Average Error: 0.0 → 0.0
Time: 4.2s
Precision: binary64
\[\frac{x - y}{z - y} \]
↓
\[\begin{array}{l} t_0 := \frac{y}{z - y}\\ \left(\frac{x}{z - y} - t_0\right) + \mathsf{fma}\left(-t_0, 1, t_0\right) \end{array} \]
(FPCore (x y z) :precision binary64 (/ (- x y) (- z y)))
↓
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ y (- z y)))) (+ (- (/ x (- z y)) t_0) (fma (- t_0) 1.0 t_0))))
double code(double x, double y, double z) {
	return (x - y) / (z - y);
}
↓
double code(double x, double y, double z) {
	double t_0 = y / (z - y);
	return ((x / (z - y)) - t_0) + fma(-t_0, 1.0, t_0);
}

function code(x, y, z)
	return Float64(Float64(x - y) / Float64(z - y))
end
↓
function code(x, y, z)
	t_0 = Float64(y / Float64(z - y))
	return Float64(Float64(Float64(x / Float64(z - y)) - t_0) + fma(Float64(-t_0), 1.0, t_0))
end

code[x_, y_, z_] := N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]
↓
code[x_, y_, z_] := Block[{t$95$0 = N[(y / N[(z - y), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(x / N[(z - y), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision] + N[((-t$95$0) * 1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]]

\frac{x - y}{z - y}
↓
\begin{array}{l}
t_0 := \frac{y}{z - y}\\
\left(\frac{x}{z - y} - t_0\right) + \mathsf{fma}\left(-t_0, 1, t_0\right)
\end{array}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original0.0
Target0.0
Herbie0.0
\[\frac{x}{z - y} - \frac{y}{z - y} \]

Derivation

  1. Initial program 0.0

    \[\frac{x - y}{z - y} \]

  2. Applied egg-rr0.0

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2022133 
(FPCore (x y z)
  :name "Graphics.Rasterific.Shading:$sgradientColorAt from Rasterific-0.6.1"
  :precision binary64

  :herbie-target
  (- (/ x (- z y)) (/ y (- z y)))

  (/ (- x y) (- z y)))