Rosa's FloatVsDoubleBenchmark

Average Error: 0.5 → 0.3

Time: 17.7s

Precision: binary64

Math

\[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

↓

\[\mathsf{fma}\left(x1, 3 \cdot \frac{x1 \cdot \left(3 \cdot {x1}^{2} - x1\right)}{{x1}^{2} + 1}, \mathsf{fma}\left(x1, x1, 1\right) \cdot \left(x1 + \left(\frac{\mathsf{fma}\left(2, x2, \mathsf{fma}\left(x1, x1 \cdot 3, -x1\right)\right)}{\mathsf{fma}\left(x1, x1, 1\right)} \cdot \left(x1 \cdot \left(\mathsf{fma}\left(6, \frac{x1}{\frac{\mathsf{fma}\left(x1, x1, 1\right)}{x1}}, \frac{4}{\frac{\mathsf{fma}\left(x1, x1, 1\right)}{x2}}\right) - \mathsf{fma}\left(2, \frac{x1}{\mathsf{fma}\left(x1, x1, 1\right)}, 6\right)\right) + 4 \cdot \left(x1 \cdot x1\right)\right) + x1 \cdot \left(x1 \cdot -6\right)\right)\right)\right) + \mathsf{fma}\left(3, \frac{x1 \cdot \left(x1 \cdot 3\right) - \mathsf{fma}\left(2, x2, x1\right)}{\mathsf{fma}\left(x1, x1, 1\right)}, x1\right) \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (+
  x1
  (+
   (+
    (+
     (+
      (*
       (+
        (*
         (*
          (* 2.0 x1)
          (/ (- (+ (* (* 3.0 x1) x1) (* 2.0 x2)) x1) (+ (* x1 x1) 1.0)))
         (- (/ (- (+ (* (* 3.0 x1) x1) (* 2.0 x2)) x1) (+ (* x1 x1) 1.0)) 3.0))
        (*
         (* x1 x1)
         (-
          (* 4.0 (/ (- (+ (* (* 3.0 x1) x1) (* 2.0 x2)) x1) (+ (* x1 x1) 1.0)))
          6.0)))
       (+ (* x1 x1) 1.0))
      (*
       (* (* 3.0 x1) x1)
       (/ (- (+ (* (* 3.0 x1) x1) (* 2.0 x2)) x1) (+ (* x1 x1) 1.0))))
     (* (* x1 x1) x1))
    x1)
   (* 3.0 (/ (- (- (* (* 3.0 x1) x1) (* 2.0 x2)) x1) (+ (* x1 x1) 1.0))))))

↓

(FPCore (x1 x2)
 :precision binary64
 (+
  (fma
   x1
   (* 3.0 (/ (* x1 (- (* 3.0 (pow x1 2.0)) x1)) (+ (pow x1 2.0) 1.0)))
   (*
    (fma x1 x1 1.0)
    (+
     x1
     (+
      (*
       (/ (fma 2.0 x2 (fma x1 (* x1 3.0) (- x1))) (fma x1 x1 1.0))
       (+
        (*
         x1
         (-
          (fma
           6.0
           (/ x1 (/ (fma x1 x1 1.0) x1))
           (/ 4.0 (/ (fma x1 x1 1.0) x2)))
          (fma 2.0 (/ x1 (fma x1 x1 1.0)) 6.0)))
        (* 4.0 (* x1 x1))))
      (* x1 (* x1 -6.0))))))
  (fma 3.0 (/ (- (* x1 (* x1 3.0)) (fma 2.0 x2 x1)) (fma x1 x1 1.0)) x1)))

C

double code(double x1, double x2) {
	return x1 + (((((((((2.0 * x1) * (((((3.0 * x1) * x1) + (2.0 * x2)) - x1) / ((x1 * x1) + 1.0))) * ((((((3.0 * x1) * x1) + (2.0 * x2)) - x1) / ((x1 * x1) + 1.0)) - 3.0)) + ((x1 * x1) * ((4.0 * (((((3.0 * x1) * x1) + (2.0 * x2)) - x1) / ((x1 * x1) + 1.0))) - 6.0))) * ((x1 * x1) + 1.0)) + (((3.0 * x1) * x1) * (((((3.0 * x1) * x1) + (2.0 * x2)) - x1) / ((x1 * x1) + 1.0)))) + ((x1 * x1) * x1)) + x1) + (3.0 * (((((3.0 * x1) * x1) - (2.0 * x2)) - x1) / ((x1 * x1) + 1.0))));
}

