}{\color{blue}{t + -1 \cdot \frac{x}{z}}} - \frac{x}{t \cdot z - x}\right)}{x + 1} \]

  5. Simplified0.4

    \[\leadsto \frac{x + \left(\frac{y}{\color{blue}{t - \frac{x}{z}}} - \frac{x}{t \cdot z - x}\right)}{x + 1} \]

  6. Taylor expanded in y around 0 0.1

    \[\leadsto \color{blue}{\left(\frac{y}{\left(t - \frac{x}{z}\right) \cdot \left(1 + x\right)} + \frac{x}{1 + x}\right) - \frac{x}{\left(t \cdot z - x\right) \cdot \left(1 + x\right)}} \]

  7. Simplified0.1

    \[\leadsto \color{blue}{\frac{y}{\left(x + 1\right) \cdot \left(t - \frac{x}{z}\right)} + \frac{x - \frac{x}{\mathsf{fma}\left(z, t, -x\right)}}{x + 1}} \]

  8. Final simplification0.1

    \[\leadsto \frac{y}{\left(x + 1\right) \cdot \left(t - \frac{x}{z}\right)} + \frac{x - \frac{x}{\mathsf{fma}\left(z, t, -x\right)}}{x + 1} \]

Reproduce

herbie shell --seed 2022181 
(FPCore (x y z t)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, A"
  :precision binary64

  :herbie-target
  (/ (+ x (- (/ y (- t (/ x z))) (/ x (- (* t z) x)))) (+ x 1.0))

  (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0)))