Diagrams.Solve.Tridiagonal:solveTriDiagonal from diagrams-solve-0.1, C

Average Accuracy: 100.0% → 100.0%

Time: 5.0s

Precision: binary64

Cost: 6656

Math

\[x - y \cdot z \]

↓

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

FPCore

(FPCore (x y z) :precision binary64 (- x (* y z)))

↓

(FPCore (x y z) :precision binary64 (fma (- y) z x))

C

double code(double x, double y, double z) {
	return x - (y * z);
}

↓

double code(double x, double y, double z) {
	return fma(-y, z, x);
}

Julia

function code(x, y, z)
	return Float64(x - Float64(y * z))
end

↓

function code(x, y, z)
	return fma(Float64(-y), z, x)
end

Wolfram

code[x_, y_, z_] := N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision]

↓

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

Target

Original	100.0%
Target	100.0%
Herbie	100.0%

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

Derivation

1. Initial program 100.0%

   \[x - y \cdot z \]

2. Applied egg-rr 36.7%

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

   [Start] 100.0	\[ x - y \cdot z \]
   sub-neg [=>] 100.0	\[ \color{blue}{x + \left(-y \cdot z\right)} \]
   flip3-+ [=>] 36.7	\[ \color{blue}{\frac{{x}^{3} + {\left(-y \cdot z\right)}^{3}}{x \cdot x + \left(\left(-y \cdot z\right) \cdot \left(-y \cdot z\right) - x \cdot \left(-y \cdot z\right)\right)}} \]
   distribute-rgt-neg-in [=>] 36.7	\[ \frac{{x}^{3} + {\color{blue}{\left(y \cdot \left(-z\right)\right)}}^{3}}{x \cdot x + \left(\left(-y \cdot z\right) \cdot \left(-y \cdot z\right) - x \cdot \left(-y \cdot z\right)\right)} \]
   distribute-rgt-neg-in [=>] 36.7	\[ \frac{{x}^{3} + {\left(y \cdot \left(-z\right)\right)}^{3}}{x \cdot x + \left(\color{blue}{\left(y \cdot \left(-z\right)\right)} \cdot \left(-y \cdot z\right) - x \cdot \left(-y \cdot z\right)\right)} \]
   distribute-rgt-neg-in [=>] 36.7	\[ \frac{{x}^{3} + {\left(y \cdot \left(-z\right)\right)}^{3}}{x \cdot x + \left(\left(y \cdot \left(-z\right)\right) \cdot \color{blue}{\left(y \cdot \left(-z\right)\right)} - x \cdot \left(-y \cdot z\right)\right)} \]
   distribute-rgt-neg-in [=>] 36.7	\[ \frac{{x}^{3} + {\left(y \cdot \left(-z\right)\right)}^{3}}{x \cdot x + \left(\left(y \cdot \left(-z\right)\right) \cdot \left(y \cdot \left(-z\right)\right) - x \cdot \color{blue}{\left(y \cdot \left(-z\right)\right)}\right)} \]

3. Applied egg-rr 9.7%

   \[\leadsto \frac{\color{blue}{\sqrt{{\left(y \cdot z\right)}^{6} + {x}^{6}}}}{x \cdot x + \left(\left(y \cdot \left(-z\right)\right) \cdot \left(y \cdot \left(-z\right)\right) - x \cdot \left(y \cdot \left(-z\right)\right)\right)} \]
   Proof

