Average Error: 2.0 → 0.2
Time: 5.7s
Precision: binary64
\[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
↓
\[\begin{array}{l} t_1 := \left(1 + t\right) - z\\ \mathsf{fma}\left(a, \frac{z}{t_1} - \frac{y}{t_1}, x\right) \end{array} \]
(FPCore (x y z t a)
 :precision binary64
 (- x (/ (- y z) (/ (+ (- t z) 1.0) a))))
↓
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (+ 1.0 t) z))) (fma a (- (/ z t_1) (/ y t_1)) x)))
double code(double x, double y, double z, double t, double a) {
	return x - ((y - z) / (((t - z) + 1.0) / a));
}
↓
double code(double x, double y, double z, double t, double a) {
	double t_1 = (1.0 + t) - z;
	return fma(a, ((z / t_1) - (y / t_1)), x);
}

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

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

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

Error

Target

Original2.0
Target0.2
Herbie0.2
\[x - \frac{y - z}{\left(t - z\right) + 1} \cdot a \]

Derivation

  1. Initial program 2.0

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

  2. Simplified1.8

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

  3. Taylor expanded in a around 0 0.2

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

  4. Simplified0.2

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2022210 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Chart.SparkLine:renderSparkLine from Chart-1.5.3"
  :precision binary64

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

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