↓

double code(double x1, double x2) {
	return fma(x1, (3.0 * ((x1 * ((3.0 * pow(x1, 2.0)) - x1)) / (pow(x1, 2.0) + 1.0))), (fma(x1, x1, 1.0) * (x1 + (((fma(2.0, x2, fma(x1, (x1 * 3.0), -x1)) / fma(x1, x1, 1.0)) * ((x1 * (fma(6.0, (x1 / (fma(x1, x1, 1.0) / x1)), (4.0 / (fma(x1, x1, 1.0) / x2))) - fma(2.0, (x1 / fma(x1, x1, 1.0)), 6.0))) + (4.0 * (x1 * x1)))) + (x1 * (x1 * -6.0)))))) + fma(3.0, (((x1 * (x1 * 3.0)) - fma(2.0, x2, x1)) / fma(x1, x1, 1.0)), x1);
}

Julia

function code(x1, x2)
	return Float64(x1 + Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(2.0 * x1) * Float64(Float64(Float64(Float64(Float64(3.0 * x1) * x1) + Float64(2.0 * x2)) - x1) / Float64(Float64(x1 * x1) + 1.0))) * Float64(Float64(Float64(Float64(Float64(Float64(3.0 * x1) * x1) + Float64(2.0 * x2)) - x1) / Float64(Float64(x1 * x1) + 1.0)) - 3.0)) + Float64(Float64(x1 * x1) * Float64(Float64(4.0 * Float64(Float64(Float64(Float64(Float64(3.0 * x1) * x1) + Float64(2.0 * x2)) - x1) / Float64(Float64(x1 * x1) + 1.0))) - 6.0))) * Float64(Float64(x1 * x1) + 1.0)) + Float64(Float64(Float64(3.0 * x1) * x1) * Float64(Float64(Float64(Float64(Float64(3.0 * x1) * x1) + Float64(2.0 * x2)) - x1) / Float64(Float64(x1 * x1) + 1.0)))) + Float64(Float64(x1 * x1) * x1)) + x1) + Float64(3.0 * Float64(Float64(Float64(Float64(Float64(3.0 * x1) * x1) - Float64(2.0 * x2)) - x1) / Float64(Float64(x1 * x1) + 1.0)))))
end

↓

function code(x1, x2)
	return Float64(fma(x1, Float64(3.0 * Float64(Float64(x1 * Float64(Float64(3.0 * (x1 ^ 2.0)) - x1)) / Float64((x1 ^ 2.0) + 1.0))), Float64(fma(x1, x1, 1.0) * Float64(x1 + Float64(Float64(Float64(fma(2.0, x2, fma(x1, Float64(x1 * 3.0), Float64(-x1))) / fma(x1, x1, 1.0)) * Float64(Float64(x1 * Float64(fma(6.0, Float64(x1 / Float64(fma(x1, x1, 1.0) / x1)), Float64(4.0 / Float64(fma(x1, x1, 1.0) / x2))) - fma(2.0, Float64(x1 / fma(x1, x1, 1.0)), 6.0))) + Float64(4.0 * Float64(x1 * x1)))) + Float64(x1 * Float64(x1 * -6.0)))))) + fma(3.0, Float64(Float64(Float64(x1 * Float64(x1 * 3.0)) - fma(2.0, x2, x1)) / fma(x1, x1, 1.0)), x1))
end

Wolfram

code[x1_, x2_] := N[(x1 + N[(N[(N[(N[(N[(N[(N[(N[(N[(2.0 * x1), $MachinePrecision] * N[(N[(N[(N[(N[(3.0 * x1), $MachinePrecision] * x1), $MachinePrecision] + N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(N[(3.0 * x1), $MachinePrecision] * x1), $MachinePrecision] + N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision] + N[(N[(x1 * x1), $MachinePrecision] * N[(N[(4.0 * N[(N[(N[(N[(N[(3.0 * x1), $MachinePrecision] * x1), $MachinePrecision] + N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(3.0 * x1), $MachinePrecision] * x1), $MachinePrecision] * N[(N[(N[(N[(N[(3.0 * x1), $MachinePrecision] * x1), $MachinePrecision] + N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(x1 * x1), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision] + N[(3.0 * N[(N[(N[(N[(N[(3.0 * x1), $MachinePrecision] * x1), $MachinePrecision] - N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x1_, x2_] := N[(N[(x1 * N[(3.0 * N[(N[(x1 * N[(N[(3.0 * N[Power[x1, 2.0], $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision]), $MachinePrecision] / N[(N[Power[x1, 2.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(x1 * x1 + 1.0), $MachinePrecision] * N[(x1 + N[(N[(N[(N[(2.0 * x2 + N[(x1 * N[(x1 * 3.0), $MachinePrecision] + (-x1)), $MachinePrecision]), $MachinePrecision] / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(x1 * N[(N[(6.0 * N[(x1 / N[(N[(x1 * x1 + 1.0), $MachinePrecision] / x1), $MachinePrecision]), $MachinePrecision] + N[(4.0 / N[(N[(x1 * x1 + 1.0), $MachinePrecision] / x2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(2.0 * N[(x1 / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision] + 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(4.0 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x1 * N[(x1 * -6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(3.0 * N[(N[(N[(x1 * N[(x1 * 3.0), $MachinePrecision]), $MachinePrecision] - N[(2.0 * x2 + x1), $MachinePrecision]), $MachinePrecision] / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision]), $MachinePrecision]

