Kahan p13 Example 1

Average Error: 99.9% → 99.9%

Time: 21.4s

Precision: binary64

Cost: 1856.00

Math

\[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]

↓

\[\frac{-1 + \frac{-4 \cdot t}{2 + \left(t + \frac{1}{t}\right)}}{-1 + \left(-1 + \frac{-4 \cdot t}{t + \left(2 + \frac{1}{t}\right)}\right)} \]

FPCore

(FPCore (t)
 :precision binary64
 (/
  (+ 1.0 (* (/ (* 2.0 t) (+ 1.0 t)) (/ (* 2.0 t) (+ 1.0 t))))
  (+ 2.0 (* (/ (* 2.0 t) (+ 1.0 t)) (/ (* 2.0 t) (+ 1.0 t))))))

↓

(FPCore (t)
 :precision binary64
 (/
  (+ -1.0 (/ (* -4.0 t) (+ 2.0 (+ t (/ 1.0 t)))))
  (+ -1.0 (+ -1.0 (/ (* -4.0 t) (+ t (+ 2.0 (/ 1.0 t))))))))

C

double code(double t) {
	return (1.0 + (((2.0 * t) / (1.0 + t)) * ((2.0 * t) / (1.0 + t)))) / (2.0 + (((2.0 * t) / (1.0 + t)) * ((2.0 * t) / (1.0 + t))));
}

↓

double code(double t) {
	return (-1.0 + ((-4.0 * t) / (2.0 + (t + (1.0 / t))))) / (-1.0 + (-1.0 + ((-4.0 * t) / (t + (2.0 + (1.0 / t))))));
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    code = (1.0d0 + (((2.0d0 * t) / (1.0d0 + t)) * ((2.0d0 * t) / (1.0d0 + t)))) / (2.0d0 + (((2.0d0 * t) / (1.0d0 + t)) * ((2.0d0 * t) / (1.0d0 + t))))
end function

↓

real(8) function code(t)
    real(8), intent (in) :: t
    code = ((-1.0d0) + (((-4.0d0) * t) / (2.0d0 + (t + (1.0d0 / t))))) / ((-1.0d0) + ((-1.0d0) + (((-4.0d0) * t) / (t + (2.0d0 + (1.0d0 / t))))))
end function

Java

public static double code(double t) {
	return (1.0 + (((2.0 * t) / (1.0 + t)) * ((2.0 * t) / (1.0 + t)))) / (2.0 + (((2.0 * t) / (1.0 + t)) * ((2.0 * t) / (1.0 + t))));
}

↓

public static double code(double t) {
	return (-1.0 + ((-4.0 * t) / (2.0 + (t + (1.0 / t))))) / (-1.0 + (-1.0 + ((-4.0 * t) / (t + (2.0 + (1.0 / t))))));
}

Python

def code(t):
	return (1.0 + (((2.0 * t) / (1.0 + t)) * ((2.0 * t) / (1.0 + t)))) / (2.0 + (((2.0 * t) / (1.0 + t)) * ((2.0 * t) / (1.0 + t))))

↓

def code(t):
	return (-1.0 + ((-4.0 * t) / (2.0 + (t + (1.0 / t))))) / (-1.0 + (-1.0 + ((-4.0 * t) / (t + (2.0 + (1.0 / t))))))

Julia

function code(t)
	return Float64(Float64(1.0 + Float64(Float64(Float64(2.0 * t) / Float64(1.0 + t)) * Float64(Float64(2.0 * t) / Float64(1.0 + t)))) / Float64(2.0 + Float64(Float64(Float64(2.0 * t) / Float64(1.0 + t)) * Float64(Float64(2.0 * t) / Float64(1.0 + t)))))
end

↓

function code(t)
	return Float64(Float64(-1.0 + Float64(Float64(-4.0 * t) / Float64(2.0 + Float64(t + Float64(1.0 / t))))) / Float64(-1.0 + Float64(-1.0 + Float64(Float64(-4.0 * t) / Float64(t + Float64(2.0 + Float64(1.0 / t)))))))
end

MATLAB

function tmp = code(t)
	tmp = (1.0 + (((2.0 * t) / (1.0 + t)) * ((2.0 * t) / (1.0 + t)))) / (2.0 + (((2.0 * t) / (1.0 + t)) * ((2.0 * t) / (1.0 + t))));
end

↓

function tmp = code(t)
	tmp = (-1.0 + ((-4.0 * t) / (2.0 + (t + (1.0 / t))))) / (-1.0 + (-1.0 + ((-4.0 * t) / (t + (2.0 + (1.0 / t))))));
end