   [Start] 36.7	\[ \frac{{x}^{3} + {\left(y \cdot \left(-z\right)\right)}^{3}}{x \cdot x + \left(\left(y \cdot \left(-z\right)\right) \cdot \left(y \cdot \left(-z\right)\right) - x \cdot \left(y \cdot \left(-z\right)\right)\right)} \]
   add-sqr-sqrt [=>] 18.2	\[ \frac{\color{blue}{\sqrt{{x}^{3} + {\left(y \cdot \left(-z\right)\right)}^{3}} \cdot \sqrt{{x}^{3} + {\left(y \cdot \left(-z\right)\right)}^{3}}}}{x \cdot x + \left(\left(y \cdot \left(-z\right)\right) \cdot \left(y \cdot \left(-z\right)\right) - x \cdot \left(y \cdot \left(-z\right)\right)\right)} \]
   sqrt-unprod [=>] 9.8	\[ \frac{\color{blue}{\sqrt{\left({x}^{3} + {\left(y \cdot \left(-z\right)\right)}^{3}\right) \cdot \left({x}^{3} + {\left(y \cdot \left(-z\right)\right)}^{3}\right)}}}{x \cdot x + \left(\left(y \cdot \left(-z\right)\right) \cdot \left(y \cdot \left(-z\right)\right) - x \cdot \left(y \cdot \left(-z\right)\right)\right)} \]
   distribute-rgt-neg-out [=>] 9.8	\[ \frac{\sqrt{\left({x}^{3} + {\left(y \cdot \left(-z\right)\right)}^{3}\right) \cdot \left({x}^{3} + {\color{blue}{\left(-y \cdot z\right)}}^{3}\right)}}{x \cdot x + \left(\left(y \cdot \left(-z\right)\right) \cdot \left(y \cdot \left(-z\right)\right) - x \cdot \left(y \cdot \left(-z\right)\right)\right)} \]
   cube-neg [=>] 9.8	\[ \frac{\sqrt{\left({x}^{3} + {\left(y \cdot \left(-z\right)\right)}^{3}\right) \cdot \left({x}^{3} + \color{blue}{\left(-{\left(y \cdot z\right)}^{3}\right)}\right)}}{x \cdot x + \left(\left(y \cdot \left(-z\right)\right) \cdot \left(y \cdot \left(-z\right)\right) - x \cdot \left(y \cdot \left(-z\right)\right)\right)} \]
   add-sqr-sqrt [=>] 4.9	\[ \frac{\sqrt{\left({x}^{3} + {\left(y \cdot \left(-z\right)\right)}^{3}\right) \cdot \left({x}^{3} + \left(-{\left(y \cdot \color{blue}{\left(\sqrt{z} \cdot \sqrt{z}\right)}\right)}^{3}\right)\right)}}{x \cdot x + \left(\left(y \cdot \left(-z\right)\right) \cdot \left(y \cdot \left(-z\right)\right) - x \cdot \left(y \cdot \left(-z\right)\right)\right)} \]
   sqrt-unprod [=>] 6.4	\[ \frac{\sqrt{\left({x}^{3} + {\left(y \cdot \left(-z\right)\right)}^{3}\right) \cdot \left({x}^{3} + \left(-{\left(y \cdot \color{blue}{\sqrt{z \cdot z}}\right)}^{3}\right)\right)}}{x \cdot x + \left(\left(y \cdot \left(-z\right)\right) \cdot \left(y \cdot \left(-z\right)\right) - x \cdot \left(y \cdot \left(-z\right)\right)\right)} \]