Error

Bits error versus x1

Bits error versus x2

Derivation

1. Initial program 0.5

   \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

2. Simplified 0.3

   \[\leadsto \color{blue}{\mathsf{fma}\left(x1, x1 \cdot \frac{\mathsf{fma}\left(2, x2, \mathsf{fma}\left(x1, x1 \cdot 3, -x1\right)\right) \cdot 3}{\mathsf{fma}\left(x1, x1, 1\right)}, \mathsf{fma}\left(x1, x1, 1\right) \cdot \left(\left(\frac{\mathsf{fma}\left(2, x2, \mathsf{fma}\left(x1, x1 \cdot 3, -x1\right)\right)}{\mathsf{fma}\left(x1, x1, 1\right)} \cdot \left(x1 \cdot \left(-6 + 2 \cdot \frac{\mathsf{fma}\left(2, x2, \mathsf{fma}\left(x1, x1 \cdot 3, -x1\right)\right)}{\mathsf{fma}\left(x1, x1, 1\right)}\right) + \left(x1 \cdot x1\right) \cdot 4\right) + x1 \cdot \left(x1 \cdot -6\right)\right) + x1\right)\right) + \mathsf{fma}\left(3, \frac{x1 \cdot \left(x1 \cdot 3\right) - \mathsf{fma}\left(2, x2, x1\right)}{\mathsf{fma}\left(x1, x1, 1\right)}, x1\right)} \]

3. Taylor expanded in x2 around 0 0.3

   \[\leadsto \mathsf{fma}\left(x1, x1 \cdot \frac{\mathsf{fma}\left(2, x2, \mathsf{fma}\left(x1, x1 \cdot 3, -x1\right)\right) \cdot 3}{\mathsf{fma}\left(x1, x1, 1\right)}, \mathsf{fma}\left(x1, x1, 1\right) \cdot \left(\left(\frac{\mathsf{fma}\left(2, x2, \mathsf{fma}\left(x1, x1 \cdot 3, -x1\right)\right)}{\mathsf{fma}\left(x1, x1, 1\right)} \cdot \left(x1 \cdot \color{blue}{\left(\left(4 \cdot \frac{x2}{1 + {x1}^{2}} + 6 \cdot \frac{{x1}^{2}}{1 + {x1}^{2}}\right) - \left(2 \cdot \frac{x1}{1 + {x1}^{2}} + 6\right)\right)} + \left(x1 \cdot x1\right) \cdot 4\right) + x1 \cdot \left(x1 \cdot -6\right)\right) + x1\right)\right) + \mathsf{fma}\left(3, \frac{x1 \cdot \left(x1 \cdot 3\right) - \mathsf{fma}\left(2, x2, x1\right)}{\mathsf{fma}\left(x1, x1, 1\right)}, x1\right) \]

4. Simplified 0.3

   \[\leadsto \mathsf{fma}\left(x1, x1 \cdot \frac{\mathsf{fma}\left(2, x2, \mathsf{fma}\left(x1, x1 \cdot 3, -x1\right)\right) \cdot 3}{\mathsf{fma}\left(x1, x1, 1\right)}, \mathsf{fma}\left(x1, x1, 1\right) \cdot \left(\left(\frac{\mathsf{fma}\left(2, x2, \mathsf{fma}\left(x1, x1 \cdot 3, -x1\right)\right)}{\mathsf{fma}\left(x1, x1, 1\right)} \cdot \left(x1 \cdot \color{blue}{\left(\mathsf{fma}\left(6, \frac{x1}{\frac{\mathsf{fma}\left(x1, x1, 1\right)}{x1}}, \frac{4}{\frac{\mathsf{fma}\left(x1, x1, 1\right)}{x2}}\right) - \mathsf{fma}\left(2, \frac{x1}{\mathsf{fma}\left(x1, x1, 1\right)}, 6\right)\right)} + \left(x1 \cdot x1\right) \cdot 4\right) + x1 \cdot \left(x1 \cdot -6\right)\right) + x1\right)\right) + \mathsf{fma}\left(3, \frac{x1 \cdot \left(x1 \cdot 3\right) - \mathsf{fma}\left(2, x2, x1\right)}{\mathsf{fma}\left(x1, x1, 1\right)}, x1\right) \]