Wolfram

code[t_] := N[(N[(1.0 + N[(N[(N[(2.0 * t), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * t), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(N[(N[(2.0 * t), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * t), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[t_] := N[(N[(-1.0 + N[(N[(-4.0 * t), $MachinePrecision] / N[(2.0 + N[(t + N[(1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-1.0 + N[(-1.0 + N[(N[(-4.0 * t), $MachinePrecision] / N[(t + N[(2.0 + N[(1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9

   \[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]

2. Simplified 100.0

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(t, \frac{4}{\frac{1}{t} + \left(2 + t\right)}, 1\right)}{\mathsf{fma}\left(t, \frac{4}{\frac{1}{t} + \left(2 + t\right)}, 2\right)}} \]
   Proof

   [Start] 99.9	\[ \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]

3. Applied egg-rr 99.9

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

4. Simplified 99.9

   \[\leadsto \color{blue}{\frac{-1 + \frac{-4 \cdot t}{2 + \left(t + \frac{1}{t}\right)}}{-2 + \frac{-4 \cdot t}{2 + \left(t + \frac{1}{t}\right)}}} \]
   Proof

   [Start] 99.9	\[ -\frac{\mathsf{fma}\left(t, \frac{4}{t + \left(\frac{1}{t} + 2\right)}, 1\right)}{-2 - \frac{t \cdot 4}{t + \left(\frac{1}{t} + 2\right)}} \]
   distribute-neg-frac [=>] 99.9	\[ \color{blue}{\frac{-\mathsf{fma}\left(t, \frac{4}{t + \left(\frac{1}{t} + 2\right)}, 1\right)}{-2 - \frac{t \cdot 4}{t + \left(\frac{1}{t} + 2\right)}}} \]
   fma-udef [=>] 99.9	\[ \frac{-\color{blue}{\left(t \cdot \frac{4}{t + \left(\frac{1}{t} + 2\right)} + 1\right)}}{-2 - \frac{t \cdot 4}{t + \left(\frac{1}{t} + 2\right)}} \]
   associate-*r/ [=>] 99.9	\[ \frac{-\left(\color{blue}{\frac{t \cdot 4}{t + \left(\frac{1}{t} + 2\right)}} + 1\right)}{-2 - \frac{t \cdot 4}{t + \left(\frac{1}{t} + 2\right)}} \]
   distribute-neg-in [=>] 99.9	\[ \frac{\color{blue}{\left(-\frac{t \cdot 4}{t + \left(\frac{1}{t} + 2\right)}\right) + \left(-1\right)}}{-2 - \frac{t \cdot 4}{t + \left(\frac{1}{t} + 2\right)}} \]
   metadata-eval [=>] 99.9	\[ \frac{\left(-\frac{t \cdot 4}{t + \left(\frac{1}{t} + 2\right)}\right) + \color{blue}{-1}}{-2 - \frac{t \cdot 4}{t + \left(\frac{1}{t} + 2\right)}} \]
   +-commutative [<=] 99.9	\[ \frac{\color{blue}{-1 + \left(-\frac{t \cdot 4}{t + \left(\frac{1}{t} + 2\right)}\right)}}{-2 - \frac{t \cdot 4}{t + \left(\frac{1}{t} + 2\right)}} \]
   *-commutative [<=] 99.9	\[ \frac{-1 + \left(-\frac{\color{blue}{4 \cdot t}}{t + \left(\frac{1}{t} + 2\right)}\right)}{-2 - \frac{t \cdot 4}{t + \left(\frac{1}{t} + 2\right)}} \]
   distribute-neg-frac [=>] 99.9	\[ \frac{-1 + \color{blue}{\frac{-4 \cdot t}{t + \left(\frac{1}{t} + 2\right)}}}{-2 - \frac{t \cdot 4}{t + \left(\frac{1}{t} + 2\right)}} \]
   distribute-rgt-neg-out [<=] 99.9	\[ \frac{-1 + \frac{\color{blue}{4 \cdot \left(-t\right)}}{t + \left(\frac{1}{t} + 2\right)}}{-2 - \frac{t \cdot 4}{t + \left(\frac{1}{t} + 2\right)}} \]
   neg-mul-1 [=>] 99.9	\[ \frac{-1 + \frac{4 \cdot \color{blue}{\left(-1 \cdot t\right)}}{t + \left(\frac{1}{t} + 2\right)}}{-2 - \frac{t \cdot 4}{t + \left(\frac{1}{t} + 2\right)}} \]
   associate-*r* [=>] 99.9	\[ \frac{-1 + \frac{\color{blue}{\left(4 \cdot -1\right) \cdot t}}{t + \left(\frac{1}{t} + 2\right)}}{-2 - \frac{t \cdot 4}{t + \left(\frac{1}{t} + 2\right)}} \]
   metadata-eval [=>] 99.9	\[ \frac{-1 + \frac{\color{blue}{-4} \cdot t}{t + \left(\frac{1}{t} + 2\right)}}{-2 - \frac{t \cdot 4}{t + \left(\frac{1}{t} + 2\right)}} \]
   associate-+r+ [=>] 99.9	\[ \frac{-1 + \frac{-4 \cdot t}{\color{blue}{\left(t + \frac{1}{t}\right) + 2}}}{-2 - \frac{t \cdot 4}{t + \left(\frac{1}{t} + 2\right)}} \]
   +-commutative [=>] 99.9	\[ \frac{-1 + \frac{-4 \cdot t}{\color{blue}{2 + \left(t + \frac{1}{t}\right)}}}{-2 - \frac{t \cdot 4}{t + \left(\frac{1}{t} + 2\right)}} \]

