fma_test1

Average Error: 61.8 → 0.2

Time: 3.4s

Precision: binary64

\[0.9 \leq t \land t \leq 1.1\]

Math

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

↓

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

FPCore

(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 (* (pow t 8.0) 2.56e-126))))

C

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((pow(t, 8.0) * 2.56e-126)));
}

Fortran

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(((t ** 8.0d0) * 2.56d-126)))
end function

Java

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((Math.pow(t, 8.0) * 2.56e-126)));
}

Python

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((math.pow(t, 8.0) * 2.56e-126)))

Julia

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((t ^ 8.0) * 2.56e-126)))
end

MATLAB

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(((t ^ 8.0) * 2.56e-126)));
end

Wolfram

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[(N[Power[t, 8.0], $MachinePrecision] * 2.56e-126), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]

Error

Bits error versus t

Target

Original	61.8
Target	50.6
Herbie	0.2

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

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. Simplified 0.4

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

3. Applied egg-rr 0.3

   \[\leadsto t \cdot \color{blue}{\sqrt{\left(t \cdot t\right) \cdot 1.6 \cdot 10^{-63}}} \]

4. Applied egg-rr 0.2

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

5. Applied egg-rr 0.2

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

6. Final simplification 0.2

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

Reproduce

herbie shell --seed 2022131 
(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)))))