   sqr-neg [<=] 6.4	\[ \frac{\sqrt{\left({x}^{3} + {\left(y \cdot \left(-z\right)\right)}^{3}\right) \cdot \left({x}^{3} + \left(-{\left(y \cdot \sqrt{\color{blue}{\left(-z\right) \cdot \left(-z\right)}}\right)}^{3}\right)\right)}}{x \cdot x + \left(\left(y \cdot \left(-z\right)\right) \cdot \left(y \cdot \left(-z\right)\right) - x \cdot \left(y \cdot \left(-z\right)\right)\right)} \]
   sqrt-unprod [<=] 2.7	\[ \frac{\sqrt{\left({x}^{3} + {\left(y \cdot \left(-z\right)\right)}^{3}\right) \cdot \left({x}^{3} + \left(-{\left(y \cdot \color{blue}{\left(\sqrt{-z} \cdot \sqrt{-z}\right)}\right)}^{3}\right)\right)}}{x \cdot x + \left(\left(y \cdot \left(-z\right)\right) \cdot \left(y \cdot \left(-z\right)\right) - x \cdot \left(y \cdot \left(-z\right)\right)\right)} \]
   add-sqr-sqrt [<=] 5.3	\[ \frac{\sqrt{\left({x}^{3} + {\left(y \cdot \left(-z\right)\right)}^{3}\right) \cdot \left({x}^{3} + \left(-{\left(y \cdot \color{blue}{\left(-z\right)}\right)}^{3}\right)\right)}}{x \cdot x + \left(\left(y \cdot \left(-z\right)\right) \cdot \left(y \cdot \left(-z\right)\right) - x \cdot \left(y \cdot \left(-z\right)\right)\right)} \]
   sub-neg [<=] 5.3	\[ \frac{\sqrt{\left({x}^{3} + {\left(y \cdot \left(-z\right)\right)}^{3}\right) \cdot \color{blue}{\left({x}^{3} - {\left(y \cdot \left(-z\right)\right)}^{3}\right)}}}{x \cdot x + \left(\left(y \cdot \left(-z\right)\right) \cdot \left(y \cdot \left(-z\right)\right) - x \cdot \left(y \cdot \left(-z\right)\right)\right)} \]
   difference-of-squares [<=] 5.3	\[ \frac{\sqrt{\color{blue}{{x}^{3} \cdot {x}^{3} - {\left(y \cdot \left(-z\right)\right)}^{3} \cdot {\left(y \cdot \left(-z\right)\right)}^{3}}}}{x \cdot x + \left(\left(y \cdot \left(-z\right)\right) \cdot \left(y \cdot \left(-z\right)\right) - x \cdot \left(y \cdot \left(-z\right)\right)\right)} \]
   cancel-sign-sub-inv [=>] 5.3	\[ \frac{\sqrt{\color{blue}{{x}^{3} \cdot {x}^{3} + \left(-{\left(y \cdot \left(-z\right)\right)}^{3}\right) \cdot {\left(y \cdot \left(-z\right)\right)}^{3}}}}{x \cdot x + \left(\left(y \cdot \left(-z\right)\right) \cdot \left(y \cdot \left(-z\right)\right) - x \cdot \left(y \cdot \left(-z\right)\right)\right)} \]

