Average Error: 61.8 → 0.2
Time: 2.3s
Precision: binary64
\[0.9 \leq t \land t \leq 1.1\]
\[\left(1 + t \cdot 2 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right) \]
↓
\[\sqrt{\sqrt{2.56 \cdot 10^{-126} \cdot {t}^{8}}} \]
(FPCore (t)
 :precision binary64
 (+ (* (+ 1.0 (* t 2e-16)) (+ 1.0 (* t 2e-16))) (- -1.0 (* 2.0 (* t 2e-16)))))
↓
(FPCore (t) :precision binary64 (sqrt (sqrt (* 2.56e-126 (pow t 8.0)))))
double code(double t) {
	return ((1.0 + (t * 2e-16)) * (1.0 + (t * 2e-16))) + (-1.0 - (2.0 * (t * 2e-16)));
}
↓
double code(double t) {
	return sqrt(sqrt((2.56e-126 * pow(t, 8.0))));
}

real(8) function code(t)
    real(8), intent (in) :: t
    code = ((1.0d0 + (t * 2d-16)) * (1.0d0 + (t * 2d-16))) + ((-1.0d0) - (2.0d0 * (t * 2d-16)))
end function
↓
real(8) function code(t)
    real(8), intent (in) :: t
    code = sqrt(sqrt((2.56d-126 * (t ** 8.0d0))))
end function

public static double code(double t) {
	return ((1.0 + (t * 2e-16)) * (1.0 + (t * 2e-16))) + (-1.0 - (2.0 * (t * 2e-16)));
}
↓
public static double code(double t) {
	return Math.sqrt(Math.sqrt((2.56e-126 * Math.pow(t, 8.0))));
}

def code(t):
	return ((1.0 + (t * 2e-16)) * (1.0 + (t * 2e-16))) + (-1.0 - (2.0 * (t * 2e-16)))
↓
def code(t):
	return math.sqrt(math.sqrt((2.56e-126 * math.pow(t, 8.0))))

function code(t)
	return Float64(Float64(Float64(1.0 + Float64(t * 2e-16)) * Float64(1.0 + Float64(t * 2e-16))) + Float64(-1.0 - Float64(2.0 * Float64(t * 2e-16))))
end
↓
function code(t)
	return sqrt(sqrt(Float64(2.56e-126 * (t ^ 8.0))))
end

function tmp = code(t)
	tmp = ((1.0 + (t * 2e-16)) * (1.0 + (t * 2e-16))) + (-1.0 - (2.0 * (t * 2e-16)));
end
↓
function tmp = code(t)
	tmp = sqrt(sqrt((2.56e-126 * (t ^ 8.0))));
end

code[t_] := N[(N[(N[(1.0 + N[(t * 2e-16), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(t * 2e-16), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 - N[(2.0 * N[(t * 2e-16), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]
↓
code[t_] := N[Sqrt[N[Sqrt[N[(2.56e-126 * N[Power[t, 8.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]

\left(1 + t \cdot 2 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right)
↓
\sqrt{\sqrt{2.56 \cdot 10^{-126} \cdot {t}^{8}}}

Error

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original61.8
Target50.6
Herbie0.2
\[\mathsf{fma}\left(1 + t \cdot 2 \cdot 10^{-16}, 1 + t \cdot 2 \cdot 10^{-16}, -1 - 2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right) \]

Derivation

  1. Initial program 61.8

    \[\left(1 + t \cdot 2 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right) \]

  2. Simplified0.4

    \[\leadsto \color{blue}{t \cdot \left(t \cdot 4 \cdot 10^{-32}\right)} \]

  3. Taylor expanded in t around 0 0.4

    \[\leadsto \color{blue}{4 \cdot 10^{-32} \cdot {t}^{2}} \]

  4. Applied egg-rr0.2

    \[\leadsto \color{blue}{\sqrt{{t}^{4} \cdot 1.6 \cdot 10^{-63}}} \]

  5. Applied egg-rr0.2

    \[\leadsto \sqrt{\color{blue}{\sqrt{2.56 \cdot 10^{-126} \cdot {t}^{8}}}} \]

  6. Final simplification0.2

    \[\leadsto \sqrt{\sqrt{2.56 \cdot 10^{-126} \cdot {t}^{8}}} \]

Reproduce

herbie shell --seed 2022165 
(FPCore (t)
  :name "fma_test1"
  :precision binary64
  :pre (and (<= 0.9 t) (<= t 1.1))

  :herbie-target
  (fma (+ 1.0 (* t 2e-16)) (+ 1.0 (* t 2e-16)) (- -1.0 (* 2.0 (* t 2e-16))))

  (+ (* (+ 1.0 (* t 2e-16)) (+ 1.0 (* t 2e-16))) (- -1.0 (* 2.0 (* t 2e-16)))))