Average Error: 61.8 → 0.4
Time: 4.8s
Precision: binary64
Cost: 6976
\[0.9 \leq t \land t \leq 1.1\]
\[\left(1 + t \cdot 2 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right) \]
↓
\[\left(t \cdot 2 \cdot 10^{-16}\right) \cdot \sqrt{4 \cdot 10^{-32} \cdot \left(t \cdot t\right)} \]
(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 (* (* t 2e-16) (sqrt (* 4e-32 (* t t)))))
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 (t * 2e-16) * sqrt((4e-32 * (t * t)));
}

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 = (t * 2d-16) * sqrt((4d-32 * (t * t)))
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 (t * 2e-16) * Math.sqrt((4e-32 * (t * t)));
}

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 (t * 2e-16) * math.sqrt((4e-32 * (t * t)))

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 Float64(Float64(t * 2e-16) * sqrt(Float64(4e-32 * Float64(t * t))))
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 = (t * 2e-16) * sqrt((4e-32 * (t * t)));
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[(N[(t * 2e-16), $MachinePrecision] * N[Sqrt[N[(4e-32 * N[(t * t), $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)
↓
\left(t \cdot 2 \cdot 10^{-16}\right) \cdot \sqrt{4 \cdot 10^{-32} \cdot \left(t \cdot t\right)}

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original61.8
Target50.6
Herbie0.4
\[\mathsf{fma}\left(1 + t \cdot 2 \cdot 10^{-16}, 1 + t \cdot 2 \cdot 10^{-16}, -1 - 2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right) \]