4. Applied egg-rr 9.7%

   \[\leadsto \frac{\sqrt{{\left(y \cdot z\right)}^{6} + {x}^{6}}}{x \cdot x + \color{blue}{\left(y \cdot z\right) \cdot \left(x + y \cdot z\right)}} \]
   Proof

   [Start] 9.7	\[ \frac{\sqrt{{\left(y \cdot z\right)}^{6} + {x}^{6}}}{x \cdot x + \left(\left(y \cdot \left(-z\right)\right) \cdot \left(y \cdot \left(-z\right)\right) - x \cdot \left(y \cdot \left(-z\right)\right)\right)} \]
   sub-neg [=>] 9.7	\[ \frac{\sqrt{{\left(y \cdot z\right)}^{6} + {x}^{6}}}{x \cdot x + \color{blue}{\left(\left(y \cdot \left(-z\right)\right) \cdot \left(y \cdot \left(-z\right)\right) + \left(-x \cdot \left(y \cdot \left(-z\right)\right)\right)\right)}} \]
   associate-*r* [=>] 9.6	\[ \frac{\sqrt{{\left(y \cdot z\right)}^{6} + {x}^{6}}}{x \cdot x + \left(\left(y \cdot \left(-z\right)\right) \cdot \left(y \cdot \left(-z\right)\right) + \left(-\color{blue}{\left(x \cdot y\right) \cdot \left(-z\right)}\right)\right)} \]
   distribute-rgt-neg-in [=>] 9.6	\[ \frac{\sqrt{{\left(y \cdot z\right)}^{6} + {x}^{6}}}{x \cdot x + \left(\left(y \cdot \left(-z\right)\right) \cdot \left(y \cdot \left(-z\right)\right) + \color{blue}{\left(x \cdot y\right) \cdot \left(-\left(-z\right)\right)}\right)} \]
   add-sqr-sqrt [=>] 4.8	\[ \frac{\sqrt{{\left(y \cdot z\right)}^{6} + {x}^{6}}}{x \cdot x + \left(\left(y \cdot \left(-z\right)\right) \cdot \left(y \cdot \left(-z\right)\right) + \left(x \cdot y\right) \cdot \left(-\color{blue}{\sqrt{-z} \cdot \sqrt{-z}}\right)\right)} \]
   sqrt-unprod [=>] 8.3	\[ \frac{\sqrt{{\left(y \cdot z\right)}^{6} + {x}^{6}}}{x \cdot x + \left(\left(y \cdot \left(-z\right)\right) \cdot \left(y \cdot \left(-z\right)\right) + \left(x \cdot y\right) \cdot \left(-\color{blue}{\sqrt{\left(-z\right) \cdot \left(-z\right)}}\right)\right)} \]
   sqr-neg [=>] 8.3	\[ \frac{\sqrt{{\left(y \cdot z\right)}^{6} + {x}^{6}}}{x \cdot x + \left(\left(y \cdot \left(-z\right)\right) \cdot \left(y \cdot \left(-z\right)\right) + \left(x \cdot y\right) \cdot \left(-\sqrt{\color{blue}{z \cdot z}}\right)\right)} \]
   sqrt-unprod [<=] 4.8	\[ \frac{\sqrt{{\left(y \cdot z\right)}^{6} + {x}^{6}}}{x \cdot x + \left(\left(y \cdot \left(-z\right)\right) \cdot \left(y \cdot \left(-z\right)\right) + \left(x \cdot y\right) \cdot \left(-\color{blue}{\sqrt{z} \cdot \sqrt{z}}\right)\right)} \]
   add-sqr-sqrt [<=] 9.6	\[ \frac{\sqrt{{\left(y \cdot z\right)}^{6} + {x}^{6}}}{x \cdot x + \left(\left(y \cdot \left(-z\right)\right) \cdot \left(y \cdot \left(-z\right)\right) + \left(x \cdot y\right) \cdot \left(-\color{blue}{z}\right)\right)} \]
   associate-*r* [<=] 9.6	\[ \frac{\sqrt{{\left(y \cdot z\right)}^{6} + {x}^{6}}}{x \cdot x + \left(\left(y \cdot \left(-z\right)\right) \cdot \left(y \cdot \left(-z\right)\right) + \color{blue}{x \cdot \left(y \cdot \left(-z\right)\right)}\right)} \]
   distribute-rgt-out [=>] 9.6	\[ \frac{\sqrt{{\left(y \cdot z\right)}^{6} + {x}^{6}}}{x \cdot x + \color{blue}{\left(y \cdot \left(-z\right)\right) \cdot \left(y \cdot \left(-z\right) + x\right)}} \]
   add-sqr-sqrt [=>] 4.8	\[ \frac{\sqrt{{\left(y \cdot z\right)}^{6} + {x}^{6}}}{x \cdot x + \left(y \cdot \color{blue}{\left(\sqrt{-z} \cdot \sqrt{-z}\right)}\right) \cdot \left(y \cdot \left(-z\right) + x\right)} \]
