Average Error: 0.0 → 0.1
Time: 16.0s
Precision: 64
\[\frac{x - y}{z - y}\]
↓
\[\frac{x}{z - y} - \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{z - y} \cdot y\right)\right)\]
\frac{x - y}{z - y}
↓
\frac{x}{z - y} - \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{z - y} \cdot y\right)\right)
double f(double x, double y, double z) {
        double r25649928 = x;
        double r25649929 = y;
        double r25649930 = r25649928 - r25649929;
        double r25649931 = z;
        double r25649932 = r25649931 - r25649929;
        double r25649933 = r25649930 / r25649932;
        return r25649933;
}
↓
double f(double x, double y, double z) {
        double r25649934 = x;
        double r25649935 = z;
        double r25649936 = y;
        double r25649937 = r25649935 - r25649936;
        double r25649938 = r25649934 / r25649937;
        double r25649939 = 1.0;
        double r25649940 = r25649939 / r25649937;
        double r25649941 = r25649940 * r25649936;
        double r25649942 = expm1(r25649941);
        double r25649943 = log1p(r25649942);
        double r25649944 = r25649938 - r25649943;
        return r25649944;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

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Results

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Target

Original0.0
Target0.0
Herbie0.1
\[\frac{x}{z - y} - \frac{y}{z - y}\]

Derivation

  1. Initial program 0.0

    \[\frac{x - y}{z - y}\]
  2. Using strategy rm

  3. Applied div-sub0.0

    \[\leadsto \color{blue}{\frac{x}{z - y} - \frac{y}{z - y}}\]
  4. Using strategy rm

  5. Applied log1p-expm1-u0.0

    \[\leadsto \frac{x}{z - y} - \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{y}{z - y}\right)\right)}\]
  6. Using strategy rm

  7. Applied div-inv0.1

    \[\leadsto \frac{x}{z - y} - \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{y \cdot \frac{1}{z - y}}\right)\right)\]

  8. Final simplification0.1

    \[\leadsto \frac{x}{z - y} - \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{z - y} \cdot y\right)\right)\]

Reproduce

herbie shell --seed 2019179 +o rules:numerics
(FPCore (x y z)
  :name "Graphics.Rasterific.Shading:$sgradientColorAt from Rasterific-0.6.1"

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

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