Average Error: 45.0 → 0
Time: 5.0s
Precision: binary64
Cost: 64
\[\mathsf{fma}\left(x, y, z\right) - \left(1 + \left(x \cdot y + z\right)\right) \]
↓
\[-1 \]
(FPCore (x y z) :precision binary64 (- (fma x y z) (+ 1.0 (+ (* x y) z))))
↓
(FPCore (x y z) :precision binary64 -1.0)
double code(double x, double y, double z) {
	return fma(x, y, z) - (1.0 + ((x * y) + z));
}
↓
double code(double x, double y, double z) {
	return -1.0;
}

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

code[x_, y_, z_] := N[(N[(x * y + z), $MachinePrecision] - N[(1.0 + N[(N[(x * y), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]
↓
code[x_, y_, z_] := -1.0

\mathsf{fma}\left(x, y, z\right) - \left(1 + \left(x \cdot y + z\right)\right)
↓
-1

Error

Target

Original45.0
Target0
Herbie0
\[-1 \]

Derivation

  1. Initial program 45.0

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

  2. Simplified0

    \[\leadsto \color{blue}{-1} \]
    Proof
    -1: 0 points increase in error, 0 points decrease in error
    (Rewrite<= metadata-eval (-.f64 0 1)): 0 points increase in error, 0 points decrease in error
    (-.f64 (Rewrite<= +-inverses_binary64 (-.f64 (fma.f64 x y z) (fma.f64 x y z))) 1): 26 points increase in error, 0 points decrease in error
    (Rewrite=> associate--l-_binary64 (-.f64 (fma.f64 x y z) (+.f64 (fma.f64 x y z) 1))): 173 points increase in error, 0 points decrease in error
    (-.f64 (fma.f64 x y z) (Rewrite<= +-commutative_binary64 (+.f64 1 (fma.f64 x y z)))): 0 points increase in error, 0 points decrease in error
    (-.f64 (fma.f64 x y z) (+.f64 1 (Rewrite=> fma-udef_binary64 (+.f64 (*.f64 x y) z)))): 1 points increase in error, 1 points decrease in error

  3. Final simplification0

    \[\leadsto -1 \]

Reproduce

herbie shell --seed 2022291 
(FPCore (x y z)
  :name "simple fma test"
  :precision binary64

  :herbie-target
  -1.0

  (- (fma x y z) (+ 1.0 (+ (* x y) z))))