   sqrt-unprod [=>] 7.5	\[ \frac{\sqrt{{\left(y \cdot z\right)}^{6} + {x}^{6}}}{x \cdot x + \left(y \cdot \color{blue}{\sqrt{\left(-z\right) \cdot \left(-z\right)}}\right) \cdot \left(y \cdot \left(-z\right) + x\right)} \]
   sqr-neg [=>] 7.5	\[ \frac{\sqrt{{\left(y \cdot z\right)}^{6} + {x}^{6}}}{x \cdot x + \left(y \cdot \sqrt{\color{blue}{z \cdot z}}\right) \cdot \left(y \cdot \left(-z\right) + x\right)} \]
   sqrt-unprod [<=] 5.1	\[ \frac{\sqrt{{\left(y \cdot z\right)}^{6} + {x}^{6}}}{x \cdot x + \left(y \cdot \color{blue}{\left(\sqrt{z} \cdot \sqrt{z}\right)}\right) \cdot \left(y \cdot \left(-z\right) + x\right)} \]
   add-sqr-sqrt [<=] 10.0	\[ \frac{\sqrt{{\left(y \cdot z\right)}^{6} + {x}^{6}}}{x \cdot x + \left(y \cdot \color{blue}{z}\right) \cdot \left(y \cdot \left(-z\right) + x\right)} \]
   +-commutative [<=] 10.0	\[ \frac{\sqrt{{\left(y \cdot z\right)}^{6} + {x}^{6}}}{x \cdot x + \left(y \cdot z\right) \cdot \color{blue}{\left(x + y \cdot \left(-z\right)\right)}} \]
   add-sqr-sqrt [=>] 4.8	\[ \frac{\sqrt{{\left(y \cdot z\right)}^{6} + {x}^{6}}}{x \cdot x + \left(y \cdot z\right) \cdot \left(x + y \cdot \color{blue}{\left(\sqrt{-z} \cdot \sqrt{-z}\right)}\right)} \]
   sqrt-unprod [=>] 7.7	\[ \frac{\sqrt{{\left(y \cdot z\right)}^{6} + {x}^{6}}}{x \cdot x + \left(y \cdot z\right) \cdot \left(x + y \cdot \color{blue}{\sqrt{\left(-z\right) \cdot \left(-z\right)}}\right)} \]
   sqr-neg [=>] 7.7	\[ \frac{\sqrt{{\left(y \cdot z\right)}^{6} + {x}^{6}}}{x \cdot x + \left(y \cdot z\right) \cdot \left(x + y \cdot \sqrt{\color{blue}{z \cdot z}}\right)} \]
   sqrt-unprod [<=] 4.9	\[ \frac{\sqrt{{\left(y \cdot z\right)}^{6} + {x}^{6}}}{x \cdot x + \left(y \cdot z\right) \cdot \left(x + y \cdot \color{blue}{\left(\sqrt{z} \cdot \sqrt{z}\right)}\right)} \]
   add-sqr-sqrt [<=] 9.7	\[ \frac{\sqrt{{\left(y \cdot z\right)}^{6} + {x}^{6}}}{x \cdot x + \left(y \cdot z\right) \cdot \left(x + y \cdot \color{blue}{z}\right)} \]

5. Taylor expanded in y around 0 100.0%

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

6. Simplified 100.0%

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

   [Start] 100.0	\[ -1 \cdot \left(y \cdot z\right) + x \]
   associate-*r* [=>] 100.0	\[ \color{blue}{\left(-1 \cdot y\right) \cdot z} + x \]
   fma-def [=>] 100.0	\[ \color{blue}{\mathsf{fma}\left(-1 \cdot y, z, x\right)} \]
   mul-1-neg [=>] 100.0	\[ \mathsf{fma}\left(\color{blue}{-y}, z, x\right) \]

7. Final simplification 100.0%

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

Alternatives

Alternative 1
Accuracy	72.8%
Cost	520

\[\begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{+51}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-46}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 2
Accuracy	100.0%
Cost	320

\[x - y \cdot z \]

Alternative 3
Accuracy	58.5%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023143 
(FPCore (x y z)
  :name "Diagrams.Solve.Tridiagonal:solveTriDiagonal from diagrams-solve-0.1, C"
  :precision binary64

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

  (- x (* y z)))