5. Applied egg-rr 99.9

   \[\leadsto \frac{-1 + \frac{-4 \cdot t}{2 + \left(t + \frac{1}{t}\right)}}{\color{blue}{\left(\frac{-4 \cdot t}{t + \left(2 + \frac{1}{t}\right)} + -1\right) - 1}} \]

6. Final simplification 99.9

   \[\leadsto \frac{-1 + \frac{-4 \cdot t}{2 + \left(t + \frac{1}{t}\right)}}{-1 + \left(-1 + \frac{-4 \cdot t}{t + \left(2 + \frac{1}{t}\right)}\right)} \]

Alternatives

Alternative 1
Error	99.4%
Cost	1737.00

\[\begin{array}{l} \mathbf{if}\;t \leq -0.6 \lor \neg \left(t \leq 0.56\right):\\ \;\;\;\;0.8333333333333334 + \left(\frac{0.037037037037037035}{t \cdot t} + \frac{-0.2222222222222222}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(1 + \left(-1 + t \cdot \frac{4}{\frac{t + 1}{t}}\right)\right)}{2 + \left(t \cdot t\right) \cdot 4}\\ \end{array} \]

Alternative 2
Error	99.9%
Cost	1728.00

\[\begin{array}{l} t_1 := \frac{-4 \cdot t}{2 + \left(t + \frac{1}{t}\right)}\\ \frac{-1 + t_1}{t_1 + -2} \end{array} \]

Alternative 3
Error	99.4%
Cost	1481.00

\[\begin{array}{l} t_1 := \left(t \cdot t\right) \cdot 4\\ \mathbf{if}\;t \leq -0.6 \lor \neg \left(t \leq 0.56\right):\\ \;\;\;\;0.8333333333333334 + \left(\frac{0.037037037037037035}{t \cdot t} + \frac{-0.2222222222222222}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{t_1}{t + 1}}{2 + t_1}\\ \end{array} \]

Alternative 4
Error	99.3%
Cost	1225.00

\[\begin{array}{l} \mathbf{if}\;t \leq -0.64 \lor \neg \left(t \leq 0.43\right):\\ \;\;\;\;0.8333333333333334 + \left(\frac{0.037037037037037035}{t \cdot t} + \frac{-0.2222222222222222}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + t \cdot \left(t \cdot 4\right)}{2 + \left(t \cdot t\right) \cdot 4}\\ \end{array} \]

Alternative 5
Error	99.3%
Cost	969.00

\[\begin{array}{l} \mathbf{if}\;t \leq -0.8 \lor \neg \left(t \leq 0.235\right):\\ \;\;\;\;0.8333333333333334 + \left(\frac{0.037037037037037035}{t \cdot t} + \frac{-0.2222222222222222}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot t + 0.5\\ \end{array} \]

Alternative 6
Error	99.1%
Cost	585.00

\[\begin{array}{l} \mathbf{if}\;t \leq -0.79 \lor \neg \left(t \leq 0.56\right):\\ \;\;\;\;0.8333333333333334 + \frac{-0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;t \cdot t + 0.5\\ \end{array} \]

Alternative 7
Error	98.6%
Cost	584.00

\[\begin{array}{l} \mathbf{if}\;t \leq -0.9:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 0.56:\\ \;\;\;\;t \cdot t + 0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \]

Alternative 8
Error	98.4%
Cost	328.00

\[\begin{array}{l} \mathbf{if}\;t \leq -0.34:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 1:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \]

Alternative 9
Error	60.4%
Cost	64.00

\[0.5 \]

Reproduce

herbie shell --seed 2023093 
(FPCore (t)
  :name "Kahan p13 Example 1"
  :precision binary64
  (/ (+ 1.0 (* (/ (* 2.0 t) (+ 1.0 t)) (/ (* 2.0 t) (+ 1.0 t)))) (+ 2.0 (* (/ (* 2.0 t) (+ 1.0 t)) (/ (* 2.0 t) (+ 1.0 t))))))