5. Taylor expanded in x2 around 0 0.3

   \[\leadsto \mathsf{fma}\left(x1, \color{blue}{3 \cdot \frac{x1 \cdot \left(3 \cdot {x1}^{2} - x1\right)}{1 + {x1}^{2}}}, \mathsf{fma}\left(x1, x1, 1\right) \cdot \left(\left(\frac{\mathsf{fma}\left(2, x2, \mathsf{fma}\left(x1, x1 \cdot 3, -x1\right)\right)}{\mathsf{fma}\left(x1, x1, 1\right)} \cdot \left(x1 \cdot \left(\mathsf{fma}\left(6, \frac{x1}{\frac{\mathsf{fma}\left(x1, x1, 1\right)}{x1}}, \frac{4}{\frac{\mathsf{fma}\left(x1, x1, 1\right)}{x2}}\right) - \mathsf{fma}\left(2, \frac{x1}{\mathsf{fma}\left(x1, x1, 1\right)}, 6\right)\right) + \left(x1 \cdot x1\right) \cdot 4\right) + x1 \cdot \left(x1 \cdot -6\right)\right) + x1\right)\right) + \mathsf{fma}\left(3, \frac{x1 \cdot \left(x1 \cdot 3\right) - \mathsf{fma}\left(2, x2, x1\right)}{\mathsf{fma}\left(x1, x1, 1\right)}, x1\right) \]

6. Final simplification 0.3

   \[\leadsto \mathsf{fma}\left(x1, 3 \cdot \frac{x1 \cdot \left(3 \cdot {x1}^{2} - x1\right)}{{x1}^{2} + 1}, \mathsf{fma}\left(x1, x1, 1\right) \cdot \left(x1 + \left(\frac{\mathsf{fma}\left(2, x2, \mathsf{fma}\left(x1, x1 \cdot 3, -x1\right)\right)}{\mathsf{fma}\left(x1, x1, 1\right)} \cdot \left(x1 \cdot \left(\mathsf{fma}\left(6, \frac{x1}{\frac{\mathsf{fma}\left(x1, x1, 1\right)}{x1}}, \frac{4}{\frac{\mathsf{fma}\left(x1, x1, 1\right)}{x2}}\right) - \mathsf{fma}\left(2, \frac{x1}{\mathsf{fma}\left(x1, x1, 1\right)}, 6\right)\right) + 4 \cdot \left(x1 \cdot x1\right)\right) + x1 \cdot \left(x1 \cdot -6\right)\right)\right)\right) + \mathsf{fma}\left(3, \frac{x1 \cdot \left(x1 \cdot 3\right) - \mathsf{fma}\left(2, x2, x1\right)}{\mathsf{fma}\left(x1, x1, 1\right)}, x1\right) \]

Reproduce

herbie shell --seed 2022151 
(FPCore (x1 x2)
  :name "Rosa's FloatVsDoubleBenchmark"
  :precision binary64
  (+ x1 (+ (+ (+ (+ (* (+ (* (* (* 2.0 x1) (/ (- (+ (* (* 3.0 x1) x1) (* 2.0 x2)) x1) (+ (* x1 x1) 1.0))) (- (/ (- (+ (* (* 3.0 x1) x1) (* 2.0 x2)) x1) (+ (* x1 x1) 1.0)) 3.0)) (* (* x1 x1) (- (* 4.0 (/ (- (+ (* (* 3.0 x1) x1) (* 2.0 x2)) x1) (+ (* x1 x1) 1.0))) 6.0))) (+ (* x1 x1) 1.0)) (* (* (* 3.0 x1) x1) (/ (- (+ (* (* 3.0 x1) x1) (* 2.0 x2)) x1) (+ (* x1 x1) 1.0)))) (* (* x1 x1) x1)) x1) (* 3.0 (/ (- (- (* (* 3.0 x1) x1) (* 2.0 x2)) x1) (+ (* x1 x1) 1.0))))))