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

    \[\leadsto \color{blue}{t \cdot \left(t \cdot 4 \cdot 10^{-32}\right)} \]
    Proof
    (*.f64 t (*.f64 t 1/25000000000000000000000000000000)): 0 points increase in error, 0 points decrease in error
    (*.f64 t (*.f64 t (Rewrite<= metadata-eval (*.f64 1/5000000000000000 1/5000000000000000)))): 138 points increase in error, 90 points decrease in error
    (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 t t) (*.f64 1/5000000000000000 1/5000000000000000))): 44 points increase in error, 51 points decrease in error
    (Rewrite<= swap-sqr_binary64 (*.f64 (*.f64 t 1/5000000000000000) (*.f64 t 1/5000000000000000))): 49 points increase in error, 75 points decrease in error
    (Rewrite<= +-lft-identity_binary64 (+.f64 0 (*.f64 (*.f64 t 1/5000000000000000) (*.f64 t 1/5000000000000000)))): 0 points increase in error, 0 points decrease in error
    (+.f64 (Rewrite<= mul0-lft_binary64 (*.f64 0 (*.f64 t 1/5000000000000000))) (*.f64 (*.f64 t 1/5000000000000000) (*.f64 t 1/5000000000000000))): 0 points increase in error, 0 points decrease in error
    (Rewrite<= distribute-rgt-in_binary64 (*.f64 (*.f64 t 1/5000000000000000) (+.f64 0 (*.f64 t 1/5000000000000000)))): 0 points increase in error, 0 points decrease in error
    (*.f64 (*.f64 t 1/5000000000000000) (+.f64 (Rewrite<= metadata-eval (+.f64 -1 1)) (*.f64 t 1/5000000000000000))): 0 points increase in error, 0 points decrease in error
    (*.f64 (*.f64 t 1/5000000000000000) (Rewrite<= associate-+r+_binary64 (+.f64 -1 (+.f64 1 (*.f64 t 1/5000000000000000))))): 256 points increase in error, 0 points decrease in error
    (*.f64 (*.f64 t 1/5000000000000000) (Rewrite=> +-commutative_binary64 (+.f64 (+.f64 1 (*.f64 t 1/5000000000000000)) -1))): 0 points increase in error, 0 points decrease in error
    (Rewrite=> distribute-rgt-in_binary64 (+.f64 (*.f64 (+.f64 1 (*.f64 t 1/5000000000000000)) (*.f64 t 1/5000000000000000)) (*.f64 -1 (*.f64 t 1/5000000000000000)))): 256 points increase in error, 0 points decrease in error
    (+.f64 (*.f64 (+.f64 1 (*.f64 t 1/5000000000000000)) (*.f64 t 1/5000000000000000)) (*.f64 (Rewrite<= metadata-eval (+.f64 -2 1)) (*.f64 t 1/5000000000000000))): 0 points increase in error, 0 points decrease in error
    (+.f64 (*.f64 (+.f64 1 (*.f64 t 1/5000000000000000)) (*.f64 t 1/5000000000000000)) (*.f64 (+.f64 (Rewrite<= metadata-eval (neg.f64 2)) 1) (*.f64 t 1/5000000000000000))): 0 points increase in error, 0 points decrease in error
    (+.f64 (*.f64 (+.f64 1 (*.f64 t 1/5000000000000000)) (*.f64 t 1/5000000000000000)) (Rewrite<= distribute-rgt1-in_binary64 (+.f64 (*.f64 t 1/5000000000000000) (*.f64 (neg.f64 2) (*.f64 t 1/5000000000000000))))): 0 points increase in error, 0 points decrease in error
    (+.f64 (*.f64 (+.f64 1 (*.f64 t 1/5000000000000000)) (*.f64 t 1/5000000000000000)) (+.f64 (*.f64 t 1/5000000000000000) (Rewrite<= distribute-lft-neg-in_binary64 (neg.f64 (*.f64 2 (*.f64 t 1/5000000000000000)))))): 0 points increase in error, 0 points decrease in error
    (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 (*.f64 (+.f64 1 (*.f64 t 1/5000000000000000)) (*.f64 t 1/5000000000000000)) (*.f64 t 1/5000000000000000)) (neg.f64 (*.f64 2 (*.f64 t 1/5000000000000000))))): 0 points increase in error, 0 points decrease in error
    (+.f64 (Rewrite=> distribute-lft1-in_binary64 (*.f64 (+.f64 (+.f64 1 (*.f64 t 1/5000000000000000)) 1) (*.f64 t 1/5000000000000000))) (neg.f64 (*.f64 2 (*.f64 t 1/5000000000000000)))): 256 points increase in error, 0 points decrease in error
    (+.f64 (*.f64 (+.f64 (+.f64 1 (*.f64 t 1/5000000000000000)) 1) (Rewrite<= +-rgt-identity_binary64 (+.f64 (*.f64 t 1/5000000000000000) 0))) (neg.f64 (*.f64 2 (*.f64 t 1/5000000000000000)))): 0 points increase in error, 0 points decrease in error
    (+.f64 (*.f64 (+.f64 (+.f64 1 (*.f64 t 1/5000000000000000)) 1) (+.f64 (*.f64 t 1/5000000000000000) (Rewrite<= metadata-eval (-.f64 1 1)))) (neg.f64 (*.f64 2 (*.f64 t 1/5000000000000000)))): 0 points increase in error, 0 points decrease in error
    (+.f64 (*.f64 (+.f64 (+.f64 1 (*.f64 t 1/5000000000000000)) 1) (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 (*.f64 t 1/5000000000000000) 1) 1))) (neg.f64 (*.f64 2 (*.f64 t 1/5000000000000000)))): 0 points increase in error, 256 points decrease in error
    (+.f64 (*.f64 (+.f64 (+.f64 1 (*.f64 t 1/5000000000000000)) 1) (-.f64 (Rewrite<= +-commutative_binary64 (+.f64 1 (*.f64 t 1/5000000000000000))) 1)) (neg.f64 (*.f64 2 (*.f64 t 1/5000000000000000)))): 0 points increase in error, 0 points decrease in error
    (+.f64 (Rewrite<= difference-of-sqr--1_binary64 (+.f64 (*.f64 (+.f64 1 (*.f64 t 1/5000000000000000)) (+.f64 1 (*.f64 t 1/5000000000000000))) -1)) (neg.f64 (*.f64 2 (*.f64 t 1/5000000000000000)))): 0 points increase in error, 0 points decrease in error
    (Rewrite<= associate-+r+_binary64 (+.f64 (*.f64 (+.f64 1 (*.f64 t 1/5000000000000000)) (+.f64 1 (*.f64 t 1/5000000000000000))) (+.f64 -1 (neg.f64 (*.f64 2 (*.f64 t 1/5000000000000000)))))): 256 points increase in error, 0 points decrease in error
    (+.f64 (*.f64 (+.f64 1 (*.f64 t 1/5000000000000000)) (+.f64 1 (*.f64 t 1/5000000000000000))) (Rewrite<= sub-neg_binary64 (-.f64 -1 (*.f64 2 (*.f64 t 1/5000000000000000))))): 0 points increase in error, 0 points decrease in error

  3. Applied egg-rr0.5

    \[\leadsto \color{blue}{{\left(t \cdot 2 \cdot 10^{-16}\right)}^{2}} \]

  4. Applied egg-rr0.5

    \[\leadsto \color{blue}{\left(t \cdot 2 \cdot 10^{-16}\right) \cdot \left(t \cdot 2 \cdot 10^{-16}\right)} \]

  5. Applied egg-rr0.4

    \[\leadsto \left(t \cdot 2 \cdot 10^{-16}\right) \cdot \color{blue}{\sqrt{4 \cdot 10^{-32} \cdot \left(t \cdot t\right)}} \]

  6. Final simplification0.4

    \[\leadsto \left(t \cdot 2 \cdot 10^{-16}\right) \cdot \sqrt{4 \cdot 10^{-32} \cdot \left(t \cdot t\right)} \]

Alternatives

Alternative 1
Error0.3
Cost6848
\[t \cdot \sqrt{\left(t \cdot t\right) \cdot 1.6 \cdot 10^{-63}} \]
Alternative 2
Error0.4
Cost448
\[t \cdot \left(2 \cdot 10^{-16} \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right) \]
Alternative 3
Error0.4
Cost320
\[4 \cdot 10^{-32} \cdot \left(t \cdot t\right) \]
Alternative 4
Error0.4
Cost320
\[t \cdot \left(t \cdot 4 \cdot 10^{-32}\right) \]

Error

Reproduce

herbie shell --seed 2022313 
(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)))))