Average Error: 1.4 → 1.3
Time: 9.8s
Precision: binary64
\[x + y \cdot \frac{z - t}{z - a} \]
↓
\[\begin{array}{l} t_1 := \frac{-z}{t - z}\\ x + \frac{y}{\mathsf{fma}\left(a, \frac{1}{t - z}, t_1\right) + \mathsf{fma}\left(t_1, 1, \frac{z}{t - z}\right)} \end{array} \]
(FPCore (x y z t a) :precision binary64 (+ x (* y (/ (- z t) (- z a)))))
↓
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z) (- t z))))
   (+ x (/ y (+ (fma a (/ 1.0 (- t z)) t_1) (fma t_1 1.0 (/ z (- t z))))))))
double code(double x, double y, double z, double t, double a) {
	return x + (y * ((z - t) / (z - a)));
}
↓
double code(double x, double y, double z, double t, double a) {
	double t_1 = -z / (t - z);
	return x + (y / (fma(a, (1.0 / (t - z)), t_1) + fma(t_1, 1.0, (z / (t - z)))));
}

function code(x, y, z, t, a)
	return Float64(x + Float64(y * Float64(Float64(z - t) / Float64(z - a))))
end
↓
function code(x, y, z, t, a)
	t_1 = Float64(Float64(-z) / Float64(t - z))
	return Float64(x + Float64(y / Float64(fma(a, Float64(1.0 / Float64(t - z)), t_1) + fma(t_1, 1.0, Float64(z / Float64(t - z))))))
end

code[x_, y_, z_, t_, a_] := N[(x + N[(y * N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]
↓
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[((-z) / N[(t - z), $MachinePrecision]), $MachinePrecision]}, N[(x + N[(y / N[(N[(a * N[(1.0 / N[(t - z), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] + N[(t$95$1 * 1.0 + N[(z / N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

x + y \cdot \frac{z - t}{z - a}
↓
\begin{array}{l}
t_1 := \frac{-z}{t - z}\\
x + \frac{y}{\mathsf{fma}\left(a, \frac{1}{t - z}, t_1\right) + \mathsf{fma}\left(t_1, 1, \frac{z}{t - z}\right)}
\end{array}

Error

Target

Original1.4
Target1.3
Herbie1.3
\[x + \frac{y}{\frac{z - a}{z - t}} \]

Derivation

  1. Initial program 1.4

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

  2. Simplified1.4

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

  3. Taylor expanded in y around 0 1.4

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

  4. Simplified1.3

    \[\leadsto \color{blue}{x + \frac{y}{\frac{a - z}{t - z}}} \]

  5. Applied egg-rr1.6

    \[\leadsto x + \frac{y}{\color{blue}{\sqrt[3]{\frac{a - z}{t - z}} \cdot {\left(\sqrt[3]{\frac{a - z}{t - z}}\right)}^{2}}} \]

  6. Applied egg-rr1.3

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

  7. Final simplification1.3

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

Reproduce

herbie shell --seed 2022210 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisLine from plot-0.2.3.4, A"
  :precision binary64

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

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